UBC Faculty Research and Publications

A single stochastic model for forecasting nurse supply and for estimating life-cycle activity patterns Kazanjian, Arminée, 1947- May 31, 1991

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Centre for Health Services and Policy Research A SINGLE STOCHASTIC MODEL FOR FORECASTING NURSE SUPPLY AND FOR ESTIMATING LIFE-CYCLE ACTIVITY PATTERNS HHRU91:4R MAY1991 Health Human Resources Unit Research Reports THE UNIVERSITY OF BRITISH COLUMBIA A SINGLE STOCHASTIC MODEL FOR FORECASTING NURSE SUPPLY AND FOR ESTIMATING LIFE-CYCLE ACTIVITY PATTERNS Prepared for Presentation at the 5th Symposium Health and Economics - Human Resources in Health Care Antwerp, November 16-17, 1989 HHRU 91:4R Health Human Resources Unit Centre for Health Services and Policy Research The John F. McCreary Health Sciences Centre The University of British Columbia Vancouver, B.C., Canada V6T 1Z3 A. Kazanjian May 1991 1. PROBABILISTIC MODELLING OF I.ABDUR MARKET BEHAVIOUR 1.1 Rationale Since microeconomic forces that equilibrate primary markets do not readily apply to the Canadian health care system, it is, therefore, important to pursue a rational, long-term planning approach where health personnel, especially nursing personnel, are concerned. In Canada, as most elsewhere, 98 percent of registered nurses are female; the majority are married with dependents. Thus, there is considerable variation among individual nurses in the amount of time spent in the labour force. Our data confirm that changes in the labour market status of individual nurses vary greatly and are made, obviously, based on personal decisions. Yet, given the unique role of the providers of hea~th care and the complexity of the financing process in Canada, a thorough understanding of the interaction dynamics of the individual (micro) and socioeconomic (macro) factors is necessary for a comprehensive analysis of the manpower situation. In the absence of reliable data on each individual nurse's decision making process over time, and on the description of the structural constraints in the health care sector in the Province of British Columbia over the same time period, this age-specific probabilistic model was deemed the most appropriate tool for a number of reasons. First, and foremost, because this model provides a degree of detail unmatched by other statistical models such as regression analysis. Age-specific attrition, reactivation and survival rates provide a wealth of information on nurses' labour market behaviour. A second reason was the model's potential for estimating detailed life-cycle activity patterns, an important yet unexplored area of investigation. The third reason was, we hoped, the model's potential contribution to our understanding of the relationship between the logic of individual action and the socioeconomic context over time. If this was shown to be changing, longitudinal data on transition probabilities would always provide the "true" picture. 1. 2 Data Utilization of data generated mainly for administrative purposes has both advantages and limitations. For our purposes, the expediency of data collection and the completeness of records far outweighs the disadvantages. The professional association is also the regulating body operating under a provincial statute; this is the case in most other Canadian provinces. The association maintains a computerized registration database containing over 60,000 records. The level of reliability of these data is generally very high and in all instances empirically verifiable. In developing the projection model we chose to use historical information on membership status which was easily retrievable. 'While mandatory registration did not exist in British Columbia until this year, membership has been synonymous to licensure. In the past few years the number of unlicensed nurses employed in the field was estimated at less than 4 percent of the 2 membership; this figure for 1989, under the amended status, was verified to be less than 2 percent. The association database contains a five-year membership history record for all nurses who have ever been licensed. Although not used in the analysis of time sequence data, the cumulative five-year information provides us with the sufficiently large number of cases generally required by a probabilistic model. 