Analysis of Atlantic Cod Data (with subsampling)

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
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