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A comparison of some robust estimates of correlations in the presence of asymmetry Lind, John Charles 1983

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A COMPARISON OF SOME ROBUST ESTIMATES OF CORRELATIONS IN THE PRESENCE OF ASYMMETRY by JOHN CHARLES LIND M.A., The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES De p a r t m e n t o f P s y c h o l o g y We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e g u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA June 1983 © J o h n C h a r l e s L i n d , 1983 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y 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 . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my 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 . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f ~Y^yjC-]/\o\o<^\j  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 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 U n i v e r s i t y -6 (3/81) A b s t r a c t Three methods f o r o b t a i n i n g robust estimates of c o r r e l a t i o n m a t r ices were compared i n c o n d i t i o n s of asymmetrically contaminated normal d i s t r i b u t i o n s . The e s t i m a t o r s examined i n c l u d e d a m u l t i v a r i a t e trimming (MVT) procedure based on the Mahalanobis d i s t a n c e measure, and two procedures d e r i v e d from the r o b u s t e s t i m a t i o n of r e g r e s s i o n c o e f f i c i e n t s . The r e g r e s s i o n c o e f f i c i e n t s were o b t a i n e d from the sample cumulant g e n e r a t i n g f u n c t i o n o f the r e s i d u a l s . Monte C a r l o r e s u l t s were ob t a i n e d f o r v a r i o u s l e v e l s o f sample s i z e and o u t l i e r contamination. C o r r e l a t i o n s o b t a i n e d from the r e g r e s s i o n procedures were observed t o be h i g h l y robust w i t h r e s p e c t t o asymmetric contamination and were able to w i t h s t a n d l a r g e r amounts of o u t l i e r contamination than the MVT e s t i m a t e s . The MVT esti m a t e s tended to be s l i g h t l y l e s s b i a s e d than c o r r e l a t i o n s o b t a i n e d from the r e g r e s s i o n procedure i n c o n d i t i o n s w i t h the s m a l l e s t amounts of contamination. The use of these e s t i m a t e s f o r o u t l i e r i d e n t i f i c a t i o n i s d i s c u s s e d . i i i T a b l e o f C o n t e n t s Page A b s t r a c t i i L i s t o f T a b l e s i v L i s t o f F i g u r e s v i Acknowledgement v i i I n t r o d u c t i o n 1 Method 2 3 D a t a G e n e r a t i o n 23 I n d e p e n d e n t V a r i a b l e s 26 MVT P r o c e d u r e 28 R e g r e s s i o n P r o c e d u r e 29 R e s u l t s and D i s c u s s i o n 35 I m p l i c a t i o n s f o r P r a c t i c e 6 4 R e f e r e n c e s 6 8 i v L i s t o f T a b l e s Page T a b l e 1 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=30 and 5 p e r c e n t c o n t a m i n a t i o n T a b l e 2 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=30 and 10 p e r c e n t c o n t a m i n a t i o n T a b l e 3 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=30 and 15 p e r c e n t c o n t a m i n a t i o n T a b l e 4 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=60 and 5 p e r c e n t c o n t a m i n a t i o n T a b l e 5 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=60 and 10 p e r c e n t c o n t a m i n a t i o n T a b l e 6 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=60 and 15 p e r c e n t c o n t a m i n a t i o n T a b l e 7 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=60 and 25 p e r c e n t c o n t a m i n a t i o n T a b l e 8 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=120 and 5 p e r c e n t c o n t a m i n a t i o n T a b l e 9 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=120 and 10 p e r c e n t c o n t a m i n a t i o n T a b l e 10 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=120 a n d 15 p e r c e n t c o n t a m i n a t i o n T a b l e 11 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=120 and 20 p e r c e n t c o n t a m i n a t i o n 36 37 38 41 42 43 44 47 48 49 50 V Page T a b l e 12 A v e r a g e b i a s o f r a n d MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=120 and 25 p e r c e n t c o n t a m i n a t i o n 51 T a b l e 13 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=120 and 30 p e r c e n t c o n t a m i n a t i o n 52 T a b l e 14 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=60 and 10 p e r c e n t c o n t a m i n a t i o n -Symmetric c o n t a m i n a t i o n c o n d i t i o n 54 T a b l e 15 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=60 and 15 p e r c e n t c o n t a m i n a t i o n - Mean v e c t o r o f c o n t a m i n a t i n g d i s t r i b u t i o n = [1,...,1] 55 T a b l e 16 A v e r a g e b i a s o f r a n d MSE o f t h e z- : t r a n s f o r m e d e s t i m a t e s f o r n=60 and 15 p e r c e n t c o n t a m i n a t i o n - Mean v e c t o r o f c o n t a m i n a t i n g d i s t r i b u t i o n = [ 5 f . . . , 5 ] 56 T a b l e 17 A v e r a g e b i a s o f r a n d MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=60 and 5 p e r c e n t c o n t a m i n a t i o n - O r d e r o f t h e v a r i a b l e s r e v e r s e d 5 8 T a b l e 18 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=60 and 10 p e r c e n t c o n t a m i n a t i o n - O r d e r o f t h e v a r i a b l e s r e v e r s e d 59 T a b l e 19 A v e r a g e b i a s o f r and MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=60 a n d 15 p e r c e n t c o n t a m i n a t i o n - O r d e r o f t h e v a r i a b l e s r e v e r s e d 60 T a b l e 20 A v e r a g e b i a s o f r a n d MSE o f t h e z-t r a n s f o r m e d e s t i m a t e s f o r n=60 and 10 p e r c e n t c o n t a m i n a t i o n - O r d e r o f t h e l a s t t h r e e v a r i a b l e s r e v e r s e d 61 L i s t o f F i g u r e s Page F i g u r e 1 S c a t t e r p l o t o f 50 o b s e r v a t i o n s w i t h 2 o u t l i e r s v i i Acknowledgement I wish t o thank my.thesis s u p e r v i s o r , Dr. A. R. Ha k s t i a n , and the members of my t h e s i s committee; Dr. J . H. S t e i g e r , Dr. R. D. Hare and Dr. S. W. Nash, f o r t h e i r v a l u a b l e suggestions and comments r e g a r d i n g the contents of t h i s study. 1 A C o m p a r i s o n o f Some R o b u s t E s t i m a t e s o f C o r r e l a t i o n s i n t h e P r e s e n c e o f Asymmetry I n t h e a n a l y s i s o f d a t a f r o m p s y c h o l o g i c a l t e s t s o r e x p e r i m e n t s , o b s e r v a t i o n s a r e f r e q u e n t l y e n c o u n t e r e d w h i c h a p p e a r t o be i n c o n s i s t e n t w i t h t h e m a j o r i t y o f t h e o b s e r v a -t i o n s i n t h e s a m p l e . T h e s e o b s e r v a t i o n s , w h i c h o f t e n become a p p a r e n t d u r i n g d a t a i n s p e c t i o n o r e d i t i n g , s t a n d o u t f r o m t h e r e s t o f t h e o b s e r v a t i o n s i n t h e sample by h a v i n g v a l u e s t h a t may seem t o be u n u s u a l l y l a r g e o r s m a l l f o r t h e v a r i -a b l e s i n v o l v e d . Commonly r e f e r r e d t o as w i l d p o i n t s o r o u t l i e r s , t h e s e o b s e r v a t i o n s may be p r e s e n t a s i s o l a t e d o c c u r r e n c e s s c a t t e r e d t h r o u g h o u t a sample o r t h e y may o c c u r a s s m a l l g r o u p s t h a t , a l t h o u g h s i m i l a r t o e a c h o t h e r , d i f f e r f r o m t h e m a j o r i t y o f t h e o b s e r v a t i o n s . W i l d p o i n t s o r o u t l i e r s f r e q u e n t l y o c c u r a s a r e s u l t o f e r r o r s i n r e c o r d i n g o r measurement. D a t a o b t a i n e d f r o m p s y -c h o l o g i c a l t e s t s , f o r ex a m p l e , may c o n t a i n o u t l y i n g o b s e r -v a t i o n s t h a t a r i s e f r o m e r r o r s t h a t o c c u r i n t h e a d m i n i s t r a -t i o n o r s c o r i n g o f t h e t e s t s . I n o t h e r s i t u a t i o n s , o u t l i e r s may o c c u r when human o b s e r v e r s make e r r o r s i n r e c o r d i n g s u b j e c t r e s p o n s e s o r e r r o r s may o c c u r when s u b j e c t s f a i l t o f o l l o w i n s t r u c t i o n s p r o v i d e d i n t h e a d m i n i s t r a t i o n o f a t e s t o r e x p e r i m e n t . O u t l y i n g o b s e r v a t i o n s a l s o f r e q u e n t l y o c c u r as a r e s u l t o f l a b o r a t o r y e q u i p m e n t t h a t f a i l s t o f u n c t i o n p r o p e r l y and c o n s e q u e n t l y g e n e r a t e s s p u r i o u s r e s u l t s . The s e n s i t i v i t y o f human s u b j e c t s t o s m a l l c h a n g e s i n e x p e r i -2 m e n t a l p r o c e d u r e , o r i n d i v i d u a l d i f f e r e n c e s i n p r i o r e x p e c t -a n c i e s a b o u t t h e p u r p o s e o f e x p e r i m e n t s i n w h i c h a s u b j e c t p a r t i c i p a t e s , a r e f a c t o r s t h a t may a f f e c t some i n d i v i d u a l s i n t h e sample d i f f e r e n t l y t h a n t h e m a j o r i t y , and c o n s e q u e n t l y r e s u l t i n t h e p r e s e n c e o f o u t l y i n g o b s e r v a t i o n s . A common s o u r c e o f o u t l y i n g o b s e r v a t i o n s i n p s y c h o l o g i c a l r e s e a r c h i s t h e r e s u l t o f i n d i v i d u a l s o t h e r t h a n t h o s e f r o m t h e popular t i o n o f i n t e r e s t t h a t a r e e r r o n e o u s l y i n c l u d e d i n t h e sample b e i n g e x a m i n e d . T h i s p r o b l e m a r i s e s i n a r e a s s u c h as c l i n i -c a l p s y c h o l o g y where m e n t a l d i s o r d e r s t h a t d i s p l a y s i m i l a r symptoms may d i f f e r i n t h e i r e t i o l o g y , r e s u l t i n g i n sam p l e s o f i n d i v i d u a l s f r o m p o p u l a t i o n s o f i n t e r e s t t h a t a r e contam-i n a t e d b y t h e p r e s e n c e o f i n d i v i d u a l s f r o m o t h e r p o p u l a t i o n s . F o r example, i n t h e d e v e l o p m e n t o f a p s y c h o l o g i c a l t e s t f o r t h e a s s e s s m e n t o f d e p r e s s i o n , a r e s e a r c h e r may c h o o s e a sample o f i n d i v i d u a l s t h a t d i s p l a y s a p a r t i c u l a r s e t o f symptoms t h a t d e f i n e t h e d i s o r d e r o f i n t e r e s t . I f a p r o p o r t i o n o f t h e sample c o n s i s t s o f i n d i v i d u a l s t h a t d i f f e r f r o m t h e m a j o r i t y o f i n d i v i d u a l s i n t h e u n d e r l y i n g c a u s e o f t h e d i s o r d e r , s u c h as u n d e t e c t e d o r g a n i c o r d r u g r e l a t e d e f f e c t s , t h e t e s t s c o r e s o f t h e s e i n d i v i d u a l s may d i f f e r f r o m t h o s e o f t h e l a r g e r g r o u p and s u b s e q u e n t l y p r o v i d e t e s t r e s u l t s t h a t may be m i s l e a d i n g . When an i n s p e c t i o n o f sample d a t a r e v e a l s e r r o r s p r o d u c e d by e q u i p m e n t f a i l u r e , m i s c l a s s i f i e d i n d i v i d u a l s , o r i n r e c o r d i n g and measurement, a s t a n d a r d p r a c t i c e i s t o remove t h e q u e s t i o n a b l e o b s e r v a t i o n s f r o m t h e sample p r i o r t o 3 a n a l y s i s . I n many s i t u a t i o n s , however, t h e s o u r c e o f anomalous o b s e r v a t i o n s may n o t be a p p a r e n t f r o m an i n s p e c -t i o n o f t h e d a t a . The r e s e a r c h e r i s t h e n f a c e d w i t h t h e p r o b l e m o f how t o d e c i d e i f t h e o b s e r v a t i o n s i n q u e s t i o n a r e v a l i d o r due t o e r r o r . I n t h e a b s e n c e o f i n f o r m a t i o n i n d i c a t i n g t h a t o u t l y i n g o b s e r v a t i o n s a r e t h e r e s u l t o f e r r o r s , a common c h o i c e i s t o assume t h a t t h e q u e s t i o n a b l e o v s e r v a t i o n s a r e v a l i d and i n c l u d e them i n t h e s a mple. I n t h e s e s i t u a t i o n s t h e sample d i s t r i b u t i o n i s o f t e n o b s e r v e d t o c l o s e l y a p p r o x i m a t e t h e assumed f o r m o f t h e p o p u l a t i o n d i s t r i b u t i o n i n s p i t e o f t h e p r e s e n c e o f a s m a l l number o f o u t l i e r s . The o u t l y i n g o b s e r v a t i o n s a r e t h e n i n c l u d e d i n t h e sample s i n c e i t i s o f t e n assumed t h a t commonly u s e d s t a t i s t i c s s u c h as means, v a r i a n c e s and c o v a r i a n c e s , w h i c h p e r f o r m o p t i m a l l y u n d e r t h e i d e a l m o d e l, w i l l a l s o p e r f o r m w e l l i n c o n d i t i o n s t h a t d e v i a t e s l i g h t l y f r o m t h e i d e a l f o r m o f t h e d i s t r i b u t i o n . A d e m o n s t r a t e d l i m i t a t i o n , however, o f t h e u s u a l e s t i m a t e s o f means, v a r i a n c e s and c o v a r i a n c e s i s t h e i r s e n s i t i v i t y t o s l i g h t d e p a r t u r e s f r o m an i d e a l i z e d f o r m o f t h e d i s t r i b u t i o n (Huber, 1977b, p . l ) . F o r example, among a p p r o x i m a t e l y 70 e s t i m a t e s o f l o c a t i o n , t h e a r i t h m e t i c mean has b e e n shown t o be one o f t h e l e a s t r o b u s t l o c a t i o n e s t i m a t e s i n t h e p r e s e n c e o f o u t l i e r s (Andrews, B i c k e l , Hampel, Huber, R o g e r s and Tukey, 1 9 7 2 ) . The u s u a l e s t i -mates o f v a r i a n c e s and c o v a r i a n c e s a r e a l s o s e n s i t i v e t o s m a l l d e v i a t i o n s i n t h e shape o f a d i s t r i b u t i o n c a u s e d by t h e p r e s e n c e o f o u t l i e r s , s i n c e t h e s e e s t i m a t e s a r e a d v e r s e l y i n f l u e n c e d b y i n f o r m a t i o n c o n t a i n e d i n t h e t a i l s o f a d i s t r i b u t i o n ( G n a n a d e s i k a n & K e t t e n r i n g , 1972; Huber, 1977b, p p . 