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Survival and growth curve analyses applied to a barnacle data set Freund, Vivien 1983-12-31

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SURVIVAL AND GROWTH CURVE ANALYSES A P P L I E D TO A BARNACLE DATA SET  By VIVIEN B.A.,  A THESIS  FREUND  The U n i v e r s i t y  SUBMITTED  of  Durham,  1981  IN P A R T I A L F U L F I L L M E N T  THE REQUIREMENTS  FOR THE DEGREE  MASTER OF  OF  OF  SCIENCE  in  THE FACULTY (Institute  of  STUDIES  A p p l i e d Mathematics  We a c c e p t to  OF GRADUATE  this  the  thesis  required  as  © Vivien  Statistics)  conforming  standard  THE U N I V E R S I T Y OF B R I T I S H December  and  COLUMBIA  1983  Freund,  1983  In presenting  this  thesis i n partial  f u l f i l m e n t of the  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e of B r i t i s h Columbia, I agree that it  freely  the L i b r a r y s h a l l  a v a i l a b l e f o r r e f e r e n c e and s t u d y .  agree t h a t p e r m i s s i o n f o r extensive for  University  s c h o l a r l y p u r p o s e s may  for  financial  of  The U n i v e r s i t y o f B r i t i s h 1956 Main M a l l V a n c o u v e r , Canada V6T 1Y3 Date  DE-6  (3/81)  Columbia  my  It is thesis  s h a l l n o t be a l l o w e d w i t h o u t my  permission.  Department  thesis  be g r a n t e d by t h e h e a d o f  copying or p u b l i c a t i o n of t h i s  gain  further  copying of t h i s  d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . understood that  I  make  written  ABSTRACT  A  data  examined.  set  Survival  nonparametric survival slope.  growth  The  distributions  to and  data,  statistical  the  to  survival  is  problems  and growth  curves  the not  are  model  are  model  fitted  tested of  using  Rao.  longitudinal,  which are  of  barnacles  c o m p a r e d by means  and a random e f f e c t s  growth  secondly  are  exponential  parameters  which  the  distributions  tests.  Polynomial  relating and Roy  relating  is  model and  to  three  fitted  is  for  the  developed  for  various  firstly  Owing this  then  of  the  the  is  hypotheses model  nature  presents  of  of  various  A/j J o h n  Pothoff  the  discussed.  Thesis  the  Petkau  Supervisor  -  i i i  T A B L E OF  -  CONTENTS Page  ABSTRACT  i i  TABLE OF CONTENTS  i i i  LIST  OF T A B L E S  LIST  OF FIGURES  v  ACNOWLEDGEMENTS  vi  1.  2.  3.  iv  INTRODUCTION  1  1.1  The E x p e r i m e n t  3  1.2  The Data  5  EXAMINATION OF THE SURVIVAL DATA 2.1  Description  2.2 2.3 2.4 2.5  E x p o n e n t i a l Models I n f e r e n c e on t h e B ' s Random E f f e c t s M o d e l Incorporating Netting  of  the  Survival  6 Data  and D o w l i n g  8  Lines  into  the M o d e l . .  19 31 36 46  EXAMINATION OF THE GROWTH DATA  52  3.1  Growth Data  52  3.2 3.3 3.4 3.5 3.6 3.7 3.8  54 56 64 67 76 83  3.9  Growth Curves f o r Items The G r o w t h C u r v e M o d e l s A p p l i c a t i o n of the Models S y s t e m One O y s t e r L i n e s S y s t e m Two O y s t e r L i n e s A p p l i c a t i o n of the Model to a l l Seventeen L i n e s E x t e n s i o n of the Model t o A l l o w D i f f e r e n t C o v a r i a n c e M a t r i c e s f o r D i f f e r e n t Items C o l l a p s e Over L i n e s W i t h i n each S y s t e m - t y p e C o m b i n a t i o n .  3.10 3.11 3.12  Misusing ° a t a Growth Curves Discussion  for  Individual  Barnacles  91 95 98 101 107  CONCLUSION  114  BIBLIOGRAPHY  116  -  LIST  iv  -  OF T A B L E S Page  Table  1  Survival  Table  2  Results at  Table  3  t  Data of  Fitting  7 the  E x p o n e n t i a l Model  Starting  = 0  R e s u l t s of at t = 5  23 Fitting  the E x p o n e n t i a l Model  Starting 25  - v -  LIST  OF  FIGURES Page  Figure  1  Growth Curves  for  the  S y s t e m One O y s t e r  Lines  73  Figure  2  Growth Curves  for  the  S y s t e m Two O y s t e r  Lines  73  Figure  3  Growth Curves  for  the  Dowling L i n e s  Figure  4  Growth Curves  for  the N e t t i n g  Lines  87 87  -  vi  -  ACKNOWLEDGEMENTS  I  would  the  time  he  Dr.  Nancy R e i d  spent  reading  the  time  help.  to  The  like  to  in  for  financial  and F e l l o w s h i p  helping  her  thesis.  express  my g r a t i t u d e  to  sympathy  Many  thanks  support  Committee  is  of  produce  this  to  Dr.  thesis.  and encouragement also  the  to Helene  Canadian  gratefully  John Petkau I  along  for  am i n d e b t e d the  way,  Commonwealth  to  and  C r e p e a u who a l w a y s  acknowledged.  all  for  had  Scholarship  INTRODUCTION  1.  The  examination  of  a particular  data  set  leads,  during  the  course  i of  this  thesis,  to  the  discussion  statistical  techniques.  headings  survival  of  investigated  part  of  some  object  their  usually that  are  the  choose  Once  attaches  it  itself  adult  glue  again  move.  The  easier  pull the  off  to the  a bad  death.  In  The  itself away  the  any  have  place, case  that  It  has  stand where  it  barnacle  shell the  object.  from  the  rock.  the to  there  The Organs  is  the not  a little  "stalk" stalk which  pr  look  is  is  first be  important  plankton  the  for  barnacle  subsequently can then  for  than  forces  the  will  never  example,  to,  the If  it  The  it  to that  barnacle  starve  to  predators. a clam covered w i t h  "neck"  a  in  attached  surf.  was  They  It  that,  of  which  which w i l l  barnacle  because  ocean  like  serves  live,  but  much f o o d ,  by  ocean.  object,  strong  general  set  a whale.  glue  of  barnacles.  bring  The  barnacle  in  to  mild  be s t r o n g  the  or  other  so  data  for  to  two  barnacles.  strong.  be e a t e n  looks a  other  is  in  a boat  the  The  a place  some  use  up t o  might  and w i t h  to  they  live  currents  or  extremely  rock  to  are  a relatively  which  barnacle.  adult  protective  glue  is  for  strong  rock,  using  which  a number  under  curves.  a rock,  there  with  chosen a  which  chip  barnacles  chooses  where  fall  of  and g r o w t h d a t a  looking  s u c h as  a place  first  and g r o w t h  creatures  spent  ocean  has  using  is  is  techniques  survival  little  lives  in  of  choose a p l a c e  they  food.  analysis  consisted  Barnacles  These  and a p p l i c a t i o n  with  a  which  it  attaches  rest  of  the  to keep  the  little  like  feet,  barnacle  protrude  from  - 2 -  the bottom of the s h e l l .  I t i s w i t h these that the b a r n a c l e e a t s .  a d u l t b a r n a c l e i s approximately h i s master's experiment  4-5  cm l o n g .  Mr.  Harry Goldberg, f o r  t h e s i s i n b i o r e s o u r c e e n g i n e e r i n g at U.B.C. conducted  r e l a t i n g t o the s u r v i v a l and  These l i n e s were arranged  i n two  systems.  l a c k of food and because of p r e d a t o r s .  line,  types of m a t e r i a l .  B a r n a c l e s were d y i n g due  At each of a number of  s h e l l l e n g t h , neck l e n g t h , s h e l l weight  and neck weight.  each Mr.  i n t e r e s t e d i n m o d e l l i n g the s u r v i v a l d i s t r i b u t i o n and i n  comparing the d i s t r i b u t i o n s f o r the t h r e e d i f f e r e n t f o r the two  systems.  He was  types of l i n e  also interested i n f i t t i n g  curves obtained a c r o s s systems and m a t e r i a l t y p e s . i n t e r e s t e d i n the length-weight  and  growth curves  f o r each l i n e f o r s h e l l l e n g t h and neck l e n g t h and i n comparing F i n a l l y he  the was  r e l a t i o n s h i p f o r s h e l l and neck.  From  our p o i n t of view the s t a t i s t i c a l methodology i s of more i n t e r e s t t h i s p a r t i c u l a r data s e t , so the length-weight  be f i t t e d  l e n g t h could be done i n an  identical  f o r neck l e n g t h . Due  was  A l s o growth  only f o r s h e l l l e n g t h , not f o r neck l e n g t h .  E v e r y t h i n g that i s done f o r s h e l l way  than  r e l a t i o n s h i p , which  i n v o l v e s s t r a i g h t f o r w a r d r e g r e s s i o n , w i l l not be d i s c u s s e d . curves w i l l  to  times  c o l l e c t e d on these b a r n a c l e s , namely the number dead on  Goldberg was  an  growth of b a r n a c l e s . The  b a r n a c l e s were a t t a c h e d t o l i n e s of t h r e e d i f f e r e n t  data was  The  to the l i m i t e d nature of the growth data - l o n g i t u d i n a l  not c o l l e c t e d - i t was  curves and The  not s t r a i g h t f o r w a r d t o f i t meaningful  t h i s presented i n t e r e s t i n g  remainder  the experiment  and  statistical  of S e c t i o n 1 i s devoted the data.  data growth  problems.  t o a f u l l e r d e s c r i p t i o n of  In S e c t i o n 2 the s u r v i v a l data i s  -  examined: lines  are  model  is  firstly  the  survival  compared u s i n g fitted  for  3  -  distributions  nonparametric  each  line  and t h e  tests. slopes  c o m p a r e d by means o f  the  likelihood  ratio  models  are  developed  for  the  of  curves  are  fitted  various  for  hypotheses  determined determining available,  characteristics.  the  curve  there  a r e a number  of  then  fitted  is  made  to  attempt but  this  1.1  The  is  found  test to  for  are  impossible  of  three  In  with  important of  order  the  between data  the to  the  3  lines  and  way i t  is  in  able  each  data  to  lines,  later).  items  growth  limited  be  are  effects  factors  seventeen  be d e s c r i b e d  the  Section  In t h i s  because  curves  random  seventeen  each item w i t h i n  differences  be  are  be made i n  (which w i l l for  Finally  the  each of  exponential  resulting  tested.  type  Within  separately  of  However to  the  different  Growth  line.  within  An a  line,  available.  Experiment  Seventeen constructed i.  models. "items"  are  have  of  the  curves.  each  parameters  assumptions  apply  curves  growth  the  for  various  Then  test.  the  s y s t e m and m a t e r i a l  growth strong  shell-length  about  whether  slopes  for  as  lines  different  types  of  material  11  were  follows:  Oyster  lines  -  ropes,  each with  10  or  ropes  each with  20  pieces  oyster  shells  attached. ii.  Dowling  lines  attached.  -  of  wood  (dowling)  -  i i i .  Netting  lines  -  10 c o m p a r t m e n t s  long  4  -  cylindrical  and a p i e c e  of  pieces  hard  of  rubber  netting in  each  with  each  compartment. The  seventeen  each  lines  system being  were  given  arranged  in  two  The to  lines the  along  items the  taken  on e a c h  coastline  barnacles  could  move.  this  In  and b e c a u s e  of  to  line.  become  1  1  the  o c e a n , where  The  lines  a natural attached  later  environment. and t h o s e  retrieved  At  already  attached  approximately  oyster  s y s t e m one line,  line,  approximately  s y s t e m one n e t t i n g  line,  and  The  actual  given  in  are  T h e r e were  oyster  the  numbers  in  collected  on t h e  1600 Table  dying the  due  to  number  approximately  500  1000 on e a c h d o w l i n g  attached  and  stage  line.  on e a c h  were  this  of  t=0)  barnacles  became  A r e c o r d was made  on e a c h  d a t a was  were  barnacles  predators.  initially  times  1  the  (at  lines  2  environment  initially  s y s t e m two  out  in  of  Netting  6  were  number  Dowling  6  System 2  the  below:  Oyster System 1  systems,  set  no new  could lack of 200  food  barnacles each  line  and  1.  a number  barnacles.  of  on  s y s t e m two n e t t i n g of  not  barnacles  on t h e At  up  on  line.  subsequent  -  1.2  5  -  The Data  S u r v i v a l Data At number  t  = 1,  of  barnacles t  2,  the  the  5,  6,  found  were  removed.  then  died  number  available  4,  barnacles  = 7 weeks w e r e  barnacles  3,  of  those  before deaths  at  8,  9,  on e a c h So f o r  each it  time  10, line  More  is  = 6 and t  dead  found  recorded  at  i n Table  a  dead  = 7 weeks.  information,  item within  given  the  The  the barnacles  detailed  on e a c h  data  14 a n d 17 w e e k s  was r e c o r d e d .  between t  was p r o b a b l y  The s u r v i v a l  11,  example  had d i e d  = 5 weeks.  t o us a l t h o u g h  experiment.  dead  that t  7,  line,  some  at No  namely was n o t  stage  during  1.  Growth Data At  each of  selected  the above  from each  item each barnacle sampled  barnacle  (i)  neck  (ii) The  shell  item  from items barnacles  ways  It  sample  of  was a s s u m e d t h a t of  measurements  being were  within  sampled.  each  For  each  not  removed  taken:  in  centimetres.  The sample  at  lines each  size  These  barnacles  was 5 e x c e p t  where  it  was 1 0 .  in  the  lines:  of  samples  number  of  2 x 6 x 10 ( o r 11) x 5 = 600 3 x 20 x 5  From n e t t i n g  lines:  2 x 10 x 10  is  cases  time was:  lines:  an e n o r m o u s  were  So t h e t o t a l  From d o w l i n g  it  b a r n a c l e s was  length  measured  i n which  live  length  on n e t t i n g  is  line.  probability  the f o l l o w i n g  From o y s t e r  So t h e r e  a haphazard  on e a c h  had e q u a l  d a t a was r e c o r d e d  from the i t e m s .  times  amount  incomplete,  of  growth  data.  =300 = 200  The g r o w t h  a r e d i s c u s s e d more  fully  data  and t h e  in Section  3.  -  order  barnacles times  to  were  (on a  obtain  removed  total  of  survival  data,  of  barnacles  these  assumption  this  number: type  in  of  material,  of  total the  dead at so  the  number  whereas  dowling  first  time  at  The  which  information, was n o t first  distributions essentially  the  reasonable  to  and a c r o s s  the  be  -  simpler  question for  lines  same.  collapse  to  the  across  have  the  the  barnacles  types  of  This  chosen  from  2969 line  at  total each  initially  6561,  the  removal  analyses.  removed  these  and at  on 2552.  the  end  1121. line  and  No b a r n a c l e s found  of  that  to  of  the died  d e a d was  deaths  number before  t  on e a c h  = 6. item  found t  = 5  More within  us.  of  this  were  was  were  number  interest  If  1.  a number  r e m o v e d were  lines  on e a c h  barnacles the  of  the  40  relating  compared t o  of  1200 a n d  Table  of  two d o w l i n g  we w o u l d  in  namely  avaiable  1668,  barnacles  given  affect  200 b a r n a c l e s number  each of  was a s s u m e d  small  and n e t t i n g  respectively  are  It  at  analyses  barnacles  very  on e a c h  approximately  material  In a l l  the  total  barnacles  time  line  since  the  of  data,  significantly  removed was  number  of  was n e g l e c t e d .  approximately  each  type  occasions).  would not  of  initial  detailed each  fact  oyster,  s t u d y was  The  length-weight  from each  five  number  total  respectively The  the  seemed r e a s o n a b l e  r a n d o m and t h e  -  EXAMINATION OF THE SURVIVAL DATA  2.  In  6  was w h e t h e r  same t y p e were the  within  found  six  to  oyster  lines  in  system  1.  three  "treatments"  be  the  survival  the  same  s y s t e m were  the  case  it  lines  within  Then the (types  of  would  each  be  system  situation  would  material)  within  - 7 -  T a b l e 1 - S u r v i v a l Data  Initial #  6  7  Number Found Dead a t Week 8 9 10 11  14  17  01  204  31  32  33  3  6  4  7  5  02  187  27  30  22  0  8  3  11  9  03  134  22  31  15  1  3  18  1  3  04  137  27  21  19  5  6  7  5  5  05  187  34  26  29  9  6  10  4  8  06  306  40  27  25  9  7  14  11  14  DI  937  128  122  113  32  29  41  44  45  D2  817  127  168  118  37  32  16  50  37  Nl  952  75  226  119  116  19  75  10  7  07  530  30  42  69  34  35  26  39  20  08  180  9  18  12  7  14  7  9  17  09  579  24  40  39  47  78  49  37  17  010  155  14  16  28  9  9  1  011  677  20  61  80  35  62  30  67  16  012  470  7  22  33  28  38  21  33  25  D3  1215  50  89  126  77  96  61  99  32  N2  1600  19  23  118  216  175  110  75  48  7  4  -  each of two  systems i n s t e a d  lines within be  order  The  s i t u a t i o n of  l i n e e f f e c t would be known to  Three non-parametric rank t e s t s were performed i n  of v a r i o u s  groups of  lines.  D e s c r i p t i o n of the Non-Parametric  a.  context  t e s t can be  derived  e i t h e r as a l i n e a r rank t e s t or i n  of Cox's p r o p o r t i o n a l hazards model.  considered The  from the  l a t t e r point  by  It w i l l  1972)  time, t i s the  i s a vector  B(p x 1) i s a v e c t o r  of  observed value  of  x 1) be  of r e g r e s s i o n  Let I ( p x p) be  T,  c o e f f i c i e n t s and function.  the v e c t o r whose j t h element i s given ologL  where L i s the  (zp)  covariates,  X o ( t ) i s an a r b i t r a r y u n s p e c i f i e d b a s e - l i n e hazard Let U(p  be  f o r f a i l u r e time i s  the hazard f u n c t i o n X ( t ; z ) = \ o ( t ) exp  where T i s f a i l u r e  first  likelihood  for  the  of view:  p r o p o r t i o n a l hazards model (Cox,  specified  x p)  Tests  Logrank (Savage) T e s t  This  z(l  system.  more complicated  to determine whether d i f f e r e n c e s e x i s t e d between the s u r v i v a l  distributions  2.1  -  of the present  treatment w i t h i n  insignificant.  8  by  j = 1  > PS  6.  the m a t r i x whose ( j , h ) t h element i s g i v e n  by  j , h = 1 ,••• , p.  - 9 -  Then  a test  statistic  of  the n u l l  U(8o).  asymptotically consistently  Ho  Under  and o t h e r  estimated  I(Bo).  by  information  mild  Note  matrix.  based  regularity  that  is  To f i n d  on t h e  score  conditions  matrix  being  which  used  U(6o)  is  c a n be  t o denote  U and I t h e l i k e l i h o o d  the must  obtained:  Let  the d i s t i n c t  Let  dj[  be t h e number  of  deaths  Let  n^ be t h e number  at  risk  death  In  t h e case where  by  (Kalbfleisch  times  ties  be t ( ^ ) < . . . at  t(±)*  just  prior  and c e n s o r i n g  k  ^ s^ i s  t h e sum o f  failures  at  individuals  subsets exact  partial  T,  at r i s k  at  specified  R  d  (  t  i  ( i )  associated  with  , 1 =  t^  (1^,... l  - 0 and  d  t h e d^  ), ^ . ( t ^ )  ( (^)) t  with  ties  arises  i  s  t  n  X(t;z)dt \<t;z)  dt  1 -  t h e s e t of  s e t of  replacement.  from a d i s c r e t e  \j(t)dt =  e  is  all  i  by  1 -  given  )  chosen from R(t(£)) without  likelihood  is  i  s^ =  o f d^ i t e m s  the l i k e l i h o o d  expCs^)  the c o v a r i a t e s  d  t^^.  are allowed  T.  i=i  where:  to  expCs^)  n  =  < t(k)*  1980):  and P r e n t i c e ,  L  all  HQ:8 = 8 0 i s  n o r m a l w i t h mean 0 a n d c o v a r i a n c e  observed F i s h e r be  hypothesis  \,<t)dt a  e  x  p  (  z  P  )  The  model  -  This  likelihood  is  suggested  ( 1 9 7 4 )  very  the  L  following  =  n  This  approximation  is  This  means  each  at  compared  to  the  The  statistic  compute.  approximation  at  R(t  provided  time  the  Peto to  (  that  i  )  is  8)  )  d±/n±  number  of  where  U.(B)  1  small,  S  failures  is  then  asymptotically  U(8 ) 0  normal  by  I(8n)  where  I.,  log  log  particular,  covariates),  to  test  U ( 0 ) ' V  -  be  small  and  U ( 0 ) is  B  j  (that  b  K  covariance  to  j  ~  1 , . . . ,  p.  matrix  L a  = 1 , . . . ,  p.  '  failure  compared  a  >  j,h 8  Ho:8 = 0  1  L  =  (8)  jn In  must  98j  w i t h mean 0  -b' estimated  i=l,...,k.  risk.  3  U(Bg)  Breslow  L:  o test  and  ( 1 9 7 2 )  exp(z  1  good  number  to  -  exp(s,8)/  i=l  a  that  laborious  1 0  times  X ( ) 2  p  are  unrelated  to  the  tables  k where  U(0) =  d  i=l  In  the  different variable: elsewhere testing  case  where  populations  the  H Q : 6 = 0  is  ith  n  of  and V . ,  3h  i  the are  z^ c o n s i s t s if  i  survival to a  be one  observation  equivalent  to  distributions  compared, in is  z(lxp)  for is  the  jth  position  in  the  jth  testing  that  = I., ( jh  samples  an and  sample.  all  the  0 ) ,  from  p + 1  indicator zeros In  this  populations  case have  - l i -  the  same  survival  distribution.  U.(0)  k E i=l  =  J  k  (d..  In  this  - n ji  J  d  i  /  case  n  i  )  d. (n. - d . )  n, 'hi  and  where  6j  is  n  n-ji and  the  = number a t  d j ^ = number This  survival  test  g(y)  where  i.e.  that  y = log all  test  Let  p.d.f.  y(i)  distinct)  deaths  at  which  n  jth  time  = 1 if  j  sample  just  t(i)  as  differ  lifetime.  the  y  is  We f i r s t  and  the  = h and 0  iti the  a linear  only  with  prior jth  rank  ith  f(y|z)  g(y -  We w i s h  test  identical.  observation 0 ). = g(y -  consider  the  ^  e  be t h e  the  of  given  z0').  n  from these  the  to  comparing  location  defined  ...  g(y -  x p)  p  p  = 0  indicator  a n d z^-j  regression  this  0 ),  = 0  be an j  by  =  Q  vector  m o d e l we w i s h  censoring: distributions.  observations  observations  for  01 = . . .  z(l  Under  to  test  from sample  c a s e w i t h no  ordered  number  that  Let  is  Then,  p  a sample y i , . . . , y  ^ y(n)  to  9i),  otherwise),  sample.  respect  (p.d.f.'s)  are  0 = (01,...,  of  ^ •••  (6j  be d e r i v e d  distributions  s u p p o s e we o b t a i n Let  in  functions  (Z-JJ = 1 i f  0 = 0 .  