1.3 Methods A simple, age-specific projection model was developed first in 1982 and further refined in 1986. It comprises two submodels: a Markovian one to monitor yearly movements from one membership state to another (Figure 1) and a linear submodel for the infusion of new members. The advantage of using an age-specific model is that it does not require arbitrary estimates of future incrementation and attrition rates based on a presumed overall age structure of future workforces. This model is designed to apply the appropriate rates, based on the specific age distribution of stock for the projected year. When there is intuitive reason to suspect changing rates of attrition or survival, the computation of new transition matrices from more recent membership data will provide adequate verification. As a baseline model we have considered a discrete-time Markov model which involves the nurse's age and membership status at predetermined equally-spaced points in time, in particular, December 31 of each year. Although we found an appreciable number of studies using stochastic models for estimating health­care requirements, we were surprised at the paucity of similar efforts in the area of health manpower supply. The model assumes that the likelihood of moving to any membership state depends only on the nurse's current age and membership state and not on the prior membership pattern for that individual. An examination of one-step transitions from each year in a five-year period indicated homogeneous age­specific transition behaviour over time. For the purpose of this computation, the many actual registration statuses used by the professional association are aggregated into three: (1) 0 - nonmembers (who were once members); (2) 1 - non-practicing members; and (3) 2 - practicing members. Thus, all possible transition probabilities in this system are conveniently expressed in the form of a "transition matrix", for a given age x as follows: 3 Pao (x) Poi (x) Po2 (x) Pio (x) P11 (x) Pzo (x) Pzi (x) P22 (x) To obtain estimates of the probabilities, we use the membership histories of 58,175 nurses (all useable records of members ever registered with RNABC). These five-year histories along with the current membership status become a six-year record of each nurse's level of professional membership activity. Each individual's history is, however, treated as five one-step transitions based on the assumption made by the first-order model, and not as a string of transitions with two or more annual moves in a sequence. Although one could compute transition probabilities and hence forecasts for individual ages, the transition probability matrices here are derived for age groups. Single-age calculations would generally be considered ideal but in many cases the cell frequencies are too small to yield useful and reliable transition probabilities. Age categories comprise five-year intervals, grouping those under the age of 25, those between 25 and 29 years of age, 30 and 34 years of age ... 55 and 59 years of age; then 60 through 63 years of age, 64 and 65 years of age, and those over the age of 66. A number of factors were considered in selecting these categories - the clear identification of child-bearing age ranges, in order to delineate the influence of this factor, was among the reasons for this grouping. The reason for the unusual groupings beyond 60 years of age was to track more closely the behaviour of a group approaching retirement age. We noted in our earlier work that many of those beyond retirement age continued to maintain practicing membership; presumably some worked during these years. After close examination of the data for each age over 60, three age groups are included comprising 60- to 63-year-olds, 64- to 65-year-olds, and members 66 years of age and older, respectively. The latter broad category was warranted by the small size of this subgroup. The calculation of the probability matrices for nurses aged 30 to 34 is illustrated in Table 1. For example, the probability of remaining active for this age group is 0.940, i.e. of the 20,711 possible two to two transitions in a one-year period, 19,467 actually are observed. To: 0 From: 4 TABLE 1 Illustrative Matrices of Frequency Counts (a) and Probabilities (b) for Membership State; Transitions, Ages 30-34 (a) 1 2 Total 0 (b) 1 2 0 12621 102 380 13103 .963 .008 .029 1 215 2622 665 3502 .061 .749 .190 2 326 918 19467 20711 .016 .044 .940 In summary, the probability of matrices are derived using age and six­year membership data in the following steps: (1) Age-specific frequency distributions of transitions to/from one of three possible states are compiled from the six-year membership records of all nurses who had ever registered with the association. Each of 58,175 cases could contribute up to five transitions to one of the 11 age-specific 3 x 3 matrices. In fact, a total of 272,389 individual membership transitions are tallied. (2) Age-specific probability matrices are then calculated so that the rows of the matrices add up to 1.0. Usually, a single nurse contributes to two, and sometimes three, age­specific