41-47; Tukey, 1960, pp. 4 4 8 - 4 8 5 ) . S i n c e o u t l i e r s a r e p r e s e n t i n most d a t a o b t a i n e d i n p r a c t i c e , t h e u s u a l e s t i m a t e s o f means, v a r i a n c e s and c o v a r -i a n c e s s h o u l d be u s e d w i t h c a u t i o n (Tukey, 1979, pp. 103-106). W h i l e t h e p r o p o r t i o n o f o u t l i e r s p r e s e n t i n t y p i c a l d a t a s e t s h a s b e e n e s t i m a t e d t o be between f i v e and t e n p e r c e n t (Hampel, 1974; Huber, 1977b, p . ,3), i n p s y c h o l o g y e r r o r s i n measurement a r e o f t e n d i f f i c u l t t o o b s e r v e d i r e c t l y , and s u b j e c t v a r i a b i l i t y may o f t e n r e s u l t i n h e a v y - t a i l e d d i s t r i b u t i o n s o r s a m p l e s i n w h i c h t h e p r o p o r t i o n o f o u t l i e r c o n t a m i n a t i o n i s g r e a t e r t h a n 10 p e r c e n t . B e c a u s e o f t h e p r e v a l e n c e o f o u t l i e r s i n a p p l i e d r e s e a r c h , a recommended p r o c e d u r e f o r t h e a n a l y s i s o f e x p e r i m e n t a l d a t a i n v o l v e s p e r f o r m i n g two s e t s o f a n a l y s e s f o r a g i v e n sample, one u s i n g s t a n d a r d s t a t i s t i c a l p r o c e d u r e s and one u s i n g r o b u s t o r o u t l i e r r e s i s t a n t methods. I f t h e r e s u l t s o f t h e two a n a l y s e s d i f f e r , t h e r e s u l t s o f t h e r o b u s t p r o c e d u r e s h o u l d be u s e d (Hogg, 1979, pp. 1-17; Tukey, 1979, pp. 103-106). B i a s e d o r m i s l e a d i n g r e s u l t s i n t h e a n a l y s i s o f e x p e r i -m e n t a l d a t a i s a common p r o b l e m i n p s y c h o l o g i c a l r e s e a r c h s i n c e d a t a a r e o f t e n a n a l y z e d w i t h o u t p r i o r i n s p e c t i o n f o r e r r o r s . T h i s i s f r e q u e n t l y due t o t h e r o u t i n e a p p l i c a t i o n o f r e a d i l y a v a i l a b l e c o mputer p r o g r a m s t o p e r f o r m t h e 5 d e s i r e d s t a t i s t i c a l a n a l y s i s . The i n c r e a s i n g use of automated data a c q u i s i t i o n and storage systems a l s o may r e s u l t i n e r r o r s t h a t are undetected p r i o r t o a n a l y s i s . Data s e t s o b t a i n e d i n t h i s way are a l s o f r e q u e n t l y l a r g e , making v i s u a l i n s p e c t i o n o f the data i m p r a c t i c a l . The r o u t i n e use of o u t l i e r r e s i s t a n t s t a t i s t i c s i n these s i t u a t i o n s p r o v i d e s a method i n which the e f f e c t s o f unknown sources of e r r o r may be reduced or removed. In the case of u n i v a r i a t e data, a comparison of the usual e s t i m a t e s o f means and v a r i a n c e s w i t h robust e s t i m a t e s of these parameters may o f t e n p r o v i d e v a l u a b l e i n f o r m a t i o n about which o b s e r v a t i o n s may be c l a s s i f i e d as o u t l i e r s and consequently i n d i c a t e sources of e r r o r . In the case of m u l t i v a r i a t e data, however, the presence of o u t l y i n g o b s e r v a t i o n s a f f e c t s not o n l y e s t i m a t e s of l o c a t i o n and s c a l e but a l s o the o r i e n t a t i o n or c o r r e l a t i o n between v a r i a b l e s . In a d d i t i o n , d i f f e r e n t types o f o u t l i e r s may be pr e s e n t , those t h a t are the r e s u l t of e r r o r s t h a t a f f e c t o n l y a subset of the components i n an o b s e r v a t i o n v e c t o r , or those t h a t a f f e c t a l l o f the components e q u a l l y . The e f f e c t o f o u t l y i n g o b s e r v a t i o n s on e s t i m a t e s o f c o r r e l a t i o n s o r c o v a r i a n c e s i s an important con-s i d e r a t i o n i n p r a c t i c e s i n c e these estimates are 6 r o u t i n e l y u s e d i n p s y c h o l o g y f o r e x a m i n i n g r e l a t i o n s h i p s among g r o u p s o f v a r i a b l e s , o r f o r i n p u t t o more c o m p l e x a n a l y s e s s u c h as p r i n c i p a l c o m p o n e n t s o r f a c t o r a n a l y s e s . The s e n s i t i v i t y o f t h e u s u a l p r o d u c t - m o m e n t e s t i m a t e s o f c o r r e l a t i o n s a n d c o v a r i a n c e s t o t h e p r e s e n c e o f o u t l i e r s may f r e q u e n t l y r e s u l t i n t h e i n t r o d u c t i o n o f b i a s i n t h e r e s u l t s o f t h e s e a n a l y s e s . As an e x a m p l e o f t h e ; s e n s i t i v i t y o f t h e s a m p l e c o r r e l a t i o n , r , t o t h e p r e s e n c e o f o u t l i e r s , F i g u r e 1 i s a s c a t t e r p l o t o f 50 o b s e r v a t i o n s , 4 8 o f w h i c h w e r e a r t i f i c i a l l y g e n e r a t e d f r o m a p o p u l a t i o n w i t h a c o r r e l a t i o n , p , e q u a l t o .98. The s a m p l e c o r r e l a t i o n o f t h e 48 o b s e r v a t i o n s i s .97. The a d d i t i o n o f t h e two o u t l i e r s t h a t a p p e a r a t t h e u p p e r l e f t o f F i g u r e 1, h o w e v e r , r e d u c e s t h e o v e r a l l c o r r e l a t i o n t o .27. The s e n s i t i v i t y o f r t o t h e p r e s e n c e o f o u t l i e r s i n d i c a t e s t h a t t h e use o f r o b u s t e s t i m a t e s o f r a r e n e c e s s a r y e v e n i n c o n d i t i o n s w i t h s m a l l amounts o f o u t l i e r c o n t a m i n a t i o n . A l t h o u g h i n i t i a l i n v e s t i g a t i o n s o f t h e r o b u s t p r o p e r t i e s o f r h a v e b e e n p r i m a r i l y c o n c e r n e d w i t h t h e e f f e c t o f n o n n o r m a l i t y on t h e d i s t r i b u t i o n o f r ( P e a r s o n , 1929; K o w a l s k i , 1 9 7 2 ) , o r w i t h r o b u s t h y p o t h e s i s t e s t s f o r r (Duncan & L a y a r d , 1 9 7 3 ) , more r e c e n t r e s e a r c h h a s i n v o l v e d t h e d e v e l o p m e n t FIGURE 1 i 5.OOOO + S c a t t e r P l o t o f 50 O b s e r v a t i o n s w i t h 2 O u t l i e r s 4 . 1 1 1 1 + R ( X , Y ) = .27 (N = 50) + R ( X , Y ) = .97 (N = 48) 3 2 2 2 2 + p(x,y) = .98 + * 2 . 3 3 3 3 + 1 .4444 + . 5 5 5 5 6 + 2 * t 3 * 2 . 3 3 3 3 3 + » » 2 * 2** 1 .2222 + 2 .1111 + 3 . 0 0 0 0 + - 5 . 0 0 0 0 - 2 . 7 7 7 8 - . 5 5 5 5 6 1.6667 3 .8889 V2 - 3 . 8 8 8 9 - 1 . 6 6 6 7 .55556 2 .7778 5 . 0 0 0 0 8 of c o r r e l a t i o n and c o v a r i a n c e e s t i m a t e s t h a t are robust with r e s p e c t to the presence of o u t l i e r s and a number of r o b u s t e s t i m a t e s have been proposed. Robust E s t i m a t i o n Methods There are c u r r e n t l y few methods f o r the treatment of o u t l i e r s t h a t are w i d e l y used i n p s y c h o l o g i c a l r e s e a r c h . Perhaps the b e s t known method i n v o l v e s the removal or w i n s o r i z i n g of extreme scores i n the a n a l y s i s of v a r i a n c e . T h i s procedure and i t s use w i t h the t - t e s t are d i s c u s s e d by Winer ( 1 9 7 1 , pp. 5 1 - 5 4 ) . M-estimates. Of c u r r e n t l y a v a i l a b l e o u t l i e r r e s i s t a n t s t a t i s t i c s , the c l a s s of M-estimates (Huber, 1 9 6 4 ) , o b t a i n e d from m o d i f i e d maximum l i k e l i h o o d e s t i m a t i o n methods, are among the most u s e f u l i n a p p l i e d r e s e a r c h . For the e s t i m a t i o n of a l o c a t i o n parameter i n the presence of o u t l i e r s o r l o n g - t a i l e d d i s t r i b u t i o n s , M-estimates are more e f f i c i e n t than many w e l l known s t a t i s t i c s such as the mean or median, and they are o n l y s l i g h t l y l e s s e f f i c i e n t than the mean f o r normally d i s t r i b u t e d e r r o r s . For a continuous random v a r i a b l e X, w i t h p r o b a b i l i t y d e n s i t y f u n c t i o n f ( x ) , and a sample o f s i z e n, M-estimates of a l o c a t i o n parameter 6 , are o b t a i n e d by s o l v i n g an e q u a t i o n o f the form 9 n (1) z f ( x . - e) = 0. i = l 1 T h i s i s e q u i v a l e n t t o m i n i m i z i n g t h e l o g l i k e l i h o o d f u n c t i o n n L (6) = T, <j> (x. - 6), i = l 1 w h e r e <f>(x. - 9) = - l n f ( x . - 0). The r e l a t i o n b e t w e e n l l t h e f u n c t i o n s V a n d cj> i s t h e n ¥(x. - 9) = <j>'(x. - 9) = - f ' (x. - 6 ) / f ( x . - 9). F o r a n o r m a l l y d i s t r i b u t e d random v a r i a b l e X, 2 <J> (x) = x /2 + c, T(x) = x, a n d t h e s o l u t i o n t o (1) a b o v e i s 9 = x = ( l / n j E x ^ (Hogg, 1979, p. 2 ) . R o b u s t M - e s t i m a t e s o f l o c a t i o n a r e o b t a i n e d by c h o o s i n g a l t e r n a t i v e f o r m s o f t h e f u n c t i o n ¥ i n (1) a b o v e s o t h a t t h e r e s u l t i n g e s t i m a t e s a r e r e a s o n a b l y e f f i c i e n t a n d a r e a b l e t o w i t h s t a n d a s m a l l amount o f o u t l i e r c o n t a m i n a t i o n . S i n c e M - e s t i m a t e s o b t a i n e d f r o m (1) a r e n o t g e n e r a l l y s c a l e i n v a r i a n t , t h e f u n c t i o n s o l v e d i n p r a c t i c e i s u s u a l l y o f t h e f o r m n .Z V ( (x. - 0 ) / s ) = 0 i = l l'6 where s i s a robust estimate of scale such as s = median Ix. - median x.I/.6 745. 1 1 1 1 The constant .6745 i s used to adjust f o r asymptotic n o r m a l i t y (Hampel, 19 74). The m u l t i v a r i a t e trimming procedure. The m u l t i v a r i a t e trimming procedure (MVT), (Devlin e t a l . , 1975; Gnanadesikan & K e t t e n r i n g , 1972) provides f o r p v a r i a b l e s , an i t e r a t i v e approximation to the mean vector m*' = r.m* ,...,m* 1, and covariance matrix S*, 1 P by c a l c u l a t i n g at each step the squared distances d^T = (y_± - m*) ,S*~ 1(y i - m*) , i = l , . . . , n where y.!. = [ y y . ] are'the o r i g i n a l observation — i u i l -'lp ^ vectors from a sample of s i z e n. At each step 10 percent of the most extreme observations are set aside and the remaining observations used to compute m* and S* i n the same manner as the usual estimates of m and S. The procedure begins by using the usual estimates of m and S as s t a r t i n g values or, i n the presence of large • amounts of contamination, robust i n i t i a l estimates are used. The procedure i s terminated when the absolute value of the F i s h e r z-transform of the r * does not change 3k -3 between successive i t e r a t i o n s by more than 10 11 M u l t i v a r i a t e M - e s t i m a t e s . M - e s t i m a t e s o f c o v a r i a n c e m a t r i c e s (Maronna, 19 76) a r e o b t a i n e d by methods s i m i l a r t o t h e MVT p r o c e d u r e . T h e s e e s t i m a t e s a r e a l s o c a l c u l a t e d i t e r a t i v e l y , b u t i n s t e a d o f d e l e t i n g a p o r t i o n o f t h e 2 d a t a w i t h t h e l a r g e s t d^ v a l u e s , w e i g h t s a r e a s s i g n e d t o e a c h o b s e r v a t i o n b a s e d on t h e s e d i s t a n c e s . E s t i m a t e s o f t h e mean v e c t o r and c o v a r i a n c e m a t r i c e s a r e c a l c u l a t e d f r o m t h e f o r m u l a s m" [zw1{ai)y_i]/ [ E w 1 ( d i ) ] a n d S* = ( l / n ) l w 2 ( d j ) (y^ - m * ) ^ - m*)'. The w e i g h t s w^ and s u g g e s t e d by Maronna a r e o f two t y p e s . The f i r s t i s d e f i n e d as w 1 ( d i ) = (p + f ) / ( f + d^) = w 2 ( d j ) , w h i c h r e s u l t s i n maximum l i k e l i h o o d e s t i m a t e s f o r a p - v a r i a t e t - d i s t r i b u t i o n w i t h f d e g r e e s o f f r e e d o m . The s e c o n d s e t o f w e i g h t s a r e s i m i l a r t o t h o s e p r o p o s e d by Huber (1977a) and a r e o f t h e f o r m w 1 ( d i ) ={ 1, d. <k ~ p 1 and w 2(.d^) = [ w 1 ( d i ) ] z / ( k/d^, o t h e r w i s e 2 The v a l u e o f k c a n be c h o s e n as a p r o p o r t i o n o f t h e c h i - s q u a r e d i s t r i b u t i o n w i t h p d e g r e e s o f f r e e d o m , i . e . , k = X a ( p d f ) f o r 0£a<_l, and 3 i s c h o s e n so as t o make S* an a s y m p t o t i c a l l y u n b i a s e d e s t i m a t e o f t h e c o v a r i a n c e m a t r i x i n a m u l t i v a r i a t e n o r m a l d i s t r i b u t i o n . 2 E s t i m a t e s b a s e d on d. m e a s u r e s have s e v e r a l 1 l i m i t a t i o n s . T h e o r e t i c a l r e s u l t s s u g g e s t t h a t t h e s e e s t i m a t o r s may b r e a k down when t h e p r o p o r t i o n o f o u t l i e r s i n a p - v a r i a t e sample i s a p p r o x i m a t e l y 1/p (Maronna, 19 7 6 ) . The r e s u l t s o f D e v l i n e t a l . (1981) have a l s o shown t h a t m u l t i v a r i a t e M - e s t i m a t o r s a r e v e r y s e n s i t i v e t o a s y m m e t r i c a l l y d i s t r i b u t e d o u t l i e r s , w i t h s t a n d i n g o n l y 1 o r 2 p e r c e n t c o n t a m i n a t i o n f o r p=20. The p r o b l e m o f i n c r e a s e d o u t l i e r s e n s i t i v i t y i n h i g h d i m e n s i o n a l c a s e s a p p e a r s t o be l e s s i m p o r t a n t f o r t h e s e e s t i m a t o r s t h a n t h e p r e s e n c e o f a s y m m e t r i c o u t l i e r s . The r e s u l t s o f D e v l i n e t a l . (19 81) have shown t h a t f o r p=20, and s y m m e t r i c a l l y d i s t r i b u t e d o u t l i e r s , m u l t i v a r i a t e M - e s t i m a t o r s c an w i t h s t a n d 10 t o 20 p e r c e n t c o n t a m i n a t i o n w h i c h i s i m p r o v e d t o 25 p e r c e n t u s i n g r o b u s t s t a r t i n g p o i n t s i n t h e c a l c u l a t i o n s . The use o f r o b u s t s t a r t i n g p o i n t s i n t h e a s y m m e t r i c c a s e d i d n o t i m p r o v e t h e p e r f o r m a n c e o f t h e M - e s t i m a t o r s b u t i t d i d improve t h e p e r f o r m a n c e o f t h e MVT p r o c e d u r e , i n c r e a s i n g i t s a b i l i t y t o w i t h s t a n d a s y m m e t r i c o u t l i e r s f r o m 4 t o 10 p e r c e n t . 13 R o b u s t e s t i m a t e s o f r e g r e s s i o n p a r a m e t e r s . R o b u s t M - e s t i m a t e s o f r e g r e s s i o n p a r a m e t e r s a r e o b t a i n e d by a s t r a i g h t f o r w a r d e x t e n s i o n o f methods f o r o b t a i n i n g M - e s t i m a t e s o f l o c a t i o n . F o r t h e g e n e r a l l i n e a r r e g r e s s i o n m odel Y i = e 0+ e 1 x i l + 3 2 x i 2 + , . . . , + e p x i p + e., e s t i m a t e s o f t h e v e c t o r o f r e g r e s s i o n c o e f f i c i e n t s 3_l= [3o - 3-i • • • • - 3 ]/ a r e o b t a i n e d by s o l v i n g t h e n o r m a l e q u a t i o n s and } *ij = °' -J = 1 " - " P ' where x|= [ l , x ^ ^ , x ^ 2 , . . . , x ^ p ] and s i s a r o b u s t s c a l e e s t i m a t e o f t h e r e s i d u a l s . I n t h e b i v a r i a t e c a s e , t h e r o b u s t p r o p e r t i e s o f r e g r e s s i o n c o e f f i c i e n t s o b t a i n e d w i t h d i f f e r e n t c h o i c e s o f Y have b e e n e x a m i n e d by Ramsay (19 7 7 ) , a n d u s e s o f r o b u s t M - e s t i m a t e s o f r e g r e s s i o n p a r a m e t e r s i n a p p l i e d r e s e a r c h a r e d i s c u s s e d by Agee and T u r n e r (1979, pp. 1 0 7 - 1 2 6 ) . O t h e r r o b u s t e s t i m a t i o n p r o c e d u r e s . As an a l t e r n a t i v e t o m u l t i v a r i a t e M - e s t i m a t o r s f o r o b t a i n i n g r o b u s t c o r r e l a t i o n s among a l a r g e number o f v a r i a b l e s , H u b e r ( 1 9 7 7 b , p p . 