delta  s u p p o s e we h a v e p + 1 d i s t r i b u t i o n s  density  otherwise). the  of  can a l s o  1982):  probability  variable  risk  distributions  (Lawless,  z,  Kronecker  = 1 ,••• , p  (assumed t o  be  from d i s t r i b u t i o n  i  ( i = l , . • •, P + D . If can  r  (1 x n ) be  based  is  the  on U ( 0 )  rank  vector  based  (p x 1) whose j t h  on t h e  y-j/s,  element  is  then given  a test by  of  0=0  to  -  U.(0)  where It  p(r;  is  0)  follows  the  12  & loy>(r;0)  =  probability  mass  function  is  z(^)  the  is  jth  the  indicator  element  of  z(jj  variable and t h e  choosing this the  the  parametric  rank  of  p.d.f.  test  will  are  will  procedure  t h e y±'s.  test  scores  a particular  a specific  p.d.f.  ranks  r.  If  test,  g(y).  Then  the  by  are  (  i  )  scores  on t h e  data  the if  o^'s the  y(^)>  given  In  g(y)  are  data  fully  actual  arises  letting  -  1,...  z  ,p  (i)j  by  )  be a s y m p t o t i c a l l y based  j  ±  ^ ( i ) )  rank  be more e f f i c i e n t .  generated  a  associated with  a-^'s  g  define  .  Z ( i )  -g'(y  To  of  that  U.(0) = £ where  -  generated actually  rather  from a d i f f e r e n t the  case  = exp(y -  of  comes  efficient  values  the  e ), v  by  relative than  p.d.f.  logrank  the  from to  the the  test  extreme  the  value  distribution. The mean and c o v a r i a n c e permutation so  that  U(0) U(0)'  Let  asymptotically -  1  This suppose  arguments  E(U(0))=0.  is V  theory  U(0)  to X  test  c a n be  that  there  2  (  P  V be  matrix  (Lawless, the  normal, )  U(0) 1982).  covariance  H :0=O O  can be  obtained  The a ^ ' s  matrix  c a n be t e s t e d  of  c a n be  U(0).  by  by chosen  Then  since  comparing  tables.  extended  are k  for  to  distinct  accommodate observed  log  c e n s o r i n g as lifetimes  follows:  and  n-k  -  censoring  times.  If  y(i),  S ( i ) be  the  let  censored  in  [y(i)>  Prentice  (1978)  v  is  is  z(±)  the  sum o f  (i+l))«  -  indicator  these Then  U ( 0 ) ( p x 1)  13  variable  vectors the  for  score  associated  all  with  individuals  statistic  suggested  by  where  k  V  0  So i n d v i d u a l s a^.  The  taking  (  )  whose  scores  g(y)  obtaining  lifetimes  the  -  e^)  as  so  1  1  n^ = t h e  number  T h e n we o b t a i n  as  at  ( D j  the  J-I.--.P-  }  are  given  different In  scores  particular,  and u s i n g P r e n t i c e ' s  method  logrank  scores  test  via  the  of  1  i .\ nT'  a  risk  )  E [ U ( 0 ) ] = 0.  1  l  i  a  censored that  to  i . ,^7 - > E  + s  before,  and a^ l e a d s  =  i  a  are  may be c h o s e n  = exp(y  a  where  ( D j  z  =  just  prior  to  t(^)  = exp(y(.jj.)  before k  Uj(0)  with  notation  obtains  a  obtained scores  before  in  the  the  equivalently, proportional  for  censoring  in  under  testing  hazards  or  all  for of  letting  is  n  j  ±  d /n ) i  g(y)  an e x t r e m e  Lehmann samples.  of  family This  U(0)  Either model  = exp(y  asymptotically  equality  of  U(0). Cox's  j=l,...,p,  1  expectation  context  test  differences  The  by  -  j ±  variance  were m o t i v a t e d  location  (d  before.  permutation  distribution,  equal  as  = -2^  fully  value  was  or  the  c a n be u s e d . -  e^),  the  for  Since  for  log  no  derived  the value  detecting  lifetimes  distributions is  variance  extreme  efficient  when t h e r e test  Prentice  0.  this  model  lifetime  is  in  a  censoring under  the  or  or  -  assumption  b.  of  no t i e s  -  may be u s e d w i t h  a small  number  of  ties.  Test  Wilcoxon  The W i l c o x o n Wilcoxon  but  14  test)  ( o r as  can a l s o  it  is  sometimes  be d e r i v e d  as a  called, linear  Prentice's  rank  test  generalised  of  the  form  k U.(0)  =  ^  for  testing  equality  are  defined  as  with  notation  Substituting  of  ( z  i  )  .  a  lifetime  (Prentice,  as  (  + s  ±  (  i  .  )  J-1.....P  a ) ±  d i s t r i b u t i ons.  In  this  case  the  scores  1978):  a,  = 1 -  2  n (n. j =1  a,  = 1 -  n j=l  (n. J  -  d. + l)/(n.  d. J  +  1)  + l ) / ( n . + 1) J  before.  these  scores  in  Uj(u)  gives n. .  U.(0) j  where  F,  An e s t i m a t e (1979))  is  of  the  =  = -E  i II  F,(d.. i j i  (n. — -  covariance  -  d. + ^  matrix  d,  )  j=l,...,p  i 1)  of  U(0)  (see P r e n t i c e  and  Marek  V where  TT V., = Jh  k y £  d. (n. „2 i i F. — 7 i (n^ -  d.) n.. i Ji rr— - — 1) n^  , 16., jh  ( f  n, . hi^ - - — I J  J  i,h=l,...,p. ' • ,  v  -  Again U(0)' The e ) , y  scores  the  2  V^lKO) in  for  detecting are  weight  to  all  terms  weight  to  earlier  c.  of  Logrank Test  In  the  because  case -  that  the  it  the  actual  falls.  thus  to  This  an i n t e r v a l .  continuous  (1 -  X, ) 1  p  hazard  (  z  p  )  ,  fully  gives  equal  gives  more  this  test  Hence  Again  it  derived  log  rank  test  was a s s u m e d  is  give  rise  not  under  is  good  at  the  a s m a l l number  of  ties.  in  of  the  to  test  assumed  that  only  represents  (1 -  X.) l  at  if  It  only  the  hazards failure  [-/  that  in  just model  in  the  covariate  1  a.  .  it  is  interval  different  occur  x-£ f o r  were p r e s e n t may i n s t e a d  model but  obtained  to  = exp  context ties  model.  a slightly than  the  that  recorded,  contribution  where  test  ones. was  +  Data  time  x±  rank  later  from a continuous  if  = e^/(l  underlying  the Wilcoxon  from a d i s c r e t e  is  g(y)  asymptotically  log  Assuming a p r o p o r t i o n a l  the  X  the  may be u s e d w i t h  different  taking  when t h e  arose  then,  by is  it  Censoring  e  tables.  )  model  would  data,  a^),  to  on.  the  arises  a slightly  of  but  p  shifts  d^n^/n^),  than  of  (  test  Whereas  Grouped  survival  described.  1 -  data  data  be  [a^-i,  for  hazards the  So t h i s  early  derivation  proportional was  events  2  -  generated  location  (dj^ -  no t i e s ,  to X  are  logistic.  differences  assumption  case  density.  distributions  detecting  compared  this  logistic  efficient  is  15  X (u) o  this  be  the  grouped  into  which  likelihood  and  a,  now  as w i l l  prior  to  the  for  the  ith  interval,  z  end  is  du]  and X  o  (t)  -  is Let  the base-line y  ±  hazard  16 -  function.  = log [-log ( l - \ ) ] ,  1=1,..,k.  ±  Then t h e l i k e l i h o o d i s k L(y,B)  =  £ i=l  where  E log{l 1€D  (  - exp[-exp(Y,  + z B)]} -  £ 16 R  1  ±  i s the s e t of l a b e l s  attached  a n d R^ i s t h e s e t o f l a b e l s  attached  or  x^.  observed t o survive  where U . ( y , 8 )  y at 6 = 0 .  past  = -rs— l o g L  to individuals  to individuals  A test  o f Ho:8=0  exp(y,  0) i s asymptotically  x  failing  a t x^  c e n s o r e d a t x^  i s based  on u ( y ( 0 ) , 0 )  a n d y(0) i s t h e maximum l i k e l i h o o d  Again u(y(0),  + z B))  1  estimate of  n o r m a l w i t h mean 0 a n d 2  covariance  Suppose for  matrix  that  there  t h e samples  UJ(Y(0),  estimated  E n /d ±  and t h e e l e m e n t s  jh  E  (  x  n7  ) n  i  ji  (  j h"  6  J  U(y(0),0)'  l  notation V  -  1  (  n  i s as b e f o r e .  U(y(0),  d . / n . i s small then i l  When z i s a n i n d i c a t o r  )  n i  variable  ( d  j  ±  - d  n  ±  j ±  /n )  j=l,...,p  ±  matrix are  i  n  h i  "  /  n  i  d  =  i  )  '  j  } r  ,  (  „ l o g ( 1  The t e s t  d  i  -j— log(l i  d  h  =  1  P  i ^ 2  -S7»  statistic  0) w h i c h c a n be compared n  If  i  ± -— a :  where  •  i  n  and o t h e r  - d./  of the covariance  =  ^JJJCYJP) =  that  log(l  ±  q  V  0 ) where  are p + 1 samples.  i t follows  0) = -  by l ( y ( 0 ) ,  is  to X ( ) p  2  tables.  i ) » -1 and t h e t e s t i s i  -  approximately  Application  In actual they are  our  the  of  the  also  Censoring  the  at  the  same  is  is  due  end of  censoring  more to  not  clear  differences  appropriate  to  The  logrank  test  the  approximation  of  d^/n^ a r e  only  only  the  to a l a r g e  to  study  pattern  in  a.  which only  has  the  number  a n d d^/n^  in  use  the -  the  all  is  of  been  grouped -  interval ties.  rarely  into  which  However  bigger  the  there  than  0.2  the  the  test  a. to  values  the of  of  the  the  d-^/n^.  I  are  censoring.  survival or  that  is  in  still  So we  is  is  in  not  more  it  have  known will  more  be  whether  our  more  data.  model. of  be  to  would  derivation  so  sensitive  grouped  a discrete  clear  test  sensitive  be u s e d  the  not  is  a s we h a v e  hand assumes  It  is  the Wilcoxon  to  c.  used  times  the Wilcoxon  logrank  described other  some b a r n a c l e s  type  test  test  likelihood  sufficiently  is  however,  logrank  on t h e  that  samples.  logrank  on w h e r e a s If  fact  this  distribution  early  later.  small  data  recorded,  risk  We do k n o w ,  differences  assumed  at  whether  powerful.  test  smaller.  The u n d e r l y i n g it  not  rise  number  much  logrank  continuous  are  gives  a large  and u s u a l l y  alive  times  This  the  Tests  c a s e we h a v e  survival  fall.  same a s  -  17  Also  this  test  values  small.  T h e W i l c o x o n was  derived  an a p p r o x i m a t i o n  when,  as  under in  the  our  assumption  case,  a  large  of  no  ties  number  of  and ties  is are  present. \  -  Each equality  test  was a p p l i e d  of the s u r v i v a l  18 -  to various  distributions  grous  of l i n e s  of the l i n e s ,  to test with  the  the  following  results:  Logrank Lines  Compared  Logrank  Wilcoxon  2 01,02,03,04,05,05  X  07,08,09.0io,0n,0i2  X  Dl,D  X  5  2 =  34.7  X  =  28.9  X  =  32.5  X  = 129.8  X  = 129.8  X  5.8  X  2 5  Dl.D2.D3  X  Ni,N  2  X  D ,N  2  X  3  All  p-values  N2.  Here  gives  2 2 2 1  2 1  =  a r e < .001 except  the logrank  test  the hypothesis  strongly large  rejected.  amount  different  systems,  X  =  36.6  X  5  =  29.3  X  1  2  = 164.2  X  2 2  = 301.6  X  that  1 2  12.6  X  1  obtained  f o r grouped gives  data  =  gives  t h e same s y s t e m ,  appear  1  =  32.9  2 2 1  = 126.7  = 196.2  1  5.1  =  t o be much more  lines  of  from  On t h e o t h e r  of d i f f e r e n t  similar  is  due t o t h e  when l i n e s  are compared.  logrank  a r e t h e same  possibly  occur  a n d N3,  o f D3 a n d  F o r each group  small  f o r D2  33.6  p = . 0 2 5 , the  are very  distributions  =  2  The p v a l u e s  Ni a n d N2,  5  2  distributions  f o r example  35.9  i n the comparison  p < .001.  p-values  =  2  survival  The s m a l l e s t  5  2  their  of d a t a .  the s u r v i v a l  within  that  31.8  2  p - .017 and t h e W i l c o x o n  lines,  hand  1  5  for data  2 =  2  2 2  grouped  judging  types  by t h e  -  relatively more  small  important  distribution. same  type  over  these The  result  to  the N  2  is  the  always  the  Having  dying  the  this.  Firstly  itself  and  to  type  is  suggests in  that  s y s t e m may be  determining  exist  clearly  the  even between  survival  lines  not  reasonable  the  two l o g r a n k  a  to  of  the  collapse  the  results  the  comparison  small  tests  but  for  than  the  other  is  better  1.4.  Later  could of  the Wilcoxon  gives  a  three  tests.  detecting lie  in  these  6 and 7 on l i n e on t h e  could  differences  Ni  is  discrepancies  it  For  23.7  are  gives  reflect  early  cases.  be  similar  comparisons This  is  This number  last  of  and N2. large  two  differences  Ni  tests  the  the  at  of  spite  comparisons  b e t w e e n weeks  is  in  of  on, example  whereas  not  the  as  on  great.  Models  nonparametrically,  having  it  quite  biggest  compared  fit  in  three  p-value  percentage  survival  except  the Wilcoxon where  to  between  first  Exponential  model  This  differences  s y s t e m and  logrank  percentage this  a  small  the  that  2.2  case  -  groups.  a much s m a l l e r  which  In any  d-j/n^ i s For  fact  than m a t e r i a l  discrepancy  ties.  value.  factor  within  surprisingly because  chi-squared  19  the  it  survival  was  then  survival  the  secondly  shape  interest  curves. of  the  for  large  survival  different amount  to  T h e r e were  under a p a r a m e t r i c  distributions compare a  of  distributions  of  lines data,  look two  curves  model  for  the  for  a  was  of  task  lines  parametric  reasons  w o u l d be namely  different  for  doing  interest of  comparing  simpler  number  in  of  -  instead  deaths  and  of  - 20  -  number at r i s k at each time, we would j u s t have to compare a s m a l l number of parameters, f o r example s l o p e . With so much data i t was  likely  t h a t any  r e j e c t e d by a g o o d n e s s - o f - f i t t e s t .  T h i s shouldn't matter p r o v i d e d  the model captures the most important Because of i t s s i m p l i c i t y model.  The  starting  e x p o n e n t i a l was  at t=0.  p a r a m e t r i c model would  f e a t u r e s of the  the e x p o n e n t i a l was  first  fitted  that  data.  the f i r s t  to each of the 17  So the f o l l o w i n g model was  be  c h o i c e of  lines  assumed:  -Bt P(T  > t) =  e  t > 0  0 where T = l i f e t i m e  otherwise  of b a r n a c l e s .  For each of the 17 l i n e s the maximum l i k e l i h o o d obtained. to=0  of 8  was  N o t a t i o n i s d e f i n e d as f o l l o w s : ti=5  d^ deaths  d  2  t2=6  ts=14  deaths  d  Under t h i s model the l i k e l i h o o d  S i n c e there were no deaths p r i o r = 0 and  the f i r s t  probability  ....  deaths  - e'**')*  to t i = 5 weeks on any  term i s i d e n t i c a l l y one.  c o n t r i b u t i o n of the censored  9  tg=17  N-Zd^  i s p r o p o r t i o n a l to  L - (1 - e ^ f l ( e ^ l - e - e ^  di  estimate  observations:  The  9  (e"^  of the  9  lines,  l a s t term i s the  under our model the  t h a t a b a r n a c l e s u r v i v e s at l e a s t  u n t i l the end  of the  study  - 21 -  at t  9  = 17 i s e ' P ^ .  The number t h a t do s u r v i v e t h i s long i s N-Edi  where N i s the i n i t i a l number on the l i n e .  To o b t a i n the maximum  l i k e l i h o o d e s t i m a t e of B, note t h a t -Bt I =  d  Z  i ^ i - l ~ * 'Bt (e e  a i 2  o g  i  L  =  2  I  . ,  E  d |-e  i TBT Bt - e +  B t  96  -Bt  L  j  t  -P<V*i-i>  )  e  9  )  ^i-i (e  2 ^ ^  o  oB  g  L  t ( N - Ed.) x  ,  }.  - e  i s c l e a r l y < 0 f o r a l l 8 s i n c e the d  )  a r e >^ 0 (and c l e a r l y at  2  least  one d  i s > 0).  So the t u r n i n g p o i n t a t the s o l u t i o n t o ^ ^ °  1  is  g  L  = 0  Op  8, the maximum l i k e l i h o o d e s t i m a t e of p.  T h i s was o b t a i n e d by doing a Newton Raphson i t e r a t i o n :  Pn+1 J.1 M  and  r  P n  - U np )/ UnB ) r  K  8 = 1/T L  where T = median s u r v i v a l time  (not exact due to d i s c r e t e n e s s ) and  ' _ ologL  _ o logL 2  When convergence to d e s i r e d accuracy i s achieved -L(B) F i s h e r i n f o r m a t i o n and leads to an estimate  of the v a r i a n c e of B:  Var 8 = [ - U p ) ] , SE(B) = [Var p ] - 1  i s the observed  1 / 2  A f t e r o b t a i n i n g 6, two goodness of f i t t e s t s were c a r r i e d out t o check  - 22 -  the f i t of the model for each l i n e .  1.  Pearson's Goodness o f F i t T e s t  X  k+1 (0 - E ) = £ —i= i i=l  E  2  i  where 0^ = observed number of deaths i n i t h i n t e r v a l and E^ = expected number of deaths i n i t h i n t e r v a l Under our model the estimate of i s the number of time i n t e r v a l s .  X  -8t i s E^ = N(e  -Bt - e ) and  2  2  i s compared to X (k- ) tables s  where s = number of parameters estimated i n the model (=1). too large the f i t i s poor.  k+1  If X i s  I t wasn't necessary to combine any i n t e r v a l s  as f o r a l l lines the expected frequency i n a l l intervals was at least 5.  2.  X  2  L i k e l i h o o d R a t i o Goodness o f F i t T e s t  = 2  k+1 0, i 2 S 0.1og(—) i s compared to X,. . tables where i=l ± Ck-s; 1  h  notation i s as for 1.  Results Results are given i n Table 2.  Comparing to chi-squared tables with  eight degrees of freedom, the p-values for the goodness-of-fit tests are a l l < .001.  Clearly the model doesn't f i t .  The largest residuals were  found to be at the beginning which i s not surprising since the model gives a positive probability of death for t € [0,5] but no deaths were  Table  2 -  Results  of  8 ± SE(8)  Fitting  Exponential  Pearson's  Model  X  2  Starting  at  t  Likelihood  0i  0.0520 + 0.0047  299.3  242.7  0  2  0.0506  + 0.0048  221.5  192.4  0  3  0.0672  + 0.0069  263.7  216.8  0.6540 + 0.0067  187.7  167.0  05  0.0623  + 0.0055  251.9  226.2  o  0.0374  + 0.0031  232.2  215.5  Dl  0.0500  + 0.0021  368.1  380.4  D  2  0.0680  + 0.0028  94.6  106.8  Ni  0.0638  + 0.0025  506.4  477.0  0  0.0444  + 0.0026  183.3  154.6  0.0391  + 0.0041  500.1  508.6  0.0455  + 0.0025  218.7  238.8  OlO  0.0475  + 0.0051  905.4  844.5  Oil  0.0432  + 0.0022  1168.2  1008.0  0l2  0.0318  + 0.0022  743.9  402.8  D3  0.0402  + 0.0016  1692.0  1419.0  N  0.0367  + 0.0013  1707.3  1976.0  7  o 0  6  8  9  2  = 0.  Ratio  X  2  - 24 -  a c t u a l l y observed  i n this period.  So the same model was then  fitted  s t a r t i n g at t=5:  -B(t-5) t > 5  P(T > t ) -  otherwise  «  L e t t i n g t ' = t - 5, we have the f o l l o w i n g s i t u a t i o n t'  - 0  t  {  d i deaths  - 1  d  t'  deaths  2  - 9  d  8  t ' - 12  deaths  N-Zd  For t h i s model the l i k e l i h o o d i s -Bt' d L = (l - e ) where t j = 1 ,  -BtJ -Bt' d ,"2 (e - e ' J  t£ = 2  t£ = 6, t j = 9 , -Bt;.,  so T _ d.C-t! ,e ologL i i i -i i - i -at-1 96 " _. . -Bt! 8  v  4  i=2  v  (e  , " [e  B  t  J  " e  t£ = 12.  -pt'  B t  8> 8 , J (e d  t  J  8 ^  M  l  Now  -BtJ  + t' e  Z|tT - e  B  J  (1 - e  - tJCN - E d . ) .  ')  The new maximum l i k e l i h o o d e s t i m a t e s of p were o b t a i n e d and the goodness-of-fit p-values .001.  t e s t s repeated.  R e s u l t s appear i n T a b l e 3.  f o r the t e s t s are much l a r g e r t h i s  The f i t i s much b e t t e r but s t i l l  b e t t e r f o r the o y s t e r l i n e s  The  time but a r e again a l l <  poor.  The model appears  than f o r the dowling  or n e t t i n g  to f i t  lines,  though t h i s might be because there i s a s m a l l e r amount of data f o r the oyster  lines.  I t was p o s s i b l e that a b e t t e r f i t might be o b t a i n e d by s t a r t i n g at some point  other than t=5.  So the same model was f i t t e d  again, t h i s  time  Table  3 -  Results  of  Fitting  B ± SE(B)  the E x p o n e n t i a l Model  2 Pearson's  X  Starting  at  Likelihood Ratio  Oi  0.0927  + 0.0084  94.5  103.0  0  0.0883 + 0.0084  62.5  68.9  03  0.1290 + 0.0130  77.2  94.3  0  0.1240  + 0.0130  40.6  45.9  05  0.1160 + 0.0100  59.3  69.7  o  0.0613  + 0.0050  69.2  70.5  0.0867  + 0.0037  80.8  79.8  2  0.1298  + 0.0054  17.9  18.3  Nl  0.1200  + 0.0047  148.1  141.9  0  0.0740  + 0.0043  58.3  59.2  0.0630  + 0.0065  127.7  136.9  0.0755  + 0.0042  54.1  53.6  OlO  0.0818  + 0.0087  221.8  235.0  Oil  0.0713  + 0.0037  242.5  253.5  0l2  0.0497  + 0.0034  171.8  187.8  D  3  0.0656  + 0.0026  495.5  599.4  N  2  0.0588  + 0.0021  681.4  639.8  D  2  4  7  o 0  6  8  9  t = 5  X  2  - 26 -  w i t h two unknown parameters - the s l o p e parameter parameter  a - both of which a r e t o be determined by maximum  Since the l i k e l i h o o d A  find  6 and the l o c a t i o n  i s different  for a i n different  likelihood.  time i n t e r v a l s , t o  A  a and 8 i t i s n e c e s s a r y t o determine a - p r i o r i i n which i n t e r v a l the  maximum l i k e l i h o o d  e s t i m a t e of a l i e s . A  F o r t u n a t e l y i t was p o s s i b l e t o determine a - p r i o r i t h a t a would have to l i e i n the i n t e r v a l one death i n ( 5 , 6 ) .  (5,6] assuming  no deaths i n (0,5] and a t l e a s t  In our case these c o n d i t i o n s are s a t i s f i e d s i n c e no  deaths a r e observed i n (0,5] but on every l i n e the f i r s t  deaths a r e  observed i n ( 5 , 6 ] . A  Proof that For  a € (5,6]  a e (0,5] and no deaths i n ( 0 , 5 ] , the l i k e l i h o o d i s 8 - 8 ( t - a ) ( N - E d.) i=l 9  i-1  0  t2=6  ti-5  0 deaths  1  e  t =17  t -14  9  8  d^ deaths  d  8  deaths  Then  l o g L j = BctN - 8 t ( N 9  8 9 E d ) + E d i=l i=2 1  = G( 8) + pctN  1  -  -Bt . log(e 1  i-1  - e  )  N-Ed.  -  w h e r e G(B) Since  i s a function  8 > 0,  So a >_ 5. a zero are  logLi  probability  a €  increasing  a <^ 6 s i n c e  of death i n  in this  -  6 independent  i s monotonic  Furthermore  observed  of  27  in a for a £  (0,5].  a > 6 w o u l d mean t h a t  (5,6],  interval,  of a .  b u t we h a v e  t h e r e w o u l d be  assumed t h a t  a ^ 5 a n d a _< 6 t o g e t h e r  deaths  imply  (5,6].  m.  Determination of a For  a £  (5,6] the l i k e l i h o o d  is  8 -8(t -a) = (1-e 2  L  2  9  J  )  n  d l  -6(t, ,-o) (e 1  1  -B(t.-a) e  1  d. )  1  .  -B(t -a)(N-  £ d,)  9  e  1  1  1  i=3  to-o  t =5  t =6  x  4  I di  Since  we h a v e  appropriate  ?,2, 9  l  o  i  i  dg d e a t h s  already  ascertained  dV<l Q  2  L  i  9  that  oc€ ( 5 , 6 ] ,  this  is  N-Ed  the  likelihood.  