matrices. It should be noted that the larger baseline of 58,175 is used to calculate the transition probabilities (instead of the 30,058 useable records of currently registered members) in order to capture as broad of a picture as possible by calculating all probabilities of reactivation, regardless of when the member's registration had expired. A detailed table of transition probabilities is included in Appendix A. 2. APPLICATION OF THE MODEL 2.1 Projection Model We took a stochastic view of the future behaviour of our population, given initial conditions. We could then project the state of our population one or more years into the future. The initial conditions are simple distributions of current membership status (coded 0 through 2) by single ages, comprising the membership state vectors. These are then multiplied by the age-specific transition matrix to complete one iteration (1 year) in the projections. 5 The aging of the population is accounted for simply by shifting the age reference at each iteration. Those aged "n" years at one iteration became n + 1 years old at the next iteration. Table 2 provides an example of this iterative scheme; the new numbers constitute the membership state vector for 31-year-old nurses for 1989. TABLE 2 Example of the Basic Projection Model for 30-Year-Old Nurses: Projection into 1989 from 1988 "Year n" State Matrix of Transition Probabilities "Year n+l" State 323 108 755 x 2.2 New Registrant Submodel .963 .061 .016 .008 .749 .044 .029 .190 .940 330 117 740 Table 2, however, does not take into account the inflow of new registrants during the period. In order to quantify the infusion of new registrants into the projection, historical data were examined for possible trends or patterns. The charted data (Figure 2) indicate that a strong linear trend has dominated from mid-1930s onward, with "peaks" in new registrations occurring sporadically. Interestingly the past five-year period, staring in 1982 and continuing until 1988, the slope of the linear trend has shifted appreciably lower. Thus, in this most recent application of our model, we have opted to disregard the historical pattern and focus on the last four year's experience. The projected number of new registrants for 1989 was calculated using the arithmetic mean for the period 1985-88. This projected figure for subsequent years was based on a moving average of the last four years (Figure 3). In order to combine data from this new registrant submodel with that of the basic projection model, it is also necessary to derive the age distribution of new registrants. This distribution is obtained from the aggregate age distributions of six successive cohorts of initial registrants, spanning the same period as the membership data (1983-1988). The underlying assumption here is that, for the projection period from 1988 to 1993, the age distribution of new registrants would remain the same. However, the model is designed to easily allow specification of new registrant age distributions for each projection period. Estimation of the age of projected new registrants, using historical proportionate age distributions, entailed the specification of how many new members would, for example, be aged 25, 26, or 27, etc., based 6 on the historical information on age of new registrants. A detailed table of new registrant age distribution is included in Appendix B. 2.3 Projections The straightforward procedural stages involved in forecasting future manpower basically are those depicted in Figure 1. Starting from the 1980 actual membership distribution vectors (by single ages) and applying the grouped age-specific transition probabilities to each vector, the Markov-based model provides age-specific projections of numbers in each of the three membership states for the subsequent year. The infusion of new (first-time member) nurses by age into the "practicing" state (coded 2) based upon an average rate of growth of new members over the past six year period completes the projection procedure for one year. 2.4 Results This then constitutes one discrete iteration of the Markov model representing one transition period, or one year's membership increment. The results of one iteration provide a basis for the next iteration quite independently of any sequence of past iterations, a key assumption of the model we have used, and characteristic of the Markovian process we have discussed. It is instruct. to study the probability matrices themselves (Appendix A) and note the ag, :ecificity of the transition probabilities. The data show, for example, ~:,at among currently practicing nurses 25 to 29 years of age, 93 percent wiL remain practicing members next year while another 5 percent will become