99-101) h a s p r o p o s e d some c o o r d i n a t e d e p e n d e n t p r o c e d u r e s . T h e s e m e t h o d s i n v o l v e a p p l y i n g a m o n o t o n i c t r a n s f o r m a t i o n t o e a c h v a r i a b l e a n d t h e n c a l c u l a t i n g t h e c o r r e l a t i o n e s t i m a t e s f r o m t h e t r a n s -f o r m e d d a t a . The use o f t h e s e p r o c e d u r e s may be l i m i t e d i n p r a c t i c e s i n c e t h e y a r e d e s i g n e d f o r s y m m e t r i c a l l y d i s t r i b u t e d o u t l i e r s a n d b y t r a n s f o r m i n g e a c h v a r i a b l e i n d e p e n d e n t l y o f t h e o t h e r , i n f o r m a t i o n p r o v i d e d b y t h e o v e r a l l s h a p e o f t h e s a m p l i n g d i s t r i b u t i o n i s u n u s e d i n c a l c u l a t i n g t h e r o b u s t e s t i m a t e s . D e v l i n , G n a n a d e s i k a n a n d K e t t e n r i n g (19 75) e x a m i n e d s e v e r a l e s t i m a t e s o f p. t h a t were d e s i g n e d t o be i n s e n s i t i v e t o o u t l i e r s . They f o u n d t h a t an e f f e c t i v e e s t i m a t o r f o r r e d u c i n g o u t l i e r i n f l u e n c e c o u l d be o b t a i n e d f r o m sums a n d d i f f e r e n c e s o f s t a n d a r d s c o r e s f o r two v a r i a b l e s . The m e t h o d i n v o l v e s c a l c u l a t i n g t h e r o b u s t s t a n d a r d s c o r e s , y*.. = ( y . . - m*.)//v*.., ( i = l , . . . , n ; j = l , 2 ) w h e r e m*, a n d v * . . a r e r o b u s t e s t i m a t e s o f t h e mean a n d 3 33 v a r i a n c e o f y_. . T h i s i s f o l l o w e d b y o b t a i n i n g r o b u s t e s t i m a t e s o f v a r ( y * + y* ) a n d v a r ( y * - y % ) u s i n g a 15 t r i m m i n g p r o c e d u r e f o r v a r i a n c e s and t h e n c a l c u l a t i n g t h e r o b u s t e s t i m a t e o f p f r o m t h e i d e n t i t y r * ( . y * l f y * 2 ) = l / 4 [ v a r ( y * 1 + y * ) - v a r ( y * 1 ~ y * 2 ) ] . A d i s a d v a n t a g e o f t h i s e s t i m a t o r i s t h a t i t r e q u i r e s t h e a c c u r a t e e s t i m a t i o n o f t h e v a r i a b l e means and v a r i a n c e s f o r t h e sums and d i f f e r e n c e s o f t h e v a r i a b l e s i n v o l v e d . O t h e r b i v a r i a t e o u t l i e r r e s i s t a n t e s t i m a t e s o f p have been b a s e d on methods o f t r i m m i n g e x t r e m e sample o b s e r v a t i o n s . T h e s e i n c l u d e methods o f c o n v e x h u l l t r i m m i n g ( B e b b i n g t o n , 1978) and e l l i p s o i d a l t r i m m i n g ( T i t t e r i n g t o n , 1 9 7 8 ) . T h e s e p r o c e d u r e s have t h e d i s a d v a n t a g e o f b e i n g c o m p u t a t i o n a l l y complex i n h i g h d i m e n s i o n a l c a s e s o r d e m o n s t r a t e s l o w c o n v e r g e n c e i n c a l c u l a t i n g a s o l u t i o n . P r o c e d u r e s t h a t p r o v i d e r o b u s t e s t i m a t e s o f b i v a r i a t e c o r r e l a t i o n s have d i s a d v a n t a g e s when u s e d f o r e s t i m a t i n g c o r r e l a t i o n m a t r i c e s . When t h e e l e m e n t s o f a c o r r e l a t i o n m a t r i x have b e e n c a l c u l a t e d i n d i v i d u a l l y , t h e r e s u l t a n t m a t r i x may n o t be p o s i t i v e d e f i n i t e , and w i l l r e q u i r e methods f o r a d j u s t i n g t h e e i g e n v a l u e s o f t h e m a t r i x t o e n s u r e t h a t t h e y w i l l a l l be g r e a t e r t h a n z e r o . I n s i t u a t i o n s where a p r o p o r t i o n o f t h e sample 16 . o b s e r v a t i o n v e c t o r s c o n t a i n m i s s i n g c o m p o n e n t s , b i v a r i a t e e s t i m a t e s o f c o r r e l a t i o n s may be p r e f e r r e d t o t h e s i m u l t a n e o u s e s t i m a t i o n o f a l l t h e e l e m e n t s o f a c o v a r i a n c e o r c o r r e l a t i o n m a t r i x . By c a l c u l a t i n g t h e c o v a r i a n c e s o r c o r r e l a t i o n s s e p a r a t e l y when m i s s i n g d a t a i s p r e s e n t , more o f t h e s a m p l e i n f o r m a t i o n i s u s e d s i n c e o b s e r v a t i o n v e c t o r s w i t h m i s s i n g c o m p o n e n t s w i l l n o t h a v e t o be d i s c a r d e d . A m e t h o d f o r o b t a i n i n g r o b u s t e s t i m a t e s o f c o v a r i a n c e m a t r i c e s t h a t may a v o i d t h e l i m i t a t i o n s o f t h e ab o v e e s t i m a t o r s h a s b e e n s u g g e s t e d b y M o s t e l l e r a n d T u k e y ( 1 9 7 7 , p p . 2 0 3 - 2 1 9 ) . The method i s b a s e d o n r o b u s t r e g r e s s i o n t e c h n i q u e s . F i r s t , f o r t h e v a r i a b l e s x 1 = ( x ^ , . . . ,Xp) , X j i s r e g r e s s e d o n ( x ^ , . . . ,x_. _^) , j = 2 , . . . , p , u s i n g an i t e r a t i v e p r o c e d u r e ( M o s t e l l e r & T u k e y , 1 9 7 7 ; p p . 3 3 3 - 3 7 9 ) . T h i s i s f o l l o w e d b y a r r a n g i n g t h e r o b u s t r e g r e s s i o n w e i g h t s i n a l o w e r t r i a n g u l a r m a t r i x B*, p x p , where t h e e l e m e n t s b*., =0 i f j l k ; a n d i f 3-K j > k , ± s t h e c o e f f i c i e n t f o r x ^ i n t h e r e g r e s s i o n o f X j on ( x ^ , . . . , x j _ ^ ) . The n e x t s t e p i s t o f o r m a m a t r i x Z, n x p , o f t r a n s f o r m e d o b s e r v a t i o n s , where t h e t r a n s f o r m e d v a r i a t e s z ! = [ z z . ] , a r e o b t a i n e d f r o m t h e o r i g i n a l — I i l ' l p ' ^ .x. , I P o b s e r v a t i o n s x ^ = [ x ^ , . . . , x ^ ] b y c a l c u l a t i n g z ± = ( I - B * ) x ± . S i n c e t h e s a m p l e c o v a r i a n c e m a t r i x S, may be o b t a i n e d f r o m s 17 = ( l / ( n - l ) ) Ex..x| = (I-B*) " [ ( l / ( n - l ) ) E z ^ ] (I-B*) x , a r o b u s t c o v a r i a n c e m a t r i x e s t i m a t e S* i s o b t a i n e d by-c o n s t r u c t i n g a d i a g o n a l m a t r i x D*, pxp, c o n t a i n i n g r o b u s t e s t i m a t e s o f t h e v a r i a n c e s o f t h e e l e m e n t s o f z., and t h e n c a l c u l a t i n g S* = (I - B * ) " 1 D * ( I - B * ' ) " 1 . The c a l c u l a t i o n o f S* by t h e above method does n o t r e q u i r e t h e r o b u s t e s t i m a t i o n o f t h e i n t e r c e p t i n e a c h r e g r e s s i o n e q u a t i o n s i n c e t h e e l e m e n t s o f z^ a r e , Z i j = X i j - B j l X i l - - " - e j - l , j X i , j - l = S i j + B 9 j where e^^ i s t h e r e s i d u a l e r r o r , and BQ_. i s t h e i n t e r c e p t c o r r e s p o n d i n g t o t h e j t h r e g r e s s i o n e q u a t i o n . A r o b u s t e s t i m a t e f o r t h e c o r r e l a t i o n m a t r i x R*, i s t h e n o b t a i n e d by r e s c a l i n g S*. R o b u s t v a r i a n c e e s t i m a t e s o f t h e e l e m e n t s o f z^  may be o b t a i n e d f r o m a p r o c e d u r e t h a t i n v o l v e s t h e use o f T u k e y ' s b i w e i g h t ( M o s t e l l e r & Tukey, 1977; p. 2 0 8 ) , o r f r o m o t h e r r o b u s t e s t i m a t e s o f s c a l e . In p r a c t i c e t h e above r e g r e s s i o n p r o c e d u r e has s e v e r a l l i m i t a t i o n s . F o r example, when s y m m e t r i c a l l y d i s t r i b u t e d o u t l i e r s a r e p r e s e n t i n s a m p l e s drawn f r o m n o n n o r m a l d i s t r i b u t i o n s , D e v l i n e t a l . (19 81) have shown t h a t c o r r e l a t i o n s o b t a i n e d f r o m t h i s r e g r e s s i o n p r o c e d u r e t e n d t o be s l i g h t l y l e s s e f f i c i e n t t h a n e s t i m a t e s o b t a i n e d 18 f r o m e i t h e r m u l t i v a r i a t e M - e s t i m a t e s o r t h e MVT p r o c e d u r e . They a l s o n o t e d t h a t t h e r e g r e s s i o n b a s e d e s t i m a t e s o f c o v a r i a n c e s a r e n o t a f f i n e i n v a r i a n t , b u t a r e i n v a r i a n t o n l y up t o c h a n g e s o f s c a l e and s i g n o f t h e o r i g i n a l v a r i a b l e s . R o b u s t P r o c e d u r e s f o r A s y m m e t r i c O u t l i e r s A m a j o r l i m i t a t i o n o f m u l t i v a r i a t e M - e s t i m a t e s and t h e MVT p r o c e d u r e i s t h e i r s e n s i t i v i t y t o asym-m e t r i c a l l y d i s t r i b u t e d o u t l i e r s . S i n c e sample o b s e r v a t i o n s w i t h e x t r e m e s c o r e s may f r e q u e n t l y r e s u l t f r o m s y s t e m a t i c e r r o r s i n d a t a c o l l e c t i o n , o r f r o m s a m p l e s t h a t c o n t a i n members f r o m o t h e r p o p u l a t i o n s , t h e a s s u m p t i o n o f s y m m e t r i c a l l y d i s t r i b u t e d o u t l i e r s may o f t e n be o v e r l y r e s t r i c t i v e . G e n e r a l methods f o r t h e r o b u s t e s t i m a t i o n o f c o v a r i a n c e s and c o r r e l a t i o n s i n t h e p r e s e n c e o f a s y m m e t r i c a l l y d i s t r i b u t e d o u t l i e r s a r e c u r r e n t l y u n a v a i l a b l e . I n t h e l o c a t i o n c a s e , t h e t r e a t m e n t o f a s y m m e t r i c o u t l i e r s h as been d i s c u s s e d by J a e c k e l ( 1 9 7 1 ) , and a c l a s s o f M - e s t i m a t e s o f l o c a t i o n r e s i s t a n t t o t h e p r e s e n c e o f asymmetry has b e e n e x a m i n e d by C o l l i n s ( 1 9 7 6 ) . The p r o b l e m o f a s y m m e t r i c o u t l i e r s h as been d i s c u s s e d by Chambers and H e a t h c o t e (1981) i n t h e c o n t e x t o f r o b u s t r e g r e s s i o n . They p r o p o s e d e s t i m a t e s o f r e g r e s s i o n p a r a m e t e r s b a s e d on t h e sample o f e m p i r i c a l c u m u l a n t g e n e r a t i n g f u n c t i o n ( c g f ) ( K e n d a l l & S t u a r t , 1977, pp. 97-126) and p r o v i d e d some r e s u l t s w h i c h 19 i n d i c a t e t h a t t h e s e e s t i m a t e s a r e l e s s s e n s i t i v e t o a s y m m e t r i c o u t l i e r s t h a n e s t i m a t e s o f r e g r e s s i o n p a r a m e t e r s o b t a i n e d f r o m T u k e y ' s b i w e i g h t p r o c e d u r e o r f r o m t h e c l a s s o f M - e s t i m a t e s p r o p o s e d b y Huber (19 73). C h a r a c t e r i s t i c . f u n c t i o n b a s e d e s t i m a t e s . The e s t i m a t i o n o f r e g r e s s i o n c o e f f i c i e n t s b a s e d on t h e e m p i r i c a l c g f h a s b e e n shown by Chambers and H e a t h c o t e (19 81) t o be an e x t e n s i o n o f l e a s t s q u a r e s e s t i m a t i o n . The method i n v o l v e s f o r a sample o f s i z e n, e s t i m a t i n g t h e r e g r e s s i o n c o e f f i c i e n t s 3/= [ 3^ r • • • , 3 ] , f r o m t h e sample c h a r a c t e r i s t i c f u n c t i o n o f t h e r e s i d u a l s i n t h e g e n e r a l l i n e a r r e g r e s s i o n model ^k = \ + . * A j e j + E k " 3 = 1. When t h e r e s i d u a l e r r o r , e, i s n o r m a l l y d i s t r i b u t e d w i t h mean 0 and c h a r a c t e r i s t i c f u n c t i o n E ( e l t e ) = e x p ( ( - 1 / 2 ) a 2 t 2 ) , m i n i m i z i n g t h e e r r o r v a r i a n c e a 2 = n~h(y^-x^)2, k i s e q u i v a l e n t t o m i n i m i z i n g , f o r a f i x e d v a l u e o f t , th e sample a n a l o g u e o f G ( 3 _ ; t ) = - t ~ 2 l o g | E ( e l t e ) | 2 , w h i c h may be w r i t t e n as 20 ( g ; t ) = - t " 2 l o g | n " 1 E exp ( i U y . - x ' 3) ) | 2 , n k = l K ~ K ~ w i t h G ( 3 ; t ) d e f i n e d a t 0 b y c o n t i n u i t y . Chambers and n — H e a t h c o t e (19 81) n o t e d t h a t f o r t=0, t h e m i n i m i z a t i o n o f G n ( j ^ ; t ) above c o r r e s p o n d s t o t h e e s t i m a t i o n o f 3_ by l e a s t s q u a r e s . They a l s o showed t h a t f o r a m a t r i x o f mean d e v i a t e d i n d e p e n d e n t v a r i a b l e s X, and a v e c t o r o f t r u e p a r a m e t e r v a l u e s B_q , . t h e d i s t r i b u t i o n o f -1/2 n (3-3 ) i s n o r m a l w i t h c o v a r i a n c e m a t r i x — — o fi(t)=a2(t) [(X'X) where ( X ' X ) ^ i s t h e a s y m p t o t i c f o r m o f t h e c o v a r i a n c e m a t r i x o f x. 2 The s c a l a r q u a n t i t y a _(t) above i s d e r i v e d f r o m t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e r e s i d u a l e r r o r , i t E i t E E ( e ) . By w r i t i n g E ( e ) i n t e r m s o f i t s r e a l and i m a g i n a r y components u ( t ) and v ( t ) , E ( e l t ? ) = u ( t ) + i v ( t ) 2 i t e a ( t ) may be shown, when E ( e )^0, t o have t h e f o r m 0 u 2 ( t ) ( l - u ( 2 t ) ) - 2 u ( t ) v ( t ) v ( 2 t ) + v 2 ( t ) ( l + u ( 2 t ) ) a ( t ) = ' 2 ? 2 2 2 t Z ( u ( t ) + v ( t ) r 2 w i t h a (t) d e f i n e d a t 0 by c o n t i n u i t y . F o r n o r m a l l y d i s t r i b u t e d e r r o r , t h e above e x p r e s s i o n r e d u c e s t o 21 2 -2 2 2 o ( t ) = t s i n h (a t ) £ w h i c h i s s y m m e t r i c w i t h a g l o b a l minimum a t t h e o r i g i n . I n p r a c t i c e t h e f o r m o f t h e sample e s t i m a t e o f 2 a ( t ) , o b t a i n e d by r e p l a c i n g u ( t ) and v ( t ) w i t h t h e i r sample e q u i v a l e n t s « -1 n u (3;t) = n E. c o s ( t ( y , -x'6) ) n k = l K _ i n and v ( j$;t) = n E s i n ( t (y,-x'£ ) ) , n k = l K _ K ~ 2 may be e x a m i n e d b y c a l c u l a t i n g e s t i m a t e s o f w (t) o v e r a r a n g e o f v a l u e s o f t . I n s i t u a t i o n s where a s y m m e t r i c e r r o r d i s t r i b u t i o n s may be t h e r e s u l t o f o u t l i e r c o n t a m i n a t i o n i n samples drawn f r o m an u n d e r l y i n g n o r m a l d i s t r i b u t i o n , Chambers and H e a t h c o t e (1981) n o t e d t h a t 2 t h e f o r m o f o ( t ) d i f f e r e d s i g n i f i c a n t l y f r o m t h a t o b t a i n e d when e r r o r s were n o r m a l l y d i s t r i b u t e d . F o r 2 example, t h e minimum o f a (t) was o b s e r v e d t o be a t a v a l u e o f t o t h e r t h a n 0 i n t h e a s y m m e t r i c c a s e , and o s c i l l a t i o n s , w h i c h were a b s e n t f o r n o r m a l l y d i s t r i b u t e d e r r o r s , were o b s e r v e d o v e r t h e r a n g e o f t . I f o u t l i e r s 2 a r e i d e n t i f i e d and removed f r o m t h e sample, o ( t ) may be r e c a l c u l a t e d i n o r d e r t o d e t e r m i n e i f t h e sample d i s t r i b u t i o n more c l o s e l y r e s e m b l e s a d i s t r i b u t i o n t h a t i s n o r m a l i n f o r m . The e x a m i n a t i o n o f t h e s a m p l e 2 e s t i m a t e o f a ( t ) i n t h i s way p r o v i d e s u s e f u l i n f o r m a t i o n a b o u t t h e amount and t y p e o f o u t l i e r c o n t a m i n a t i o n t h a t may be p r e s e n t i n a s a m p l e . The p u r p o s e o f t h e p r e s e n t s t u d y was t o e x a m i n e t h e r o b u s t p r o p e r t i e s o f c o r r e l a t i o n s o b t a i n e d f r o m t h e r e g r e s s i o n p r o c e d u r e when e s t i m a t e s o f t h e r e g r e s s i o n c o e f f i c i e n t s a r e o b t a i n e d f r o m t h e s a m p l e c g f . The m a j o r o b j e c t i v e o f t h i s i n v e s t i g a t i o n was t o o b s e r v e t h e s e n s i t i v i t y o f c o r r e l a t i o n s o b t a i n e d i n t h i s way t o t h e p r e s e n c e o f a s y m m e t r i c o u t l i e r s a n d t o compare t h e e f f i c i e n c y o f t h e s e e s t i m a t e s w i t h t h o s e o b t a i n e d f r o m t h e MVT p r o c e d u r e . The MVT p r o c e d u r e was c h o s e n f o r c o m p a r i s o n s i n c e t h e r e s u l t s o f D e v l i n e t a l . (1981) h a v e shown i t t o be more r e s i s t a n t t o t h e i n f l u e n c e o f a s y m m e t r i c o u t l i e r s t h a n o t h e r c u r r e n t l y a v a i l a b l e e s t i m a t e s o f c o r r e l a t i o n s . 