T g  t =17  8  deaths  ^ i £ | i 2 = B (N -  and  t =14  2  2  = "P  2  , d  ^  da  -  l _  e - P  (  T  2  _  A  )  ) )  -p(t2~a) .  e  e  -8(t2-a)j2  2 Since  &  l  a  o  da  g  2  L  2  < 0 f o r a l l a and 8 > 0,  the s o l u t i o n  to  a i  °g 2 L  ba  = 0  i  -  yields  a  local  maximum.  terms  of  So i f  we m a x i m i s e L  28  = 0  -  c a n be  solved  explicitly  for  a  in  8:  over  2  a and  8 simultaneously  1 ,_rN  -  we  obtain  d  P where  8 is  t h e maximum l i k e l i h o o d  l o g f — <  Case  0,  we h a v e  a  estimate  of  6.  Since  t2=6,  8 > 0  and  < 6.  1 If  a  m  e  (5,6],  (<===>  *  ln(  "  N  d  N  *) 6  (-1,0])  since  it  then  a  m  is  the  8 maximum l i k e l i h o o d  SlogL ^ da  k  = 0,  0  '  estimate,  2. logL - 2  ?  ^ Q  a n (  a, j  of L  a, is  2  the  is  the  appropriate  solution  to  likelihood  for  a  oa in  this  Case  interval.  2 If  a  m  < 5 (<===> v  — *  ln(  N  „ N  d  J-)  < -11, '  then  the  solution  a  m  is  not  the  maximum  P likelihood  not  the  estimate  appropriate  likelihood  estimate  of  a.  It  likelihood is  a = 5.  is  f or  a K 5.  In  to  &  this  1  °f 9a  L  2  case  = 0 but  the  true  L  2  is  maximum  - 29 -  Proof For g i v e n p, ^ ^ ^ 2 . = Q to l i e i n ( 0 , 5 ] .  n  a  s  a  u n  ique  s o l u t i o n , a , which i s assumed m  We have shown t h a t the t u r n i n g p o i n t i s a maximum and  t h e r e f o r e logL>2 i s monotonic d e c r e a s i n g i n a t o the r i g h t of a A  particular  f o r a € (5,6],  m  and i n  /\  A  So a <^5.  But we know t h a t a &  (5,6],  A  So a = 5.  D e t e r m i n a t i o n of 8 logL2 was i n i t i a l l y maximised over a and 8 s i m u l t a n e o u s l y to o b t a i n and 8 .  T h i s was done v i a a Newton Raphson i t e r a t i o n as f o l l o w s :  m  Let L U e Z  a  =  5 1  °g 2 5a ' L  L  8  Q logl2 2 »  =  a i  2  =  L  Q a  ap  °S 2. 58 » L  =  a 2 l  °g 2 5a5p L  L  '  = pp  a 2 l  og 2 2  .  L  b?  I n i t i a l e s t i m a t e s were g i v e n f o r a and p.  Then the i t e r a t i o n -1  (a  ,p ) = ( a ,8 ) - (L ,L ) - n+1 n+1 n' n a p L * n' n K  fl  r  ( a  P  L  ) L  L aa ap ap pp  /\  A  L  V n>  (  B  was used u n t i l convergence t o d e s i r e d accuracy was a c h i e v e d .  Case 1 Suppose we o b t a i n a  m  £ (5,6].  Then a and p are the t r u e maximum l i k e l i h o o d e s t i m a t e s a and p. m m In  t h i s case an e s t i m a t e of the c o v a r i a n c e m a t r i x of ( a , p) i s g i v e n by:  -  30  -  _1 L  L  the  inverse  of  the  o"  L  aa aB  L  aB BB  observed F i s h e r  (a ,P)  information.  In  our  case,  since  A  = 0 c a n be s o l v e d  2  have  substituted  had j u s t  for  explicitly  a in  one e q u a t i o n  in  terms  of  to  find  8 in  one u n k n o w n .  a = a ( B ) , we c o u l d  ^°f^^  =0.  op  This  could  instead  T h e n we w o u l d  have  been  have  s o l v e d by  a  A  Newton R a p h s o n A  iteration  to  find  A  a = a(B)  and  the  B8  "  variance  L  ( L  Case  standard  error.  Then  aa  L  L  of  a would have  been e s t i m a t e d  by  ' ) | * * 2  ~ 88  8 and i t s  A  _1  p0  (a,6)  2 S u p p o s e we o b t a i n  a_ < 5 . m  r  maximum l i k e l i h o o d  Then  a  and m  estimates.  As  already  8 are "m  shown,  not  the  a = 5.  true  Knowing  that  A  a = 5, that t  we c a n t h i n k  a = 5.  = 5,  This  of  a as  means  we a l r e a d y  fixed.  that  had the  We m a x i m i s e w i t h  when we f i t t e d  best  possible  fit  the as  respect  exponential t  = 5 was  the  to  8  given  starting best  at  point  A  at for  which case  to  start  1 the  .  So  variance  8 is of  as  6 is  obtained estimated  in  this  by - L  values  starting shouldn't  at  t  are  the  = 5.  affect  the  same a s  We h a v e  analysis.  (L  -  aa  K  The X  previous  those  obtained  one d e g r e e  conclusions.  of  in  the  L„„ aa  pp  previous  freedom fewer  but  L „  As  2N-1 )"  ap  (a,B)  analysis this  -  31  -  Results A  In  those  maximising slightly  cases  over  that  t  these  lines  which  X  values.  = 5 was n o t are  as  This  For  the  was  slightly  improvement  seven  ideal  fit  of  place  the to  was  reflected  seventeen  start.  improved  The  lines  A  6 ± SE(8)  in  it  results  for  Likelihood X  A  a  ± SE(a)  Pearson's  X  76.1  77.8  0.0646 + .0071  5.21 ± .279  16.9  17.8  0.0805 + .0046  5.47 ± .112  129.2  130.8  Oil  0.0774 + .0041  5.61 ± .089  99.7  114.2  O12  0.0541 + .0038  5.72 ± .107  37.3  36.7  Ni  0.1281 + .0054  5.36 ± .079  454.2  584.8  N  0.0624 + .0019  5.81 ± .152  682.1  580.3  o  8  0  9  2  Values for  of  the  dowling  on t h e s e relates  2.3  a are  lines to  generally lines start  the  rate  smaller  than dying  at  I n f e r e n c e on t h e  for  for  other  sooner.  which  the  the  s y s t e m one  lines.  It  However  the  barnacles  die  oyster  appears scale off,  Ratio  2  2  5.24 ± .147  7  the  was  0.0763 + .0047  o  by  follows:  A  A  a > 5 the  two p a r a m e t e r s .  smaller  found  for  is  that  lines  barnacles  parameter of  and  more  8,  which  interest.  p's  A  Although statistic  and  the model  contains  a  doesn't lot  of  fit  very  information  well, about  8 is  a good  the  lines.  summary So many  -  analyses  relating  analyses  were b a s e d  two p a r a m e t e r s .  to the 8 values  then  carried  of 6 obtained  P r e v i o u s l y we h a d c o m p a r e d  certain  groups  to  compare  the d i s t r i b u t i o n s  by  testing  whether  as  were  on t h e e s t i m a t e s  of  underlying  32 -  of l i n e s  8 value.  This  lines  over  distributions  Now we were  o f t h e same g r o u p s  was done b y means  A l l of these  by m a x i m i s i n g  the s u r v i v a l  nonparametrically.  a l l s y s t e m one o y s t e r  out.  of l i n e s  i n a position  parametrically  ( s a y ) h a d t h e same  of the l i k e l i h o o d  ratio  test  follows: To t e s t  81=82=83=64=65=  H : 0  8  6  o  2 [ l o g L ( j 3 , a) rejected slope  -  l o g L ( j 3 , SLO^  i f this  statistic  f o r the*jth  provided  that  i  0  oyster  samples  s  compared  i s too large.  line  are large,  which  they  A  imposed by H o , l o g L ( j 3 ,  a) = v a l u e A  maximum l i k e l i h o o d at  estimates  t h e maximum l i k e l i h o o d  obtained  L(j?,  a) i s t h e l i k e l i h o o d  of obtaining  each  line  follows  Bj  i f the j t h l i n e  a n d otj w h i c h may be d i f f e r e n t A  This  test  a r e . Here  Q  is  the true  is  valid  q = number o f  of l o g l i k e l i h o o d  at the  =  value  of l o g l i k e l i h o o d  u n d e r HQ.  In our case  t h e o b s e r v e d numbers  f o r each  o f d e a t h s on  model w i t h  parameters  separately  ( w h i c h we  line.  A  logL(j3, jx) we m a x i m i z e  have a l r e a d y  done)  f o r each  a n d a d d t h e maximum v a l u e s  L(_B_o» J*o) I s t h e l i k e l i h o o d  parameters  B j denotes  the exponential  So t o o b t a i n  on e a c h l i n e  and H  A  a n d logL(j3o> jxo)  estimates  tables  Here  on s y s t e m o n e .  A  constraints  to X  i f the j t h l i n e  8 and a * .  of o b t a i n i n g follows  So a l l l i n e s  line  of the l o g l i k e l i h o o d .  t h e o b s e r v e d numbers  the exponential  must  have  model  of deaths  with  a common B b u t t h e i r  -  location cannot  parameters  0  ~Bt _ 1  £ i=3 +  d,  d^j  Nj  = initial  = number  of  ,  1  time  -Bt.  1  log(e  -  the  six  parts  8  e  ) -  6t9(N  - E d . ) i=l  1  1  B[ai(Ni -  + d where  This  separately.  9 <xo) =  -  ai,...,06 may be d i f f e r e n t .  be m a x i m i z e d  logL(8 ,  33  log(l  n  number  deaths  -  of  at  + a (N  d u )  2  e -  e  (  t  2  _  a  i  barnacles  time  t^  -  2  )  )  d +  i 2  )  ...  + d  on l i n e  on l i n e  + ...  1 6  + a (N 6  log(l  6  -  e "  die)]  B  (  t  2  "  a  6  )  )  j ,  j ,  6 d^=  In  £ j=l  d  this  = total  number  case,  of  deaths  0 c a n be  =  at  solved  time  to  i  and N=Nj^ + N  + ..  2  +  N . 6  give  i  a.  = t  2  , + j  N. log(  -  d, . )  J  j=l,  J  N  2,...,6.  2  This  is  clearly  This  can  be  a maximum s i n c e —  substituted  in  9  ^°  g  L  op  <  to  0 for  all  a,  i  8  L  -^f-= °P  -t (N 9  1  =  9  E i  d  ) +  oB  2  e  i~-  E  8 =  A Newton R a p h s o n i t e r a t i o n  <^Sk  d . , ( t ,  1 i  =  I i=3  d l  3  8 > 0.  give  -Bt a  and  (e  t  i_l  1  -Bt -  -  t  e  e Pt  1  A  , ) + t (N 2  1  )  using  h-i  -e  -  1  e  6  t  -  c  i  i - i  -  e  B  t  i  -  d  l  ) .  - 34 -  which i s  clearly  < 0 then y i e l d s  8.  It  is  then  straightforward  to  A  obtain  aj, j=l,  2,...,6  and t o s u b s t i t u t e  t h e maximum A  estimates  into  logL(J3o >ato) t o o b t a i n  likelihood  A  logL(8_o,  ap).  Results 1.  Test  HQ:  that  all  A  2[logL(£, Refer  2.  to X  5  T e s t Ho:  that  3. P  to X  Test  5  HQ:  p ^  =  D  p ^  lines  have  t h e same 8 v a l u e .  A  0  «0)1  = 2(-4740 -  (-4753))  = 26.  .001. that  = Pn > 3  all  the dowling  lines  have  t h e same  value. A  A  2[logL(j3, Refer  4.  = 135.  .001.  logL(j5 ,  P «  t h e same 8 v a l u e . (-2257))  0  s y s t e m two o y s t e r  o) -  have  o ) ] = 2(-2190 -  A  tables.  2  0  A  2[logL(p,  lines  A  logL(£ ,  P «  all  A  Refer  A  a) -  tables.  2  s y s t e m one o y s t e r  A  A  a) -  to X 2 t a b l e s .  T e s t Ho:  p  = p  n u  Since  used,  that  provided  ,  n  u  l  same P v a l u e .  logL(po,  P «  2  A  oo)] = 2(-5648  large.  (-5759))  = 222.  .001.  that  t h e s y s t e m one d o w l i n g  lines  have  the  2  we a r e c o m p a r i n g the sample  only  from which  two p v a l u e s , p,, u  reasonably  -  In our case  it  is  very  a n d p_ u  l  a t-test  c a n be  are estimated  is  2  large.  The use of a  if  a maximum  t-test  t  c a n be j u s t i f i e d  as f o l l o w s :  in general,  9 is  likelihood  A  estimate  of  9,  then  9 is  asymptotically  n o r m a l w i t h mean 9 a n d  - 35  9  -  2  covariance matrix  {E[  2  ]} 90  which can be e s t i m a t e d by  -1  2  2  T h i s a r i s e s from the c e n t r a l l i m i t  theorem a p p l i e d t o the sum A  c o n t r i b u t i o n s to the l o g l i k e l i h o o d .  and  S i n c e £L.  8 /(SE(8  T e s t Hn:  5.  8  Ni XT  and  6_  l  D  are maximum  2  ""8  ^1  ^2 A  rejected with p <  the  s i n c e t h e i r v a r i a n c e s are e s t i m a t e d , under HQ  f o r l a r g e samples,  compared to t - t a b l e s .  of  A  D  l i k e l i h o o d e s t i m a t e s and  {—5—i£li. 50 0  D i  )  2  +  might reasonably  be  A  s*%y)  T h i s s t a t i s t i c works out to 6.6  and HQ i s  .001. = 8„ , that the two  N  n e t t i n g l i n e s have the same 8  2  value.  = 11.5 A  A  /(SE(8  N i  )  2  SE(8  +  N 2  i s r e f e r r e d to a t - d i s t r i b u t i o n .  Again,  )^  and HQ i s r e j e c t e d .  p < .001  D i f f e r e n c e s were found the system two  i n every  set of 8 v a l u e s .  The  6 values f o r  o y s t e r l i n e s have a much s m a l l e r range of v a l u e s than  the  2 v a l u e s f o r the system one value.  The  non-parametric  t o be d i f f e r e n t l i n e s and  o y s t e r l i n e s as i n d i c a t e d by the s m a l l e r X t e s t s a l s o found  for different  the system two  l i n e s but found  well.  distributions  the system one  o y s t e r l i n e s to be approximately  homogenous i n t h e i r s u r v i v a l d i s t r i b u t i o n s . a r i s e from the f a c t  the s u r v i v a l  equally  This discrepancy  that our p a r a m e t r i c model doesn't  oyster  could  f i t the data  very  -  Other Parametric  The next.  Models  t h i s model has  parameter i s i n c l u d e d ) and  was  that we  not  two  i s a g e n e r a l i z a t i o n of the e x p o n e n t i a l ,  reasons.  difficult  Weibull  the  ( i n f a c t any model would  - i t would not be c l e a r which parameter t o focus on,  be  and  to o b t a i n with so many parameters  and  Random E f f e c t s Model  Instead be  be s l i g h t  it is  Secondly, the subsequent i n t e r p r e t a t i o n would  meaningful r e s u l t s would be hard  2.4  However the  F i r s t l y , w i t h so much data,  improvement i n f i t would probably be r e j e c t e d ) .  of a model to f i t  parameters (or three i f a l o c a t i o n  would o b t a i n a somewhat b e t t e r f i t .  pursued f o r two  probably  -  W e i b u l l would have been the n a t u r a l choice  Since  likely  36  of l o o k i n g at these p a r t i c u l a r  of more i n t e r e s t  to t h i n k of these  p o p u l a t i o n of l i n e s . lines  of each type  subpopulations interest  So we  l i n e s as a sample from an  infinite  could t h i n k of an i n f i n i t e p o p u l a t i o n  ( m a t e r i a l ) w i t h i n each system.  t h e r e would be a t r u e u n d e r l y i n g  to compare the u n d e r l y i n g 8 v a l u e s  "treatments"  seventeen l i n e s , i t might  (types of m a t e r i a l ) and  For each of  8 value.  f o r the  f o r the two  of  these  I t would be  of  different  systems, by  examining  A.  the  8 values  obtained  f o r our sample of seventeen l i n e s .  n e t t i n g l i n e i n each system and would be d i f f i c u t  to o b t a i n an estimate  s e p a r a t e l y f o r these was  only one  treatments.  With only  one  dowling l i n e i n system two i t of l i n e - t o - l i n e  So i n i t i a l l y  variance  a random e f f e c t s model  developed i n c o r p o r a t i n g only the o y s t e r l i n e s .  Throughout  this  section,  for  those  lines  for  which  a = 5,  the  estimate  of  the  variance  A.  of  8 that  was u s e d was  that  obtained  when t h e  exponential  model  starting A  at  t = 5 was  fitted.  s h o u l d have 2.2,  Although,  been e s t i m a t e d  strictly  instead  any  resulting  inaccuracies  Let  Bi  be  and  hypothetical  of  that  six  true  of  We w i s h t o  suggest  sample  the  population  respectively. would  82  were  the  manner  likely  to  the  variance  described  be  in  parameters  for  s y s t e m one and  s y s t e m two  oyster  test  H  Bi  :  0  = 82.  If  system e f f e c t  from each p o p u l a t i o n .  is  HQ is  Let  of  8  Section  negligible.  underlying  a significant  lines  in  speaking,  the lines  rejected,  present.  B^j  be  the  this  We h a v e true  a  value  8  A  for  the  jth  line  in  the  ith  s y s t e m and  be  8^  the  maximum  likelihood  2 estimate (not was  of  .  8^  = SECB^j)  unreasonable  considering  is  the  assumed t o  large  amount  be a known of  constant  data with which  8jj  estimated). A  Then,  provided  that  the model  likelihood  estimate  based  fits,  on a l a r g e  since  sample,  is  8-jj  it  seems  a maximum reasonable  to  assume P  We assume  further,  that  the  8  i  j  | B  Bjj  i ; j  ~  i  N(  r  are  B l J  ,  V  i  . ) .  independently  N(B ,  distributed  and  that  a ) 2  i  2 where values clear  o"j_ for that  is  follows  line-to-line  the  infinite  this  second  proceed while It  the  bearing  that  variance  population assumption  in  mind  that  of is  (unknown) lines  in  entirely  a strong  of  the  true  system i . reasonable.  assumption  is  It  p^j is  not  We c a n being  only  made.  -  i j ~  P  Naive  (  i ' °i  p  +  i j  V  }  Analysis Initially  strong  N  38 -  a naive  assumptions  were  analysis  was c a r r i e d  out i n which  two  fairly  made: A  1.  I t was a s s u m e d t h a t  the standard e r r o r s  of t h e B i j were  negligible  2 compared It  to the l i n e - t o - l i n e  seemed r e a s o n a b l e  was e s t i m a t e d w i t h in  effect,  Since  to expect  a large  treating  standard  A  a  2  = s  2  amount  i.e.,  t h e V^j w o u l d  of d a t a .  A  o f t h e B^-j's.  1  )  X  where  2  A  =  S  2  We o b t a i n range  =  5  E  (  8  The p r e v i o u s  P  2  E  o^ = . 0 2 6 a n d  from  .0050  to  2  w  h  e  r  e  6  2  = .011.  results  by t h e yield  = 0.1019, 3  A  )  Bjj  A  pj = |  A  2j "  since  i s provided  3  A  2  of  A  8  be s m a l l  exactly.  A  = 3- S ( B . -  •  By i g n o r i n g V^j we w e r e  3  a  V^j «  t h e V^j a n e s t i m a t e  deviation  A  that  8-y a s known  we a r e n e g l e c t i n g  sample  variance,  =  A  6  S  P  2j  The v a l u e s  °'  =  0  7  2  4  *  of  . 0 1 3 0 a n d i n s y s t e m two f r o m  i n s y s t e m one .0034  to  .0087;  while  2 the  V^j a r e s m a l l e r  entirely 2.  than  our e s t i m a t e s  of  t h e a -p  they  are not  negligible.  T h e s e c o n d a s s u m p t i o n was t h a t  same f o r b o t h  systems,  i.e.  a\  the l i n e - t o - l i n e  = o^.  Since  we h a v e  variance  was t h e  assumed t h a t  the  - 39 -  P i j ' s are normally distributed we can test the hypothesis c± = 0 by 2  2 2 comparing S^ /S to F 2  tables. o2 •51- = 5.67 ===> p = .05. c 2  55  2  s  So the assumption o"i = o appears to be a f a i r l y strong one. 2  Under assumptions 1. and 2. we have a sample of s i x 8 values from each of two normal populations with equal variance.  So H Q :  8i = 82 can  be tested by a two sample t - t e s t :  6 1  S  where S = / p  5  S  ^ * 10  5  S  4  "  6  = 2.57 ===> p = 0.015  2  p 6 + 7T 6  ^ = 0.020.  This seems to suggest that there i s a difference between the underlying B values for the two systems. f a i r l y strong assumptions.  However this analysis r e l i e d on two  The second assumption, a\ - 02  w  a  s  tested by  means of an F-test, which i s p a r t i c u l a r l y sensitive to our assumption that the 8 i j '  Second  s  &  re  normal.  A more careful analysis i s called f o r .  Analysis A  This time the standard error of the P i j ' s was not neglected. This analysis was done both assuming 01 = a and also without this 2  assumption.  - 40 -  Case 1 :  Assume Q\ = OY - O  If we assume that the l i n e - t o - l i n e variance i s the same i n both systems then there are three parameters - Bi, B A  since 8 ^  2  and a to be estimated. A  2  NC^,  a + V^)>  the l i k e l i h o o d of obtaining the twelve B's  that we obtained i s 6  2  n  n  L =  exp -1/2  i=l j=l /2* 2 / o  +  Our estimate of the variance of B-jj was made with a large amount of data, so i t can be assumed to be close to the true variance V J J . we replace V ^ j with this estimate, thus treating V ^ j as a known constant.  2  logL  Z E log(o i j  ologL 58i  ologL ap2  + V  2  i J  h  y  ii " ) +—iJ i (a + V ) ( P  ( P i , - Pi) = E j ( a + Vx )  o logL  Cp2j " P2)  5 logL  » E j  (<r + v  »i2Bt- - I  E  2 j  + constant  2  2  o6i  = -E  «  0)  = -E « j (a + V )  0)  j (a  2  + Vj )  2  )  ap2  2  2 j  ( P  E  i J (o +V .) 2  ±  i1  (a  2  - Pl>'  + V  )  2  So  - 41 -  To f i n d  solve simultaneously  =  = ^2Sk  o8  °Pi  Let  w. . = 1  e s t i m a t e s of 61, 82 and a we need t o  the maximum l i k e l i h o o d  a  J  2  + V, .  2  S o  = . 0  2  i = 1,2.  Ij  A  T  h  e  n  ^ k , „ „ . > 1  .  h  i  — • l j  1.1.2.  3  Substituting  (81* and 8 2 * y )  the above v a l u e s  s a  i  n  S  l  o  g  ,  L  9a 2  — « 0 ===> Z Z[w 9a  i  2  a logL(8i*, 2  j  i  8?*, a )  -w j  1  I  2  =  s  (6. .  3  E  1  w  ii  w  8  i i  =—  J  2  W  ) ] = 0 and  i j  2  A  A  E w. . 8., j ^ ^ i  j  J  .  J  E w.  E w. . 8. . j ^ *J  * .  i j  J  A 2  2  j  J  i  j  A  p. ±  £ w  ±j  - E w  J  + (J (E w  )  ±j  p  E w. 2  ± j  1  )} .  2  3  aiogL — - —  -  2  = 0 can be s o l v e d by a Newton Raphson i t e r a t i o n  A n u m e r i c a l check can be done t o v e r i f y  that d l o g L = 0 does i n f a c t  aa 2 yield  a maximum.  2  2  Furthermore s i n c e ^ l°gk  j  0  a  m  l°gL  2 api  t o o b t a i n cr .  2 ap  2  a  r  e  both < 0,  -  d^  g  L  =0 and  Qjjj°  = 0 will  gL  42  certainly  -  yield  A  it  is  straightforward  to  obtain  To  for  test  we n e e d  them i n  whether  t o know t h e  terms  6i and  the  negative  matrix  of  the  of  Alternatively  Suppose  =  of  a s we h a v e  a  .  our  o ) is  second  as  parameters  and  81  estimates.  82  are  An e s t i m a t e  the of  same,  the  provided  derivatives  by of  the  inverse  logL  of  evaluated  the at  the  follows:  A  Z w. . 6, . ij ij .3 ^  A  1  where w . = 12  t  were known.  i  T h e n we w o u l d  =  1,2.  have  A  E w,. , ij g  i  =  K  B. . ij and v a r  L w. .  * 8  ij  l  i  = •=  L w. .  i  ij  =  0  1,2.  A  Substituting  approximation  our  estimate  to  the  var  8  ±  w„  variance  =  of  of  — j  a  explicit  2  P2>  we c a n p r o c e e d  t h e w^j  of  A  62  estimates.  A  8 i  (81,  matrix  maximum l i k e l i h o o d  A  obtained  2  underlying  variance A  covariance  of  Having  A  A  expressions  maxima.  w^.. we o b t a i n  the  following  p^:  i  =  1,2.  rough  2  -  43 -  Results  /v  A.  Bi = 0.0983,  8  = 0.0722,  2  a = 0.0158.  A  Estimated These  A  SE(pi)  lead  = 0.0076,  estimated  SE(B )  -  2  0.0068.  to A  A  PI - P2 A  A  /(SE(Bi) which  i s almost  naive  analysis.  