non-practicing members; 2 percent will not renew their membership. In comparison, among currently practicing nurses from 45 to 49 years of age, 97 percent will renew their membership as practicing members next year, while 2 percent will renew as non-practicing members and less than 1 percent will drop their membership altogether. Thus, the data clearly show that the attrition rate among practicing nurses (the 2-to-O transition probability) varies appreciably among the age groups between a low of 1 percent (for those from 45 to 54 years of age) and a high of 14 percent (for those approaching retirement age). It is perhaps more instructive to note that among the former (i.e., 64- to 65-year-old currently practicing nurses), only 62 percent will remain practicing members the following year. Presumably, an appreciable proportion among this group will remain employed. The detailed transition probabilities for 11 age groups and for all ages - as well as providing in-depth analyses of various types of attrition (e.g., practicing to non-practicing vs. practicing to non-member or, practicing to non-member vs. non-practicing to non-member) - provide extensive information on reactivation rates and patterns by age group. An interesting feature of the probability matrices pertains to the O-to-2 transition statistic, indicating the likelihood of returning the subsequent year to a practicing 7 state once a member has dropped out. That data show that this is inversely proportional to age with, at best, 4 percent returning among those from 18 to 24 years of age, 3 percent among those from 35 to 39 years of age, and expectedly, only 0.1 percent returning at 60 to 63 years of age. The transition matrix also shows that from 30 to 34 years of age, a nurse whose membership is expired is three times more likely to return as a practicing member (P - 0.029, or almost 3% of those with expired memberships will return as practicing members) than as a non-practicing member (probability of 0.008, or less than 1% will return as non-practicing members). The details of the manpower projections over time show a particular age cohort's characteristic behaviour, and include the effect of child-bearing and aging on the supply of nurses as well as the effect of the socioeconomic conditions of the period (e.g. cultural change) which define the parameters of individual behaviour. This is illustrated best by transition probabilities computed from databases from different years. 3. PROFESSIONAL LIFE EXPECTANCY As well as projecting the future lapour force the age-specific transition matrices can be used to estimate professional life expectancy of RNs. For example, one may wish to know how many years a nurse who is presently aged 30 will practice, on average, before retiring. Alternatively, noting that a nurse may cease to practice and then resume practicing at a later date, one may wish to know how many years a 30-year-old nurse will practice, one average, before being deleted from the sy~tem at age 75. Answers to both of these questions are provided (indirectly) ·by the transition probability matrices. 3.1 Computation of Professional Life Expectancies Let eij(x) denote the expected (average) number of years spent in membership state j from now until the age of 75 by a nurse who is presently aged x and who is in membership state i. (The age of 75 is selected because our data indicate many nurses remain in practice beyond the normal retirement of 65, but few beyond the age of 75.) In addition, let Ci(x) denote the expected number of years spent continuously in membership state i from now until the age of 75 by a nurse who is presently aged x and who is in membership state i. Appendix C shows how the associated Ecx> matrix and the Ccx> vector are derived mathematically from the Pcx> matrices, and the basic reasoning follows. A 30-year-old nurse who is currently practicing has a 0.940 probability of remaining in practice until the following year (Table 1). Thus, if we start with 1,000 30-year-old practicing nurses, the number will fall to 940 by the end of one year. The average number of nurses during the year is therefore about (1,000 +940)/2 - 970, so the average amount of time worked per nurse is (1.000 + 0.940)/2 - 0.970 years. Similarly, of these 1,000 nurses, none are non-practicing at the beginning of the year, but 44 will be by the 8 end of the year, so that the average amount of time spent as a non-practicing nurse is (0.0 + 0.044)/2 - 0.022 years per nurse. Consider now the second year. The probability that the nurse will be active at the end of another year, assuming that she is active at the end of the first year, is again 0.940; the probability of being active at the end of both years is (0.940) 2 - 0.884. During the second year we would therefore expect (0.940 + 0.884)/2 - 0.912 years of work, so that over the two years we would expect 0. 