2 3 M e t h o d D a t a G e n e r a t i o n Monte C a r l o m e t h o d s were u s e d t o g e n e r a t e m u l t i v a r i a t e s a m p l e s f r o m a p o p u l a t i o n w i t h a known c o r r e l a t i o n m a t r i x , P. The number o f v a r i a b l e s p, u s e d i n a l l c o n d i t i o n s o f t h e p r e s e n t s t u d y was e q u a l t o 6, w i t h p c h o s e n t o r e p r e s e n t a r a n g e o f c o r r e l a t i o n s t y p i c a l l y e n c o u n t e r e d i n p r a c t i c e . One p o p u l a t i o n c o r r e l a t i o n m a t r i x was u s e d i n t h e p r e s e n t c a s e a n d was o b t a i n e d f r o m G o r s u c h ( 1 9 7 4 , p. 6 ) . F o r e a c h r e p l i c a t i o n o f t h e p r e s e n t p r o c e d u r e , i n d e p e n d e n t , n o r m a l l y d i s t r i b u t e d random v a r i a b l e s w i t h mean 0 and a v a r i a n c e o f 1 w e r e g e n e r a t e d a n d a r r a n g e d i n a nxp d a t a m a t r i x Y. The random number g e n e r a t o r RANDN, i m p l e m e n t e d on t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a ' s Amdahl 470-V/8 c o m p u t e r , was u s e d t o o b t a i n n o r m a l l y d i s t r i b u t e d o b s e r v a t i o n s . The a l g o r i t h m e m p l o y e d f i r s t g e n e r a t e s , on t h e i n t e r v a l ( 0 , 1 ) , u n i f o r m l y d i s t r i b u t e d r andom v a r i a b l e s w h i c h a r e s u b s e q u e n t l y u s e d t o o b t a i n n o r m a l l y d i s t r i b u t e d r andom s a m p l e s f r o m t h e a p p l i c a t i o n o f M a r s a g l i a ' s r e c t a n g u l a r - w e d g e - t a i l m e t h o d ( K n u t h , 1 9 6 8 ) . I n o r d e r t o s i m u l a t e s a m p l e s d r a w n f r o m a m u l t i -v a r i a t e n o r m a l p o p u l a t i o n w i t h a mean v e c t o r y.1 = [y y ] 24 and c o r r e l a t i o n m a t r i x P, (MVN (_p:, P)') , an nxp d a t a m a t r i x X, was o b t a i n e d f r o m t h e m a t r i x Y by t h e t r a n s f o r m a t i o n X = Y C . The pxp m a t r i x C, was o b t a i n e d f r o m t h e p r o d u c t C' = A 1 / 2 v -where A i s a d i a g o n a l m a t r i x c o n t a i n i n g t h e e i g e n v a l u e s o f P and V i s t h e pxp m a t r i x o f c o r r e s p o n d i n g n o r m a l i z e d e i g e n v e c t o r s . The d a t a c o n t a i n e d i n t h e m a t r i x X may be shown t o r e p r e s e n t a sample drawn f r o m t h e r e q u i r e d d i s t r i b u t i o n s i n c e P = C C and, by t a k i n g e x p e c t a t i o n s E ( X , X ) = E ( C Y , Y C ) =C E(Y'Y) C* = C C =P. I n a l l c o n d i t i o n s o f t h e p r e s e n t s t u d y , s a m p l e s were g e n e r a t e d by f i r s t d r a w i n g n^ o b s e r v a t i o n s f r o m a d i s t r i b u t i o n o f t h e f o r m MVN(J3,P), w i t h mean v e c t o r _P_'= [0, . .. , 0 ] and p o p u l a t i o n c o r r e l a t i o n m a t r i x P. O u t l i e r s were s i m u l a t e d by d r a w i n g o b s e r v a t i o n s f r o m a d i s t r i b u t i o n o f t h e f o r m MVN , 91) , r e p r e s e n t i n g o b s e r v a t i o n v e c t o r s w i t h i n d e p e n d e n t components c e n t e r e d a t , and h a v i n g g r e a t e r v a r i a b i l i t y t h a n v a r i a b l e s s a m p l e d f r o m t h e p o p u l a t i o n o f i n t e r e s t . The two s a m p l e s were t h e n c o m b i n e d f o r a t o t a l o f n,+n =N o b s e r v a t i o n s , w i t h p r o p o r t i o n of contamination n^/N. The values of n^ and n^ were h e l d constant w i t h i n each c o n d i t i o n and the degree o f asymmetry, or the d i s t a n c e between the two d i s t r i b u t i o n s , was c o n t r o l l e d by v a r y i n g the magnitude of _y . The r e p r e s e n t a t i o n of o u t l i e r s as o b s e r v a t i o n s from a normal d i s t r i b u t i o n w i t h a standard d e v i a t i o n t h a t i s a m u l t i p l e of the standard d e v i a t i o n of the p o p u l a t i o n o f i n t e r e s t , has been f r e q u e n t l y used to model o u t l i e r contamination t h a t may a r i s e from a wide v a r i e t y of sources encountered i n p r a c t i c e (Andrews, e t a l . , 1972; D e v l i n , e t a l . , 1981; Ramsay, .'. 1977; Tukey, 1960, pp. 4 4 8 - 4 8 5 ) . An a l t e r n a t i v e model f o r the g e n e r a t i o n of o u t l i e r contaminated samples c o n s i s t s of drawing o b s e r v a t i o n s from the mixture of d i s t r i b u t i o n s (l-e)MVN ( V > 1,P 1) + £MVN(JJ 2,P 2) . An o b s e r v a t i o n i s drawn from MVNCjJ-^P^) wit h p r o b a b i l i t y (1-e) and from MVN ( £ 2,P 2) wi t h p r o b a b i l i t y e. T h i s model d i f f e r s from t h a t used i n the p r e s e n t study by r e p r e s e n t i n g contamination of the p o p u l a t i o n from which samples are drawn, r a t h e r than r e p r e s e n t i n g a s p e c i f i c l e v e l of contamination i n each sample. The method of sample g e n e r a t i o n used i n the p r e s e n t study was chosen s i n c e a major o b j e c t i v e was to observe the l e v e l of 26 o u t l i e r c o n t a m i n a t i o n i n sample d a t a a t w h i c h t h e c o r r e l a t i o n e s t i m a t e s t e n d e d t o breakdown. T h i s i s more e a s i l y o b s e r v e d i n c o n d i t i o n s w i t h a c o n s t a n t p r o p o r t i o n o f o u t l i e r s i n e a c h sample, r a t h e r t h a n i n c o n d i t i o n s where t h e amount o f c o n t a m i n a t i o n v a r i e s b e tween s a m p l e s . S t u d i e s t h a t have s i m u l a t e d o u t l i e r c o n t a m i n a t i o n i n a manner s i m i l a r t o t h a t o f t h e p r e s e n t s t u d y i n c l u d e t h o s e c o n d u c t e d b y D e v l i n e t a l . ( 1 9 7 5 ) , H i n k l e y ( 1 9 7 8 ) , and J o h n s o n , M c G u i r e and M i l l i k e n ( 1 9 7 8 ) . I n d e p e n d e n t V a r i a b l e s Sample s i z e . The t h r e e l e v e l s o f sample s i z e e x a m i n e d were 30, 60 and 120. T h e s e were c h o s e n t o a p p r o x i m a t e s m a l l , m o derate and l a r g e sample s i z e s t h a t commonly o c c u r i n p r a c t i c e . O u t l i e r c o n t a m i n a t i o n . T h r e e l e v e l s o f o u t l i e r c o n t a m i n a t i o n were common t o e a c h l e v e l o f sample s i z e . T hese r e p r e s e n t e d t h e p r o p o r t i o n s o f .05, .10 and .15. I n o r d e r t o o b s e r v e t h e p e r f o r m a n c e o f t h e s e p r o c e d u r e s i n c o n d i t i o n s w i t h l a r g e amounts o f c o n -t a m i n a t i o n , an a d d i t i o n a l c o n d i t i o n i n w h i c h t h e p r o p o r t i o n o f o u t l i e r s was e q u a l t o .25 was o b s e r v e d f o r a sample s i z e o f 60, and f o r a sample s i z e o f 120 a d d i t i o n a l l e v e l s o f o u t l i e r c o n t a m i n a t i o n were o b s e r v e d f o r p r o p o r t i o n s o f .20, .25 , and .30. 27 O u t l i e r asymmetry. With, t h e e x c e p t i o n o f t h r e e c o n d i t i o n s , t h e mean v e c t o r o f t h e c o n t a m i n a t i n g d i s t r i b u t i o n , was e q u a l t o _3. In o r d e r t o compare t h e c o r r e l a t i o n e s t i m a t e s when o u t l i e r s were s a m p l e d f r o m a d i s t r i b u t i o n w i t h l a r g e r v a r i a n c e s t h a n t h e p o p u l a t i o n o f i n t e r e s t , b u t w i t h t h e same means, one c o n d i t i o n was e x a m i n e d i n w h i c h t h e mean v e c t o r o f t h e c o n t a m i n a t i n g d i s t r i b u t i o n was s e t e q u a l t o 0_. I n t h i s c o n d i t i o n t h e sample s i z e was e q u a l t o 60, w i t h t h e p r o p o r t i o n o f o u t l i e r s i n t h e sample e q u a l t o .10. Two c o n d i t i o n s were o b s e r v e d i n w h i c h t h e amount o f o v e r l a p between t h e c o n t a m i n a t i n g d i s t r i b u t i o n and t h e d i s t r i b u t i o n o f i n t e r e s t was v a r i e d . I n e a c h c o n d i t i o n t h e sample s i z e was e q u a l t o 60 and t h e p r o p o r t i o n o f o u t l i e r c o n t a m i n a t i o n was e q u a l t o .15. I n t h e c o n d i t i o n o f most o v e r l a p , was e q u a l ;to 1_, and i n t h e c o n d i t i o n o f l e a s t o v e r l a p , was e q u a l t o 5. V a r i a b l e o r d e r . S i n c e t h e r e g r e s s i o n p r o c e d u r e i s n o t a f f i n e i n v a r i a n t , t h r e e c o n d i t i o n s were e x a m i n e d i n w h i c h t h e o r d e r among t h e v a r i a b l e s was a l t e r e d . F o r a sample s i z e o f 60 and two c o n d i t i o n s o f 10 a n d 15 p e r c e n t o u t l i e r c o n t a m i n a t i o n , r e s u l t s were o b t a i n e d when t h e o r d e r o f t h e v a r i a b l e s was r e v e r s e d . I n a 28 t h i r d c o n d i t i o n w i t h a sample s i z e o f 6 0 and 10 p e r c e n t o u t l i e r c o n t a m i n a t i o n , t h e o r d e r among h a l f o f t h e v a r i a b l e s was r e v e r s e d w h i l e t h e o r d e r among t h e r e m a i n i n g v a r i a b l e s was l e f t u n c h a n g e d . MVT P r o c e d u r e C o m p u t a t i o n o f t h e MVT p r o c e d u r e was p e r f o r m e d i n t h e same manner as d e s c r i b e d by D e v l i n e t a l . ( 1 9 8 1 ) . A t e a c h i t e r a t i o n , t h e s q u a r e d d i s t a n c e d 2=(x.-m*) ,S*~ 1(x.-m*) I — l — — l — was o b t a i n e d f o r e a c h o b s e r v a t i o n i n t h e s a m p l e . T h i s was f o l l o w e d by r e m o v i n g 10 p e r c e n t o f t h e o b s e r v a t i o n s 2 w i t h t h e l a r g e s t d^ v a l u e s and c a l c u l a t i n g new e s t i m a t e s o f t h e mean v e c t o r , m*, and c o v a r i a n c e m a t r i x , S*, f r o m 2 t h e r e d u c e d s a mple. V a l u e s o f d^ were t h e n r e c a l c u l a t e d f o r t h e e n t i r e sample u s i n g t h e new e s t i m a t e s o f m* and S*. The p r o c e d u r e was i t e r a t e d 5 t i m e s w i t h c o n v e r g e n c e t o a s o l u t i o n u s u a l l y o c c u r r i n g w i t h i n 2 o r 3 i t e r a t i o n s . A r o b u s t e s t i m a t e o f t h e c o r r e l a t i o n m a t r i x R* was o b t a i n e d a f t e r t h e l a s t i t e r a t i o n by r e s c a l i n g t h e e l e m e n t s o f S*. R o b u s t e s t i m a t e s o f m* and S* were u s e d as s t a r t i n g p o i n t s f o r t h e MVT p r o c e d u r e . S i m i l a r t o D e v l i n e t a l . ( 1 9 8 1 ) , t h e v e c t o r o f v a r i a b l e m e d i a n s was u s e d as an i n i t i a l e s t i m a t e o f m*, w i t h t h e i n i t i a l e s t i m a t e s o f t h e e l e m e n t s o f S* o b t a i n e d f r o m s * . . = r . . s * s * . ID ID i D R o b u s t e s t i m a t e s o f t h e s t a n d a r d d e v i a t i o n s s | a n d sjjf were o b t a i n e d f r o m s? = m e d i a n |x. - m e d i a n x. I / -6745 . R e g r e s s i o n P r o c e d u r e To o b t a i n r o b u s t e s t i m a t e s o f B* i n t h e e q u a t i o n S* = (I-B*) X D * ( I - B * 1 ) x , t h e e l e m e n t s o f B* were f i r s t o b t a i n e d by m i n i m i z i n g t h e f u n c t i o n G (£;t) = - t " 2 l o g | n _ 1 E exp ( i t (y k~x£e_)) | 2 . i = l The m i n i m i z a t i o n o f G n ( £ ; t ) r e q u i r e d two s t e p s . A v a l u e o f t was f i r s t c h o s e n s u c h t h a t e s t i m a t e s o f t h e r e g r e s s i o n c o e f f i c i e n t s s a t i s f i e d t h e c r i t e r i o n o f minimum e s t i m a t e d a s y m p t o t i c v a r i a n c e . (Chambers & 2 H e a t h c o t e , 19 8 1 ) . S i n c e a (t) i s s y m m e t r i c a b o u t t h e o r i g i n 2 an e s t i m a t e o f a minimum v a l u e o f a ( t ) was o b t a i n e d by 2 c a l c u l a t i n g e s t i m a t e s o f a ( t ) a t e q u a l l y s p a c e d i n t e r v a l s o v e r a r a n g e (0<t<T). The v a l u e o f T was d e t e r m i n e d by t h e s c a l e o f t h e r e s i d u a l s , and f o r t h e v a r i a b l e s i n t h e p r e s e n t s t u d y , a v a l u e o f T=2 was c h o s e n . U s i n g an i n t e r v a l w i d t h o f 0.1 o v e r t h e r a n g e (0<t<_2) r e q u i r e d t h e c a l c u l a t i o n o f 21 2 e s t i m a t e s o f a ( t ) and t h e v a l u e o f t c o r r e s p o n d i n g 30. t o t h e mininium o f t h e o b t a i n e d v a l u e s o f a ( t ) was s e l e c t e d . F o r a f i n i t e s a m p l e s i z e , Chambers a n d A 2 H e a t h c o t e (19 81) n o t e d t h a t b ( t ) may h a v e l i m i t a t i o n s 2 as an e s t i m a t e o f a ( t ) when T i s l a r g e . The c a l c u l a t i o n 2 o f o ( t ) r e q u i r e d e s t i m a t e s o f t h e q u a n t i t i e s u ( . t ) , v ( t ) , u(.2t). a n d v ( 2 t ) w h i c h w e r e o b t a i n e d f r o m -1 n un= n E c o s ( t ( y k - x k j i ) ) k = l 1 n a n d v (,£;t)=n £ s i n ( . t (y k -3 t^ | _ ) ) . k = l T h e s e e s t i m a t e s a l s o r e q u i r e d an i n i t i a l e s t i m a t e o f A 2 _g p r i o r t o t h e e v a l u a t i o n o f a ( t ) . F r o m p r e l i m i n a r y r e s u l t s , l e a s t s q u a r e s e s t i m a t e s o f j _ b a s e d on t h e e n t i r e s a m p l e w e r e f o u n d t o be i n a d e q u a t e f o r t h i s p u r p o s e due t o t h e i r s e n s i t i v i t y t o o u t l i e r s . I n o r d e r t o o b t a i n a l e s s b i a s e d i n i t i a l e s t i m a t e o f _p_ a m o d i f i e d t r i m m i n g p r o c e d u r e was u s e d t o remove a p r o p o r t i o n o f t h e s a m p l e o b s e r v a t i o n s w i t h t h e m o s t e x t r e m e v a l u e s . The method i n v o l v e d c a l c u l a t i n g a m o d i f i e d d i s t a n c e o f t h e f o r m d * 2 = ( x . - m * ) ' D * _ 1 ( x . - m * ) , I — l — — l — f o r e a c h o b s e r v a t i o n w i t h t h e e l e m e n t s o f m*1=[m*,...,m*] l p c o n s i s t i n g o f 10 p e r c e n t t r i m m e d means a n d t h e p x p d i a g o n a l m a t r i x D* c o n t a i n i n g t h e c o r r e s p o n d i n g 10 p e r c e n t 31 trimmed v a r i a n c e s . T h i s was f o l l o w e d by r e m o v i n g 20 2 p e r c e n t o f t h e o b s e r v a t i o n s w i t h t h e l a r g e s t d? v a l u e s f r o m t h e sample and c a l c u l a t i n g an i n i t i a l e s t i m a t e o f B_ f r o m t h e r e m a i n i n g o b s e r v a t i o n s by l e a s t s q u a r e s . The s e c o n d s t e p i n o b t a i n i n g r o b u s t e s t i m a t e s o f 3_ i n v o l v e d m i n i m i z i n g G n ( £ ; t ) u s i n g t h e v a l u e o f t o b t a i n e d by t h e method d e s c r i b e d a b o v e . W r i t i n g G n ( j [ ; t ) i n terms o f i t s r e a l and i m a g i n a r y components as G n ( 3 ; t ) = - t " 2 l o g ( u 2 (3_;t)+v 2(£;t) ) and by t a k i n g d e r i v a t i v e s w i t h r e s p e c t t o t h e e l e m e n t s o f _3_, t h e n o r m a l e q u a t i o n s may be w r i t t e n -1 -1 n t u n ( 3 . ; t ) n x E ^ x j k s i n ( t ( y j - x \ B _ ) ) -1 n - v ( 3 ; t ) n E x c o s ( t (y .