identical  that  there  However  this  test  2  +  A  SE(8 ) ) 2  2  to the value  Comparing  suggest  2 > 5 6  A  the value  of the s t a t i s t i c  2.56 to normal  obtained  tables  i n the  would  is a significant  difference  between B i and  i s not c o m p l e t e l y  legitimate  f o r two r e a s o n s :  again  6 . 2  2 Firstly and s e c o n d l y  a n d 82 a r e n o t i n d e p e n d e n t  8^  it  would  seem more r e a s o n a b l e  since  t o compare A  statistic  to t-tables  The p r o b l e m i s appropriate. linear  If  of  it  i s not c l e a r  the square  combination  approximation degrees  that  as the v a r i a n c e s  of  p±  and p  (Satterthwaite,  1946) c o u l d  of  a  r  e  only  a  of the  estimates.  of freedom a r e  our s t a t i s t i c  variables,  be u s e d  were  a  Satterthwaite's  to estimate  the  freedom. A  However  involve  the value  2  how many d e g r e e s  chi-squared  both  A  of the denominator  of independent  they  our e s t i m a t e s  complicated  of t h e v a r i a n c e s  form and S a t t e r t h w a i t e ' s  of  A  81 a n d 82  approximation  a r e o f a more i s not  applicable  here. Also, for  t h e whole  t h e B-LJ'S.  given.  However  Because  a n a l y s i s was b a s e d on o u r a s s u m p t i o n of these  the value  problems  of our t e s t  an e x a c t  statistic  is  p-value  of  normality  cannot  sufficiently  be  large  to  - 44 -  suggest  a s i g n i f i c a n t d i f f e r e n c e between 61 and 82> even a l l o w i n g f o r  slight inaccuracies.  C a s e 2:  o~i a n d a ? n o t a s s u m e d  equal  I f a\ and o"2 a r e not assumed t o be e q u a l then we need e s t i m a t e s of  f o u r parameters - 6 1 , 6 2 , cri, 0 2 .  F o r the system one o y s t e r l i n e s A  A  A  the l i k e l i h o o d under our model of o b t a i n i n g 6 n , 6 1 2 , . . . , P 1 6 i s  L =  n j  exp ^2TC  -1/2  , 2 __ , „ • a + Vi  (Pi, " Pi)' ^ V)  (o^ + 1  U  J  and  l o g L = - Y E In  I f we l e t w,  ,  (cr +  ,  Vj.)  +  (Pi, - Pi)' J (<T + V i . ) 1 3  +  constant.  then  £  "1.1 "'.1  E wj .  As before we can s u b s t i t u t e i n d l o g l  Then  - = 0 can be s o l v e d by boi'  bai'  Newton Raphson t o o b t a i n a\ .  Then i t i s easy t o f i n d 81•  Analogously  A  to the p r e v i o u s case, an estimate of the v a r i a n c e of 6 l I s g i v e n by A  var B j *  j j  oi  2  +  m  V  L  - 45 -  Similarly  the l i k e l i h o o d  of o b t a i n i n g 6 2 1 , . . ' 8 2 6 i s  6 L  =  (62, " 8 ) ' 2 2  exp - 1 / 2  n  (cr + V .) 2  i=l  2  2  /2n / c r + Vo. 2 3  J  Z  and in  o"2 , 82 as w e l l as an e s t i m a t e of the v a r i a n c e o f 82 can be o b t a i n e d the same way.  Results A  A  Bi = 0.0997, p A  2  = 0.0719.  2  = 0.0089.  A  a i = 0.0219, a A  Estimated  A  S E C p ^ = 0.0098, e s t i m a t e d S E ( p ) = 0.0043. 2  Pi " P2 A  A  / ( S E ( p i ),2 +, 2  The except  same problems a r i s e  t h a t t h i s time  p-value  cannot  3.  be quoted,  2  N  2  but a g a i n a s i g n i f i c a n t Our t e s t  02,  Again an exact d i f f e r e n c e between  s t a t i s t i c w i l l be found  a t the 5% l e v e l i f i t i s compared to t f t a b l e s f o r any f >_  the r e s u l t  i s almost  A  for  „ p 2) - ) S, E, (  o r  Pi and P2 a r e n ' t c o r r e l a t e d .  I t seems to make very l i t t l e  = o"2;  A  as f o r the case where we assumed ai =  Pi and P2 i s s t r o n g l y suggested. significant  = 2.59 A  d i f f e r e n c e whether or not we assume o"i  identical  case 1, Pi and P2 a r e c o r r e l a t e d  a n a l y s i s t o o much.  f o r both c a s e s .  So the f a c t  probably doesn't  a f f e c t the  that  A  However the f a c t  remains t h a t our a n a l y s i s depends  on our assumption  of n o r m a l i t y f o r the P i j '  s  t h i s assumption.  A summary  i s g i v e n below:  of the r e s u l t s  ^  an  0 1 3 7  he s e n s i t i v e  to  - 46 -  Estimate  ± standard  error  Second A n a l y s i s Parameter  in  assuming  l  ~  a 2  02  0.0983 ± .0076  0.0997 ± .0098  62  0.0724 ± .0024  0.0722 ± .0068  0.0719 ± .0043  analyses  c a n be  Significant  when l i n e s  suggest  interpreted the  tests  that  were  from d i f f e r e n t  81  and  as meaning  survival  system e f f e c t  nonparametric  is  are  82  that  system i s  distribution, certainly  carried  out  at  not  the  systems were  significantly  an i m p o r t a n t  least  of  surprising  biggest  different  the  factor  oyster  a s when  differences  lines.  the occurred  compared.  I n c o r p o r a t i n g N e t t i n g and Dowling L i n e s i n t o the Model  It also  of  is  each  not  interest  determining  to  o"i = a  without  0.1019 ± .0041  determining  2.5  assuming  Analysis  Bi  Our which  Naive A n a l y s i s  Second  the  only  the  system e f f e c t  t o know w h e t h e r survival  an e s t i m a t e  of  type  distribution.  s y s t e m a n d one d o w l i n g  obtain  the  line  in  which of  With  is  of  material only  s y s t e m two i t  line-to-line  interest.  variance  for  plays  It  a role  is in  one n e t t i n g  line  will  possible  each  not  be  type  of  in  material  - 47  w i t h i n each system.  It w i l l  -  have to be assumed t h a t  line-to-line 2  v a r i a n c e i s independent of type of m a t e r i a l . the l i n e - t o - l i n e v a r i a n c e s f o r systems one  cr^  As b e f o r e  and  two  2 and  o"2  are  respectively  ( r e g a r d l e s s of m a t e r i a l t y p e ) . The  p r e v i o u s model i s extended to i n c o r o r a t e n e t t i n g and  dowling  l i n e s as f o l l o w s : Let P ^ j k be the t r u e 8 v a l u e f o r the k t h l i n e of the j t h type i n system i.  L e t B^jk be our maximum l i k e l i h o o d  estimate  of B-jjk and l e t  V\£jk = v a r i a n c e of S ^ j ^ . assumed to be a known c o n s t a n t .  ^kKjk and  8  where f t.  ±  j  ^ijk'W  N(8  k  +  f  = system e f f e c t  i = l , 2 , and  = type e f f e c t  j = l , 2 , 3 , and  t  +  ±  £ f i  =  j due  to type 2  (dowling),  t3  = e f f e c t due  to type 3  (netting).  *  s  °i  are assuming no 2  2  0, = 0.  to type 1 ( o y s t e r ) ,  = e f f e c t due  Pijk  o )  j  2  We  t.,  E t  1  So t i = e f f e c t  system-type i n t e r a c t i o n .  2 i n s t e a d of a±^ as we  don't have  The  assumption of n o r m a l i t y f o r the 8 i j k '  s  *  s  the assumption of n o r m a l i t y f o r the 8 i j k '  v a r i a n c e of  sufficient  r e p l i c a t i o n to a l l o w the v a r i a n c e to depend on type.  cannot be v e r i f i e d .  Assume  As b e f o r e  the  c e r t a i n l y reasonable,  but  ^  s  s  P  u r e ±  y  a  n  assumption  It follows that Bijk  N(8  +  f  i  +  t., o  2 +  V  i  j  k  ) .  In order to compare the s l o p e s of the s u r v i v a l curves f o r the systems and  and  f o r the t h r e e types  of m a t e r i a l , we  need e s t i m a t e s  two  of 8, f i ,  -  2 f ,  tj,  2  t ,  t3,  2  covariances  of  oi fj,  48  -  2 and  o  f ,  ti,  2  and e s t i m a t e s  2  t , 2  of  the  t3.  The  that  we o b t a i n e d  variances  likelihood  under  and our model  of  A  obtaining  the  seventeen  L  nnn  =  i  j  Pijk's  •  VZ%  k  / a  2  exp  +  l Substituting  f  = -fy  2  ,  (01  ijk = -ti  3  o  2  ,  f^,  t^,  t ,  the  t , 2  iteration,  the  maximum  likelihood  obtained  by Newton  Raphson.  2 ,  0^  are  P)  2  2 During  -  2  2.  estimates  and t  is  and  o  are  2  constrained  to  be  nonnegative.  Results A  8  = 0.0862.  A  f  x  = 0.0176  A  ===> f  A  = 0.00245,  ai  = .0234,  0  t  2  A  A  P^jk  =  = -  0.0176.  A  ti  The  2  p +  2  = 0.00105,  t  3  » -  .00350.  = .0072. A  A  f^ + t j  are  given  below:  Type System  Note  Oyster  Netting  1  0.1063  0.1003  0.1049  2  0.0711  0.0651  0.0697  that  the  estimates  of  the  slope  for  Dowling  respectively  the  s y s t e m one  and  -  the  s y s t e m two  identical oyster  to  oyster  the  lines  using  Raphson A  SE  8  A  A  A  second  0.0997 of  and  these  derivatives  under  the  model  but  not  for  the  0.0719).  estimates  evaluated  w h i c h was a l r e a d y  similar  at  calculated  are  obtained  the  as  usual  maximum  during  the  Newton  A  = SE f  = .0031.  2  A  A  It  then  was  between  t^,  var(fi  l  t  and  "  t  test  for  =  .0039.  differences  between  fi  and f  2  and  t : 3  =  5.62  2  2  A  =  0.25  =  1.02  A  -  A  t ) 2  A  "  1 : 1  t  3  A  A  var(ti ti  before,  2  to  3  f )  A  var(ti  since  interest  A  SE t  A  A  Since  = .0038,  2  of  -  A  t  SE t  ~ f;  fi  /  of  before  are  process.  = .0030,  /  obtained  variances  estimates,  SE ti  /  (0.1063 and 0.0711)  (respectively the  -  = .0035.  SE f i A  of  the m a t r i x  likelihood  A  estimates  alone  Estimates  lines  49  -  + t it  the  t ) 3  2  + t  seems  3  = 0 it  is  reasonable  variances  are  unnecessary to  compare  estimated.  also  these  Again  it  to  compare  quantities is  not  t to  clear  2  to  t . 3  t-tables how many  As  -  degrees the  of  freedom  significance  significance comparison the a  level  of  significant  was  found  values the  is  difference  be  be 5%,  three  no  types  which the between  but  the  the  8 values  of m a t e r i a l .  two  die  systems  above  level  off, but  for  the  not  the  is  a  between  overall  of  each whatever  that  there  two s y s t e m s between  suggests  there  if  namely  difference This  comparisons  comparisons,  sameconclusion,  the  three  accordingly:  significant  barnacles  doing  significance  For  the  between  we a r e  adjusted  be 5/3% = 1.66%.  difference  -  Since  then  f r e e d o m we r e a c h  the  at  be u s e d .  should to  previously),  for  rate  level  could  degrees  should  50  that  the  is  (as 8  regarding  significant the  three  types  of  material.  Discussion Inspecting one a r e any  generally  particular  than  those  above  any  analyses  variable  of  than  suggest It  is  those  seen  for  show no  type.  effect.  in  it  material  other  which  type  8 values,  larger  type  for  significant more  the  These  that  seen  s y s t e m one t h a n  in  values  for  system two,  whereas  tendency  be b i g g e r  to  observations  a significant is  the  are borne  system e f f e c t  f r o m a\ a n d system  a  2  that  but  system  p values or  for  smaller  out  by  not  a  8 values  the  are  two. A.  Statistical  analyses  an i n s p e c t i o n  of  type  The  effect.  values than  for  values  the  values  a values  s y s t e m two for  the  relating  suggests for  types  the that  a values  were  there  both a system  s y s t e m one a r e  and v a l u e s  other  to  for of  dowling  line.  The  is  generally lines  are  physical  not  carried  smaller generally  but  and than smaller  interpretation  of  - 51 -  these tend one  o b s e r v a t i o n s i s that b a r n a c l e s on dowling to s t a r t d y i n g e a r l i e r  and system one l i n e s  than other b a r n a c l e s .  tend t o d i e a t a f a s t e r r a t e than those  B a r n a c l e s i n system  i n system two.  Furthermore  the r a t e a t which the system one b a r n a c l e s d i e shows more v a r i a b i l i t y from l i n e t o l i n e . The  final  on the l i n e s .  o b s e r v a t i o n r e l a t e s t o the i n i t i a l  There are more b a r n a c l e s i n i t i a l l y  number o f b a r n a c l e s  on a l i n e of a g i v e n  type i n system two than on a l i n e of the same type i n system one. A l s o A  we found  A  the 8 v a l u e s to be g e n e r a l l y s m a l l e r and the a v a l u e s g e n e r a l l y  l a r g e r f o r system two l i n e s than f o r system one l i n e s .  Formal  s t a t i s t i c a l t e s t s a r e not a p p r o p r i a t e and even the simple coefficient  i s not very meaningful  a g i v e n type.  correlation  w i t h such a s m a l l number of l i n e s of  However i t i s worth n o t i n g t h a t f o r the system one o y s t e r  l i n e s , the c o r r e l a t i o n c o e f f i c i e n t between the i n i t i a l number of b a r n a c l e s and 6 i s - .93.  T h i s c o r r e l a t i o n suggests  fewer b a r n a c l e s i n i t i a l l y ,  b a r n a c l e s tend t o s t a r t d y i n g e a r l i e r and at  a faster rate. variable related  to both  system i s more important  6 and i n i t i a l  number.  than type as regards  T h i s w i l l be d i s c u s s e d  A l l i n a l l i t appears t h a t survival  normality  distribution.  c o n c l u s i o n i s somewhat t e n t a t i v e s i n c e i t i s based on the  a n a l y s i s of parameters p and a o f a model which doesn't the d a t a .  with  However the c o r r e l a t i o n may a l s o be due t o a t h i r d  a g a i n l a t e r i n r e l a t i o n t o the growth d a t a .  However t h i s  t h a t on l i n e s  adequately f i t  Furthermore the subsequent a n a l y s e s r e q u i r e d a s t r o n g assumption.  -  3.  3.1  At  Growth  Data  Growth  data  each  of  these  on e a c h o y s t e r haphazard rubber  was  sample  equally  It  will  likely  to  s a m p l e d was n o t and n e c k  line  to  In  number  were  total of  not  there  were  s y s t e m 2. barnacles happened  For  each  removed  on e a c h d o w l i n g  of  from  measured at  the  10 l i v e  was  barnacle  of  in  items  to which  they  time  was:  2 x 6 x 1 0 x 5 =  lines:  2 x  = 200.  s e v e r a l ways:  only  on b a r n a c l e s  Also  at  t  that  fewer  item.  = 10 a n d of  the  than  This  was  from 3 o y s t e r 11 w e e k s  lines  five  at  but  (or  because  the  = 1 week,  lines  all  of  were  them.  barnacles population  were  were  the  observations  and 1 d o w l i n g  observations  not  ten)  t  The  600,  From n e t t i n g  10  length  cm.  growth d a t a :  = 300,  in  being  shell  recorded  of  of  barnacle  live  3 x 20 x 5  incomplete  a  was  were  10 x  line  barnacles  probability  amount  each  lines:  the  shell  From e a c h p i e c e  sampled  Measurements  17 w e e k s .  from each  lines:  f r o m some  particular  sample  14,  From d o w l i n g  d a t a was taken  wood  example  11,  follows:  selected.  was a n e n o r m o u s  barnacles  From o y s t e r  was  10,  on e a c h i t e m e v e r y  so f o r  size.  as  of  a haphazard  sampled,  2,...,9,  taken  barnacles  be a s s u m e d t h a t be  = 1,  were  l e n g t h were m e a s u r e d .  attached.  t  from each p i e c e  related  sampled b a r n a c l e s  The  at  samples  5 live  on e a c h n e t t i n g  selected.  total  collected  and  of  -  EXAMINATION OF THE GROWTH DATA  times  line  52  It  taken  line  in  on  sometimes  sampled  from which  from the  a  -  barnacles dying.  were  Sometimes,  s y s t e m one  oyster  than  five  this  case a l l The  at  the  at  t  s a m p l e d was  (or  ten)  remaining  analyses  same t i m e  points  to  data w i l l  14,  is  be d i s c u s s e d  relatively  not  included  were  Only  relating  shells,  neck  length  Again  could  17 l i n e s  wood o r  because  proportion  for  somewhat  these  of  -  not to  only  and m e t h o d s  this  would  the  fewer  shell  the  are  than  In  in  each point  are  taken  collected have  Also (or  items  lost  were  barnacles by  doing  The  an i d e n t i c a l  in  missing  neglected  ten)  items  data way.  is  time,  measurements The  continually  dying. sampled  on a p a r t i c u l a r  a n d 20-30% t o w a r d s  s y s t e m one o y s t e r  approximately  lines  the  end  where  25% i n i t i a l l y  of the  In  (oyster  from each i t e m .  sampled  been  the  data  was  of  taken  handling  of  five  a number  barnacles  barnacles  fewer  data  could  of  amount  treated  are At  the  item.  l e n g t h was a n a l y z e d .  been  there  for  were  h a v e made  much i n f o r m a t i o n  have  barnacles  on t h e  larger  of  barnacles  So  The a n a l y s e s  since  time  rubber).  10% i n i t i a l l y  items  in  barnacles  small)  measurements  considered worthwhile.  relating  from which  approximately  are  point  data  However  wasn't  were  on a p a r t i c u l a r  that  used.  and  to  sampled.  assumed  the m i s s i n g  on a r a n d o m s a m p l e  diminishing  except  of  remained  due  (particularly  sizes  individuals.  laborious  any  data  all  smaller  study  sample  were  used  later.  it  on e a c h o f  population  The  for  getting the  initial  barnacles  from them.  pieces  made  at  the  to  summary:  are  if  sampled  this.  small  of  17 weeks was  accommodate  c o m p u t a t i o n much more  end  barnacles  w h i c h were  = 2,3,4...,9,  extended  the  where  living  -  continually  towards lines  53  item the  is study  proportions  and c l o s e  to  100% by  - 54  the end  of the study.  -  These p r o p o r t i o n s are only e s t i m a t e s  as we  a c t u a l l y have the s u r v i v a l data f o r each item, only f o r each I n d i v i d u a l b a r n a c l e s cannot be i d e n t i f i e d . growth curves curves  3.2  f o r the s h e l l  f o r the two  We  f o r the t h r e e types  line.  would l i k e to f i t  l e n g t h of the b a r n a c l e s and  systems and  don't  to compare  these  of m a t e r i a l .  Growth Curves f o r Items  S o p h i s t i c a t e d growth curve models have been developed  for  l o n g i t u d i n a l data - f o r the s i t u a t i o n where an o b s e r v a t i o n i s obtained at each time p o i n t on each of a number of i n d i v i d u a l s . don't have l o n g i t u d i n a l data f o r the b a r n a c l e s :  In our case  a t each p o i n t we  measurements on a sample of u n i d e n t i f i e d b a r n a c l e s - each b a r n a c l e or may  not have been measured at the p r e v i o u s time p o i n t .  however, can be i d e n t i f i e d and obtained  i f we  from each item at each time  as l o n g i t u d i n a l data f o r the items:  The  were to average over the p o i n t , we  we  have may  items, sample  c o u l d t h i n k of our  data  on each l i n e we would have a number  of items and we would have a measurement r e p r e s e n t i n g each item at each time.  T h i s measurement would a c t u a l l y be the average s h e l l l e n g t h  a sample of b a r n a c l e s p i c k e d from the remaining i t e m at t h a t  l i v i n g b a r n a c l e s on t h a t  time.  S i n c e each b a r n a c l e has equal p r o b a b i l i t y of being r e g a r d l e s s of s i z e , and  the remaining  sampled,  s i n c e the p r o p o r t i o n of b a r n a c l e s sampled i s  q u i t e l a r g e , i t should be reasonable all  over  to a l l o w t h i s average to r e p r e s e n t  l i v i n g b a r n a c l e s on t h a t items  at that time.  So  -  instead  of  following  individual  individual  effectively items  items  fit  of  a line  the  average  growth  the  have  But  polynomial per  the  be  following  have  same p a r a m e t e r s .  in  any  population  curve  growth c u r v e s .  i t e m we w i l l  case  We may a l l o w  for  growth curve  typical  item  from t h i s  average  over  all  shell  of  of  items  barnacles  be w e l l - r e p r e s e n t e d  from  that  item at  that  firstly  because  the  living  barnacles  represents would the  the  expect  So  the main each of  the  assume  of  lies  17 l i n e s  in  to  line  line.  line.  that  comparison  on e a c h  on t h a t  on e a c h  to  With  interest  the  items  items  particular then  of  by  changes  dying  is  change  due  This  time.  have  as  a  We may  What w i l l  average  rather  for  the  unrelated are  to  then  this  natural  size.  If  not  for  So t h e  individual  provided the  to  that  barnacles  affect  big nor the  taken  reasons  population  growth  curve  barnacles.  population  a  time  two  the  on  the  barnacles  because  particularly  population w i l l  of  will  barnacles that  changes  dying.  than  This  a particular  sample  secondly  barnacles,  neither  (We hope  size  barnacles  line. living  item at  over a  grow and to  oyster  remaining  with  individual  in  the  average  population  time-interval the  the  barnacles  for  of  first  on a p a r t i c u l a r  growth curves  growth curves  probability  small,  the  the  changes  time).  natural  for  length  line  will  a  we w o u l d  represent?  average  to  fit  and we may t h i n k  show how t h e  of  to  17 l i n e s .  parameters  Consider  -  barnacles  observation  from a c o n c e p t u a l  the  curve  one  like  be i m p o s s i b l e .  comparison  sample  only  within  will  different  -  items.  S u p p o s e we w o u l d  all  55  be  We  similar  the that  die  in  particularly  average  size  of  -  the  living  barnacles.  The changes  entirely  due t o t h e g r o w t h  there  is  a tendency  while  each i n d i v i d u a l  of  in this  barnacle  grows  barnacles  growth  characteristics  of  growth  characteristics  of i n d i v i d u a l  individual  probability addressed  of  later  a p p r o a c h and of So u s i n g curves were  systems  is  bigger,  the n a t u r a l  it  related  is  the assumptions  f o r each of  the f i r s t  t o Rao ( 1 9 6 5 ,  that data  would  were  the  the to the  be  of the made.  polynomial  growth  curve  models  and the  f o r t h e two then  able  whether  and Roy ( 1 9 6 4 )  were  of the  being  being  Two g r o w t h  obtained  then  the case,  question w i l l  for items,  of m a t e r i a l  hand,  not r e f l e c t  Without  are i m p l i c i t l y  The c u r v e s  types  this  size  the l e g i t i m a c y  t h e 17 l i n e s .  