The data are lengths of 287 Atlantic cod from 6 age groups (ages 3 to 8). The subsample was obtained by randomly selecting a maximum of 3 fish from each size class and determining their ages, for a total of 55 fish aged.

Because all of the largest and smallest fish had been aged and any that were older than 8 years or younger than 3 years were removed from the sample, we were confident in assuming that there were exactly k = 6 age groups in the sample.

Initial values for the parameters were found by letting the proportions be equal and setting each standard deviation to 2. The initial means were chosen by inspecting the mixture histogram (Plot #001) and the columns of the subsampling data. Plot #002 shows that the initial values cover the data reasonably well, even though the proportions have not been fitted.

The first fit (Plot #003) is an unconstrained mixture of normals. The second fit (Plot #004) is a mixture of gamma components, constrained to have a constant coefficient of variation, with the means of successive age groups lying on a von Bertalanffy growth curve.

MACDONALD & PITCHER MIXTURE ANALYSIS Reference: J. Fish. Res. Board Can. 36:987-1001 Program MIX Copyright (c) 1985-1996 by ICHTHUS DATA SYSTEMS. Release 3.1.3, March 1996. Portions of this program include material Copyright (c) 1987-1996 by Absoft Corp. WORKSPACE (22750) OPEN A MIXTURE DATA FILE Title: Atlantic cod Number of intervals: 22 Do you want to continue reading this file (Y/N) ? y INTERVAL OBSERVED COUNT RIGHT BOUNDARY Atlantic cod 1 1.0000 27.5000 2 3.0000 29.5000 3 2.0000 31.5000 4 5.0000 33.5000 5 6.0000 35.5000 6 9.0000 37.5000 7 11.0000 39.5000 8 16.0000 41.5000 9 29.0000 43.5000 10 22.0000 45.5000 11 29.0000 47.5000 12 28.0000 49.5000 13 35.0000 51.5000 14 30.0000 53.5000 15 29.0000 55.5000 16 15.0000 57.5000 17 8.0000 59.5000 18 5.0000 61.5000 19 1.0000 63.5000 20 .0000 65.5000 21 2.0000 67.5000 22 1.0000 Any errors to correct (Y/N) ? n