-x\ 3) ) = 0, k = l , . . . , p . n — j = i J K 3 ~J— S i n c e G n(3_;t) i s n o t a c o n v e x f u n c t i o n , a p r o b l e m common t o some o t h e r r o b u s t e s t i m a t e s o f l o c a t i o n s u c h as T u k e y ' s b i w e i g h t o r Andrew's s i n e (Andrews, 1 9 7 4 ) , a m o d i f i e d Newton-Raphson p r o c e d u r e was u s e d t o o b t a i n t h e e s t i m a t e s o f g_. The a l g o r i t h m and i t s p r o o f o f c o n v e r g e n c e a r e p r o v i d e d by Chambers and H e a t h c o t e (19 8 1 ) . The p r o c e d u r e i n v o l v e s s i m p l i f y i n g t h e above n o r m a l e q u a t i o n s t o o b t a i n e x p r e s s i o n s o f t h e f o r m -2 -1 n n L, (3; t ) = n t I E x. , s i n ( t (y .-y • - ( x l - x ! ) 3) , k = l , . . . , p . * - i = 1 j = 1 UK 3 1 - 3 - 1 -32 U s i n g t h e i n i t i a l e s t i m a t e o f _3_ o b t a i n e d a b o v e a s a s t a r t i n g v a l u e and c a l c u l a t i n g L 1 (_§_; t ) = [ L ^ ( 3_; t ) ,. . . , L (3_;t)] , a s o l u t i o n f o r §_ was o b t a i n e d b y p e r f o r m i n g P t h e i n t e r a t i o n s g (m+1) = g (m) +;<[.u2 (B ; t ) + v 2 (£; t ) ] - 1 (X' X) _ 1 L (|; t ) f o r m=l,2,..., w i t h (X'X) p r o v i d e d by t h e c o v a r i a n c e m a t r i x o f t h e s a m p l e o b s e r v a t i o n s . The i t e r a t i o n s w e r e c o n t i n u e d u n t i l s u c c e s s i v e v a l u e s o f G ( 3 ; t ) n — -4 d i f f e r e d b y l e s s t h a n 1x10 . I f t h e p r o c e d u r e f a i l e d t o c o n v e r g e a f t e r 40 i t e r a t i o n s , t h e r e s u l t s f o r t h e r e p l i c a t i o n i n v o l v e d w e r e r e m o v e d f r o m t h e s t u d y . F a i l u r e t o c o n v e r g e was i n f r e q u e n t , and i n c o n d i t i o n s w i t h l a r g e amounts o f c o n t a m i n a t i o n , o c c u r r e d i n l e s s t h a n 1 p e r c e n t o f t h e r e p l i c a t i o n s . The amount o f b i a s i n t r o d u c e d b y r e p l a c i n g t h e s a m p l e s i n t h e s e r e p l i c a t i o n s was assumed t o be n e g l i g i b l e . The p-1 s e t s o f r e g r e s s i o n c o e f f i c i e n t s o b t a i n e d i n t h e a b o v e manner w e r e a r r a n g e d i n t h e m a t r i x B* and t h e e s t i m a t e o f t h e s a m p l e c o r r e l a t i o n m a t r i x R, was t h e n o b t a i n e d b y r e s c a l i n g t h e e l e m e n t s o f S* = ( I - B * ) ~ 1D* ( I - B * * The p x p d i a g o n a l m a t r i x D* c o n t a i n e d r o b u s t v a r i a n c e e s t i m a t e s o f v a r i a b l e s o b t a i n e d f r o m t h e t r a n s f o r m a t i o n Z± = ( l - B * - ) x i # i = l , . . . ,n. R o b u s t e s t i m a t e s o f t h e v a r i a n c e s o f were o b t a i n e d b y s q u a r i n g t h e r o b u s t e s t i m a t e s o f s c a l e , s* = median| z. -median z., I/.6745 , l<i<;n l<_i<n f o r k = l , . . . , p . A s e c o n d method f o r o b t a i n i n g r o b u s t e s t i m a t e s o f c o r r e l a t i o n s , b a s e d on a m o d i f i c a t i o n o f t h e above r e g r e s s i o n p r o c e d u r e , was a l s o e x a m i n e d i n t h e p r e s e n t s t u d y . T h i s method i n v o l v e d r e m o v i n g a p r o p o r t i o n o f t h e sample o b s e r v a t i o n s w i t h t h e l a r g e s t r e s i d u a l s d e t e r m i n e d by t h e e s t i m a t e o f £_ o b t a i n e d above. The p r o p o r t i o n o f t h e o b s e r v a t i o n s removed was d e t e r m i n e d A 2 by o b s e r v i n g t h e f o r m o f a ( t ) c a l c u l a t e d f r o m t h e r e d u c e d s a m p l e . F i v e p e r c e n t o f t h e o b s e r v a t i o n s w i t h t h e l a r g e s t r e s i d u a l s were removed a t e a c h s t e p o f t h e - 2 p r o c e d u r e and i f t h e minimum v a l u e o f a ( t ) o b t a i n e d f r o m t h e r e d u c e d sample was o b s e r v e d t o o c c u r f o r a v a l u e o f t~->0, i n d i c a t i n g t h e p r e s e n c e o f asymmetry, an a d d i t i o n a l 5 p e r c e n t o f t h e o b s e r v a t i o n s were removed. A 2 The p r o c e s s was r e p e a t e d u n t i l g ( t ) was a minimum a t a v a l u e o f t j ^ . l o r a maximum o f 2 0 p e r c e n t o f t h e o b s e r v a t i o n s were removed. A l e a s t s q u a r e s e s t i m a t e o f was t h e n c a l c u l a t e d on t h e r e m a i n i n g o b s e r v a t i o n s and u s e d a s a s t a r t i n g p o i n t f o r m i n i m i z i n g G (j$_;t) on 3.4 t h e r e d u c e d s a m p l e . The p u r p o s e o f t h i s p r o c e d u r e was t o d e t e r m i n e i f e s t i m a t e s o f B* and c o n s e q u e n t l y S* 2 c o u l d be i m p r o v e d u s i n g e s t i m a t e s o f o ( t ) t o i d e n t i f y and remove o u t l i e r s f r o m sample d a t a . To d i s t i n g u i s h t h e above two r e g r e s s i o n methods t h e y w i l l be s u b s e q u e n t l y r e f e r r e d t o as t h e R E G 1 and R E G 2 p r o c e d u r e s . I n a l l c o n d i t i o n s o f t h e p r e s e n t s t u d y , 2 0 0 r e p l i c a t i o n s were p e r f o r m e d . F o r e a c h o f t h e e s t i m a t e s e x a m i n e d , t h e p r o d u c t moment c o r r e l a t i o n s , t h e c o r r e l a t i o n s d e r i v e d f r o m t h e MVT p r o c e d u r e , t h e c o r r e l a t i o n s o b t a i n e d f r o m t h e two r e g r e s s i o n p r o c e d u r e s , t h e a v e r a g e b i a s o f r , and t h e mean s q u a r e d e r r o r (MSE) o f t h e F i s h e r z - t r a n s f o r m o f r were t a b u l a t e d . The b i a s o f r was c a l c u l a t e d as t h e mean d e v i a t i o n o f t h e sample c o r r e l a t i o n f r o m t h e p o p u l a t i o n p a r a m e t e r . The MSE was o b t a i n e d f r o m t h e mean s q u a r e d d i f f e r e n c e between t h e z - t r a n s f o r m s o f r and t h e c o r r e s p o n d i n g p a r a m e t e r v a l u e . 35 R e s u l t s arid D i s c u s s i o n The r e s u l t s f o r c o n d i t i o n s w i t h a s a m p l e s i z e o f 30 a r e p r e s e n t e d i n T a b l e s 1 t o 3. S i n c e c o n d i t i o n s o f 5 a n d 15 p e r c e n t c o n t a m i n a t i o n c o u l d n o t be r e p r e s e n t e d e x a c t l y f o r t h i s s a m p l e s i z e , two o u t l i e r s o r 6.7 p e r c e n t o f t h e s a m p l e r e p r e s e n t e d t h e c o n d i t i o n w i t h t h e l e a s t amount o f c o n t a m i n a t i o n , a n d f i v e o u t l i e r s , o r 16.7 p e r c e n t o f t h e s a m p l e r e p r e s e n t e d t h e c o n d i t i o n w i t h t h e l a r g e s t amount o f c o n t a m i n a t i o n . I n a l l t a b l e s t h e c o l u m n t i t l e UNCORR r e f e r s t o t h e u n c o r r e c t e d o r p r o d u c t moment c o r r e l a t i o n s . The s e n s i t i v i t y o f r t o t h e p r e s e n c e o f a s y m m e t r i c o u t l i e r s i s a p p a r e n t f r o m t h e r e s u l t s i n T a b l e s 1 t o 3. The amount o f b i a s p r e s e n t i n t h e e s t i m a t e s i s o b s e r v e d t o d e p e n d p r i m a r i l y on t h e l o c a t i o n o f t h e o u t l y i n g o b s e r v a t i o n s . The c o o r d i n a t e s o f t h e mean v e c t o r o f t h e c o n t a m i n a t i n g d i s t r i b u t i o n w e r e a l l p o s i t i v e i n t h e s e c o n d i t i o n s , c o n s e q u e n t l y e s t i m a t e s o f l a r g e n e g a t i v e c o r r e l a t i o n s c o n t a i n e d t h e most b i a s . I n T a b l e 1 f o r e x a m p l e , t h e b i a s o f r f o r p=-.5 0 a n d P=-.48 w e r e .48 a n d .53 r e s p e c t i v e l y , h o w e v e r , t h e b i a s o f r f o r p=-.47 was .02. F o r t h e l e v e l s o f o u t l i e r c o n t a m i n a t i o n r e p r e s e n t e d b y T a b l e s 1 a n d 2, and MVT p r o c e d u r e i s o b s e r v e d t o be TABLE 1 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=30 and 5 p e r c e n t c o n t a m i n a t i o n BIAS p UNCORR. MVT REG1 REG 2 67 -.1120 -.0022 -.0167 -.0130 47 .0196 -.0223 -.0185 -.0283 43 .0330 -.0144 -.0133 -.0141 40 .0538 -.0042 -.0822 -.0367 12 .2058 -.0129 -.0390 -.0099 11 .1847 -.0294 -.0490 -.0234 03 .2793 -.0034 -.0221 -.0090 07 .2594 -.0022 .0016 .0058 10 .3190 .0169 .0593 .0369 14 .3149 .0084 .0565 .0185 14 .3418 .0141 .0232 .0233 17 .3104 .0143 .0589 .0236 41 .4710 .0166 .0419 .0321 48 .5289 .0364 .1278 .0746 50 .4773 .0125 .0265 .0225 MEAN SQ ERROR Z P UNCORR. MVT REG1 REG2 .81 .1429 .0413 .0809 .0854 .51 .1113 .0376 .0615 .0679 .46 .1348 .0426 .0749 .0783 .42 .1316 .0393 .0794 .0642 .12 .1540 .0416 .0623 .0637 .11 .1535 .0381 .0511 .0525 .03 .1894 .0443 .0620 .0602 -.07 .1797 .0366 .0603 .0587 -.10 .2399 .0381 .0625 .0566 -.14 .2111 .0361 .0687 .0583 -.14 .2318 .0442 .0585 .0647 -.17 .2155 .0348 .0622 .0545 -.44 .3972 .0414 .0690 .0648 -.52 .4498 .0396 .0788 .0657 -.55 .3997 .0432 .0726 .0716 MEANS: .2228 .0399 .0670 .0645 ay TABLE 2 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=30 and 10 p e r c e n t c o n t a m i n a t i o n BIAS MEAN SQ ERROR p UNCORR. MVT REG1 REG 2 Z P ^UNCORR. MVT REG1 REG 2 67 -.1451 -.0143 -.0183 -.0440 .81 .1980 .0417 .0596 .0829 47 -.0010 -.0057 .0007 -.0159 .51 .1184 .0387 .0681 .0746 43 .0605 .0113 -.0017 -.0117 .46 .1185 .0401 .0651 .0782 40 .0531 .0044 -.0482 -.0361 .42 .1268 .0517 .0799 .0840 12 .2444 .0149 -.0024 .0359 .12 .1945 .0417 .0900 .0921 11 .2523 .0061 -.0014 .0178 .11 .2085 .0446 .0727 .0745 03 .3266 .0228 .0083 .0295 .03 .2398 .0374 .0615 .0645 07 .3564 .0373 .0430 .0591 -.07 .2466 .0422 .0678 .0784 10 .3686 .0019 .0265 -.0049 -.10 .2497 .0403 .0651 .0615 14 .3465 -.0074 .0424 -.0065 -.14 .2328 .0440 .0656 .0689 14 .3954 .0245 .0159 .0203 -.14 .2647 .0434 .0724 .0645 17 .4023 .0039 .0376 -.0004 -.17 .2645 .0444 .0590 .0717 41 .5848 .0344 .0434 .0561 -.44 .5157 .0534 .0727 .0813 48 .6206 .0184 .0726 .0449 -.52 .5570 .0478 .0768 .0795 50 .6264 .0265 .0630 .0410 -.55 .5711 .0423 .0805 .0732 MEANS: .2738 .0436 .0705 .0753 TABLE 3 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=30 and 15 p e r c e n t c o n t a m i n a t i o n BIAS MEAN SQ ERROR p UNCORR. MVT REG1 REG 2 Z P UNCORR. MVT REG1 REG 2 67 -.1241 -.0479 -.0295 -.0731 .81 .1426 .1256 .0764 .1113 47 .0172 -.0132 -.0124 -.0244 .51 .1217 .1173 .0717 .0831 43 .0528 .0468 .0079 -.0125 .46 .1268 .1215 .0651 .0818 40 .0798 .0303 -.0560 -.0395 .42 .1213 .1014 .0735 .0685 12 .2245 .0987 -.0017 .0051 .12 .1762 .1226 .0648 .0616 11 .2669 .1224 -.0268 -.0135 .11 .1958 .1066 .0564 .0605 03 .2904 .1215 .0110 .0222 .03 .2171 .1232 .0596 .0579 07 .4137 .2085 .0267 .0594 -.07 .2973 .1454 .0600 .0794 10 .4048 .1798 .0218 -.0027 -.10 .2913 .1256 .0681 .0722 14 .4396 .1815 .0397 .0069 -.14 .3174 .1410 .0635 .0667 14 .4489 .2112 .0127 .0082 -.14 .3176 .1607 .0648 .0653 17 .4564 .1948 .0271 -.0064 -.17 .3222 .1391 .0675 .0687 41 .6018 .3055 .0073 .0260 -.44 .5078 .2432 .0652 .0631 48 .6306 .2898 .0813 .0406 -.52 .5809 .2300 .0807 .0715 50 .7064 .3036 .0382 .0266 -.55 .6951 .2498 .0611 .0682 MEAN: .2954 .1502 .0666 .0720 CO 39 t h e l e a s t a f f e c t e d b y t h e p r e s e n c e o f a s y m m e t r i c o u t l i e r s . The b i a s o f t h e MVT e s t i m a t e s i s s m a l l r e l a t i v e t o t h e b i a s o f r i n t h i s c o n d i t i o n , a n d f o r t h e r a n g e o f c o r r e l a t i o n s i n t h e p r e s e n t s t u d y , a p p e a r s t o be i n d e p e n d e n t o f t h e m a g n i t u d e o f p . E s t i m a t e s o b t a i n e d f r o m t h e REG1 and REG2 p r o c e d u r e s a r e s l i g h t l y more b i a s e d t h a n t h e MVT e s t i m a t e s i n t h e s e c o n d i t i o n s , w i t h e s t i m a t e s o f n e g a t i v e c o r r e l a t i o n s t h a t a r e s l i g h t l y more b i a s e d t h a n e s t i m a t e s o f p o s i t i v e c o r r e l a t i o n s . The c o r r e l a t i o n e s t i m a t e s w i t h t h e l a r g e s t amount o f b i a s h a v e c o r r e s p o n d i n g l y l a r g e MSE's o f t h e z-t r a n s f o r m e d e s t i m a t e s . I n T a b l e 1, f o r e x a m p l e , t h e b i a s o f r f o r P=-.50 a n d P=-.4 8 a c c o u n t f o r more t h a n h a l f o f t h e m a g n i t u d e o f t h e s q u a r e d d e v i a t i o n s . v . . r e p r e s e n t e d by t h e MSE. The a v e r a g e MSE's o f t h e MVT e s t i m a t e s i n T a b l e s 1 a n d 2 a r e .040 a n d .044 r e s p e c t i v e l y , w h i c h f o r t h i s s a m p l e s i z e c l o s e l y a p p r o x i m a t e s t h e v a r i a n c e o f .037 f o r z - t r a n s f o r m e d c o r r e l a t i o n s . I n t h e c o n d i t i o n s r e p r e s e n t e d b y T a b l e s 1 an d 2 t h e MSE's o f t h e REG1 a n d REG2 e s t i m a t e s a r e o b s e r v e d t o be s l i g h t l y l a r g e r t h a n t h e MSE's o f t h e z - t r a n s f o r m e d MVT e s t i m a t e s . The r e s u l t s o f T a b l e 3 d e m o n s t r a t e t h a t t h e MVT e s t i m a t e s were u n a b l e t o w i t h s t a n d a c o n t a m i n a t i o n l e v e l o f 17 p e r c e n t . I n t h i s c o n d i t i o n t h e MVT e s t i m a t e s were s t r o n g l y b i a s e d and l i k e r , e s t i m a t e s o f t h e l a r g e s t n e g a t i v e c o r r e l a t i o n s c o n t a i n e d t h e most b i a s . The r e g r e s s i o n b a s e d e s t i m a t e s were r e l a t i v e l y u n a f f e c t e d by t h i s l e v e l o f c o n t a m i n a t i o n . The b i a s and MSE 1s o f t h e z - t r a n s f o r m e d REG1 and REG2 e s t i m a t e s i n T a b l e 3 were s i m i l a r t o t h e i r l e v e l s i n th e c o n d i t i o n s w i t h l e s s o u t l i e r c o n t a m i n a t i o n ' r e p r e s e n t e d by T a b l e s 1 and 2. A c r o s s t h e c o n d i t i o n s r e p r e s e n t e d b y T a b l e s 1 t o 3, t h e a v e r a g e MSE o f t h e REG1 e s t i m a t e s were .067, .071 and .067. F o r t h e REG2 e s t i m a t e s t h e c o r r e s p o n d i n g a v e r a g e M S E 1s were .065, .0 75 and .072. A l i m i t e d number o f r