t h e n be  on t h e o t h e r  to a s c e r t a i n  This  was due t o P o t h o f f  1966).  and f o r the t h r e e  If  difficult  of  will  the average  barnacles.  to s i z e .  size  say, to die f i r s t ,  population  along with a discussion  fitted  If,  get s m a l l e r .  the " l o n g i t u d i n a l "  employed -  second  3.3  were  actually  barnacles  dying  average  barnacles,  living  identify  -  the b a r n a c l e s .  f o r the l a r g e r  might  56  different  compared.  The Growth Curve Models  P o t h o f f and Roy's Model It  is  assumed  measurements individual divided It  is  longitudinal  on a number  are then  into  that  groups  assumed  of  clearly which that  data  individuals. correlated.  is  available  -  successive  The measurements Typically  f o r any one  the i n d i v i d u a l s  are  a r e t o be c o m p a r e d .  each  individual  has measurements  taken  at the  - 57 -  same q p o i n t s i n time.  The s e t of q measurements f o r any one i n d i v i d u a l  form one row of the data m a t r i x Yo. E[Y ] nxq  The model i s as  = 'A nxm  0  I mxp  follows:  P pxq  where n = number of i n d i v i d u a l s , m = number of groups, q = number of time  points,  p = number of parameters f i t t e d and  A i s the d e s i g n m a t r i x a c r o s s  within  each group (p _< q ) ,  individuals,  £ i s a m a t r i x of parameters t o be e s t i m a t e d , P i s a m a t r i x of known c o n s t a n t s r e l a t e d of It  full  t o time (assumed t o be  rank, p ) .  i s assumed that  the rows of YQ a r e m u t u a l l y  (measurements on d i f f e r e n t i n d i v i d u a l s elements i n any one row f o l l o w  independent  are independent) and that  the m u l t i v a r i a t e  the q  normal d i s t r i b u t i o n w i t h  unknown c o v a r i a n c e m a t r i x T.Q (q x q ) , the same f o r every i n d i v i d u a l . i s not are  diagonal since  correlated.  So  s u c c e s s i v e measurements on the same i n d i v i d u a l  I t would be p o s s i b l e  t o extend the model to a l l o w the  covariance matrices f o r d i f f e r e n t i n d i v i d u a l s  t o be m u l t i p l e s  other and a l s o  t o have measurements taken  to allow d i f f e r e n t i n d i v i d u a l s  at d i f f e r e n t times. and  will To  These e x t e n s i o n s are a p p r o p r i a t e i n our s i t u a t i o n  be c o n s i d e r e d illustrate  later.  the model c o n s i d e r the f o l l o w i n g  m groups of i n d i v i d u a l s w i t h n j i n the j t h group. for  individuals  of each  situation:  we have  The growth curves  i n a l l groups a r e from the same f a m i l y ,  f o r example  - 58 -  polynomials different  of  degree  groups.  p-1.  So t h e  But  the  parameters  growth curve  may be d i f f e r e n t  associated  with  the  jth  for group  0 Then A =  {  0  n {  n { m •  0  1  ^m p-1  ^m0  =  .P-1  Alternatively coefficients This more  1  '1 p-1  '10  P  0  if  .P-1  the  time  for  P.  c a n be g e n e r a l i s e d  to  factors  may be u s e d  points  e a c h w i t h a number  are  the of  equally  spaced,  situation  where  levels  or  to  the  orthogonal  there  are  two  multi-response  or  the is  - 59 -  situation  w h e r e more  the  factorial  usual  We want hypotheses  For  of  example,  i.e.  that  all  to  or  one g r o w t h  block  estimate  the  if  than  the  designs  measured.  c a n be a c c o m m o d a t e d  parameters  the  C  I  s x m  m x p  hypothesis  parameters  for  (  is  and  E,^  all  hypothesis  is  V  that  to  test  by  Any  this  of  model.  various  I  that  C =  all  and  curves  equal,  ) and V =  are  are  the  same,  then  (m-l)xl  curves  I mxm  0.  m growth  are  -1  all  =  p x u  groups  (m-l)x(m-l) the  is  form  C -  If  characteristic  I  .  pxp  of  degree  p-2  or  less  then  V (pxl)  Solution Pothoff a more  a n d Roy  suggest  a transformation  s t a n d a r d model w h i c h has  literature.  The  transformation  been  treated  -  1  1  0  an a r b i t r a r y  PG  of  - 1  P» i s  full  symmetric  rank.  (a)  its  rows  are mutually  (b)  the  p elements  distribution  in  with  any  reduces  extensively  in  the  model  the  is  X = Y G P nxp n x q qxq qxp where G i s  which  (P G  1  P')  -  1  pxp  positive  Then X i s  -  such  definite  matrix  such  that  that:  independent, one  row f o l l o w  unknown p o s i t i v e  the  multivariate  definite  covariance  normal matrix  to  - 60 -  E = [P(G')pxp (c)  P']"  1  1  P(G')  - 1  E  G"  0  1  P' (P G"  1  P»)  ,  _ 1  E[X] = A ? . T h i s i s the u s u a l m u l t i v a r i a t e a n a l y s i s of v a r i a n c e  model.  Under t h i s model a t e s t of C£V = 0 i s based  (MANOVA)  on S^(u x u) and  S ( u x u) where e  S  = (C I V ) ' [CjCA* A j ) - ^ ' ]  h  =  V  X'  AJ(AJ  and  S  where £ i s the l e a s t A^  and  g  A^  -  1  = V  - 1  CJICJCAJ  (C I V) A j )  -  1  C|]  X'[I - A^Aj^ A ^  -  1  _ 1  CjCA' A ^  -  A^  1  X  AJ] X V,  squares e s t i m a t e of £,  = first  r columns o f A,  = first  r columns of C,  rank A = r ( _< m, < n) , rank C = s ( <m, <_ r ) , rank V = u (< p ) .  Several possible tests are a v a i l a b l e : (1)  Roy's t e s t i s based on the l a r g e s t  c h a r a c t e r i s t i c root of S, S n e  (2)  H o t e l l i n g ' s t e s t i s based on the t r a c e of S, S h e  -  -  1  ,  | e S  (3)  W i l k ' s t e s t i s based on the r a t i o of determinants In  X = Y G 0  and S . e  order t o t e s t a h y p o t h e s i s C£V = 0 under - 1  p' (p G  - 1  S  h  S  e  the o r i g i n a l model,  P ' ) " i s s u b s t i t u t e d i n t o the e x p r e s s i o n s f o r S 1  In a l l a n a l y s e s which f o l l o w , Roy's t e s t was used.  proceeds as f o l l o w s :  +  This  h  1  - 61  Let X = tables  Reject  largest  c h a r a c t e r i s t i c root of S  (e.g. M o r r i s o n ,  1976)  with  m*  = 1/2  (|s-u|-l),  n* = 1/2  (n-r-u-1).  A*  _ 1  x  best l i n e a r unbiased  Confidence  Choice  to Heck  +  A.  +1)  (m* +  1)  X can be  compared to F t a b l e s w i t h The  least  G  P•(P  squares  estimator  by  I = (AJ A )  The  T  (2n* + 2) degrees of freedom.  of £ i s g i v e n  Compare ^  I f . ^ .. i s too l a r g e .  (n* I f s* = 1, the s t a t i s t i c  S "~ .  parameters  (s, u),  the n u l l h y p o t h e s i s  + 2) and  h  s* = min  1  (2m*  -  X = (AJ A j ) "  estimator  1  A^ Y  q  _ 1  of the f u n c t i o n C£V  bounds can a l s o be found  G  _ 1  P')  _ 1  i s then  .  C£V.  f o r these f u n c t i o n s .  of G  The  analysis i s valid  f o r any  G s a t i s f y i n g the g i v e n c o n d i t i o n s .  However the c h o i c e of G a f f e c t s the power of the t e s t s and of the e s t i m a t o r s .  The  (A! A . ) A! Y E l l l o o  P'  - 1  squares e s t i m a t o r  - 1  Q  Q  (P S  P')  - 1  o  - 1  .  P o t h o f f and Roy  of E  0  t h a t the  obtained  from Y  Q  may  However E  not be used.  least  optimal  t h a t the more G  the worse the power of the t e s t w i l l be and  g r e a t e r the v a r i a n c e of the e s t i m a t o r s . estimate  suggest  variance  e s t i m a t o r of £ i s  Comparing t h i s w i t h the  of £ g i v e n above, would suggest  c h o i c e of G i s G = E . d i f f e r s from E  minimum v a r i a n c e unbiased  the  0  the  i s unknown and  I f the  an  62  -  experimenter is  to  but  use  has  G=I.  If  independent  Alternatively, run,  this  the  experiment if  a guess  variance  is  it of  For  between any  proportional  about  the  an e s t i m a t e  may be u s e d .  correlation that  no i d e a  of  form of E  Unfortunately,  ~ ii  p  p  p  1  p  available  Q  however,  obtained  the  similar  this  estimate.  experiment  be a s s u m e d t h a t apart  time.  the  is  Then E  is  and is  Q  2  3  p....  p....  , p  p....  1  ..  2  ,  p  p  p  the  choice  of  p is  f r o m an i n d e p e n d e n t  problem with Pothoff  arbitrarily.  E  from a  to use  d periods to  procedure  I....  arbitrary  unless  an  estimate  experiment.  Model The  of  simplest  to  .. .. ..  Rao's  might  respect  3  c a n be  it  with  2  it  the  c a n be made b e f o r e  Q  two o b s e r v a t i o n s  p  of  Q  be p r e f e r a b l e  example,  constant  E  c a n be o b t a i n e d  0  would E  -  Rao p o i n t s  information  obtained  alternative construct  unless  from the  way o f  out  data  reducing  and R o y ' s that  their  G = E  Q  to the  a q x q nonsingular  model  is  model  doesn't  and t h e y  be u s e d .  do n o t  So Rao  growth model t o  matrix  that  H = ( H . H„)  G must utilize  allow  an  suggests  an  the  be  chosen  all  the  estimate  u s u a l MANOVA m o d e l :  such  that  the  columns  - 63 -  of  B-i f o r m a b a s i s  PH}  = I,  PH2  for  not  G is  affect Let  E[Z] and  tests.  = Y  H  o  the  T (q-p)xp  is  = Y  1  G"  o  Pothoff  a matrix  claims  efficient  than  information  that that  S is  the  is  precisely  we s e t  G = S,  as  does.  Rao To  again Let  test  thus  of  E  found.  B = [S  _ 1  -  - 1  matrix.  AJ  Y  - (A- V  The  choice  Z  = Y  , ,  of  S  time  they  S  P'(P  S  - 1  have P')"  1  (P S "  Y  under  f o r m C£V  of  E  = 0,  a slightly P  S  -  1  ].  and  G does  H„.  o 2  N  in  of  addition  q-p  covariables  this model  model  is  as  uses  it  neglected The b e s t  more  under linear  is  A']  obtain  of  under  true.  from the  1  P  coefficients.  which i s  P'  - 1  an e s t i m a t e  the  a matrix  not  model  q  but  = A£,  and R o y ' s  Z,  is  A i  allowing  E[X]  Pothoff  this  of  P  and  - 1  £ obtained  obtained  0  This - 1  of  Rao's  what we w o u l d  hypotheses  = I -  2  Z is  covariables  However  s = Y;[I This  H  regression  under  E, u n d e r  an e s t i m a t e  ,  model,  unknown  I = (A{ A ^ where  rows  choose  P*)  - 1  ZT where  +  obtained  of  (PG  estimate  model.  estimator  - 1  the  nx(q-p)  of  in  s p a n n e d by  definite  P'  1  P*)  - 1  and R o y ' s  the  contained  and R o y ' s  unbiased  or  H e n c e E[x|z] = A£  Rao  G  estimates  for  =0.  Pothoff  P'(P  1  positive  X  as  = G"  x  space  P = p we c a n  an a r b i t r a r y  nxp Then,  vector  When r a n k  = 0. H  where  the  1  P')"  data:  O  Pothoff  Q  1  and R o y ' s  obtained  matrices different  model  from the  S  h  and S  form:  g  data  are  if  -  Then  S  S  -  h  e  V  X'  A  = V  (P  S"  where  R  = (A.'A. ) 11  and  X  = Y  The  expression  Pothoff same.  Roy's  S"  q  for  and R o y ' s The  tests  model.  parameter  -  (Aj  1  A ^  P')  1  S  C^Cj  1  is  e  S"  and S  now s l i g h t l y  to  the  equal are  e  expression to  as  use  t h a n q-p  fewer  Application  3.4  The for  discussed  two  items.  entry  was  taken  The  were  items  the  on.  In  their  (1)  The  rows  (2)  The  set  row o f  average  barnacles  at  t  the  1  . ^ ..  S  g  in is  for  t o Heck  i A.  I  for  However  same a s  not  Pothoff  tables  the  and  the  (q-p).  and A l l e n  (1969)  length  groups  it  is  the  any  to  the  preferable  contained  longitudinal  nine  entries.  o v e r a random s a m p l e item  corresponding  according  models  data matrix in  applied  data matrix  form the  with  were  from the into  entries  distribution  models  the  =  present  of  -  to  Models  shell  divided  of  A{ X V  1  different,  by G r i z z l e  growth curve  the  -  covariables.  of  Each  S.  the  n * = n - s - u - l Sometimes  (A^ A ^  1  1  H o w e v e r when c o m p a r i n g  n* i s  -  P')" .  1  identical  on  R C p  A.' Y B Y ' A. (A.' A . ) l o o l l l  - 1  model w i t h G s e t based  -  V  1  + (A.'A.) 11  1  P'(P  1  -  -  64  require  are mutually  row f o l l o w s  same c o v a r i a n c e  to  the  the  of  data The  living  to  that  row.  line  that  they  following  matrix  E  n  for  were  assumptions:  independent.  the m u l t i v a r i a t e  ith  normal  each  row.  - 65  Provided  t h a t we  are prepared  b a r n a c l e s from d i f f e r e n t data m a t r i x w i l l  to assume t h a t o b s e r v a t i o n s  b a r n a c l e s i n common.  on  items are independent, then the rows of  c e r t a i n l y be m u t u a l l y  data m a t r i x corresponds  -  to a d i f f e r e n t  indepndent  s i n c e each row  item and no  Furthermore, i f we  two  our of  items have  are prepared  our  any  to assume t h a t  the set of o b s e r v a t i o n s on an i n d i v i d u a l b a r n a c l e i s m u l t i v a r i a t e normal then  the f i r s t  p a r t of assumption 2 would a l s o be  satisfied  s i n c e each  e n t r y i n our data m a t r i x would then be the average of a number of v a r i a b l e s assumed to be n o r m a l l y reasonable row  distributed.  Whether or not i t i s  to assume that the c o v a r i a n c e m a t r i x i s the same f o r e v e r y  of our data m a t r i x w i l l be d i s c u s s e d Subsequently  another  and more r e a s o n a b l e corresponding  a n a l y s i s was  assumption was  to items  later.  c a r r i e d out i n which the weaker  made t h a t a set of  from o y s t e r or dowling  l i n e s had  m a t r i x -jr E and a set of o b s e r v a t i o n s c o r r e s p o n d i n g n e t t i n g l i n e s had  covariance matrix  In order to do t h i s the model had i n S e c t i o n 3.8. reasonable. which every  I t i s not  clear  jfi  ^ f°  r  i n c l u d e d i n the In t o t a l  covariance  to items  from  some f i x e d , unknown £.  t o be m o d i f i e d s l i g h t l y as d e s c r i b e d that even t h i s weaker assumption i s  T h i s too w i l l be d i s c u s s e d l a t e r . o b s e r v a t i o n was  observations  an average over  five  Only  those  items f o r  ( t e n ) b a r n a c l e s were  analysis.  141  items were used i n the a n a l y s e s .  used from each l i n e was  as f o l l o w s :  The  number of  items  -  for  5  Dl  11  010  6  02  8  D2  11  Oil  10  03  5  Nl  8  012  9  04  6  07  9  D3  13  05  6  08  8  N2  10  06  8  09  8  a  and  the  first  to  same  was  to  determine  three step  system-type would  it  was  of  types  could  have  growth  differences of  materials to  over  a two  lines factor  curves  to  each of  the  existed  between  these  and f o r  the  systems.  determine  be i g n o r e d .  collapse  s y s t e m a n d we w o u l d of  polynomial  interest  combination to  fit  whether  different  be r e a s o n a b l e  number  -  01  The main o b j e c t i v e lines  66  whether  If  this  of  the  two  lines  were  curves  within  the  case  each it  same t y p e w i t h i n  situation  with  the  As  the  following  items. System 1  System 2  38 22 8  50 13 10  Type Oyster Dowling Netting In  order  growth  curves  curves  were  similar  to  determine  for  fitted  analysis  whether  the  six  oyster  to  the  six  was  carried  differences  lines  lines out  for  in  system one,  and the the  existed  six  parameters oyster  between  the  polynomial were  lines  in  growth  compared. system  A  two.  - 67 -  3.5  System One O y s t e r L i n e s  The a n a l y s e s  were  carried  out u s i n g :  (a)  Rao's  model,  (b)  Pothoff  and R o y ' s  model w i t h G = I ,  (c)  Pothoff  and Roy's  model w i t h G = an independent  namely  an estimate  of E  Q  G - Y' [I 2  where Y for  Q  For  a l l models  Y  (j,k)  cannot  t =  Z t  = 58/9  use orthogonal  t  1  A;] Y  the data matrix  Q  2  =  Q  lines,  H  and t h e d e s i g n  matrix  lines.  (38x9)  Q  of £  from the system 2 o y s t e r  - AjCA} A p "  2 and A a r e r e s p e c t i v e l y  t h e s y s t e m two o y s t e r  where  obtained  estimate  = data m a t r i x and  element  h  (since  of P = ( t  t h e time  f c  points  -  t )  J  _  1  a r e not equally  s p a c e d we  coefficients).  1  1  1  1  1  1  1  1  1  -40 9  -31 9  -22 9  -13 9  -4 9  5 9  14 9  23 9  68 9  •  •  •  •  •  •  •  •  •  So P =  •  •  •  •  •  •  •  •  •  •  •  •  •  *  •  •  •  •  ,-40xp-l  f  68.p-l  - 68  where  p is  (j,k)th  curve  the  element  for  the  The  A  P  parameters  to  be  fitted.  (t^ -  group w i l l  t)  Since  * rather  3  be  £ j  +  into  six  groups  8.  So  Q  (t  we h a v e  than  -  taken  t^  the  the  growth  t)  p-1 j  items  group:  of  jth  +... + V i each  number  -  (  t  "  are  5,  °  divided  8,  5,  6,  6,  0  . . . .  0  .  .  .  0  1  .  .  .  0  .  .  .  0  with  the  following  number  in  0  = 5{  (38x6)  8{ 1  .  0  . . .  0 1  0  . . .  0 1  8{ 0  n  = 38  (number  of  In S e c t i o n C£V  = 0 under  following  1.  It  data. and  the  individuals),  3.3,  it  was d e s c r i b e d  each model.  analyses  were  was done  hypothesis  By  how t o  choosing  carried  was d e t e r m i n e d what This  m = 6 (number  out  degree  in  the  following  that  all  curves  groups),  test  q = 9,  hypotheses  C and V a p p r o p r i a t e l y  under  of  of  each  polynomial way:  were  of  of  r = 6. the  form  the  model:  adequately  initially degree  2 or  fitted  cubics less  were was  the fitted  tested.  -  This C£V  required  a test  = 0 with C = I  of  £13  ( 6 x 6 )  V  Z23  =  69  -  ~ •••  ~  Zs3  0 which  =  c a n be  written  and  0 0 0  -  s = 6,  u = 1.  1  If  quadratics  order  to  were  found  determine  quadratics  whether  were n o t  found  determine  whether  until  appropriate  2.  the  The  tested.  cubics  hypothesis This  to  linear to  C£V  that  all  were  would  be a d e q u a t e .  six  c a n be  This  curves  then  process  were  written •••  =  ?60  =  £ll  =  £21  =  •••  =  ^1 p-1  "  ^2 p-1  "  *•*  "  C  were  then  fitted  continued  1  0  0  0  0  0  1  0  0  0  0  0  1  0  0  0  0  0  1  0  0  0  0  0  1  (5x6)  ^61  ^6  p-1  identical  was  in  If  fitted  determined.  growth  £20  =  also  quartics  adequate.  p o l y n o m i a l was  hypothesis  = 0 where  curves  quadratics  be a d e q u a t e ,  w o u l d be  ?10  or  be a d e q u a t e ,  to  -  and V = I ( p x p ) , where  3.  p-1  is  s = 5,  the degree  The h y p o t h e s i s  additive written  constant  -  u = p,  of  that  70  the polynomial  the curves  (i.e. parallel)  C£V = 0 w h e r e C i s  were  found  t o be a p p r o p r i a t e  identical  was t e s t e d .  except  This  possibly  hypothesis  in  1.  f o r an  c a n be  a s i n 2. a n d  V =  0 , (p-1)  s = 5,  u = p-1.  x (p-1)  Results 1.  ( i ) Cubics  were  degree  a.  fitted  2 or l e s s  Pothoff  and R o y ' s  and the h y p o t h e s i s  that  a l l curves  were  of  was t e s t e d .  model w i t h G =  Z o  S  b.  h  S  _  Rao's S  h  S  = .115, refer  1  -  = .138, refer  1  e  possible  that  linear  were  and R o y ' s  p > .1  6 > l t 9  ,  p > .1  quadratics  curves  fitted  with  -1 S ~" = 1 . 8 4 , r e f e r 1  h  ,  are also  are adequate.  and the h y p o t h e s i s  that  G = I 1 6 . 6 t o T ,5h> 6  It  is  adequate.  was t e s t e d .  Pothoff S  1.12 t o F  to both models,  Quadratics linear  a.  6 j 5 1 +  model  According  (ii)  1.04 t o F  p < .001  a l l curves  were  -  b.  Pothoff  and R o y ' s  71  -  model w i t h G = E  J  S  c.  h  S  -  1  e  Rao's S  h  S  = 1.41,  refer  12.7  to  F  = 1.86,  refer  14.3  to  F  linear  curves  are  _  1  appropriate. parameters,  Pothoff  and Roy,  2.00 1.95 1.93 1.99 2.05 2.00  b.  c.  Pothoff  o  4,  P <  .001  ,  p <  .001  model  Clearly  a.  6 > 5  So f o r which are  each l i n e as  .113  G = E  -  .0043 .0050 .0063 .0054 .0069 .0049  -  .0042 .0047 .0057  -  .0051 .0066 .0044  Q  2.00 1.95 1.91 1.98 2.09 2.01  .114 .116  1.97  .112  -  .0043  1.92  .119  -  .0046  1.94 2.02 2.02  .132  -  .0074  .126 .104  -  .0070 .0060  1.99  .114  -  .0045  .123 .120 .102 .107  adequate we h a v e  follows:  G = I  .119 .127 .117 .106 .112  and Roy,  not  6 > t | 6  Rao  and q u a d r a t i c s estimates  of  are  three  -  All  three  choice  models  of  little. curves  G in So,  that  give  very  Pothoff  letting  and R o y ' s  y(t)  we o b t a i n  similar  -  estimates  of  model a f f e c t s  denote  under,  72  shell  say,  the  length  Rao's  £.  at  In p a r t i c u l a r , estimate  time  model  for  + .167  t +  1.07  + .178  t  +  .96  + .227  t +  .78  + .216  t  +  .92  + .181  t  +  1.10  + .172  t +  1.07  t,  01,  the  very  the  growth  02,...,  06  are  respectively:  According over 4,  a sample  5,  6,  7,  y(t)  = -  .0043  t  y(t)  = -  .0046  t  y(t)  = -  .0074  t  y(t)  -  -  .0070  t  y(t)  = -  .0060  t  y(t)  = -  .0045  t  to of  8,  Rao's living  9,  2.03,  2.13,  under  Rao's  model a r e  although  linear  not and  were  tested  2.  that  hypothesis  given  a.  Pothoff  in  very  t h e y were the  that  2  2  2  2  estimates  of  the  Pictures  Figure similar The  six  model,  1. to  1.39,  shell  1.53,  the  six  The  six  curves  each  except  other that for  length  on 01 a t  of  hypotheses  identical  average  f r o m an i t e m  respectively:  following  all  and R o y ' s  taken  cm.  identical.  