970 + 0. 912 - 1. 88 "continuous" years of work. A continuation of this reasoning over a 45-year period would yield the value of C2 (30). However, to obtain the total probability that the nurse is practicing at the end of two years, we must also add the probability that she becomes non­practicing at the end of the first year but resumes practicing status by the end of the second year (0.044 x 0.190 - 0.008), and that she drops out by the end of the first year but re-registers by the end of the second year (0.016 x 0.029 - 0.0005). The overall probability that she will be practicing at the end of the second year, given that she is practicing at the outset, is therefore 0.884 + 0.008 + 0.0005 = 0.893. The expected amount of time spent practicing during the second year is therefore (0.940 + 0.893)/2 = 0.916 years, and the total amount of time spent practicing over the two years is 0.970 + 0.916 = 1.886 years. A continuation of this reasoning to the age of 75 would yield the value of professional life expectancy at age 30 (E22 (30)). Fortunately, the mathematics given in Appendix C provides some shortcuts in performing these calculations, and the computer finishes the task. Table 3 shows the results for E(30) and C2 (30), indicating that a 30-year-old nurse who is currently practicing will spend, on average, a total of 24.3 years as a practicing nurse, 7.0 years as a non-practicing nurse, and 13.7 years unregistered over the 45-year period up to the age of 75, and that she will practice for an average of 17.l continuous years before leaving "practicing" status for the first time. (Detailed, age-specific professional life expectancy estimates are available upon request.) Current Status 0 1 2 TABLE 3 Professional Life Expectancy for 30-Year-Old Nurses Total Number of Years in Status (E (30)) 0 32.6 16.8 13.7 1 3.6 9.4 7.0 2 8.8 18.8 24.3 Continuous Number of Years (C (30)) 17.1 9 4. CONCLUDING REMARKS It is only recently that life-cycle activity patterns are being taken into consideration when estimating future supply. Most often, the increase in the number of female physicians has triggered more serious thinking about life-cycle activity patterns and different levels of productivity. Our Markovian model provides ample evidence for the careful consideration of these patterns, especially for a predominantly female workforce. The application of the model to health human resources planning provides a powerful tool to the decision-makers in this sector. Given the descriptive character of this model, its usefulness extends across planning and policy-making levels and would provide equally useful information to healthcare executives and to state or national level health planners. 4.1 Sensitivity to Socioeconomic Context The forecasting capability developed from this study attempts to address two questions central to human resources planning: (1) how many nurses will there be in this jurisdiction (province, state, country, etc.) in the next five years? and (2) what is the time frame involved in the nurses' life-cycle activity patterns? Our analysis indicates that our model yields accurate answers to both of these questions, at any point in time. But more importantly, the model is highly sensitive to changing demographic (Figure 4) and socioeconomic conditions occuring either gradually over time or abruptly. To illustrate, we examined key statistics from the model, applied to 1980 membership data (Figure 5) and 1988 data (Figure 6). Not only is the comparison of results between the two time periods interesting in itself, but the model's capability to accurately reflect changing labour market conditions is an additional feature unique to the method. It is unlikely that any model could anticipate sudden fluctuations in labour market conditions. The next best alternative is then to have a model that most accurately captures changing conditions. 'While other models often fail because underlying assumptions are no longer held true, our model is almost assumption-free. 