e p l i c a t i o n s were o b s e r v e d f o r 20 p e r c e n t c o n t a m i n a t i o n and n=30; however, t h e REG1 and REG2 e s t i m a t e s were u n a b l e t o w i t h s t a n d t h i s l e v e l o f c o n t a m i n a t i o n and were s t r o n g l y b i a s e d . T a b l e s 4 t o 7 c o n t a i n t h e r e s u l t s f o r n=60 and c o n d i t i o n s o f 5, 10, 15 and 25 p e r c e n t c o n t a m i n a t i o n . I n t h e 5, 10 and 15 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n s , t h e r e s u l t s a r e s i m i l a r t o t h o s e o b t a i n e d f o r n=30. O v e r a l l t h e MVT e s t i m a t e s were l e s s b i a s e d and had s m a l l e r MSE's t h a n t h e REG1 and REG2 e s t i m a t e s i n t h e TABLE 4 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=60 and 5 p e r c e n t c o n t a m i n a t i o n BIAS MEAN SQ ERROR p UNCORR. MVT REG1 REG2 Z P UNCORR. MVT REG1 REG 2 67 -.0729 -.0011 -.0135 -.0180 .81 .0805 .0211 .0373 .0385 47 .0036 -.0064 -.0164 -.0100 .51 .0649 .0207 .0381 .0309 43 .0373 .0017 -.0047 .0010 .46 .0653 .0188 .0295 .0279 40 .0443 .0001 -.0598 -.0043 .42 .0662 .0210 .0349 .0249 12 .1707 .0041 .0061 .0247 .12 .0905 .0174 .0244 .0273 11 .1839 .0106 .0094 .0237 .11 .0939 .0204 .0267 .0299 03 .1988 .0002 .0017 .0258 .03 .1029 .0229 .0273 .0319 07 .2403 -.0047 .0067 .0084 -.07 .1182 .0194 .0250 .0264 10 .2685 .0120 .0370 .0204 -.10 .1282 .0173 .0250 .0216 14 .2935 .0007 .0388 .0105 -.14 .1486 .0179 .0262 .0245 14 .2686 .0075 .0158 .0047 -.14 .1300 .0198 .0253 .0233 17 .3148 .0053 .0447 .0219 -.17 .1663 .0185 .0303 .0266 41 .3576 -.0057 -.0046 -.0053 -.44 .2110 .0195 .0259 .0257 48 .4021 .0032 .0769 .0143 -.52 .2677 .0192 .0426 .0283 50 .4391 .0076 .0106 .0037 -.55 .2899 .0229 .0342 .0306 MEAN: .1349 .0198 .0302 .0279 TABLE 5 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=60 and 10 p e r c e n t c o n t a m i n a t i o n BIAS MEAN SQ ERROR p UNCORR. MVT REG1 REG 2 Z P UNCORR. MVT REG1 REG2 67 -.2656 -.0096 -.0123 -.0175 .81 .0946 .0201 .0320 .0382 47 .0107 -.0015 -.0102 -.0101 .51 .0605 .0221 .0280 .0315 43 .0363 -.0043 -.0153 -.0158 .46 .0607 .0226 .0292 .0347 40 .0357 -.0021 -.0588 -.0247 .42 .0660 .0181 .0325 .0316 12 .2203 .0001 -.0067 .0169 .12 .1072 .0209 .0244 .0284 11 .2422 .0031 .0055 .0277 .11 .1233 .0176 .0222 .0280 03 .2521 .0017 -.0064 .0269 .03 .1202 .0190 .0278 .0302 07 .3587 .0128 .0162 .0199 -.07 .1938 .0202 .0212 .0259 10 .3793 .0249 .0437 .0300 -.10 .2105 .0207 .0263 .0250 14 .3885 .0228 .0539 .0269 -.14 .2146 .0222 .0324 .0278 14 .4002 .0199 .0177 .0144 -.14 .2188 .0252 .0339 .0331 17 .4095 .0152 .0433 .0195 -.17 .2356 .0205 .0293 .0283 41 .5484 .0224 .0156 .0264 -.44 .3903 .0229 .0282 .0302 48 .5784 .0217 .0716 .0391 -.52 .4599 .0229 .0406 .0361 50 .6296 .0235 .0278 .0206 -.55 .5218 .0261 .0306 .0349 MEAN: .2052 .0214 .0292 .0309 TABLE 6 A v e r a g e b i a s o f r and MSE o f o r n=60 and 15 pe: BIAS p UNCORR. MVT REG1 REG2 67 -.1479 -.0306 .0025 -.0190 47 -.0126 .0373 .0053 -.0041 43 .0624 .0659 .0047 .0040 40 .0648 .0631 -.0114 .0056 12 .2437 .1754 .0049 .0407 11 .2655 .1799 -.0015 .0291 03 .3346 .2199 .0013 .0470 07 .3895 .2254 -.0065 .0167 10 .4266 .2667 -.0016 -.0190 14 .4454 .2875 .0037 -.0209 14 .4293 .2621 .0016 .0005 17 .4691 .2929 .0058 .0003 41 .6434 .3752 .0151 .0139 48 .6862 .4253 .0536 .0357 50 .6855 .4023 .0102 .0095 t h e z - t r a n s f o r m e d e s t i m a t e s c e n t c o n t a m i n a t i o n MEAN SQ ERROR Z P UNCORR. MVT REG1 REG2 .81 .0968 .0661 .0362 .0524 .51 .0487 .0637 .0276 .0342 .46 .0629 .0725 .0273 .0367 .42 .0516 .0622 .0290 .0356 .12 .1304 .0894 .0244 .0261 .11 .1357 .0903 .0228 .0303 .03 .1810 .1151 .0240 .0315 -.07 .2226 .1112 .0207 .0296 -.10 .2462 .1349 .0261 .0312 -.14 .2658 .1404 .0261 .0321 -.14 .2452 .1267 .0236 .0266 -.17 .2875 .1415 .0216 .0290 -.44 .5081 .2180 .0238 .0273 -.52 .5936 .2806 .0298 .0324 -.55 .6057 .2742 .0275 .0318 MEAN: .2455 .1325 .0260 .0325 (LO TABLE 7 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=60 and 25 p e r c e n t c o n t a m i n a t i o n B IAS MEAN SQ ERROR p UNCORR. MVT REG1 REG2 Z P UNCORR. MVT REG1 REG2 67 -.1908 -.1065 -.0120 -.0480 .81 .1134 .0813 .0449 .0649 47 -.0197 .0449 -.0123 -.0212 .51 .0389 .0604 .0440 .0496 43 .0071 .0558 -.0225 -.0351 .46 .0430 .0704 .0436 .0511 40 .0112 .0539 .0083 -.0263 .42 .0312 .0554 .0408 .0387 12 .2561 .2601 -.0028 .0420 .12 .1159 .1363 .0272 .0335 11 .2695 .2775 .0215 .0611 .11 .1381 .1555 .0332 .0414 03 .3222 .3221 .0064 .0445 .03 .1536 .1728 .0330 .0369 07 .4286 .4101 .0067 .0469 -.07 .2384 .2417 .0403 .0497 10 .4445 .4381 .0355 .0326 -.10 .2556 .2689 .0337 .0408 14 .4717 .4515 .0259 .0151 -.14 .2874 .2818 .0297 .0368 14 .4606 .4261 .0135 .0218 -.14 .2658 .2537 .0340 .0433 17 .5182 .5000 .0430 .0493 -.17 .3326 .3244 .0299 .0445 41 .6876 .6238 .0480 .0632 -.44 .5662 .4936 .0351 .0391 48 .7759 .6963 .0568 .0724 -.52 .7365 .6236 .0363 .0437 50 .7783 .7107 .0444 .0499 -.55 .7412 .6383 .0438 .0494 MEAN: .2705 .2572 .0366 .0442 45 5 a n d 10 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n s . The b i a s a n d MSE's o f t h e MVT e s t i m a t e s w e r e r e l a t i v e l y c o n s t a n t a c r o s s t h e r a n g e o f pi i n t h e 5 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n , h o w e v e r e s t i m a t e s o f n e g a t i v e c o r r e l a t i o n s w e r e s l i g h t l y more b i a s e d i n t h e p r e s e n c e o f 10 p e r c e n t c o n t a m i n a t i o n . The R E G l e s t i m a t e s o f n e g a t i v e c o r r e l a -t i o n s w e r e s l i g h t l y more b i a s e d t h a n t h e REG2 e s t i m a t e s i n t h e 5 a n d 10 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n s , h o w e v e r , i n t h e c o n d i t i o n s w i t h more t h a n 10 p e r c e n t c o n t a m i n a t i o n t h e R E G l a n d REG2 e s t i m a t e s c o n t a i n e d s i m i l a r amounts o f b i a s . The a v e r a g e MSE o f t h e z - t r a n s f o r m e d MVT e s t i m a t e s w e r e .020 a n d .021 i n t h e 5 a n d 10 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n s w h i c h c l o s e l y a p p r o x i m a t e s f o r n=60, a v a r i a n c e o f .017 f o r z - t r a n s f o r m e d c o r r e l a t i o n s . The MSE's o f t h e R E G l a n d REG2 e s t i m a t e s w e r e s l i g h t l y l a r g e r i n t h e s e c o n d i t i o n s a n d w e r e .030 and .029 f o r t h e R E G l e s t i m a t e s a n d .02 8 and .031 f o r t h e REG2 e s t i m a t e s . I n t h e c o n d i t i o n s o f 15 and 25 p e r c e n t o u t l i e r c o n t a m i n a t i o n , t h e MVT e s t i m a t e s w e r e o b s e r v e d t o b r e a k down a n d p r o v i d e d e s t i m a t e s t h a t w e r e s t r o n g l y b i a s e d w i t h l a r g e MSE's ( s e e T a b l e s 6 a n d 7 ) . I n t h e s e c o n d i t i o n s t h e b i a s o f t h e R E G l and REG2 e s t i m a t e s was 46 a p p r o x i m a t e l y t h e same o r s m a l l e r t h a n t h e i r v a l u e s i n t h e 5 and 10 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n s . I n t h e 25 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n t h e R E G l a n d REG2 e s t i m a t e s w e r e l e s s s t a b l e a n d h a d l a r g e r MSE's t h a n i n t h e 5 t o 15 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n s . I n t h e 15 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n f o r e x a m p l e , t h e a v e r a g e MSE's o f t h e R E G l a n d REG2 e s t i m a t e s w e r e .026 a n d .033 w h i c h i n c r e a s e d t o .037 and .044 i n t h e p r e s e n c e o f 25 p e r c e n t c o n t a m i n a t i o n . The r e s u l t s f o r c o n d i t i o n s w i t h n=120 a p p e a r i n T a b l e s 8 t o 13. The r a n g e o f c o n t a m i n a t i o n e x a m i n e d i n t h e s e c o n d i t i o n s v a r i e d f r o m 5 t o 30 p e r c e n t i n i n t e r v a l s o f 5 p e r c e n t . F o r t h i s s a m p l e s i z e t h e r e s u l t s w e r e s i m i l a r t o t h e n=30 a n d n=60 c o n d i t i o n s . The MVT e s t i m a t e s were t h e l e a s t b i a s e d a n d h a d t h e s m a l l e s t a v e r a g e MSE's i n t h e 5 a n d 10 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n s b u t were o b s e r v e d t o b r e a k down i n c o n d i t i o n s w i t h more t h a n 20 p e r c e n t J c o n t a m i n a t i o n . The R E G l e s t i m a t e s o f n e g a t i v e c o r r e l a t i o n s i n t h e 5 and 10 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n s w e r e o b s e r v e d t o be s l i g h t l y more b i a s e d t h a n t h e REG2 e s t i m a t e s , h o w e v e r t h e REG2 e s t i m a t e s o f p o s i t i v e c o r r e l a t i o n s w e r e s l i g h t l y more b i a s e d t h a n t h e R E G l e s t i m a t e s . I n c o n d i t i o n s w i t h n=120, t h e R E G l a nd REG2 e s t i m a t e s TABLE 8 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=120 and 5 p e r c e n t c o n t a m i n a t i o n BIAS MEAN SQ ERROR p UNCORR. MVT REG1 REG 2 Z P UNCORR. MVT REG1 REG2 67 -.0739 .0028 -.0004 -.0022 .81 .0499 .0102 .0197 .0220 47 .0259 -.0042 -.0069 -.0088 .51 .0321 .0105 .0131 .0144 43 .0375 .0012 -.0013 .0053 .46 .0359 .0093 .0137 .0142 40 .0387 -.0028 -.0629 -.0077 .42 .0377 .0103 .0216 .0147 12 .1786 .0053 -.0027 .0131 .12 .0644 .0091 .0092 .0101 11 .1891 .0016 -.0009 .0135 .11 .0685 .0112 .0119 .0126 03 .2167 .0036 -.0049 .0149 .03 .0823 .0110 .0130 .0131 07 .2748 .0022 .0064 .0095 -.07 .1093 .0108 .0123 .0121 10 .2908 .0023 .0186 .0071 -.10 .1185 .0084 . .0110 .0105 14 .2988 -.0028 .0306 .0047 -.14 .1191 .0095 .0136 .0109 14 .2916 .0112 .0263 .0226 -.14 .1190 .0106 .0133 .0143 17 .3404 .0013 .0213 .0065 -.17 .1466 .0088 .0109 .0102 41 .4206 .0007 .0083 .0055 -.44 .2336 .0090 .0112 .0130 48 .4493 .0063 .0704 .0126 -.52 .2766 .0085 .0208 .0126 50 .4732 .0045 .0134 .0103 -.55 .3036 .0102 .0150 .0152 MEAN: .1198 .0098 .0140 .0133 TABLE 9 A v e r a g e b i a s o f r a n d MSE o f f o r n=120 and 10 pe BIAS p UNCORR. MVT REGl REG 2 67 -.1073 .0040 -.0015 -.0045 47 .0168 -.0027 -.0079 .0061 43 .0352 -.0004 -.0090 .0017 40 .0475 -.0015 -.0484 -.0028 12 .2412 .0023 -.0097 .0342 11 .2579 .0054 -.0063 .0348 03 .3011 -.0005 -.0101 .0337 07 .3479 .0046 .0010 -.0036 10 .3986 .0122 .0227 .0058 14 .4220 .0116 .0365 .0028 14 .3941 .0070 .0052 -.0114 17 .4293 .0124 .0302 .0112 41 .5710 .0166 .0139 .0069 48 .6285 .0254 .0685 .0333 50 .6394 .0125 .0107 .0002 t h e z - t r a n s f o r m e d e s t i m a t e s r c e n t c o n t a m i n a t i o n MEAN SQ ERROR Z P UNCORR. MVT REGl REG2 .81 .0572 .0093 .0168 .0186 .51 .0304 .0088 .0128 .0180 .46 .0354 .0080 .0124 .0154 .42 .0305 .0094 .0152 .0160 .12 .0998 .0091 .0123 .0165 .11 .1056 .0102 .0123 .0158 .03 .1284 .0097 .0117 .0155 -.07 .1584 .0101 .0104 .0124 -.10 .2004 .0101 .0128 .0124 -.14 .2145 .0115 .0147 .0159 -.14 .1890 .0092 .0111 .0132 -.17 .2206 .0105 .0123 .0139 -.44 .3828 .0106 .0139 .0139 -.52 .4820 .0104 .0200 .0152 -.55 .5017 .0114 .0145 .0149 MEAN: .1891 .0099 .0136 .0152 00 TABLE 10 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=120 and 15 p e r c e n t c o n t a m i n a t i o n BIAS MEAN SQ ERROR p UNCORR. MVT REG1 REG2 Z P UNCORR. MVT REG1 REG2 67 -.1471 -.0474 .0057 .0003 .81 .0729 .0448 .0154 .0176 47 .0004 .0381 .0052 .0117 .51 .0253 .0370 .0144 .0171 43 .0391 .0527 -.0002 .0085 .46 .0252 .0366 .0139 .0164 40 .0426 .0397 -.0266 -.0099 .42 .0265 .0298 .0135 .0138 12 .2433 .1521 .0017 .0446 .12 .0954 .0573 .0118 .0155 11 .2623 .1686 .0005 .0452 .11 .1060 .0656 .0114 .0155 03 .3035 .1956 -.0084 .0421 .03 .1294 .0724 .0109 .0167 07 .3949 .2454 -.0048 .0012 -.07 .1931 .0959 .0108 .0134 10 .4103 .2720 .0110 .0034 -.10 .2066 .1064 .0110 .0138 14 .4532 .2734 .0267 .0108 -.14 .2475 .1088 .0142 .0155 14 .4563 .2949 .0012 -.0006 -.14 .2507 .1235 .0113 .0135 17 .4635 .2783 .0222 .0152 -.17 .2519 .1118 .0149 .0173 41 .6364 .3531 .0044 .0039 -.44 .4711 .1802 .0096 .0133 48 .6773 .3870 .0243 .0152 -.52 .5511 .2236 .0128 .0135 50 .7162 .4080 .0121 .0097 -.55 .6159 .2468 .0135 .0167 MEAN: .2179 .1027 .0126 .0153 TABLE 11 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=120 and 20 p e r c e n t c o n t a m i n a t i o n BIAS MEAN SQ ERROR p UNCORR. MVT REGl REG2 Z P UNCORR. MVT REGl REG 2 67 -.1777 -.0705 -.0097 -.0125 .81 .0908 .0495 .0155 .0179 47 -.0164 .0377 .0022 .0090 .51 .0205 .0314 .0149 .0165 43 .0113 .0688 -.0004 .0130 .46 .0233 .0424 .0145 .0197 40 .0342 .0814 -.0071 .0026 .42 .0218 .0462 .0131 .0183 12 .2764 .2440 -.0017 .0548 .12 .1139 .0992 .0132 .0206 11 .2763 .2500 .0162 .0705 .11 .1161 .1070 .0134 .0248 03 .3340 .3057 .0085 .0695 .03 .1472 .1328 .0138 .0232 07 .4046 .3640 .0042 .0072 -.07 .1966 .1712 .0109 .0153 10 .4434 .3868 -.0030 -.0097 -.10 .2388 .1964 .0119 .0172 14 .4710 .4015 .0028 -.0145 -.14 .2603 .2063 .0113 .0155 14 .4786 .4234 -.0054 -.0081 -.14 .2710 .2199 .0101 .0125 17 .4913 .4215 -.0028 -.0050 -.17 .2791 .2203 .0120 .0167 41 .6912 .5786 .0286 .0264 -.44 .5459 .4022 .0129 .0160 48 .7372 .6025 .0186 .0171 -.52 .6502 .4531 .0118 .0174 50 .7545 .6225 .0155 .0141 -.55 .6852 .4858 .0138 .0152 MEANS: .2440 .1909 .0129 .0178 U l o TABLE 12 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=120 and 25 p e r c e n t c o n t a m i n a t i o n BIAS MEAN SQ ERROR p UNCORR. MVT REGl REG 2 Z P UNCORR. MVT REGl REG 2 67 -.2050 -.1249 -.0005 -.0126 .81 .1086 .0718 .0222 .0288 47 -.0288 .0196 -.0005 .0046 .51 .0255 .0340 .0264 .0219 43 .0077 .0576 .0145 .0259 .46 .0183 .0337 .0228 .0222 40 .0208 .0732 .0159 .0002 .42 .0202 .0385 .0201 .0171 12 .2619 .2720 .0166 .0801 .12 .0987 .1188 .0184 .0232 11 .2593 .2689 .0651 .1149 .11 .0960 .1091 .0258 .0338 03 .3347 .3419 .0510 .1042 .03 .1465 .1589 .0229 .0303 07 .4160 .4047 .0124 .0288 -.07 .2072 .2058 .0226 .0251 10 .4407 .4329 .0323 .0304 -.10 .2327 .2314 .0204 .0217 14 .4825 .4654 .0213 .0127 -.14 .2659 .2608 .0191 .0189 14 .4789 .4687 .0326 .0356 -.14 .2693 .2665 .0270 .0215 17 .4944 .4810 .0246 .0308 -.17 .2805 .2771 .0199 .0255 41 .7088 .6663 .0612 .0693 -.44 .5754 .5176 .0223 .0244 48 .7624 .7104 .0565 .0744 -.52 .6857 .6093 .0251 .0322 50 .7856 .7312 .0664 .0587 -.55 .7342 .6495 .0300 .0263 MEAN: .2510 .2388 .0230 .0249 H 1 TABLE 13 Av e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=120 and 30 p e r c e n t c o n t a m i n a t i o n BIAS MEAN SQ ERROR p UNCORR. MVT REG1 REG2 Z P UNCORR. MVT REG1 REG2 67 -.2176 -.1512 -.0224 -.0332 .81 .1189 .0798 .0344 .0336 47 -.0529 -.0037 -.0127 -.0138 .51 .0192 .0279 .0272 .0236 43 -.0077 .0466 -.0003 .0035 .46 .0149 .0270 .0307 .0296 40 .0092 .0497 .0251 .0055 .42 .0165 .0290 .0263 .0203 12 .2586 .2738 .0655 .1068 .12 .0956 .1139 .0294 .0341 11 .2718 .2799 .1067 .1457 .11 .1004 .1160 .0432 .0474 03 .3391 .3418 .1025 .1438 .03 .1486 .1600 .0416 .0460 07 .4252 .4401 .0872 .1434 -.07 .2171 .2407 .0553 .0534 10 .4586 .4681 .1145 .1363 -.10 .2484 .2691 .0408 .0458 14 .4809 .4779 .0938 .1110 -.14 .2655 .2707 .0310 .0348 14 .4838 .4892 .1061 .1531 -.14 .2702 .2849 .0508 .0506 17 .5099 .5108 .1237 .1508 -.17 .2976 .3104 .0527 .0612 41 .7142 .6976 .1611 .1994 -.44 .5853 .5634 .0810 .0896 48 .7782 .7477 .1568 .2006 -.52 .7151 .6676 .0784 .0945 50 .7966 .7661 .1706 .2181 -.55 .7525 .7087 .0856 .1042 MEAN: .2577 .2579 .0472 .0512 Ln 53 were r e l a t i v e l y u n a f f e c t e d by l e v e l s o f c o n t a m i n a t i o n up t o 20 p e r c e n t . The b i a s and a v e r a g e MSE's o f t h e s e e s t i m a t e s were o f s i m i l a r m a g n i t u d e f o r t h i s r a n g e o f c o n t a m i n a t i o n ; however, i n c o n d i t i o n s o f 25 and 30 p e r c e n t c o n t a m i n a t i o n ( T a b l e s 12 and 1 3 ) , a s h a r p i n c r e a s e i n t h e b i a s and MSE's was o b s e r v e d . The t h r e e c o n d i t i o n s t h a t i n v o l v e d v a r y i n g t h e l o c a t i o n o f t h e o u t l y i n g o b s e r v a t i o n s a r e r e p r e s e n t e d i n T a b l e s 14 t o 16. T a b l e 14 r e p r e s e n t s t h e c o n d i t i o n w i t h t h e c o n t a m i n a t i n g and p o p u l a t i o n d i s t r i b u t i o n s c e n t e r e d a t t h e same l o c a t i o n , i . e . , t h e mean v e c t o r s were e q u a l . The b i a s o f t h e MVT and r e g r e s s i o n b a s e d e s t i m a t e s was s m a l l e r i n t h i s c o n d i t i o n t h a n i n c o n d i t i o n s w i t h a s y m m e t r i c c o n t a m i n a t i o n . The r e l a t i v e p e r f o r m a n c e o f t h e r e g r e s s i o n b a s e d e s t i m a t e s and MVT e s t i m a t e s i n t h i s c o n d i t i o n was s i m i l a r t o t h e a s y m m e t r i c c o n d i t i o n s w i t h t h e REG1 and REG2 e s t i m a t e s h a v i n g s l i g h t l y l a r g e r l e v e l s o f b i a s and MSE. The MVT e s t i m a t e s were o b s e r v e d t o be l e s s s e n s i t i v e t o s y m m e t r i c a l l y d i s t r i b u t e d o u t l i e r s i n t h i s c o n d i t i o n , h a v i n g l e s s b i a s and e q u i v a l e n t MSE's i n c o m p a r i s o n t o t h e e q u i v a l e n t c o n d i t i o n o f a s y m m e t r i c c o n t a m i n a t i o n shown i n T a b l e 5. The REGl and REG2 were a l s o l e s s b i a s e d i n t h e s y m m e t r i c c o n d i t i o n , however MSE's o f TABLE 14 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=60 and 10 p e r c e n t c o n t a m i n a t i o n -Symmetric c o n t a m i n a t i o n c o n d i t i o n BIAS MEAN SQ ERROR p UNCORR. MVT REG1 REG 2 Z P UNCORR. MVT REG1 REG 2 67 -.3232 -.0080 -.0183 -.0249 .81 .2487 .0206 .0380 .0417 47 -.2163 .0073 .0120 -.0056 .51 .1169 .0194 .0258 .0276 43 -.2115 -.0069 -.0111 -.0193 .46 .1097 .0213 .0233 .0274 40 -.1995 -.0053 -.0514 -.0199 .42 .0932 .0203 .0309 .0275 12 -.0298 .0032 .0061 -.0020 .12 .0513 .0212 .0234 .0271 11 -.0274 .0069 .0068 .0050 .11 .0497 .0224 .0259 .0274 03 -.0199 .0126 .0176 .0196 .03 .0450 .0198 .0221 .0248 07 .0165 .0026 .0043 .0057 -.07 .0455 .0185 .0212 .0230 10 .0603 .0015 .0193 .0080 -.10 .0532 .0188 .0261 .0265 14 .0771 -.0090 .0151 -.0065 -.14 .0516 .0173 .0322 .0263 14 .0808 -.0023 -.0102 -.0085 -.14 .0562 .0195 .0284 .0264 17 .0651 -.0085 .0122 -.0010 -.17 .0401 .0165 .0241 .0212 41 .2133 .0023 .0059 .0082 -.44 .0955 .0245 .0297 .0320 48 .2146 .0050 .0600 .0293 -.52 .1140 .0214 .0361 .0333 50 .2680 .0087 .0021 .0046 -.55 .1393 .0197 .0294 .0331 MEAN: .0873 .0201 .0278 .0283 Ln TABLE 15 A v e r a g e b i a s o f r a n d MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=60 a n d 15 p e r c e n t c o n t a m i n a t i o n -Mean v e c t o r o f c o n t a m i n a t i n g d i s t r i b u t i o n = [ 1 , . . . , 1 ] BIAS MEAN SQ ERROR p UNCORR. MVT REGl REG2 Z P UNCORR. MVT REGl REG2 67 -.3609 -.1496 -.0177 -.0325 .81 .2805 .1033 .0327 .0416 47 -.2661 -.1084 -.0188 -.0353 .51 .1376 .0750 .0273 .0335 43 -.2236 -.0908 -.0205 -.0301 .46 .1091 .0516 .0277 .0339 40 -.2158 -.0937 -.0559 -.0414 .42 .1018 .0548 .0320 .0330 12 -.0091 -.0064 -.0007 -.0031 .12 .0451 .0395 .0237 .0233 11 -.0170 -.0034 .0028 .0063 .11 .0483 .0396 .0203 .0213 03 .0264 .0005 -.0024 .0005 .03 .0500 .0367 .0261 .0265 07 .1174 .0611 .0065 .0221 -.07 .0592 .0420 .0207 .0272 10 .1032 .0465 .0087 .0059 -.10 .0547 .0438 .0265 .0280 14 .1559 .0779 .0370 .0279 -.14 .0650 .0471 .0288 .0294 14 .1305 .0608 .0043 .0109 -.14 .0632 .0427 .0221 .0260 17 .1570 • 0732 .0253 .0237 -.17 .0667 .0442 .0230 .0290 41 .3015 .1201 .0102 .0207 -.44 .1623 .0704 .0261 .0271 48 .3731 .1272 .0518 .0343 -.52 .2127 .0675 .0260 .0300 50 .3718 .1587 .0073 .0176 -.55 .2210 .0790 .0298 .0316 MEAN: .1118 .0558 .0262 .0294 (Jl TABLE 16 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=60 and 15 p e r c e n t c o n t a m i n a t i o n Mean v e c t o r o f c o n t a m i n a t i n g d i s t r i b u t i o n = [ 5 , . . . , 5 ] BIAS MEAN SQ ERROR p UNCORR. MVT REGl REG2 Z P UNCORR. MVT REGl REG 2 67 .0223 .0500 .0107 -.0300 .81 .0663 .1092 .0337 .0748 47 .1921 .1732 -.0029 -.0449 .51 .1554 .1437 .0317 .0467 43 .2309 .2029 .0172 -.0311 .46 .1748 .1683 .0291 .0512 40 .2528 .2193 -.0336 -.0148 .42 .1947 .1697 .0325 .0305 12 .4994 .3833 .0018 .0462 .12 .4298 .2641 .0224 .0327 11 .4987 .3882 .0137 .0634 .11 .4161 .2580 .0292 .0384 03 .5764 .4420 .0102 .0423 .03 .5116 .2999 .0218 .0278 07 .6582 .5066 -.0159 -.0013 -.07 .6238 .3653 .0286 .0471 10 .6901 .5172 .0166 .0042 -.10 .6678 .3708 .0243 .0261 14 .7188 .5294 .0146 -.0181 -.14 .6948 .3747 .0224 .0251 14 .7224 .5484 .0136 .0207 -.14 .7207 .4219 .0282 .0312 17 .7379 .5533 .0069 -.0202 -.17 .7366 .4139 .0244 .0290 41 .9518 .6891 .0001 .0047 -.44 1.1474 .5919 .0280 .0324 48 .9995 .7384 .0736 .0269 -.52 1.2776 .7029 .0417 .0335 50 1.0197 .7446 .0250 .0032 -.55 1.3178 .7090 .0293 .0337 MEAN: .6090 .3576 .0285 .0373 U l 57 t h e s e e s t i m a t e s were s i m i l a r i n b o t h t h e s y m m e t r i c and a s y m m e t r i c c o n d i t i o n s . I n t h e c o n d i t i o n s o f 15 p e r c e n t c o n t a m i n a t i o n and n=60 r e p r e s e n t e d by T a b l e s 15 and 16, t h e REG1 . e s t i m a t e s were r e l a t i v e l y u n a f f e c t e d by t h e d i s t a n c e between t h e c o n t a m i n a t i n g and p o p u l a t i o n d i s t r i b u t i o n s , w i t h s i m i l a r l e v e l s o f b i a s and MSE i n t h e s e c o n d i t i o n s . The REG2 e s t i m a t e s were a d v e r s e l y a f f e c t e d b y t h e d i s t a n c e between t h e d i s t r i b u t i o n s b e i n g more b i a s e d and h a v i n g l a r g e r MSE's i n t h e c o n d i t i o n w i t h t h e l a r g e s t s e p a r a t i o n b e t w e e n d i s t r i b u t i o n s r e p r e s e n t e d b y T a b l e 16. The r e s u l t s o f c o n d i t i o n s w h i c h i n v o l v e d r e o r d e r i n g t h e v a r i a b l e s a p p e a r i n T a b l e s 17 t o 20. T a b l e s 17 t o 19 c o n t a i n t h e r e s u l t s f o r c o n d i t i o n s i n w h i c h t h e o r d e r o f t h e v a r i a b l e s was r e v e r s e d . F o r n=60, t h e l e v e l s o f c o n t a m i n a t i o n e x a m i n e d i n t h e s e c o n d i t i o n s were 5, 10 and 15 p e r c e n t . C o r r e s p o n d i n g c o n d i t i o n s w i t h t h e o r i g i n a l v a r i a b l e o r d e r a r e r e p r e s e n t e d by T a b l e s 4 t o 6. The REG1 and REG2 e s t i m a t e s were o b s e r v e d t o be r e l a t i v e l y u n a f f e c t e d by t h e o r d e r o f t h e v a r i a b l e s i n t h e s e c o n d i t i o n s . The b i a s and MSE's were s i m i l a r f o r b o t h v a r i a b l e o r d e r s w i t h t h e a v e r a g e MSE o f t h e REG1 e s t i m a t e s b e i n g .030, .029 and .026 i n t h e o r i g i n a l v a r i a b l e o r d e r and .028, .028 and .026 i n t h e r e v e r s e d TABLE 17 A v e r a g e b i a s o f r and MSE o f z - t r a n s f o r m e d e s t i m a t e s f o r n=60 and 5 p e r c e n t c o n t a m i n a t i o n -O r d e r o f v a r i a b l e s r e v e r s e d BIAS MEAN SQ ERROR p UNCORR. MVT REG1 REG 2 Z P UNCORR. MVT REG1 REG 2 .67 -.0897 -.0036 -.0481 -.0223 .81 .0878 .0221 .0426 .0345 .47 -.0178 -.0182 -.0163 -.0207 .51 .0671 .0212 .0261 .0275 .43 -.0033 -.0102 -.0585 -.0321 .46 .0696 .0251 .0338 .0297 .40 .0224 .0004 -.0020 -.0059 .42 .0635 .0179 .0217 .0229 .12 .1618 -.0015 -.0045 .0029 .12 .0963 .0193 .0248 .0231 .11 .1661 -.0337 -.0154 -.0055 .11 .0938 .0180 .0248 .0235 .03 .2004 .0078 -.0024 .0138 .03 .1094 .0208 .0288 .0262 -.07 .2373 .0118 .0319 .0162 -.07 .1158 .0244 .0304 .0302 -.10 .2401 .0088 .0182 .0185 -.10 .1248 .0227 .0255 .0268 -.14 .2777 .0211 .0183 .0245 -.14 .1342 .0222 .0266 .0277 -.14 .2827 .0050 .0149 .0133 -.14 .1434 .0185 .0223 .0230 -.17 .2819 .0082 .0379 .0091 -.17 .1390 .0222 .0271 .0257 -.41 .4018 .0098 .0129 .0214 -.44 .2444 .0230 .0264 .0280 -.48 .4482 .0128 .0193 .0279 -.52 .3079 .0184 .0266 .0265 -.50 .4097 .0149 .0235 .0242 -.55 .2661 .0249 .0332 .0327 MEAN: .1375 .0214 .0280 .0272 o o TABLE 18 A v e r a g e b i a s o f r and MSE o f o r n=60 and 10 per* O r d e r o f v a r i BIAS p UNCORR. MVT REGl REG2 67 -.1141 -.0095 -.0458 -.0408 47 -.0078 -.0049 -.0109 -.0097 43 .0095 -.0011 -.0340 -.0176 40 .0499 -.0069 -.0061 .0052 12 .2353 .0123 -.0030 .0158 11 .2399 .0202 -.0023 .0047 03 .2713 .0202 .0103 .0309 07 .3402 .0102 .0205 -.0024 10 .3367 -.0006 .0002 .0149 14 .3883 .0080 .0052 .0180 14 .3894 .0148 .0159 .0117 17 .4136 .0117 .0214 .0077 41 .5552 .0029 -.0041 .0094 48 .6113 .0051 .0035 .0159 50 .6228 .0095 .0069 .0063 t h e z - t r a n s f o r m e d e s t i m a t e s e n t c o n t a m i n a t i o n -b l e s r e v e r s e d MEAN SQ ERROR z P UNCORR. MVT REGl REG 2 .81 .0909 .0222 .0412 .0419 .51 .0632 .0190 .0291 .0310 .46 .0574 .0142 .0288 .0289 .42 .0709 .0195 .0266 .0309 .12 .1209 .0210 .0320 .0323 .11 .1200 .0178 .0252 .0263 .03 .1414 .0206 .0246 .0267 -.07 .1784 .0217 .0289 .0345 -.10 .1698 .0210 .0246 .0270 -.14 .2058 .0223 .0278 .0318 -.14 .2211 .0200 .0255 .0302 -.17 .2329 .0214 .0256 .0327 -.44 .4039 .0202 .0264 .0313 -.52 .4869 .0208 .0272 .0345 -.55 .5203 .0192 .0264 .0282 MEANS: .2056 .0201 .0280 .0312 c n t o TABLE 19 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s n=60 and 15 p e r c e n t c o n t a m i n a t i o n O r d e r o f t h e v a r i a b l e s r e v e r s e d BIAS MEAN SQ ERROR p UNCORR. MVT REG1 REG2 Z P UNCORR. MVT REG1 REG 2 67 -.1543 -.0691 -.0300 -.0384 .81 .1031 .0803 .0333 .0423 47 -.0032 .0215 -.0230 -.0317 .51 .0519 .0603 .0268 .0288 43 .0252 .0407 -.0323 -.0384 .46 .0505 .0704 .0227 .0328 40 .0336 .0372 -.0171 -.0041 .42 .0595 .0652 .0301 .0308 12 .2810 .1810 -.0253 -.0139 .12 .1600 .0967 .0274 .0332 11 .2570 .1657 -.0195 -.0353 .11 .1267 .0911 .0257 .0308 03 .3440 .2164 -.0074 .0173 .03 .1894 .1062 .0268 .0306 07 .3936 .2563 .0142 -.0139 -.07 .2123 .1279 .0250 .0296 10 .4431 .2677 -.0030 .0228 -.10 .2695 .1339 .0223 .0283 14 .4333 .2685 .0035 .0230 -.14 .2553 .1350 .0276 .0321 14 .4580 .2818 .0022 .0100 -.14 .2771 .1417 .0246 .0293 17 .4523 .2841 -.0048 -.0233 -.17 .2699 .1436 .0199 .0289 41 .6705 .4064 .0185 .0376 -.44 .5519 .2474 .0252 .0318 48 .6991 .4176 .0198 .0375 -.52 .6170 .2768 .0274 .0351 50 .7041 .4017 .0059 .0148 -.55 .6374 .2809 .0260 .0285 MEANS: .2554 .1371 .0261 .0315 C3 TABLE 20 A v e r a g e b i a s o f r and MSE o f t h e z - t r a n s f o r m e d e s t i m a t e s f o r n=60 and 10 p e r c e n t c o n t a m i n a t i o n -O r d e r o f t h e l a s t t h r e e v a r i a b l e s r e v e r s e d BIAS MEAN SQ ERROR p UNCORR. MVT REG1 REG2 Z P UNCORR. MVT REG1 REG 2 67 -.1074 .0005 -.0133 -.0165 .81 .0824 .0213 .0371 .0427 47 .0275 .0041 -.0023 .0091 .51 .0636 .0195 .0247 .0305 43 .0328 -.0033 -.0136 .0031 .46 .0528 .0181 .0236 .0262 40 .0402 -.0043 -.0205 -.0286 .42 .0623 .0252 .0337 .0335 12 .2374 -.0121 -.0422 -.0390 .12 .1190 .0243 .0301 .0298 11 .2311 -.0105 -.0268 -.0276 .11 .1134 .0233 .0255 .0277 03 .2905 -.0035 -.0173 -.0143 .03 .1474 .0234 .0256 .0269 07 .3465 .0138 .0161 .0303 -.07 .1838 .0198 .0246 .0295 10 .3733 .0033 -.0052 .0262 -.10 .2073 .0193 .0216 .0272 14 .4134 .0119 .0101 .0468 -.14 .2420 .0200 .0272 .0347 14 .4036 .0040 .0023 .0057 -.14 .2291 .0215 .0252 .0312 17 .4001 .0090 -.0025 .0330 -.17 .2307 .0175 .0232 .0319 41 .5833 .0251 .0681 .0499 -.44 .4386 .0233 .0385 .0329 48 .6106 .0218 .0722 .0398 -.52 .4801 .0213 .0389 .0338 50 .6522 .0215 .0180 .0258 -.55 .5601 .0223 .0332 .0360 MEANS: .2142 .0213 .0288 .0316 62 o r d e r . The a v e r a g e MSE o f t h e REG2 e s t i m a t e s were .028, .031.and .033 i n t h e o r i g i n a l o r d e r and .027, .031 and .0 32 i n t h e c o r r e s p o n d i n g c o n d i t i o n s w i t h r e v e r s e d o r d e r . The r e s u l t s f o r a t h i r d c o m b i n a t i o n o f v a r i a b l e s w i t h n=6 0 and 10 p e r c e n t c o n t a m i n a t i o n , a r e p r e s e n t e d i n T a b l e 20. I n t h i s c o n d i t i o n t h e o r d e r o f t h e f i r s t 3 v a r i a b l e s was unc h a n g e d and t h e o r d e r o f t h e r e m a i n i n g 3 v a r i a b l e s r e v e r s e d . The REGl and REG2 a l s o a p p e a r t o be u n a f f e c t e d b y t h i s c o m b i n a t i o n o f v a r i a b l e s h a v i n g l e v e l s o f b i a s and MSE's s i m i l a r t o t h a t o b t a i n e d i n c o n d i t i o n s o f e q u i v a l e n t l e v e l s o f c o n t a m i n a t i o n and sample s i z e . From an e x a m i n a t i o n o f t h e r e s u l t s a c r o s s c o n d i t i o n s i n t h e p r e s e n t s t u d y some g e n e r a l f e a t u r e s o f t h e e s t i m a t e s were a p p a r e n t . The p r e d o m i n a n t f e a t u r e o f r was i t s s e n s i t i v i t y t o s m a l l amounts o f o u t l i e r c o n t a m i n a t i o n . I n a d d i t i o n t o t h e l e v e l o f c o n t a m i n a t i o n , r was s e n s i t i v e t o t h e l o c a t i o n o f t h e o u t l y i n g o b s e r v a t i o n s and t h e c o r r e l a t i o n s among t h e v a r i a b l e s i n t h e c o n t a m i n a t i n g d i s t r i b u t i o n . F o r example, i n t h e s y m m e t r i c c o n t a m i n a t i o n c o n d i t i o n r e p r e s e n t e d b y T a b l e 14, r was l e s s b i a s e d f o r v a l u e s o f p n e a r 0 t h a n f o r l a r g e r v a l u e s o f p , s i n c e t h e c o r r e l a t i o n s among t h e 63 v a r i a b l e s i n the. c o n t a m i n a t i n g d i s t r i b u t i o n were e q u a l t o 0. The d e m o n s t r a t e d s e n s i t i v i t y o f r t o v a r i o u s f o r m s o f o u t l i e r c o n t a m i n a t i o n i n d i c a t e t h a t f o r many k i n d s o f d a t a f r e q u e n t l y e n c o u n t e r e d i n p s y c h o l o g i c a l r e s e a r