with  The  are  2.57  and a r e  quite  identical then  2.23,  2  barnacles  14 weeks  1.92,  from being  model,  2  t  1.67,  curves are  = 2, 1.80,  obtained not  G = I  were  too  far  -  almost  parallel,  the  curves  were  an a d d i t i v e  constant  results:  curves  3,  identical  was  tested,  Figure 1.  1 .0  1  1 2.0  1  1 4.0  Growth Curves for the System 1 Oyster Lines  1  1 6.0  1  1 8.0  r — i 10.0  1  1 12.0  r  TIMEIVEEK5)  Figure 2.  Growth Curves for the System 2 Oyster Lines  -  Largest  root  Refer  with  of  S  = .29  s* = 3 ,  _  =  1  e  t o Heck  m* = 1/2,  74  -  .419  tables  n* = 2 5 .  P * ,.04 b.  Pothoff  and R o y ' s  model,  G = E o  Largest  root  Refer  613 _ = .38  ,*  of  S  h  -  = .613  1  e  to Heck  tables  1 • Dl J  with  s* = 3 ,  m* = 1/2,  n* » 2 5 .  p = .'01 c.  Rao's model Largest root  p «  c o  s* = 3,  growth curves  Figure  1.  t o Heck  tables  n* = 2 5 .  amount  data  There  are is  a  choice  and hence  and R o y ' s  to  reject  although they  considerable  seems  the  of  identical,  Pothoff  estimators  = .452  1  m* = 1/2,  o b t a i n e d under that  -  .05  We h a v e a s u f f i c i e n t six  S, S h e  452 ^*. - = . 3 1 1.452  Refer  with  of  the  appear  G greatly  the  p-values.  very  d i s c r e p a n c y between  model w i t h G = I ,  of  hypothesis  affects  the  the  that  the  similar p-values  and w i t h G = T. .  variance  Q  of  in  the  It  -  3.  The  hypothesis  constant a.  was  the  and R o y ' s  Largest  root  p  »  curves  -  were  identical  except  for  an  additive  tested,  Pothoff  Refer  b.  that  75  .164 1.164  model,  of  S  = .14  _  G = I =  1  e  .164  t o Heck  tables  .05  Pothoff  and R o y ' s  Largest  root  of  S  model,  S  h  G =  =  _ 1 e  Z  Q  .224  .224 Refer p c.  »  ^ * ~  * ^  t  o  H e c  ^  tables  .05  Rao's  model  Largest  root  of  S, S n e -  =  1  .212  .212 Refer p Not is  »  ^ * 212  observed  in  similar  the  spite  between  with G = S  q  t  ^  o  e c  ^  t  a  ° l  e  s  .05  surprisingly clear  =  of the  curves the  will  found  to  considerable  p-values  and w i t h G = I .  results  are  now be  obtained  parallel.  discrepancy  obtained Before  be  under  for  the  which  Pothoff  discussing  the  This is  conclusion again  and R o y ' s  results  model  further,  s y s t e m two o y s t e r  lines.  - 76 -  3.6  System Two O y s t e r L i n e s  The  same a n a l y s e s were repeated f o r the system two o y s t e r  lines.  We had a new data m a t r i x Y(50 x 9) and a new d e s i g n m a t r i x  1  0 . . . .  0  A - 9{ (50x6) . . . 0  . . . 0 1 9{ 0  . . . . 0 1  w h i l e P was unchanged. Again  the a n a l y s e s were c a r r i e d  out u s i n g Rao's model and u s i n g  P o t h o f f and Roy's model both w i t h G = I and w i t h G = an e s t i m a t e of E o b t a i n e d from the system one o y s t e r l i n e s ,  where Y for  and A are r e s p e c t i v e l y  q  the system one o y s t e r  E  = Y ' [ I - A.(A! A . ) o o i l l  - 1  AJ]Y l o  the data m a t r i x and the d e s i g n m a t r i x  lines.  Results 1.  ( i ) C u b i c s were f i t t e d and the h y p o t h e s i s t h a t a l l curves were of degree 2 or l e s s was t e s t e d . a.  P o t h o f f and Roy's model w i t h G = E J  S b.  h  S  _ 1 e  = .066, r e f e r  .59 t o F  6 > 5 1 t  o , p > .1  Rao's model S  h  S  1 g  = .196, r e f e r  1.6 t o F g ^ g , p > .1  Q  -  Clearly  quadratics  despite  the discrepancy  77  -  are adequate. between  The c o n c l u s i o n t h e two v a l u e s  of  is  clear  the  test  statistic.  (ii)  Quadratics were  a.  were  linear  then  fitted  and Roy, G = I  S  e 8.37, refer  S  -  1  e  all  curves  was t e s t e d .  Pothoff  h  and t h e hypothe s i s t h a t  75.3 to F  6 > 5 1 |  ,  P «  .001  ,  p «  .001  ,  p «  .001  A  b.  Pothoff  a n d Roy G = E  S  = 4.51, refer  o  c.  h  S  _  1  Rao's S  h  The  S  a.  Pothoff  e  1  = 8.26, refer  hypothesis  Pothoff  linear  So q u a d r a t i c s  .094 .104 .076 .125 .097 .106  6 > l t 6  curves  are again  and R o y , G = I  2.17 2.28 2.16  b.  that  62.9 to F  % are:  2.23 2.15 2.28  6 ) 5 1 t  model  rejected. of  40.5 to F  -  .0052 .0070 .0053  -  .0099 .0063 .0082  and R o y , G = £  2.21 2.16 2.29 2.18 2.27  .091 .101 .077 .128 .095  - .0042 - .0069 - .0059 - .0105 - .0063  2.14  .103  -  .0080  are adequate appropriate.  is  very  strongly  The e s t i m a t e s  -  c.  -  Rao  Again,  2.24  .094  -  .0054  2.16 2.25 2.17  .107  -  .0074  .081 .122  2.28  .099  2.15  .107  -  .0055 .0096 .0064 .0084  the  estimates  identical. 08  012 a r e  over  a sample  5,  6,  obtained  The g r o w t h  According  4,  78  7,  curves  y(t)  = -  .0054 t  y(t)  = -  .0074 t  y(t)  = -  .0055 t  y(t)  = -  .0096 t  y(t)  = -  .0064 t  y(t)  = -  .0084 t  to Rao's  9,  living  2.29,  2.38,  2.45,  under  Rao's  model  are  those  linearity oyster  obtained  homogeneous  than than  methods  we o b t a i n  under  for  for the  2  +  .164  t  +  1.41  2  +  .202  t  +  1.16  2  +  .152  t  +  1.50  2  +  .246  t  +  0.98  2  +  .181  t  +  1.38  2  +  .215  t  +  1.11  estimates  barnacles are  2.65  given  was r e j e c t e d  lines  model,  14 weeks  2.20,  than  that  three  are  almost  Rao's  model  for  07,  respectively:  of  8,  from a l l  the  Pictures  Figure  2.  1.72,  of The  s y s t e m one o y s t e r  obtained  for  the  the  shell  on 07  1.85, six  curves lines  p-value  s y s t e m one o y s t e r  curves  average  f r o m an i t e m  w i t h a much s m a l l e r the  the  respectively:  cm. in  taken  of  lines).  at  are  = 2,  obtained linear  hypothesis  the  They  3,  2.10,  less  (the  s y s t e m one  t  1.98,  curves  for  length  system are  oyster  also  of  two less  lines  -  whereas  the  parallel,  curves  these by  for  the  s y s t e m one  curves  are  far  was  confirmed  the  results  2.  The  hypothesis  a.  Pothoff  and R o y ' s  Largest  root  that  of  all  S  from b e i n g  of  the  six  _  were  parallel.  were  found  This  to  be  observation  tests:  identical  was  tested.  G = I =  1  L  lines  following  curves  model, S h e  oyster  .801  801 Refer  ^'gQ^  =  1% c r i t i c a l  «44  value  t o Heck  = .32,  tables  p <  .01 At  b.  c.  Pothoff  and R o y ' s  Largest  root  of  S,S h e  -  G = E =  1  .894  Refer  = .47  t o Heck  1% c r i t i c a l  value  = .32,  p <  S  .660  Rao's  root  of  S,  °  Refer  o  tables,  1% c r i t i c a l  value  =  .32,  .01  model  Largest  3.  model,  -  =  1  h e  ^ * =  .40  1% c r i t i c a l  value  As  the  expected  The  hypothesis  a.  Pothoff Largest  that  to Heck  = .36,  curves the  tables  p <  are  curves  .01  not were  and R o y ' s m o d e l , G = I r o o t o f S, S = .648 h e -  1  found  to  parallel  be  identical.  was  then  tested.  -  b.  Refer  = .39  to Heck  1% c r i t i c a l  value  = .30,  p <  Pothoff  and R o y ' s  model,  G = E  Largest  root  of  S  n L  678 . *,-. = .40 1. o / o  Refer  Rao's  S  _  =  1  e  value  = .30,  -  tables  .01  .678  to Heck  Q  1% c r i t i c a l c.  80  tables,  p <  .01  model  Largest  root  of  S, S h e  -  =  1  .490  .49  The  Refer  ~ .33  t o Heck  1% c r i t i c a l  value  = .33,  p =  .01  somewhat  larger  than  in  p-values  hypothesis  are  that  the  curves  are  tables,  parallel  2.,  but  must  be  nevertheless,  even  the  rejected.  Discussion  As a l r e a d y lines  are  lines. won't  even  for  reasonable  combination. curves  the  more homogenous  But be  observed,  than  The for  s y s t e m two a r e  the to  than  the  ignore  s y s t e m one  generally  lines  lines  term i s  larger  curves  curves  s y s t e m one  constant  the  growth  for  for  which those  s y s t e m two were  a particular  generally  than  the  s y s t e m one  differences  within  curves  the  larger suggests  for  oyster  oyster  found,  so  it  system-type the  that  on s y s t e m o n e .  system  barnacles The  two on  system  -  one  curves  a r e more  of  barnacles  to  start  of  barnacles  more  on i t e m s  with  We c a n n o t  barnacles. growth dying  at  as  the It  patterns  for  the  two  tendency  for  the  bigger  that  interval  in  size  of  very  any  quickly  after more  that,  for  the  more  The  Models  be  whereas rate  it  smaller  pointed  greatly  affect  of  out, the  p-values  the  of  the  to  for  die  if  the  the  the  that  of £,  tests.  survival  there  for  in  of  is  a  number average  the  of size  increase  barnacles  die  off  oyster  lines  at  = 17 w e e k s .  This  G in but  different  rate  similar  it  at  individual  a much s l o w e r  s y s t e m one  but  quickly  large  increase  lines  grow  probability  different  then a  size rate.  to  not  reflect  example,  oyster  = 8 weeks,  very  of  does  first,  cause  = 5 and t  average  less  size  quickly  constant  population,  but  reflect  on s y s t e m two  choice  estimate  a more  t h a n w o u l d be e x p e c t e d  for  J; a r e  fairly  the  average  on s y s t e m two t e n d  systems,  would  = 5 and t  curves  increase  at  difference  On s y s t e m one  between t  The  whereas  natural  Suppose,  barnacles  curves.  on s y s t e m one and  the  the  interval  on,  barnacles  two  to  increases  may a l s o  barnacles  linear  estimates  the  the  individual.  already  hence  to  that  systems.  between t  constant  The  size,  a particular  in  that  represent  for  to  in  s y s t e m one  possible  related  deaths  later  than barnacles  curves  is  s y s t e m two  rapidly  however,  first  the  -  s y s t e m two a p p e a r s  in  characteristics is  than  and much l e s s  conclude  on,  in  on i t e m s  quickly  later  linear  81  die  off  could  a  account  lines.  for  all  Pothoff does  The p - v a l u e  three  methods.  and R o y ' s  affect  its  obtained  model  As doesn't  variance  using G = I  and  - 82 -  was u s u a l l y  considerably different  The  o b t a i n e d w i t h G = I tended  p-value  from that obtained u s i n g G = E . Q  t o agree more c l o s e l y w i t h the A  p-value  obtained under Rao's model.  So the p-value  obtained with G = E o  r  may be suspect, p a r t i c u l a r l y  s i n c e the estimate  of the c o v a r i a n c e  obtained  from the system one o y s t e r l i n e s turned out t o be very  different  from t h a t obtained from the system two o y s t e r l i n e s .  The  estimates  of the v e c t o r of v a r i a n c e s o b t a i n e d  respectively  the system one o y s t e r l i n e s and the system two o y s t e r l i n e s were: [.015, .018, .016, .014, .0096, .026, .035, .051, .040] and [.025, .024, .041, .020, .012, .0094, .018, .014, The  estimates  of the c o r r e l a t i o n  m a t r i x were  .024]. respectively:  1 .62  1  .48  .61  1  .44  .57  .55  1  .23  .32  .29  .43  1  .13  .27  .47  .41  .40  1  .20  .25  .40  .37  .23  .77  1  .19  .23  .19  .36  .07  .62  .69  1  .03  .09  .17  .28  .09  .19  .19  .26  and  matrix  from  -  83  -  1 .72  1  .74  .73  1  .52  .59  .61  1  .54  .61  .58  .68  1  .47  .54  .49  .55  .63  1  .24  .22  .13  .30  .27  .31  1  .35  .35  .35  .50  .57  .61  .47  1  .25  .22  .15  .29  .32  .43  .45  .59  Observations the  end o f  for  the  the  study  s y s t e m two o y s t e r than  later  on.  The  although  8 weeks  and b e t w e e n  In £  0  other  3.7  of  the  the  a r e more  to  lines  beginning, variable  correlation  observations  whereas  at  matrices  between at  t  a r e more  the have  differences system as dubious  the  it  may be  choice  of  of  a roughly at  = 9 weeks  that  high using  G in  towards  measurements  observations  = 8 and t  variable  beginning  s y s t e m two a n d p a r t i c u l a r l y  these  led  at  oyster  correlations  f r o m one  system,  for  t  for  the  the  study  similar = 7 and t  are system  one.  an e s t i m a t e  the model  for  of  the  p-values.  A p p l i c a t i o n o f the Model t o A l l Seventeen L i n e s  The model  two  low f o r  view  obtained  than  lines  pattern  particularly  s y s t e m one  and  next to  s t e p was test  for  to  incorporate  differences  not  the only  netting between  and d o w l i n g the  two  into  the  systems  but  =  - 84  -  a l s o between the t h r e e types of m a t e r i a l . found between the system  S i n c e d i f f e r e n c e s had been  one o y s t e r l i n e s and between the system  o y s t e r l i n e s , each of the seventeen l i n e s was parameters. lines,  After f i t t i n g  two  allowed d i f f e r e n t  growth curves f o r each of the seventeen  c o n t r a s t s were examined.  T h i s time only Rao's model was available  used.  from a s i m i l a r but independent  S i n c e no e s t i m a t e of E experiment  i n this  0  was  case, the  only r e a s o n a b l e c h o i c e of G i n P o t h o f f and Roy's model would be I , which could d i f f e r Y  Q  system  substantially  from  £ . 0  (141 x 9) i s the data m a t r i x f o r a l l 17 l i n e s . one appears  1  first,  0  then the data f o r system two.  The data f o r P i s unchanged.  . 0  A = 5{ (141x17) 0  . .  1 . .  8{  1 . .  . 0 . 0  . 0  0 . 1  . 0  0  . 0  1  . 0 10f 0 141 q = 9  ....  o 1  (number of i n d i v i d u a l s ) ,  m  (number of time p o i n t s ) , r  17 (number of g r o u p s ) , 17 (rank A ) .  - 85 -  At  t h i s stage i t was assumed t h a t each s e t of n i n e measurements had the  same c o v a r i a n c e m a t r i x , whether measurements were averages over f i v e as for  o y s t e r and dowling, or over t e n as f o r n e t t i n g . Suppose we wanted an o v e r a l l t e s t f o r d i f f e r e n c e s between the  growth  curves f o r the two systems.  T h i s could be done by t e s t i n g the  hypothesis: Ll  8  where l is  ±  =  [?  +  8  5  1Q  ±1  l2  +  •••  +  . . . l  8  ±  l9  -  9  I l O " ... ~ 9 J L i  7  = 0  ^1  the s e t of p parameters f o r group i .  S i n c e the f i r s t  n i n e groups  c o r r e s p o n d t o system one l i n e s , and the other e i g h t t o system two l i n e s , t h i s i s e q u i v a l e n t t o t e s t i n g t h a t the average of the  over the  s  system one l i n e s i s the same as the average of the C j / s over the system two l i n e s .  A r e j e c t i o n of t h i s h y p o t h e s i s would  indicate a  d i f f e r e n c e between the two systems i n the average v a l u e of a t l e a s t one of  the parameters.  C£V = 0 where  C  The h y p o t h e s i s can be w r i t t e n = (8 8... 8 -9 -9 ... -9) and V = I (p x p ) .  (1x17) Other c o n t r a s t s were examined  i n order t o determine whether  differences  e x i s t e d between the t h r e e types of m a t e r i a l . First needed  of a l l , i t was determined what degree of p o l y n o m i a l was  t o f i t the growth curves f o r a l l 17 l i n e s a d e q u a t e l y .  done i n the same way as f o r the o y s t e r l i n e s p r e v i o u s l y . t h a t q u a r t i c s were needed. D2, D3 were r e s p e c t i v e l y :  The f i t t e d  growth  T h i s was  I t was found  curves o b t a i n e d f o r D l ,  - 86 -  and  y(t) =  - .0002  t  H  + .0067 t  3  - .086  t  2  + .578 t +  .44  y(t) =  .0005  t  k  - .0143 t  3  + .126  t  2  - .224 t +  1.06  y(t) -  .00001 t  + .0009 t  3  - .030  t  2  + .387 t +  .51  4  f o r Nl and N2: y(t) =  .0001  t  - .0023 t  3  + .014  t  2  + .100 t +  1.24  y(t) -  .0002  t * - .0059 t  3  + .058  t  2  - .113 t +  1.62  H  1  P i c t u r e s of the curves f o r the dowling l i n e s appear the n e t t i n g almost  l i n e s i n F i g u r e 4.  linear.  The curves f o r the n e t t i n g l i n e s a r e  A t e s t of the h y p o t h e s i s of l i n e a r i t y f o r these curves  alone y i e l d e d a p-value as l a r g e as 7%. t h a t q u a d r a t i c s a r e adequate because  i n F i g u r e 3 and f o r  We have a l r e a d y a s c e r t a i n e d  f o r the o y s t e r l i n e s .  of the dowling l i n e s , and because  Clearly i t i s  of D2 i n p a r t i c u l a r ,  that  I.  q u a r t i c s a r e needed.  The c o e f f i c i e n t  of t  i s much l a r g e r f o r D2 than  f o r any other l i n e and the growth curve f o r D2 looks different  remarkably  from any of t h e o t h e r growth curves (see F i g u r e 2 ) .  discussing this,  r e s u l t s of the t e s t s comparing  Before  the curves f o r the two  systems and f o r the t h r e e m a t e r i a l types w i l l be p r e s e n t e d :  1.  The growth curves f o r the two systems were compared by t e s t i n g CIV = 0 where  C =(8 (1x17)  L a r g e s t r o o t of S , S ~ h e R e f e r 16.9 t o F  5  1 1 6  1  8  = .732,  , p < .001.  8 -9 -9 ... -9) and V = I (5x5),  - 87 -  ^  i  0.0  I  Figure 3 .  Growth Curves for the Dowling Lines  Figure 4.  Growth Curves for the Netting Lines  1 2.0  1  I  4.0  1  1 6.0  1  1 8.0  T I M E (WEEKS)  1  1 10.0  1  1 12.0  1  1  14.0  88 -  -  2.  The growth curves f o r the o y s t e r and dowling  l i n e s were compared by  t e s t i n g C£V = 0 where C = (1 1 1 1 1 1 -4 -4 0 1 1 1 1 1 1 -4 0) and V = I- (5x5). Largest  root of S , S h e  R e f e r 25.9 t o F  3.  5  )  1  The growth curves  1  6  -  1  = 1.121-,  , p < .001.  f o r the o y s t e r and n e t t i n g  t e s t i n g C^V = 0 where C = (1 1 1 1 1 1  0  l i n e s were compared by  0 - 6 1 1 1 1 1 1 0 - 6 )  and V = I (5x5). L a r g e s t r o o t of S . S h e R e f e r 9.9 t o F  4.  5  >  1  1  6  -  1  = .426, '  , p < .001.  The growth curves f o r the dowling t e s t i n g C£V = 0 where C = ( 0  0 0 0  and n e t t i n g  l i n e s were compared by  0 0 2 2 - 3  00  0 0 0 0 2 - 3 )  and V = I ( 5 x 5 ) . L a r g e s t r o o t of S R e f e r 12 t o F The  5  }  n  6  h  S  _ 1 e  = .524.  , p < .001.  same a n a l y s e s were then repeated t h i s time a l l o w i n g the curves t o  differ  by an a d d i t i v e c o n s t a n t .  So t h i s time £  Q  was not i n c l u d e d ;  f o r each t e s t C remained unchanged, but V was r e p l a c e d by 0 1 0 0 0  0 0 1 0 0  0 0 0 1 0  0 0 0 0 1  -  89  -  Results 1.  The  two  systems  Largest Refer p  2.  root  12.2  < .001  Oyster  Refer p  3.  21.7  Oyster  p  4.  root  1  1  -  1  -  1  -  1  .412  =  .730  9  S,S h e  F  4  >  1  1  of  to  9  Y  S  h  1  1  = .35 7  9  .  Largest  root  13.9  < .001  )  =  netting  and  p  of  to  Dowling  Refer  4  1  .  10.7  < .001  F  -  dowling  and  Largest  S, S h e  .  root  < .001  Refer  to  and  Largest  of  netting  to  of  S.S h e  = .471  Fi+  .  Discussion  Differences three  types  of  are  found  material.  both between This  is  in  the  two  contrast  systems  and b e t w e e n  t o what was  found  for  the the  -  90 -  s u r v i v a l d i s t r i b u t i o n s , namely t h a t  only system was an important  With such a l a r g e amount of data i t i s not d i f f e r e n c e s a r e found everywhere. statistics and  surprising  factor.  that  The r e l a t i v e s i z e of the t e s t  i n d i c a t e s that the b i g g e s t d i f f e r e n c e s l i e between dowling  the other two types of m a t e r i a l .  T h i s was t o be expected i n view of  the unusual growth curve f o r D2. For the comparisons  of the two systems and of o y s t e r and dowling  l i n e s the p - v a l u e s were much l a r g e r when the curves were a l l o w e d t o differ  by an a d d i t i v e c o n s t a n t .  comparisons  On the other hand f o r the other two  the p - v a l u e s d i d not change s u b s t a n t i a l l y when the curves  were a l l o w e d t o d i f f e r by a c o n s t a n t .  So the d i f f e r e n c e between n e t t i n g  and dowling and between n e t t i n g and o y s t e r i s ( a l m o s t ) s o l e l y attributable  t o the d i f f e r e n t  shapes  of the growth c u r v e s .  The  d i f f e r e n c e between the two systems and between o y s t e r and dowling i s due partly  t o the d i f f e r e n t  locations.  shapes  of the curves and p a r t l y  We conclude that b a r n a c l e s on system  b i g g e r than those on system  reflect  two a r e g e n e r a l l y  one and t h a t b a r n a c l e s on o y s t e r l i n e s are  g e n e r a l l y b i g g e r than those on dowling The  to d i f f e r e n t  lines.  smoothness of the growth curves f o r the n e t t i n g l i n e s may the a v e r a g i n g over t e n i n s t e a d of f i v e ,  r a t h e r than a growth  characteristic. The  average  a t an almost  s i z e of b a r n a c l e s on n e t t i n g  constant r a t e except  increases a l i t t l e  to increase  towards the end of the study when i t  f a s t e r , whereas the average  dowling l i n e s i s p a r t i c u l a r l y  l i n e s appears  small i n i t i a l l y ,  s i z e of b a r n a c l e s on increases rapidly to  -  start of  w i t h and l e s s  barnacles  on o y s t e r  The a v e r a g e t  = 2 weeks,  remain  size  increases  This  could  so d i f f e r e n t ,  sampled at t  later  lines  lies  on.  fairly  The b e h a v i o r  these  rapidly  until  t = 8 weeks, to increase  be a g r o w t h c h a r a c t e r i s t i c , be t h a t  = 8 a n d 9 weeks a n d p a r t i c u l a r l y  small  small  large  at  appears  to  extremely  but since  particularly  size  two e x t r e m e s .  