OPEN A SUBSAMPLING DATA FILE Title: Atlantic cod Number of intervals: 22 Number of components: 6 Do you want to continue reading this file (Y/N) ? y INTVL RGT BND SUM SUBSAMPLING DATA... Atlantic cod 1 27.500 1 1 0 0 0 0 0 2 29.500 3 3 0 0 0 0 0 3 31.500 2 2 0 0 0 0 0 4 33.500 3 3 0 0 0 0 0 5 35.500 3 2 1 0 0 0 0 6 37.500 3 0 1 2 0 0 0 7 39.500 3 0 1 2 0 0 0 8 41.500 3 0 2 1 0 0 0 9 43.500 3 0 0 3 0 0 0 10 45.500 3 0 0 2 1 0 0 11 47.500 3 0 0 2 1 0 0 12 49.500 3 0 0 1 0 2 0 13 51.500 3 0 0 1 1 1 0 14 53.500 3 0 0 1 0 1 1 15 55.500 3 0 0 1 1 1 0 16 57.500 3 0 0 0 2 1 0 17 59.500 3 0 0 0 1 1 1 18 61.500 3 0 0 1 0 1 1 19 63.500 1 0 0 0 1 0 0 20 65.500 0 21 67.500 2 0 0 0 0 2 0 22 1 0 0 0 0 0 1 READ A FULL SET OF PARAMETER VALUES How many components? NOTE: Must be at least 1, at most 15 6 Enter the 6 proportions: 1 1 1 1 1 1 Enter the 6 means: 30 38 43 50 55 60 Enter the 6 sigmas: 2 2 2 2 2 2 Proportions do not sum to 1. Do you want to re-scale (Y/N) ? y Proportions .16667 .16667 .16667 .16667 .16667 .16667 Means 30.0000 38.0000 43.0000 50.0000 55.0000 60.0000 Sigmas 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 FIT BY QUASI-NEWTON ALGORITHM AND/OR COMPUTE STANDARD ERRORS >> Distribution selected: Normal >> Subsampling data will be used Enter iteration limit [or 0 to get standard errors of current parameters]: 30 Display observed and expected counts (Y/N) ? n Display variance-covariance matrix (Y/N) ? n Constraints on proportions: 0 (NONE), 1 (SPECIFIED PROPORTIONS FIXED). Enter choice: 0 Constraints on means: 0 (NONE), 1 (SPECIFIED MEANS FIXED), 2 (MEANS EQUAL), 3 (EQUALLY SPACED), 4 (GROWTH CURVE). Enter choice: 0 Constraints on sigmas: 0 (NONE), 1 (SPECIFIED SIGMAS FIXED), 2 (FIXED COEF. OF VARIATION), 3 (CONSTANT COEF. OF VARIATION), 4 (SIGMAS EQUAL). Enter choice: 0 Do you want to abort (Y/N) ? n Number of iterations = 18 Fitting Normal components Proportions and their standard errors .05365 .08139 .43178 .20584 .18538 .04196 .01498 .03075 .07179 .06278 .05995 .02731 Means and their standard errors 31.8358 39.7396 46.1947 51.8763 52.5712 54.9786 .9382 1.1127 .8288 1.1280 1.3936 3.5755 Sigmas and their standard errors 2.8551 2.7257 5.0369 4.0950 5.0538 6.6290 .7339 .7406 .4816 .6529 .7548 2.0990 Degrees of freedom = 22 - 1 + 126 - 21 - 17 - 21 = 88 Chi-squared = 50.7649 (P = .9995) * WARNING * GOODNESS-OF-FIT TEST MAY BE INVALID; 85 EXPECTED COUNTS ARE < 1 CHOOSE A DISTRIBUTION Distribution chosen: Gamma FIT BY QUASI-NEWTON ALGORITHM AND/OR COMPUTE STANDARD ERRORS >> Distribution selected: Gamma >> Subsampling data will be used Enter iteration limit [or 0 to get standard errors of current parameters]: 30 Display observed and expected counts (Y/N) ? n Display variance-covariance matrix (Y/N) ? n Constraints on proportions: 0 (NONE), 1 (SPECIFIED PROPORTIONS FIXED). Enter choice: 0 Constraints on means: 0 (NONE), 1 (SPECIFIED MEANS FIXED), 2 (MEANS EQUAL), 3 (EQUALLY SPACED), 4 (GROWTH CURVE). Enter choice: 4 Is Kth mean different (Y/N) ? n Constraints on sigmas: 0 (NONE), 1 (SPECIFIED SIGMAS FIXED), 2 (FIXED COEF. OF VARIATION), 3 (CONSTANT COEF. OF VARIATION), 4 (SIGMAS EQUAL). Enter choice: 3 Is Kth sigma different (Y/N) ? n Do you want to abort (Y/N) ? n Number of iterations = 15 Fitting Gamma components Proportions and their standard errors .05675 .08988 .39432 .20700 .19580 .05625 .01525 .03280 .06919 .06280 .05408 .02897 Means (ON A GROWTH CURVE) and standard errors (Linf = 58.988; t1-t0 = 2.0795; k = .377258) (s.e.: 3.990 .4117 .103497) 32.0693 40.5287 46.3297 50.3076 53.0354 54.9060 .9836 .7564 .6671 Sigmas (CONSTANT COEF. OF VAR. = .0996) and standard error 3.1942 4.0368 4.6146 5.0108 5.2825 5.4688 .2536 Degrees of freedom = 22 - 1 + 126 - 21 - 9 - 18 = 99 Chi-squared = 56.6878 (P = .9998) * WARNING * GOODNESS-OF-FIT TEST MAY BE INVALID; 91 EXPECTED COUNTS ARE < 1