c h t h e use o f r f o r e x a m i n i n g t h e r e l a t i o n s h i p s among v a r i a b l e s i s o f t e n i n a p p r o p r i a t e . I n a l l c o n d i t i o n s o f 5 and 10 p e r c e n t c o n t a m i n a t i o n i n t h e p r e s e n t s t u d y , t h e MVT e s t i m a t e s were t h e l e a s t b i a s e d and h a d t h e s m a l l e s t MSE's o f t h e e s t i m a t e s e x a m i n e d . The b i a s o f t h e MVT e s t i m a t e s i n c o n d i t i o n s w i t h more t h a n 10 p e r c e n t c o n t a m i n a t i o n i n d i c a t e s t h a t a l i m i t a t i o n o f t h e MVT p r o c e d u r e i s i t s dependence on t h e amount o f t r i m m i n g t h a t must be i n i t i a l l y s p e c i f i e d . I n p r a c t i c e , f o r example, i f t h e amount o f o u t l i e r c o n t a m i n a t i o n i n a sample i s u n d e r e s t i m a t e d when s p e c i f y i n g t h e amount o f i n i t i a l t r i m m i n g , t h e MVT e s t i m a t e s w i l l be b i a s e d . C o n v e r s e l y , i f t h e amount o f i n i t i a l t r i m m i n g i s o v e r e s t i m a t e d , t h e r e m o v a l o f v a l i d o b s e r v a t i o n s f o r t h e sample w i l l t e n d t o b i a s r t o w a r d 0. The a b i l i t y o f t h e r e g r e s s i o n e s t i m a t e s t o r e s i s t l a r g e amounts o f a s y m m e t r i c c o n t a m i n a t i o n was t h e p r i n c i p a l f i n d i n g o f t h e p r e s e n t s t u d y . T h e s e e s t i m a t e s were a b l e t o w i t h s t a n d up t o 25 p e r c e n t c o n t a m i n a t i o n i n sample s i z e s o f 60 and 120 and were o n l y s l i g h t l y more b i a s e d t h a n t h e MVT e s t i m a t e s i n t h e 5 and 10 p e r c e n t c o n t a m i n a t i o n c o n d i t i o n s . The REG2 p r o c e d u r e d i d n o t p r o v i d e s i g n i f i c a n t l y b e t t e r e s t i m a t e s t h a n t h e R E G l p r o c e d u r e i n any o f t h e c o n d i t i o n s e x a m i n e d . I m p l i c a t i o n s f o r P r a c t i c e The r e s u l t s o f t h e p r e s e n t s t u d y , and t h a t o f p r e v i o u s r e s e a r c h , d e m o n s t r a t e t h a t t h e r e c u r r e n t l y e x i s t s no s i n g l e r o b u s t e s t i m a t e o f a c o r r e l a t i o n m a t r i x t h a t o u t p e r f o r m s o t h e r r o b u s t c o r r e l a t i o n e s t i m a t e s a c r o s s c o n d i t i o n s o f commonly e n c o u n t e r e d f o r m s o f o u t l i e r c o n t a m i n a t i o n . I n p r a c t i c e , t h e r e f o r e , t h e c h o i c e o f r o b u s t e s t i m a t o r o f c o r r e l a t i o n s w i l l d epend p r i m a r i l y on t h e amount o f p r i o r i n f o r m a t i o n a v a i l a b l e a b o u t t h e f o r m o f o u t l i e r c o n t a m i n a t i o n p r e s e n t i n t h e s a m p l e . A r e s e a r c h e r , f o r e x a m p l e , may become aware o f c o n d i t i o n s o r p r o c e s s e s w h i c h have r e s u l t e d i n t h e g e n e r a t i o n o f e r r o r s by human o b s e r v e r s o r l a b o r a t o r y i n s t r u m e n t s . I n s i t u a t i o n s where e r r o r s a r e known t o be s y m m e t r i c a l l y d i s t r i b u t e d , m u l t i v a r i a t e M - e s t i m a t o r s o f t h e f o r m p r o p o s e d by Maronna (1976) have b e e n shown t o be p r e f e r a b l e t o t h o s e o b t a i n e d f r o m t h e MVT p r o c e d u r e ( D e v l i n e t a l . , 1 9 8 1 ) . I f a s m a l l o r m o d e r a t e amount o f a s y m m e t r i c c o n t a m i n a t i o n i s known t o be p r e s e n t i n sample d a t a , t h e r e s u l t s o f D e v l i n e t a l . (1981) and o f the p r e s e n t study i n d i c a t e t h a t the MVT procedure would p r o v i d e r e l a t i v e l y unbiased e s t i m a t e s . In o t h e r s i t u a t i o n s , i f a l a r g e amount of asymmetric contamination i s known to e x i s t i n sample data, the r e s u l t s of the p r e s e n t study i n d i c a t e t h a t e s t i m a t e s o b t a i n e d by the c h a r a c t e r i s t i c f u n c t i o n based r e g r e s s i o n method would be l e s s b i a s e d than those o b t a i n e d from the MVT procedure. For most ki n d s of data o b t a i n e d i n p r a c t i c e , however, a r e s e a r c h e r o f t e n l a c k s i n f o r m a t i o n r e g a r d i n g the presence or absence of o u t l y i n g o b s e r v a t i o n s . In these s i t u a t i o n s the use of g r a p h i c a l methods such as sample histograms, s c a t t e r p l o t s and oth e r e x p l o r a t o r y data a n a l y s i s techniques (Tukey, 19 77) p r o v i d e an important step p r i o r to the a n a l y s i s o f data t h a t w i l l a i d i n the i d e n t i f i c a t i o n o f e r r o r s , or i n d i c a t e the form of o u t l i e r contamination. In a d d i t i o n to g r a p h i c a l techniques, the MVT procedure p r o v i d e s a method f o r d e t e c t i n g the presence of o u t l y i n g o b s e r v a t i o n s . Using robust i n i t i a l e s t i m a t e s of the mean v e c t o r and corresponding c o v a r i a n c e matrix, c o r r e l a t i o n s o b t a i n e d from the MVT procedure may be compared to the usua l ^ p r o d u c t moment c o r r e l a t i o n s . I f the c o r r e l a t i o n s o b t a i n e d by the two methods are observed t o d i f f e r s i g n i f i c a n t l y , the 'presericecdf o u t l i e r s 66 w o u l d be i n d i c a t e d . A l t h o u g h t h e MVT p r o c e d u r e may i n d i c a t e t h e p r e s e n c e o f o u t l i e r s , a l i m i t a t i o n o f t h i s method i s t h a t i t p r o v i d e s l i t t l e i n f o r m a t i o n a b o u t t h e symmetry o r asymmetry o f t h e sample d i s t r i b u t i o n . The d e t e c t i o n o f o u t l i e r s may a l s o be a c c o m p l i s h e d by t h e a p p l i c a t i o n o f t h e c h a r a c t e r i s t i c f u n c t i o n b a s e d r e g r e s s i o n p r o c e d u r e . A m a j o r a d v a n t a g e o f t h i s method A 2 i n v o l v e s t h e use o f t h e f u n c t i o n a (t) t o d e t e c t t h e p r e s e n c e o f asymmetry i n t h e sample d i s t r i b u t i o n . The A 2 r o u t i n e i n s p e c t i o n o f a ( t ) o v e r a r a n g e o f v a l u e s o f t , p r i o r t o t h e a n a l y s i s o f d a t a may i n d i c a t e , i n a d d i t i o n t o t h e p r e s e n c e o r a b s e n c e o f asymmetry, d e p a r t u r e s f r o m n o r m a l i t y (Chambers & H e a t h c o t e , 1 9 8 1 ) . T h i s may s u g g e s t a l t e r n a t i v e a n a l y s i s s t r a t e g i e s , o r i n d i c a t e t h e p r e s e n c e o f p r e v i o u s l y unknown s u b g r o u p s 2 i n t h e p o p u l a t i o n . When i n s p e c t i o n o f 3 ( t ) i n d i c a t e s t h a t t h e sample d i s t r i b u t i o n i s s y m m e t r i c and c o n t a i n s a s m a l l o r m o d e r a t e amount o f c o n t a m i n a t i o n , c o r r e l a t i o n e s t i m a t e s o b t a i n e d f r o m t h e MVT p r o c e d u r e o r f r o m m u l t i v a r i a t e M - e s t i m a t e s may be p r e f e r r e d . I f a l a r g e amount o f o u t l i e r c o n t a m i n a t i o n i s i n d i c a t e d o r s u s p e c t e d t o be p r e s e n t i n sample d a t a , t h e c h a r a c t e r i s t i c f u n c t i o n b a s e d r e g r e s s i o n p r o c e d u r e s h o u l d be a p p l i e d . Of c u r r e n t l y a v a i l a b l e methods f o r o b t a i n i n g r o b u s t e s t i m a t e s o f c o r r e l a t i o n m a t r i c e s , t h e MVT and r e g r e s s i o n p r o c e d u r e s , i n a d d i t i o n t o m u l t i v a r i a t e M - e s t i m a t e s , p r o v i d e r o b u s t methods f o r most t y p e s o f d a t a e n c o u n t e r e d i n p r a c t i c e . By r e d u c i n g t h e :.' f r e q u e n c y o f m i s l e a d i n g r e s u l t s and making e x p e r i m e n t a l f i n d i n g s e a s i e r t o r e p l i c a t e , t h e r o u t i n e use o f t h e s e p r o c e d u r e s i n c o m b i n a t i o n w i t h s t a n d a r d methods o f a n a l y s i s w i l l s e r v e t o i m p r o v e t h e q u a l i t y o f r e s e a r c h i n p s y c h o l o g y . R e f e r e n c e s Agee, W. S., & T u r n e r , R.H. A p p l i c a t i o n o f r o b u s t r e g r e s s i o n t o t r a j e c t o r y d a t a r e d u c t i o n . I n R. L. L a u n e r & G. N. W i l k e n s o n (Eds.), R o b u s t n e s s i n s t a t i s t i c s . New Y o r k : A c a d e m i c P r e s s , 1979. Andrews, D.F. A r o b u s t method f o r m u l t i p l e l i n e a r r e g r e s s i o n . T e c h n o m e t r i c s , 1974, 16, 523-531. A n d r e w s , D.F., B i c k e l , P.J., H a m p e l , F. R., H u b e r , P.J., R o g e r s , W.H. & Tukey, J . W. R o b u s t e s t i m a t e s o f l o c a t i o n . P r i n c e t o n : P r i n c e t o n U n i v e r s i t y P r e s s , 1972. B e b b i n g t o n , A.C. A method o f b i v a r i a t e t r i m m i n g f o r r o b u s t e s t i m a t i o n o f t h e c o r r e l a t i o n c o e f f i c i e n t . A p p l i e d S t a t i s t i c s , 1978, 27(3), 221-226. Chambers, R.L. & H e a t h c o t e , CR. On t h e e s t i m a t i o n o f s l o p e and t h e i d e n t i f i c a t i o n o f o u t l i e r s i n l i n e a r r e g r e s s i o n . B i o m e t r i k a , 1981, 68(1), 21-33. C o l l i n s , J.R. R o b u s t e s t i m a t i o n o f a l o c a t i o n p a r a m e t e r i n t h e p r e s e n c e o f asymmetry. The A n n a l s o f S t a t i s t i c s , 1976, 4 ( 1 ) , 6 8 - 8 5 . D e v l i n , S.J., G n a n d e s i k a n , R. & K e t t e n r i n g , J.R. R o b u s t e s t i m a t i o n and o u t l i e r d e t e c t i o n w i t h c o r r e l a t i o n c o e f f i c i e n t s . B i o m e t r i k a , 1975, 62(3), 531-545. D e v l i n , S.J., G n a n a d e s i k a n , R. & K e t t e n r i n g , J.R. R o b u s t e s t i m a t i o n o f d i s p e r s i o n m a t r i c e s and p r i n c i p a l c omponents. J o u r n a l o f t h e A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n , 1981, 76(374), 354-362. 69 D u n c a n , G.T. & L a y a r d , M.W.J. A M o n t e C a r l o s t u d y o f a s y m p t o t i c a l l y r o b u s t t e s t s f o r c o r r e l a t i o n c o e f f i c i e n t s . B i o m e t r i k a , 1973, 60(3), 551-558. G n a n a d e s i k a n , R. & K e t t e n r i n g , J.R. R o b u s t e s t i m a t e s , r e s i d u a l s , and o u t l i e r d e t e c t i o n w i t h m u l t i r e s p o n s e d a t a . B i o m e t r i c s , 1972, 28, 81-124. G o r s u c h , R.L. F a c t o r A n a l y s i s , P h i l a d e l p h i a : S a u n d e r s , 1974. Hampel F. R. The i n f l u e n c e c u r v e and i t s r o l e i n r o b u s t e s t i m a t i o n . J o u r n a l o f t h e A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n , 1974, 69(346), 383-393. H i n k l e y , D.V. I m p r o v i n g t h e j a c k k n i f e w i t h s p e c i a l r e f e r e n c e t o c o r r e l a t i o n e s t i m a t i o n . B i o m e t r i k a , 1978, 65(1), 13-21. Hogg, R.V. An i n t r o d u c t i o n t o r o b u s t e s t i m a t i o n . I n R.L. L a u n e r & G.N. W i l k e n s o n (Eds.), R o b u s t n e s s i n s t a t i s t i c s . New Y o r k : A c a d e m i c P r e s s , 1979. Huber, P.J. R o b u s t e s t i m a t i o n o f a l o c a t i o n p a r a m e t e r . A n n a l s o f M a t h e m a t i c a l S t a t i s t i c s , 1964, 35, 73-101. Huber, P.J. R o b u s t r e g r e s s i o n : A s y m p t o t i c s , c o n j e c t u r e s and Monte C a r l o . A n n a l s o f S t a t i s t i c s , 1973, _1(5), 799-821. H u b e r , P.J. R o b u s t C o v a r i a n c e s , I n S.S. G u p t a & D.S. M o o r e (Eds.), S t a t i s t i c a l d e c i s i o n t h e o r y and r e l a t e d t o p i c s I I . New Y o r k : A c a d e m i c P r e s s , 1977a. Huber, P.J. R o b u s t s t a t i s t i c a l p r o c e d u r e s . P h i l a d e l p h i a : S o c i e t y f o r I n d u s t r i a l and A p p l i e d M a t h e m a t i c s , 1977b. J a e c k e l , L.A. R o b u s t e s t i m a t e s o f l o c a t i o n : Symmetry and a s y m m e t r i c c o n t a m i n a t i o n . A n n a l s o f M a t h e m a t i c a l S t a t i s t i c s , 1971, 42 ( 3 ) , 1020-1034. J o h n s o n , D.E., M c G u i r e , S.A. and M i l l i k e n , G.A. E s t i m a t i n g a i n t h e p r e s e n c e o f o u t l i e r s . T e c h n o m e t r i c s , 1978, 20 ( 4 ) , 4 4 1 - 4 5 5 . K e n d a l l , M.G. & S t u a r t , A. The a d v a n c e d t h e o r y o f s t a t i s t i c s ( V o l . 1, 4 t h e d . ) . New Y o r k : M a c M i l l a n , 1977. Knuth, D.E. The a r t o f c o m p u t e r p r o g r a m m i n g ( V o l . 2 ) :  S e m i n u m e r i c a l a l g o r i t h m s . R e a d i n g , Mass.: A d d i s o n -W e sley, 1968. K o w a l s k i , C.J. On t h e e f f e c t s o f n o n - n o r m a l i t y on t h e d i s t r i b u t i o n o f t h e s a m p l e p r o d u c t moment c o r r e l a t i o n c o e f f i c i e n t . A p p l i e d S t a t i s t i c s , 1972, 21.(1), 1-12. Maronna, R.A. R o b u s t M - e s t i m a t o r s o f m u l t i v a r i a t e l o c a t i o n and s c a t t e r . The A n n a l s o f S t a t i s t i c s , 1976, 4 ( 1 ) , 51-67. M o s t e l l e r , F. & Tukey, J.W. D a t a a n a l y s i s and r e g r e s s i o n . R e a d i n g , Mass.: A d d i s o n - W e s l e y , 1977. P e a r s o n , E.S. Some n o t e s on s a m p l i n g t e s t s w i t h two v a r i a b l e s . B i o m e t r i k a , 1929, _2_1, 337-360. Ramsay, J.P. A c o m p a r a t i v e s t u d y o f s e v e r a l r o b u s t e s t i m a t e s o f s l o p e , i n t e r c e p t , and s c a l e i n l i n e a r r e g r e s s i o n . J o u r n a l o f t h e A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n , 1 9 7 7 , 7 2 ( 3 5 9 ) , 608-615. T i t t e r i n g t o n , D.M. E s t i m a t i o n o f c o r r e l a t i o n c o e f f i c i e n t s b y e l l i p s o i d a l t r i m m i n g . A p p l i e d S t a t i s t i c s , 1978, 27(3), 71 227-234. Tukey, J.W. A s u r v e y o f s a m p l i n g f r o m c o n t a m i n a t e d d i s t r i b u t i o n s . I n I. O l k i n (Ed.), C o n t r i b u t i o n s t o p r o b a b i l i t y a n d s t a t i s t i c s . S t a n f o r d : S t a n f o r d U n i v e r s i t y P r e s s , 1960. Tukey, J.W. E x p l o r a t o r y d a t a a n a l y s i s . R e a d i n g , Mass.: A d d i s o n - W e s l e y , 1977. Tukey, J.W. R o b u s t t e c h n i q u e s f o r t h e u s e r . I n R.L. L a u n e r & G.N. W i l k e n s o n (Eds.), R o b u s t n e s s i n s t a t i s t i c s . New Y o r k : A c a d e m i c P r e s , 1979. W i n e r , B.J. S t a t i s t i c a l p r i n c i p a l s i n e x p e r i m e n t a l d e s i g n (2nd ed.). New Y o r k : M c G r a w - H i l l , 1971. 

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