'When there is sufficient evidence of changing conditions: social, demographic, economic, etc., repeated applications of the model to current membership data will define the magnitude of these changes and their impact on nurse human resouces (see, for example, Figure 7). Continued application of the model will provide longitudinal data on the relationship between the logic of individual decisions and the structural parameters which define the limits of individual action. REFERENCES Aiken, L.H., Blendon, R.J., Rogers, D.E., et al (1981), "The Shortage of Hospital Nurses: a New Perspective", Am J Nurs, 81:1612. Benham, L. (1971), "The labour market for registered nurse: a three­equation model", Rev Econ Stat, 1:246. Bognanno, M.F. (1970), An Economic Study of the Hours of Labour Offered by the Registered Nurse, Ph.D. thesis, University of Iowa. University Microfilms, Inc., Ann Arbour, Michigan. Evans, R.G. (1984), Strained Mercy: the Economics of Canadian Health Care. Toronto: Butterworths. Feldstein, P.J. (1979), Health Care Economics. New York: John Wiley & Sons, Inc. Giovannetti, P. (1982), "Primary Nursing - the Number One Solution". In: Zilm, G., Hilton, A., Richmond, M., eds Nursing Research: a Base for Practice. Service and Education. Vancouver, B.C.: Proceedings of National Nursing Research Conference, April:l6. Hacon, W.S. (1974), The Health Manpower Situation in Canada. In: Pan­American Conference on Health Manpower Planning, September 10-14, 1973, Ottawa, Canada, PAHO and WHO Scientific Publication, 279:28-34. Health Manpower Research Unit (1988), ROLLCALL 87. A Status Report on Health Personnel in the Province of British Columbia, HMRU Report 88:4, University of British Columbia, March. Health Manpower Research Unit (1989), ROLLCALL UPDATE 88. A Status Report of Selected Health Personnel in the Province of British Columbia, HMRU Report 89:1, University of British Columbia, March. Jenny, J.L. (1982), Issues Affecting Nurses' Hospital Employment of the 80's, Ottawa: Canadian Hospital Association. Kazanjian, A., Wong. G.W. (1982), Nurse Requirements in British Columbia: a Report on the Supply Situation. Report S:ll. Vancouver: Division of Health Services Research and Development, University of British Columbia. Kazanjian, A., Chan, S. (1984), Nurse Requirements in British Columbia: an Analysis of the 1979-82 Trends, Report S:l6. Vancouver: Division of Health Services Research and Development, University of British Columbia. Kazanjian, A., Brothers, K., Wong, G.W. (1986), "Modeling the Supply of Nurse Labor. Life-Cycle Activity Pattern of Registered Nurses in One Canadian Delivery System", Medical Care, 24(12). Kolesar, P. (1970), "A Markovian Model for Hospital Admission Scheduling", Management Science, 16:384. Lalonde, M. (1974), A New Perspective on the Health of Canadians: a Working Document by the Minister of National Health and Welfare. Ottawa: Information Canada. Marshall, A., Goldhammer, H. (1955), "An Application of Markov Processes to the Study of the Epidemiology of Mental Diseases", J Arn Stat Assoc, 50:99. Moussa, M.A., Hathout, H., Kasraoui, R., et al (1982), "Measuring the Change in Contingency Tables Using Markov Models - Application to the Effect of Preceding Conception on the Next One", Methods Inf Med, 21:31. Registered Nurses' Association of British Columbia (1986), RNABC News, 18:32. Trinkl, F. (1974), "A Stochastic Analysis of Programs for the Mentally Retarded", Operations Research, 22:1175. US Department of Health and Human Services, Division of Nursing (1982), Nurse Supply, Distribution and Reguirements. Third Report to the Congress, February 17, 1982. DHHS pub. no HRA 82-7. Washington, DC: U.S. Government Printing Office. US Department of Health and Human Services, Office of Data Analysis and Management (1984), Report to the President and Congress on the Status of Health Personnel in the United States. Volumes 1 & 2. DHHS pub. no. HRS-P­OD-84-4. Washington, DC: US Government Printing Office. Ward, R.A. (1975), The Economics of Health Resources. Reading, MA: Addison-Wesley Co. CJ) -IZ om ()~ ~ () CJ) c -I JJ OJJ om ~z -I z c: -, (,/) CD (f) c: "'O "'O -'< ,, '"tJ cE' -, ~ Q CD '--• CD ~ 0 ..+ 5· :J ~ 0 c. CD -Figure 2 Historical Data on New Registrants Number of New Registrants 250o.-~~~~~~~~~~~~~~~~~~~~~~-. • • I 2000 • •• • •• • • •• 1500 -i • • •• • • • • • • • • •• • • • 1000 -i .. • • • •• • •• •• •• • • 500 -i. • • •• • • • • • •• • •• • •• • •• • • • ••• I 0 1920 1930 1940 1950 1960 1970 1980 1988 Year -Figure 3 New Registrant Infusion Estimation By 4-Year "Moving Average" Figure 4 Nurse Age Distribution, 1980 and 1988 Number of Nurses (Thousands) 9 8 7 6 5 4 3 2 1 0 ~__,.--~~r----~-.--~~--.-~~-.--~~-.-~~-,-~~--.~~~..-----~-.---' 18-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-63 64-65 Age Group Figure 5 Nurse Transition Probabilities December 1980 Probability 1~~~~~~~~~~~~~~~~~~~~~~~~~~-----, 0.9~ 0.8 0.7 0.6 0.5 0.4 0.3 0.2 --~---~ 0.~ [ ; ; j I 0~2 1 ~ ! J 18-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-63 64-65 Age Group r Probability Figure 6 Nurse Transition Probabilities December 1988 1~~~~~~~~~~~~~~~~~~~~~~~~---. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 -1 I I I 1>2 I-_ 0>2 Qt j j T T r t + =:::t J 18-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-63 64-65 Age Group Figure 7 Nurse Working Life Comparison 1980 and 1988 Expected Continuous Years 20r--~~~~~~~~~~~~~~~~~~~~~~~~~~~--. 15 10 5 . . . . . . . . . . . . . . . . . . . . . . . . . . +++++++++++ .. + +++·· .. ·•····· + . + . ++ ++ . + + . + ++. + + .. + + + + ++······ ···-·-··--· .. ++++++++ ... - ~~++ + 0'--~~~__._~~~~_._~~~~--'-~~~~---~~~___, 15 25 35 45 55 65 Present Age of Nurse . 1988 + 1980 APPENDIX A AGE-SPECIFIC TRANSITION PROBABILITY CALCULATIONS TRANSITIONS TRANSITION PROBABILITIES --------------------------- -----------------------Status Status at End of Year Status at End of Year Age at Start Range of Year 0 1 2 All 0 1 2 ----------------------------------- -----------------------0 941 4 36 981 0.959 0.004 0.037 18-24 1 8 97 52 157 0.051 0.618 0.331 2 101 201 5011 5313 0.019 0.038 0.943 0 6594 42 210 6846 0.963 0.006 0.031 25-29 1 98 1187 414 1699 0.058 0.699 0.244 2 286 800 15225 16311 0.018 0.049 0.933 0 12621 102 380 13103 0.963 0.008 0.029 30-34 1 215 2622 665 3502 0 .061 0.749 0.190 2 326 918 19467 20711 0.016 0.044 0.940 0 15340 117 398 15855 0.968 0.007 0.025 35-39 1 183 2822 642 3647 0.050 0. 774 0 .176 2 214 680 18601 19495 0.011 0.035 0.954 0 18609 103 420 19132 0.973 0.005 0.022 40-44 1 172 2128 405 2705 0.064 0.787 0. 1 so 2 178 454 17244 17876 0.010 0.025 0.965 0 16787 65 267 17119 0.981 0.004 0.016 45-49 1 82 1469 236 1787 0.046 0.822 0 .132 2 110 304 12809 13223 0.008 0.023 0.969 0 15235 56 164 15455 0.986 0.004 0.011 50-54 1 62 1090 157 1309 0.047 0.833 0.120 2 66 251 9126 9443 0.007 0.027 0.966 0 14511 24 79 14614 0.993 0.002 0.005 55-59 1 60 886 76 1022 0.059 0.867 0.074 2 73 272 5752 6097 0.012 0.045 0.943 0 10886 14 15 10915 0.997 0.001 0.001 60-63 1 58 600 20 678 0.086 0.885 0.029 2 72 195 1788 2055 0.035 0.095 0.870 0 5704 5 2 5711 0.999 0.001 0.000 64-65 1 30 237 3 270 0. 111 0.878 0.011 2 47 84 214 345 0.136 0.243 0.620 0 24016 7 6 24029 0.999 o.ooo 0.000 66-75 1 45 692 12 749 0.060 0.924 0.016 2 12 38 185 235 0.051 0.162 0.787 0 141244 539 1977 143760 0.982 0.004 0.014 18-75 1 1013 13830 2682 17525 0.058 0.789 0.153 2 1485 4197 105422 111104 0.013 0.038 0.949 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 69 70 71 APPENDIX B RNABC 1983-1988 New Registrant Proportionate Age Distribution 2 5 ======= ================= =============================== 80 203 370 520 647 737 708 560 466 411 372 362 360 315 288 268 253 228 243 181 171 152 125 129 109 112 =========================================== ====================================================== ============================================================= =========================================================== =============================~================= ======================================= ================================== =============================== ============================== ============================== --------------------------======================== ----------------------===================== =================== ==================== =============== ============== ============= ----------=========== ========= ---------98 84 77 50 45 39 39 27 18 = 24 26 25 28 13 = 11 ======== ======= ====== ==== == = 6 5 8 = 3 2 1 2 1 SCALE: MEAN: MEDIAN: VARIANCE: STD.DEV: TOTAL N = 12.0 32.3 3o.e 67. 1 8.2 9009 per division APPENDJX C ln order to calculate the expected time spent by a member in each of the membership states, let us first generalize P(x), the one-year transition matrix at age x, to t1x ,x ), the transition matrix from ·1 2 age x to age x. (i.e., a period of x -x years)*. If we let p(x) denote the vector (of length 3) of l 2 2 l probabilities that an individual is in one of the three membership states, then Then the expected time spent in each of the states, given an initial probability distribution of p(x 0 ) at age x0 , is 75 75 'x pT(t)dt"/x 0 0 T T 75 T p (x )t(x ,t)dt=p (x )/x t(x ,t)dt • p (x )E(x ), 0 0 0 0 0 0 0 75 where E(x) = /x t(x,t)dt. A rec~rsive approximation to E(x) may be obtained as follows: 75 E(x) = /x+l~(x,t)dt + J t\x,t)dt x x+J "/x+J t(x,t)dt + t(x,x+!) J 75 t(x+l ,t)dt x = /~+J t(x,t)dt + P(x)E(x+:) x x+J = ~[t(x,x) + t(x,x+l)] + P(x)E(x+l) =~[I + P(x)] + P(x)E(x+l) •~I + P(x)[~I+E(x+1)] 75 Since £(75) = J t(7s,t)dt • 0, we may readily compute the E matrices from the P matrices by working from 75 £(75) backwards. The following approximation to Ci(x), the expected amount of time spent continuously in state 1 for a member already in state i, is obtained similarly: * An i11111ediate consequence of this definition is that t(x ,x ) • t(x ,x )t(x ,x ). Note also that P(x) • l s l 2 2 3 t(x,x+1), and that t(x,x)•I, the identity matrix. 


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