on D2 i s e x c e p t i o n a l l y  t = 12 weeks a n d t h e n  i t may a l s o  of the average  somewhere b e t w e e n  of barnacles  t h e same u n t i l  rapidly. is  rapidly  91 -  this  behavior  barnacles  barnacles  were  were  s a m p l e d a t t = 14 w e e k s .  3.8  E x t e n s i o n o f t h e Model t o A l l o w D i f f e r e n t C o v a r i a n c e M a t r i c e s f o r Different  Items  An e s t i m a t e the  dowling  obtained  the  lines  since  was o b t a i n e d  and t h e n e t t i n g  covariance  lines  measurements  ten instead  some  Q  from the n e t t i n g  estimates over  of E  of f i v e .  matrix  individuals 1968).  seem more  on n e t t i n g  that  there  smaller  on n e t t i n g  on o y s t e r  t o t h e c a s e where  lines  reasonable  or dowling lines.  the covariance  are N individuals  lines,  the estimate  than  the other  are averages t o assume  lines  is  The growth c u r v e matrices  for  t h e 9^  a  r  e  scalars  2  E, e  and E i s  3  models  different  of each other ( I t o , who a r e a s s u m e d t o h a v e  E,...,  fixed  that  (1/5)E f o r  matrices  G l E, e where  was c o n s i d e r a b l y  I t would  f o r items  from the o y s t e r  Not s u r p r i s i n g l y  a r e a s s u m e d t o be known m u l t i p l e s  Suppose  covariance  lines.  from items  E, a n d ( 1 / 1 0 ) E f o r i t e m s  c a n be e x t e n d e d  separately  but unknown.  Suppose t h e  - 92 -  i n d i v i d u a l s are d i v i d e d i n t o groups w i t h nj i n the j t h group. I n d i v i d u a l s i n the same group are assumed t o have the same c o v a r i a n c e m a t r i x , so 0i = 02 = .... = Q ^ and s i m i l a r l y  f o r the other  n  groups.  D e f i n e a d i a g o n a l m a t r i x of weights  1/0!  W (NxN)  i/e  2  i/e  N  I n s t e a d of choosing £ t o minimise minimise  (X - A £ ) ' (X - A £ ) i t i s now chosen t o  (X - A £ ) W (X - Ac;) (weighted f  least  squares).  i s given to i n d i v i d u a l s with smaller covariance matrix. t h a t £ = (A^ W A T  So more weight I t follows  A{ W X.  1  T e s t s of C£V = 0 a r e based on  = ( q I V)« ( C i R C i ) ~ ( d I V)  S where R = (A^ W A j ) " and  S = V'(P S  - 1  l  + (AJ W A j ) "  1  P*)  1  AJ W X B X' W A j ( A { W  ^ )-1  V.  _ 1  I f W = I ( t h i s means equal weight  i s g i v e n t o a l l o b s e r v a t i o n s ) we  o b t a i n the same e x p r e s s i o n f o r R and £ and hence f o r S^ as p r e v i o u s l y . The  e s t i m a t e of E i s g i v e n by S, where  S=-g  Sx+g1  and  S  t  +  9"  S  3  + ....  ni+n2 l +  i s the e s t i m a t e of the c o v a r i a n c e m a t r i x f o r group t , namely 1  o  2  n^+1  S^ = Y ' [ I - A,(A» A , ) " A. ] Y t  s  l  l  l  l  1  o  where Y  i s the d a t a m a t r i x f o r group t O  a  r  - 93 -  and A, the d e s i g n m a t r i x f o r group t , i s a ( n  t  x 1) v e c t o r of l ' s .  I n d i v i d u a l s i n group 2 have c o v a r i a n c e m a t r i x 9 have c o v a r i a n c e m a t r i x 0 , ,,£ and so on. n]+n +l 2  (all  ^ £> those i n group 3  n i +  I f 0i = 0o = ... = 9„ = 1 N 1  i n d i v i d u a l s have the same c o v a r i a n c e m a t r i x )  for  the standard model, namely Y ' [ I - A.(A! A . ) o i l  then S w i l l  be as  A'] Y where Y and A l 1 o o  - 1  are the data and design m a t r i c e s f o r the complete s e t of d a t a . t e s t s based fitted  and S  e  are then as b e f o r e .  Q u a r t i c s were a g a i n  and a l l the t e s t s of the p r e v i o u s s e c t i o n were repeated w i t h the  adjusted In  on  The  and S . e  our case S = y [ c o v a r i a n c e m a t r i x f o r o y s t e r s ] +  •^•[covariance m a t r i x f o r dowling]  + [covariance matrix  for netting]  0^ = 2, i = 1,...,60 and 69,...,131 (items on o y s t e r and dowling  and  lines), 0^ = 1, i = 61,...,68 and 132,...,141 (items on n e t t i n g A, Y  lines).  and P a r e unchanged.  Q  Results The curves  1.  same t e s t s as i n the p r e v i o u s s e c t i o n , t o compare the growth  f o r v a r i o u s p a i r s of subgroups were c a r r i e d out:  The two systems L a r g e s t r o o t of S R e f e r 13.9 t o F  h  S -  5 > 1 1 6  1  = .595  , p < .001  -  2.  Oyster  3.  root  S , S h e  _  5  ,  1  1  6  root  ,  o f S, S h e  14.2 to F  5  ,i  1  6  Netting  and D o w l i n g  Largest  root  Refer  = .902  1  p < .001  and N e t t i n g  Largest  4.  of  20.9 to F  Oyster  Refer  -  and D o w l i n g  Largest Refer  94  of  -  ,  p < .001  S, S n e  -  14.8 to F  5  >  1  1  6  = .610  1  ,  = .640  1  p < .001  Comments The since (the  estimate  within same  of £ i s  each group  covariance  unaffected  (line)  matrix  is  by i n c l u d i n g  t h e same w e i g h t  W i n the a n a l y s i s ,  i s given  assmed f o r i n d i v i d u a l s  to a l l  items  i n t h e same  group). However involving analysis too  Q  netting,  it  or dowling  of  the t e s t s  the p-value  was l a r g e r .  b i g because  netting £ .  the p - v a l u e s  This  lines,  F o r comparisons  obtained  Is because  was o b t a i n e d  are a f f e c t e d .  i n the o r i g i n a l  the estimate  from a l l i t e m s ,  and t h e s e were  not i n v o l v i n g  In  of  netting  unweighted  t h e c o v a r i a n c e was  whether  a l l assumed  comparisons  from  t o have  the o r i g i n a l  oyster, t h e same  p v a l u e was  -  smaller.  This  small  because  lines  but a l s o  the  weighted  types  both  remain  the c o v a r i a n c e  from items  lines.  be e x p e c t e d  between  with  of  not only  on n e t t i n g  should  conclusions  exist  on o y s t e r  The p - v a l u e s  t o be more  t h e same h o w e v e r ,  t h e two s y s t e m s  the biggest  matrix  difference  was t o o  and  dowling  obtained  in  reliable. namely  that  and between being  the  between  three  the  oyster  lines.  C o l l a p s e Over L i n e s W i t h i n E a c h System-Type Combination  Although within then  differences  a system,  carried  lines of  from items  -  the estimate  was o b t a i n e d  of m a t e r i a l s  and d o w l i n g  3.9  it  because  analysis  The main differences  is  95  out  (particularly i n which  we g e t a t w o - f a c t o r  items  i n each  had been  category  within  these  given  analysis  all  individuals  item  ignored.  as p r e v i o u s l y  Netting  22  13  Dowling  8  10  of  line  t h e same  covariance  was a s s u m e d in  type  analysis  was  Ignoring The number  below:  50  It  t h e same  described.  38  (items). type  were  Oyster  -  of  system two), another  System 2  was u n w e i g h t e d  on t h e j t h  lines  System 1  Type  The  between  differences  situation is  found  the i t h  that  matrix  the growth  s y s t e m was  was a s s u m e d curve  for  an  for  - 96 -  +  So E[Y ] = A Q  (  ^oj  +  ^lj  fc  I P,  +  ••*  +  0  V l j  t  P  _  1  )  0  where A (141 x 5) = 38  22  50  13  10  £(5xp) =  and Y  Q  HOI  Hll•  -  H02  M-12 •  V-i  HO 3  Hi 3 •  H  £oi  ^11 *  'p-1 1  ^02  Si2 •  1  p-1i 1i 2  p-1 3  Vl  2  and P are as before.  Quartics were f i t t e d and the usual hypotheses were tested, namely that the growth curves for various pairs of subgroups were i d e n t i c a l :  -  1.  Refer  root  root  C I  Refer  Netting  2  ,  8  P < '001  S  h  5  1  2  8  -  ,  and V = 1 ( 5 x 5 ) .  = 4.561  1  e  V = 0 where root  of  P  «.001  Largest  S, S h e 5  root  comparisons  1  of  2  8  1  ,  = .874 p < .001  is  C = (0 1 -1 0 0 ) a n d V = 1 ( 5 x 5 ) .  S, S n e -  5  )  differences  between  difference  >  -  where  128.5 t o F  Again  C = (1 0 - 1 0 0 ) a n d V = 1 ( 5 x 5 ) .  and D o w l i n g  C H = 0  collapsing  1  of S  22.4 to F  Refer  >  = .752.  1  and N e t t i n g  Largest  being  5  116.7 t o F  Oyster  Test  -  V = 0 where C = (1 - 1 0 0 0 )  C I  Refer  4.  S, S n e  and V = 1 ( 5 x 5 ) .  and D o w l i n g  Largest  Test  of  19.2 t o F  Oyster Test  V = 0 w h e r e C = (0 0 0 1 - 1)  C I  Largest  3.  -  T h e two s y s t e m s Test  2.  97  1  2  8  ,  1  = 5.022 P «  .001  are found  dowling  and each  between  t h e two s y s t e m s .  involving over  lines  dowling is  of  everywhere w i t h the other  two t y p e s .  However  are s u s p i c i o u s l y  suspect.  the b i g g e s t The  differences smallest  the p-values small  and t h i s  f o r t h e two method  of  -  3.10  generalisation  namely  the g e n e r a l i s a t i o n  data.  In t h i s  same  times  for  this  model  it  taken.  all is  assumed  is  individuals  same  qj  Individuals same t i m e  Let we h a v e  that  in  into  number Aj  are subdivided  groups  they  belong  matrix  Bj  matrix the  (q x q j )  to). of O's  observations for  for  to allow  been  random it  would  have measurements i n Sj  at  the  is  Under or n o t  have The n  S  such  u  taken  at  the  nj.  measurements  taken  at  whether  Si, S2,...,  individuals  missing  relaxed.  as f o l l o w s :  groups  used,  are taken  the value  may n o t h a v e  for every  of  time  set  points.  taken  at  the  t h e same number  Then  ( n j x m) ( s i n c e w i t h i n according  Also  to which  f o r each group  and l ' s  indicating  individuals  the i n d i v i d u a l s  f o r a complete  same  of  at  been  of  = 9j')*  q be t h e t o t a l  data matrix  Sj  have  (1973)  have  proceeds  u disjoint  groups  individuals  the  model  to  b u t may h a v e m e a s u r e m e n t s  a design matrix  missing  could  unrelated  T h e number  also  observations  is missing  t h e same g r o u p  in different  (qj  is  that  data  could  by K l e i n b a u m  (Items)  generalised  points.  points  points  suggested  missing  are divided  that  time  the model  individuals  Kleinbaum's  individuals  of  way t h e a s s u m p t i o n  observation  time  -  M i s s i n g Data  Another  an  98  in  Sj•  i n Sj  then  each group  each group  an  the p o s i t i o n s If  Sj,  incidence of  ^ j ( j x qj) n  Q  is  the  is  covariance  on a n y i n d i v i d u a l  f o r each j  Sj  the m "treatment"  S j we h a v e  and £  of q o b s e r v a t i o n s  individual),  of  for  = 1,  2,...,u  (assumed we h a v e :  -  E[Y.] 3 and  for  P are  e a c h row o f  as  Section As  for  before  growth  estimates  of  properties  of  Yj  are  assumed  to  have  Z*  as be  the  Yj  univariate  1  B. J  matrix  is  B.' 3  complete  not  assumed  possible  closed  (BAN)  to  form,  column v e c t o r  finds  weighted  S  B., 3  °  data  where  E  described  but  Z 0  normal  and  in  which  have  yj  be  squares  where M  indicates  the  unbiased  and c o n s i s t e n t  - 1  B. j  P'  the  estimator  some  best  same l a r g e  sample  estimates.  the  the  vector  the  model  a  in  the  form of  turns  out  to  A.]" 3  ^* general  A  [P B , ( B ! j J  Z  of  Z  o  the m a t r i x  B.) j  _ 1  ®A!]  M and Z , Q  found as  follows:  pooled  estimate  using  Q  the  be:  is  Z,  o f E,  f o r m e d by p u t t i n g of  inverse of  columns  T h e n a BAN e s t i m a t o r  model)  ®A! 3  generalized  likelihood  finds  u B.) 3  independent  distribution.  t h e maximum  Kleinbaum  A  P B.(B! 3 3  be m u t u a l l y  f o r m e d by p u t t i n g  by w r i t i n g  least  find  estimators  and l e t  to  a multivariate  underneath each other.  u =  is  other  (which Kleinbaum  j  P  for  t h e maximum l i k e l i h o o d  columns  of  it  normal  each  [ Z  covariance  rows  £ and Z i n  underneath  £*=  3  models  model  asymptotically  A  I  = A.  curve  the  row i s  this  Let  the  -  3.3.  and each Under  the  Y. J  99  y. 3 3  an  The  A  (r,s)  element  those  individuals  t .  In  order  x w)  is  of  g  unbiased  a  r  to  full  of  g  for test  £  q  is  the  which measurements a hypothesis  r a n k w a n d H'£*  estimator  usual  linear  is  of  the  were  f o r m H'F,*  estimable  i n y where y  is  obtained  the  at  = 0,  ( w h i c h means vector  only both  t  r  and  w h e r e H (mp that  f o r m e d by  it  has  putting  an  -  the  W  n  columns  = (H'  of  £*)'[H'[  I  is  definite  estimator  P B.(B'  W is n  described which  is  E  B.)  a n y BAN e s t i m a t o r  of  £ .  in the  the  W  Q  of  3.3.  analogue  of  B,  _ 1  £* a n d  Under H ,  analogue  Section  of  the  E  S  S h e  -  is  n  S  -  in  1  statistic  A , ]  any  estimators  Pothoff  (Aj'  generalized  Aj)  in W  inverse  is  n  positive  by  distributed model  W * may be  used  n  W * is  obtained  by  n  by a more  analogous  complicated  to R of  J  matrix  Rao's  by  £*)  obtained  3  model.  given  (H'  1  and R o y ' s  statistic  Rao's  -  is  consistent  h e replacing  H]  -  asymptotically  An a l t e r n a t i v e S,  is  q  the  in  1  test  P» ® A '  In p a r t i c u l a r  0  K l e i n b a u m may be u s e d . 2  -  Y underneath each o t h e r ) ,  where  as X . w  100  Qj  whose  model.  A p p l i c a t i o n of the Model t o our Data This because  generalisation  data  at  on a l l  taken  Under items  have  some d a t a was n e g l e c t e d  observations the  might  at  available  at  2,  11 weeks was  10 and  model t h i s  taken  3,...,9,  10,  data  into  on t h e m . 14,  For  group  and so o n .  each group  Thus  the  A^'s  would  3,...,9,  have  first  be d e t e r m i n e d .  the  our  data  set,  assumption  same t i m e s .  that  Namely  been i n c l u d e d :  according  items  f o r m one  11,  items  the  for  ignored.  example  17 w o u l d  at  at  could  groups  d a t a was a v a i l a b l e Within  2,  taken  satisfy  = 1,  which  line.  to  were  been d i v i d e d  d a t a was  order  individuals  Kleinbaum's  would have  which  t  in  been a p p r o p r i a t e  14,  would  for  to  17 w o u l d  the  which  group,  form  Then p o l y n o m i a l s  times  data  those  be d i v i d e d  The  was  for  another  according would  to  have  - 101 -  been  fitted  as  However the  total  before  since  amount  been q u i t e  and the  the  of  amount  data,  laborious  it  parameters of  data  and s i n c e  was n o t  for  different  neglected  application  was of  so the  considered worthwhile  lines  compared.  small  compared  model would to  pursue  to  have  this  model.  3.11  Growth Curves f o r I n d i v i d u a l Barnacles  In  all  averaging at a  each  the  over  time  different  contained  fit  for  two  the  the  of  five  item.  curve  components  within  analyses  sample  each  growth  variability to  previous  items  For  for of  each item. variability  and i t e m  to  -  the  different  with  the  limited  investigated  in  this  Consider  just  the  first  oyster  be a v a i l a b l e ,  e a c h row o f  There  would  rows  that to  observations  five is  individuals  one  was n o t  observation  possible  covariance  to  fit  be  nice  matrix  It  would  on e a c h i t e m a n d t h e n  items  data  line.  within  that  is  but  is  a  to  line.  available  at  to  within  each  different  a data matrix  correspond  to  one  e a c h i t e m on t h e  a data matrix  a row may o r may n o t  observations  Ideally  which would  corresponding  available  each item, in  only  after  will  be  section.  would  be  was  item v a r i a b i l i t y .  for  possible  growth d a t a ,  barnacle-to-barnacle  curves  is  for  it  So o u r  growth  this  Furthermore  reason  curve  Whether  correspond  this  the  there  growth  the  all  (ten),  to  average  compare  fact  relating  Y .  set  of  correspond time  In Y ,  2  2  five to  points  rows,  the in  same  barnacle.  line.  five  Y^  In  rows  successive barnacle.  different  rows  may  -  correspond  But E [ Y ] J = E [ Y ] 2  would have used Y^  estimate it  correspond  we  one  = A£P.  time  So we  are  independent.  can use  to o b t a i n a l e a s t  However,  our data m a t r i x Y  squares  estimate  just  2  of £.  matter whether or not the o b s e r v a t i o n s at s u c c e s s i v e to the same b a r n a c l e s - our e s t i m a t e i s s t i l l  i f we  as  This  i n v o l v e s only the average of the o b s e r v a t i o n s a t each time  doesn't  course  -  to the same b a r n a c l e , so the rows are dependent.  the f i v e o b s e r v a t i o n s at any  we  102  and  times  valid.  Of  were to assume t h a t a l l items have the same growth  curve,  would o b t a i n the same growth curve f o r the o y s t e r l i n e as p r e v i o u s l y  when we  first  averaged  d e s i g n m a t r i x A, we  w i t h i n each item.  can now  But by choosing a  obtain a d i f f e r e n t  growth curve  different f o r each  item. T e s t i n g f o r d i f f e r e n c e s between the growth curves i s not s t r a i g h t f o r w a r d as f i t t i n g hypotheses r e l a t i n g S~. c  the growth c u r v e s :  to the parameters we  These i n v o l v e S and  S, an e s t i m a t e  S i n c e each row  of Y  2  of the c o v a r i a n c e m a t r i x  E . Q  barnacle.  We  and S  o  have a l r e a d y  difficult.  does ot n e c e s s a r i l y correspond  b a r n a c l e , Y ' [ I - A^CA^' A ^ ) 2  test  need to f i n d m a t r i c e s S^  f o r the set of o b s e r v a t i o n s on an i n d i v i d u a l o b t a i n e d £, but S i s more  i n order to  as  - 1  A]/] Y  2  to the same  does not p r o v i d e an e s t i m a t e  However i f a s t r o n g assumption i s made about the form of E , i t  i s p o s s i b l e to o b t a i n an estimate  Q  of i t .  of  - 103 -  O b t a i n i n g an E s t i m a t e of If  we assume  correlation then  E  is  Q  that  between  E  n  variance  is  observations  proportional  constant  d weeks  over  apart  time  and  that  the  on an i n d i v i d u a l  is  to:  12  P  P  1  P  P  1  P  P  1  P  P  1  P  P  1  P  P  1  P 1  P 1  Under This  this is  strong  obtained  A number towards  the  on them. the  for  end  It  the  clear  the  were  study,  that  few t i m e  that  is  at  from  needed  each at  barnacles  of  the  fewer  than  these  barnacles  of  p was  is  an e s t i m a t e  Then  of  get  times  analyses  barnacles were  item,  times.  successive cannot  five  clearly  these  previous  obtained  on a p a r t i c u l a r  points.  observations  (since  that  excluded  and a n e s t i m a t e  barnacles  which  barnacles  of  all  p.  follows:  items  Suppose  last  same f o u r  as  was a s s u m e d  barnacles  barnacles.  of  assumption  were  using four  only  be  this  of  all  remained  sampling  corresponded  alive  these  barnacles  Furthermore  If  still  representative  we w o u l d  smaller).  because  it to  was the  wasn't  the usually same obvious  -  it  was a s s u m e d t h a t  corresponded  to  the  Items w i t h was n e e d e d . used.  only  Eleven  barnacles  obtained.  likelihood  items  Using  this  Koopmans  estimate  of  Our  assumption about  y _l  are  t-1,  and u  t  a /(l 2  The  are  t  these  E  = E[y ],  time,  be u s e d ,  s y s t e m one  oyster  at  14  t  = 11,  the  time  total  how t o  at  three  implies  t  - n ) = P'iy^  so  number  the  were  weeks  of  on  estimate  of  ) +z  -  distributed  a s N(0,  we l e t  Since p'  = p .  same i n d i v i d u a l  follows  that  the  t  a ), 2  y  t  respectively  variance  of  and at  t  the y ' s  and is  p' ). 2  likelihood  of  obtaining  (1 " P ' ) 2  L  y's  is,  N / 2  2 NT/2  =  the  e x  P  [  "  "  ( A  2  p  '  B  +  ( 1  +  P'  2  )C)/2a ]  2  (2itcf ) where T i s  the  individuals  total  number  of  time  points,  N is  the  total  number  and  A = E £ [ ( y ^ - H^) + ( y * - i4> ], 2  i  -  uJ)  (y*  -  \L\)  - u*) +  J  .  .  . + (yj^  -  \^_ ) ],  j  C = E E[(y^ i  2  j  B = E S[(y« i  j  j  j  2  +  .  .  .  (yj^  ]  2  -  ^ )  p  maximum  that  t  2,  barnacles  correlation.  week i n t e r v a l s  on.  replication  lines)  a n d 17  find  serial  and as  t h e maximum l i k e l i h o o d  Q  It  t  first  next  not  our model of  independently  the  the  could  (1942) d e s c r i b e s  on t h e  at  at  items;  data,  d a t a was t a k e n  observations  t  left  from the  p under  (y Z  (all  of  longitudinal  the  observation  d a t a was a v a i l a b l e  our  where  observation  one b a r n a c l e  on e a c h  i n v o l v e d was 34. was  smallest  smallest  Longitudinal  or 4  3  the  -  104  (y^  -  p£)],  of  -  where  item  y  and  leads  is  J t  the  observation  p.* = E [ y ^ ] .  to  an e s t i m a t e  at  105  time  t  -  on t h e  jth  barnacle  on t h e  p^  Maximising  this  likelihood  over  as  root  of  equation  of  p'  the  p'  turned  a cubic  the  a  n  d  ith  P  1  (Koopmans,  1942). Results  The e s t i m a t e correlation times the  is  of  between  seen t o  observations  be v e r y  high.  barnacle-to-barnacle  check time  our t  assumption  is  out  estimated  of  to  on t h e It  variance constant  is  The  estimates  turned  and  time  was a f a i r l y  to  an e s t i m a t e  serial S  0  .053  number  .043  unlikely  to  be  one.  a and  only  at  using  form.  a good  the  only  last  items over  at  to  obtain  at  each  time.  successive  estimates  time The  and  of  thus  variance  at  2  N - 1 J  .054, our  to  the  these  this  be o b t a i n e d  estimate  .056,  an e s t i m a t e  Although  time  .069,  assumption  for  points.  is  from  several  form of  a s m a l l number  three  vb  -  j  Combining  thus  a s s u m p t i o n was made a b o u t obtained  possible  variance  E  be  that  can r e a l i s t i c a l l y to  same i n d i v i d u a l  The  items.  out  weak  correlation  that  of  suggesting  of  So p = . 9 6 5 .  by  ? i the  .899.  within  (y*  where 1 i s  be  of  F,  0  of  .054,  constant  estimates of  Z  0  of  probably our  limited  secondly  barnacles  would  the  .049,  variance  the  reasons:  and  .048,  then  .060, over lead  assumed best  data  firstly the  and u s i n g  estimate it a  is strong  estimate  was  observations  of  -  106  -  = 0  T e s t i n g Hypotheses of the Form CgV A  Having  obtained  an e s t i m a t e  E  of  q  E ,  is  q  it  possible  to  A  hypotheses  of  the  f o r m C£V  = 0 by  substituting  test A  E  for  q  S and £  obtained  A  from Y  for  2  The  £ in  tests  the  of  expressions  hypotheses  for  which  S,  and S ?  h  This  e  have  been  is  discussed  not  clear.  rely  on t h e  fact  A  that  if  distribution' with k u'  D  —1  is  u  estimate data),  where  n'v-NCu, CE)  of  known  £ is  c  degrees  is  of  constant,  freedom,  (namely H o t e l l i n g ' s  obtained  from the  and D ~  then T  the  w p  (k,  £)>  Yi  (the  Wishart  e  of  Suppose  _  data matrix  n  distribution  k 1 = — u D u).  2  t  an  true  longitudinal  Then  the  namely A  t w h e r e we h a v e  A i )  (AJ  =  taken G = I i n  -  Y  AJ  1  Pothoff  P*(P  l  P')"  and R o y ' s  1  model.  A  covariance  matrix  of  (P P ' ) "  where  £ is  the  £  is P(A! A . )  1  1  covariance  -  1  E  1  matrix  P'(P  o  for  P')"  one  = (A! A , )  1  -  1  1 1  row o f  X = Y P'  (P  E  P')  -  1  .  Further A  £~  N(£,  (AJ  A j ) "  1  (1)  E)  and X'[I  Together  (1)  and  forms  this  and  (2)  the  Now s u p p o s e  -  A (A^ A j )  imply  basis  that  -  AJ]  1  X  the  that  for  the  the  W(k,  distribution  tests  estimate  X~  of  of £ is  C£V  (2)  E).  of  S^  1  Y2  (let  this  estimate  obtained  be d e n o t e d  known  = 0. instead  A  data matrix  is  £1)  and  that  from  our  -  S = Y'[I o Then  - A.(A! i l  there  are  A.) l l  A'] o  two r e a s o n s  Y  is  107  -  replaced  why t h e  by  our  estimate  same d i s t r i b u t i o n  of O  E  theory  no  E  O  .  longer  applies: A,  1. mean 5,  Although its  particular 2. the  E  covariance is  is  have  is  to  the  obtain  longitudinal  that  distribution  was a s s u m e d f o r to  same a s  distribution of  E, a n d  with  in  E.  a Wishart  possible  without  not  normal  as  because  of  it. growth curves  data,  S does  it  does  not  for  individual  seem p o s s i b l e  to  test  Discussion The  growth curve models  longitudinal  mentioned  in  p r o b l e m was  data  Section the  (item).  assumption  even i f  every 1.  cases  and 2.  3.4  At  s a m p l e d was  items the  the  ten  each of  only  of  applied  a number  assumption  equal  the  that  covariance  two r e a s o n s matrices  why were  previous of  assumptions.  was  likely  matrices this  sections  for  might  assumed  to  to  cause  every  be a be  strong  the  were  the  a different  in  averages other  population size  for  over  five  barnacles  in  same  some  cases. from which  the  each d i f f e r e n t  barnacles item.  to  As  barnacle:  barnacles  time  were  required  covarance  observations  over  that  T h e r e were  individual Our  for  assumption  individual  for  matrix  a multivariate  differences.  3.12  our  it  has  proportional  form that  So a l t h o u g h  for  not  does not  Q  special  barnacles  still  Ci  were  a  -  In  Section  3.8  observations barnacles.  were a v e r a g e s  obtained  -  was m o d i f i e d over  were  to  sometimes  allow  five  for  However  the  fact  and sometimes  o b t a i n e d w h i c h were  previously.  the  slightly  broad  that  ten  different  conclusions  remained  same. It  is  not  population time  oyster  In  especially  Section  sizes. 3.5,  from the  view  involved In  alive  on t h e  Ideally growth  3.5,  of  3.6  each  at  The  be  the  point  was  again  like  individual  fit  and  same  for  curves  growth c u r v e s  barnacle  at  natural  between  £ was any  were  fitted. of  growth  the curves  obtained  from  particular  population.  that t  error  represented  of  the  However  a number  our  be a l i v e  Sections  conclusions.  fitting  estimate  two  population  the  items  over  each  somewhat  reliable.  we d i s c u s s e d  to  show  in  even i f  the  on  each  obtained  different  that  at  system  indeed  obtained  curves  growth  number  from the  the  reach  alive  matrices  did  an a v e r a g e  represented to  total  completely  growth  which happened  curves  the  p-values  still  3.11  to  small,  So o u r  In S e c t i o n  we w o u l d  a typical  and 3 . 7 ,  But  due  considered  time.  barnacles.  lines  different  barnacles  covariance  the  so  the  of  two e s t i m a t e s  we w o u l d  data  that  barnacles  So a g a i n  the  difficulty,  large,  population.  individual  number  oyster  we o b t a i n e d w e r e  sections  barnacles  of  s y s t e m one  this  quite  Sections  the  (we know o n l y  estimates  s h o u l d not  that  were  these  time.  known  accommodate  w h i c h may be p a r t l y  of  and 3.7  p-values  data  3.6  patterns  In  3.6  natural  not  to  since  l i n e s "were p r e s e n t e d .  different  for  straightforward  sizes,  respectively  the  so  on e a c h i t e m i s  line).  In  analysis  New p - v a l u e s  from those the  the  108  would  = 2 and  represent 17 w e e k s .  the It  -  is  not  clear  that  Ideally  the  this  is  possible  following  109  -  with  the  limited  i n f o r m a t i o n would  be  data  available.  available  for  each  barnacle: (a)  Growth  data  at  each  death whichever (b)  Time  of  death  is  if  survived  past  (c)  Cause  death -  (d)  Number If  of  this  depending 1. while  of  the  were  on t h e  in  be  truncated  would  have  data,  is  obtain  food  shortage  case  the  at  its  model this  would  of  If  = 17 weeks  t  = 17 weeks  or  predator.  to which  several  it  is  approaches  to  fit  or  until  its  or  knowledge  that  it  attached.  would  growth curves  censoring.  one w a y ,  is  namely death  the time  between  represent  we w e r e  could  be  possible  covariate.  data;  barnacles  of  general  be u s e d  to  natural  the  our  between  instead  survive  each  before  to  fit  t  the  individual = 17 w e e k s ;  t  this  pattern  growth curves  and  growth curves  population,  = 2 and whether  I  allow  The  survival  type  past  t  in  d a t a as have  is  for  affect  barnacles  we  observations.  not  We w o u l d which  this  individual  would  data  groups.  interested  dependent  growth if  for  deaths  various  barnacle  think  The  sufficiently  model  a typical  g r o w t h we c o u l d  for  before  i t e m and l i n e  differences  experience  to  so  t  earlier.  growth d a t a w i t h m i s s i n g  Kleinbaum's  2.  until  = 17.  for  would  for  point  objectives:  only  would  this  the  analysis  test  the  S u p p o s e we w a n t e d  accounting  missing  t  time  but  to  that  the  of  we  growth  17 w e e k s . s u r v i v a l was  data with  censoring  = 17 w e e k s ,  in the  the  a  related  time survival  exact  lifetime  - 110  would not  be known.  The  -  time dependent c o v a r i a t e c o u l d be s h e l l  Time dependent c o v a r i a t e s can  be i n c o r p o r a t e d i n t o Cox's  p r o p o r t i o n a l hazards model, which i s d e s c r i b e d i n S e c t i o n 2.1. hazard  f u n c t i o n would then be \(t;  length.  z ( t ) l = X (t) e o  2  ^ ^  The  where i n our  case: \ ( t ) i s an unknown b a s e l i n e hazard  function,  Q  B i s an unknown parameter, z ( t ) I s our time dependent c o v a r i a t e - s h e l l In t h i s context c h a r a c t e r i s t i c s and testing H :  and  of 8 = 0 would be a t e s t  s u r v i v a l time are not  related.  8 = 0 i s d e s c r i b e d i n S e c t i o n 2.1.  Q  hypothesis  a test  l e n g t h at time t . t h a t growth The  method of  A r e j e c t i o n of  would suggest some dependency between growth  this  characteristics  s u r v i v a l time, f o r example a tendency f o r l a r g e r b a r n a c l e s  to  live  longer. 3.  There are two  namely they may shortage.  be eaten  Instead  time, i t may  p o s s i b l e causes of death f o r the by p r e d a t o r s  "do  predators?"  be of i n t e r e s t  to t e s t  t o food survival  s e p a r a t e l y whether each cause of  Then we  the b i g g e r b a r n a c l e s and  d i e due  of t e s t i n g whether growth i s r e l a t e d to  death i s r e l a t e d to growth. such as  or they may  barnacles,  tend  would be able to address to be eaten  first  by  questions  the  " i s i t the s m a l l e r b a r n a c l e s which tend t o d i e of  food  shortage?" T h i s can be done i n the by  f i n d i n g the hazard  context  of Cox's p r o p o r t i o n a l hazards model  function specific  to each cause of death.  Growth  -  is  again  over  included  the  as  two t y p e s  of  obtained  and a t e s t  death  unrelated  is  failures  other  observations,  to  like: only  at  those  so i t  correspond  to  point  is  the  not  these  only  reasons  for  the  on a p a r t i c u l a r  explained are  as  neither  average solely  With  of  only  limited  for  the big  that  the  ith  data is  the  it  or n o t which  the  of  items  the  line is  in  barnacles  that  or p a r t i c u l a r l y  not is  over to  the  not to  this  its  die  time  to  would  barnacles,  barnacles  cannot  to  which  Survival the  data  lines.  above.  between  = 2 and  by  t  the  provided size.  then  interval  is  For  described  growth that  This  the  can  two  time  the  change  will  be  be  points in  due  population.  a s c e r t a i n whether  since  be  observations  between  small,  changing  easy  size:  to  we  on a l l  represented  unrelated  used.  that  within  3.7,  of  all  8^,  are  known.  barnacle  Section  be  cause  correspond  approaches  be w e l l  vary  censored  successive  not  8 may  would  2  of  methods  deaths  is  will  related lines,  for  as  individual  a typical  barnacles  growing,  dying  P  information  of  dying  if  living  barnacles  probability available  of  and  growth data  death  any  that  a barnacle  particularly  our  line  for  follows:  size to  fitted  have  time  8^  treated  the  nor  of  not  growth e x p e r i e n c e  of  all  this  an e s t i m a t e  Furthermore  pursue  17 weeks  probability  have  cause  lines  we c a n n o t  was  are  we do n o t  The  the  that  i  known w h e t h e r  However,  curve  type  but  of  be a t e s t  To o b t a i n  same b a r n a c l e s ,  growth measurements. available  .  from each i t e m .  identified  covariate  u s u a l maximum l i k e l i h o o d  we do n o t  each time  on f i v e  of  -  So e s t i m a t e s  B^ = 0 w o u l d growth  the  Unfortunately  dependent  failure.  of  than and  a time  Ill  survival  g r o w t h d a t a was a v e r a g e d  data over  the  is items  within  -  each  line.  So  percentage  of  Using at  all  where  tended this  to  The  to  survival  start  dying  barnacles  at  dying  line  later  material) and  It  on s y s t e m o n e .  was  large,  smaller  to a t h i r d  effect  -  are  To e l i m i n a t e  on  a possible  for  example,  just  However  there  only  six  of  so a c o r r e l a t i o n  these  lines  available dying case  at it  using  wouldn't  for  every  each time would  just  certainly  item, for  items  time  f r o m one  between  suggest  lifetime  only  assume One  there  that  to  point  is is  If  s y s t e m one  s y s t e m one  size  and  size  and  percentage  between  the  probability  of  the  experimenter  view  Without  detailed  to  lines. just  from  percentage In  this  effect  by  correlation  dying of  or  were  line  Then a s i g n i f i c a n t  related  treatment  lines.  a possible  die  tend  oyster  data  oyster  to  than  obtained  survival  average  eliminate  line.  be u n r e l a t e d .  this  strong  point  have  from  average  dependency  would  other  was a  we w o u l d  60 i t e m s  From a b i o l o g i c a l and  meaningful.  e v e n be p o s s i b l e  the  particular  be v e r y  these  bigger  system or  use,  from  a system  s y s t e m two  we s h o u l d  size  those  factor  example,  generally  the  tend  underlying for  On  conclude  barnacles  barnacles  they  the  percentage  H o w e v e r we c a n n o t  the  and  found between  a smaller  effect,  are  length  time.  may r e f l e c t ,  furthermore  shell  correlation  5 and 6 .  and g r o w t h . of  each  week 5 was  a particular  average  b e t w e e n weeks 5 and 6 .  c o r r e l a t i o n may be due  (type  at  a significant  size  -  17 l i n e s  available  b e t w e e n weeks  within  treatment  the  percentage  average  first. both  were  17 l i n e s the  die  that  each of  deaths  week 5 and  lines  for  112  at  would  dying  expected  survival  and  size.  that  d a t a we  size can  true. of  interest:  correlation  between  in  Section  the  2.5  initial  it  was n o t e d  number  a  on a  that  line  and  -  the  estimates  on l i n e s  of  with  barnacles  8 and a f o r  fewer  tended  are  generally  speculate for  the  they  have  have  a better  to  compete chance  tend  barnacles  isn't for of  the  fact  growth  differences  in  than  a place. finding  a  The  are  a fast  it  a poorer is  curves the  on d o w l i n g  the  here  environment  difficult to  lines  -  it  that  the  small is  to  environments  start  to  explain  with to  the  on  Also  quickly  attribute or  may  and  population  difficult to  and  barnacles  initially  diminishes  t o know w h e t h e r  different  barnacles.  are  which would  rate.  fact  system  barnacles  a faster small  to  site  on s y s t e m 2  The  one  could  ideal  bigger.  particularly The  We  attached  healthier  on s y s t e m 2,  lines),  on s y s t e m  a more  barnacles  and a t  rate.  barnacles  the  that  rate.  barnacles  become  initially  are  oyster  on s y s t e m t w o .  bigger  place  lines  to  that  s y s t e m two i s  all  dowling  the  a faster  why more b a r n a c l e s room f o r  suggested  s y s t e m one  and a t  that  earlier  that  correlation  (e.g.  those  dying  and a t  The  observation  observations  on s y s t e m one a n d  Similarly the  initially  sooner  start  early  be a t t r i b u t a b l e  in  additional  on s y s t e m two  to  on t h e  dying  to  there  line.  initially  dying  which i s  barnacles  barnacles  due  smaller  maybe  s y s t e m one  start  the  barnacles  But  the  start  from these  two.  why  barnacles  to  We now h a v e  that  -  113  say  of  may or  be may  which.  differences initial  -  114  -  CONCLUSION  A data growth  of  logrank  provided  barnacles  than material was  fitted  the  estimated  The  type  for  lines.  only  comparison  of  system but  not  important The  the  that  estimates  tests,  existed  the  the  the  within  influence  The e x p o n e n t i a l  each of  and  between  type  s y s t e m had more  of  survival  namely  same m a t e r i a l  distribution.  model  17 l i n e s  and  compared.  lines  but  estimates  lines  the  of  for  obtained  each type  type.  Again  8 values,  subsequently  were  suggested  the  within  that  initially  including  of  the  all  true  each system.  A  s l o p e was d e p e n d e n t  s y s t e m was f o u n d  17  to  on  be t h e  more  factor.  at  polynomial  data  parameters  less  straightforward  We a v e r a g e d  the  items.  growth curves  for  items  the  case  model and R a o ' s of  indicated  over  the  to  examine  set  f r o m e a c h i t e m and t r e a t e d for  and R o y ' s in  data.  each time  longitudinal  except  for  on m a t e r i a l  longitudinal  Pothoff  of  distributions  oyster  g r o w t h d a t a was  obtained  lines  to  differences  m o d e l was d e v e l o p e d  the  these  that  suggested  8 were  relating  Nonparametric  survival  Maximum l i k e l i h o o d 8 value  of  survival  slopes,  underlying  as  even tests  A random e f f e c t s incorporating  Goldberg  indicated  on t h e  the  H.  was e x a m i n e d .  distributions  same s y s t e m .  have  by M r .  and t h e W i l c o x o n ,  survival the  set  the that  Using were  model.  second dowling differences  this  fitted  of the  set to  line.  existed  not  measurements resulting of  each  Quadratics  a s we d i d  data  set  averages, line  were  using  both  adequate  A comparison  of  the  b o t h between systems  and  - 115  between m a t e r i a l dowling  and t h e  types, other  somewhat  unreliable  matrices  for  didn't  have  to  the  but  fit not In  smaller  to  every  as  with  two t y p e s the  average  growth  for  conclusion, amount  of  biggest of  strong  individual  longitudinal  test  the  data  for  differences  differences  material. assumption  (item)  curve  -  had  to  barnacles  for  The p - v a l u e s of  equal  be m a d e . it  individual  was  between may  Although,  found  barnacles  to  obtained.  a l t h o u g h we had a l a r g e  amount  of  have  be  we possible  on e a c h  curves  data would  be  covariance  the  longitudinal  between  being  item  growth d a t a ,  b e e n more  useful.  a  -  116  -  BIBLIOGRAPHY  Breslow, N.E. (1974). Covariance B i o m e t r i c s 30, 89-99. Cox,  D.R. (1972). discussion). 187-202.  of  censored  survival  R e g r e s s i o n models and l i f e t a b l e s ( w i t h J o u r n a l o f the R o y a l S t a t i s t i c a l S o c i e t y B,  G r i z z l e , J . and A l l e n , response curves. Ito,  analysis  D.M. (1969). B i o m e t r i c s 25,  Analysis 357-381.  of  g r o w t h and  data.  34,  dose  K. (1968). On t h e e f f e c t o f h e t e r o s c e d a s t i c i t y and n o n n o r m a l i t y u p o n some m u l t i v a r i a t e t e s t p r o c e d u r e s . In: K r i s h n a i a h , P.R. ed.; Multivariate Analysis II. New Y o r k : Academic P r e s s , 87-120.  Kalbfleisch, of  J.G.  failure  and P r e n t i c e ,  time  data.  R.L.  (1980).  New Y o r k :  K h a t r i , C.G. (1966). A note growth c u r v e . A n n a l s of 18, 7 5 - 8 6 .  The  statistical  analysis  Wiley.  on a manova m o d e l a p p l i e d t o the I n s t i t u t e of S t a t i s t i c a l  problems i n Mathematics  Kleinbaum, D.G. (1973). A g e n e r a l i z a t i o n of the growth c u r v e model which allows missing data. J o u r n a l of M u l t i v a r i a t e A n a l y s i s 3, 117-124. Koopmans, T . (1942). S e r i a l c o r r e l a t i o n and q u a d r a t i c f o r m s variables. A n n a l s of M a t h e m a t i c a l S t a t i s t i c s 13, 14-33. L a i r d , N.M. and Ware, J . H . (1982). Random e f f e c t s longitudinal data. B i o m e t r i c s 38, 963-974. Lawless, J . F . Wiley. Lee,  (1982).  Statistical  models  lifetime  Y.K. (1974). A n o t e on R a o ' s r e d u c t i o n o f P o t h o f f generalized l i n e a r model. Biometrika 61, 349-351.  Lehmann, E . L . (1917). Nonparametrics: ranks. H o l d e n Day, San F r a n c i s c o . M a c h e n , D. (1975). 31, 749-753.  On a d e s i g n  Morrison, D.F. (1976). McGraw-Hill. Peto,  for  models  R.  (1972).  Journal  of  I  the  Royal  problem i n  Multivariate  Contribution  statistical  to  the  Statistical  discussion Society  B,  data.  of  New  and  Roy's  methods  based  methods.  34,  normal  for  growth s t u d i e s .  statistical  in  paper  York:  on  Biometrika  New  York:  by D . R .  205-207.  Cox.  -  Pothoff,  R.F.  analysis  and R o y , of  problems.  Biometrika  P r e n t i c e , R.L. Biometrika Prentice,  R.L.  censored  S.N.  variance  (1964). 51,  and M a r e k , rank  P.  tests.  -  A generalized  model u s e f u l  especially  multivariate  for  growth  curve  313-326.  (1978). Linear 65, 167-179.  data  117  rank  (1979).  tests  with  right-censored  A qualitative  Biometrics  35,  data.  discrepancy  between  861-886.  Rao,  CR. (1965). T h e t h e o r y o f l e a s t s q u a r e s when t h e p a r a m e t e r s a r e s t o c h a s t i c and i t s a p p l i c a t i o n t o t h e a n a l y s i s o f g r o w t h c u r v e s . Biometrika 52, 447-458.  Rao,  CR. (1966). C o v a r i a n c e a d j u s t m e n t and r e l a t e d p r o b l e m s i n multivariate analysis. In: K r i s h n a i a h , P.R. e d . ; M u l t i v a r i a t e Analysis. New Y o r k : Academic P r e s s , 87-103.  Satterthwaite, F.E. (1946). of v a r i a n c e components. Timm,  An a p p r o x i m a t e d i s t r i b u t i o n of e s t i m a t e s B i o m e t r i c s B u l l e t i n 2, 1 1 0 - 1 1 4 .  N.H. (1980). M u l t i v a r i a t e a n a l y s i s of repeated In: K r i s h n a i a h , P.R. e d . ; A n a l y s i s of V a r i a n c e . H o l l a n d , 41-87.  measurements. New Y o r k : North  

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