Heming Lake pike

Source:

Macdonald, P.D.M. and T.J. Pitcher (1979). Age-groups from size-frequency data: a versatile and efficient method of analysing distribution mixtures. Journal of the Fisheries Research Board of Canada 36, 987-1001.

Macdonald, P.D.M. (1987). Analysis of length-frequency distributions. In R.C. Summerfelt and G.E. Hall [editors], Age and Growth of Fish, Iowa State University Press, Ames, Iowa. pp 371-384.

The data give the lengths of 523 pike (Esox lucius), sampled in 1965 from Heming Lake, Manitoba, Canada. There are known to be five age-groups in the sample.

Click here to see the original data: lengths to the nearest half-cm, grouped data, given as a mixed sample and as separate samples for each age group determined by scale reading.

Analysis 1:

The data are fitted by five gamma distributions with constant coefficient of variation.

Remarks:

The components are heavily overlapped and cannot be fitted from the mixed data alone without constraining the parameters in some way. Constant coefficient of variation is a biologically reasonable constraint for this population.

 Fitting Gamma components
 
 Proportions and their standard errors
    .09684    .49464    .24084    .11729    .05039
    .01477    .09961    .06781    .05954    .03311
 
 Means and their standard errors
   22.9483   33.3271   40.4566   49.2933   60.3671
     .4424     .8025    2.9075    4.3763    3.4292
 
 Sigmas (CONSTANT COEF. OF VAR. =   .0993) and standard error
    2.2777    3.3079    4.0155    4.8926    5.9917
     .2297
 
 Degrees of freedom = 25 - 1 +   0 -  0 - 10 -   0 =  14
 
 Chi-squared =  11.7257            (P =  .6283)

Analysis 2:

The mixed data have been supplemented by the following subsampling data. Three fish in the size range 33.75 cm to 35.75 cm were aged and all three were found to be from age-group 2; five fish in the size range 35.75 cm to 37.75 cm were aged, three were found to be from age-group 2 and two were from age-group 3; etc.

 INTVL RGT BND  SUM   SUBSAMPLING DATA...     Heming Lake Pike 1965    
   1    19.750    0
   2    21.750    0
   3    23.750    0
   4    25.750    0
   5    27.750    0
   6    29.750    0
   7    31.750    0
   8    33.750    0
   9    35.750    3   0   3   0   0   0
  10    37.750    5   0   3   2   0   0
  11    39.750    0
  12    41.750    0
  13    43.750    7   0   0   7   0   0
  14    45.750    6   0   0   4   2   0
  15    47.750    0
  16    49.750    0
  17    51.750    0
  18    53.750    0
  19    55.750    0
  20    57.750    0
  21    59.750    0
  22    61.750    4   0   0   0   1   3
  23    63.750    3   0   0   0   1   2
  24    65.750    2   0   0   0   1   1
  25              0

Remarks:

With the extra information provided by the subsampling data, there is no problem fitting all the parameters of the mixed distribution, without the need for constraints. Note that normal, lognormal and gamma distributions all fit these data equally well.

 

 Fitting Gamma components    
 
 Proportions and their standard errors
    .09856    .48379    .26991    .11410    .03364
    .01815    .06149    .06738    .03743    .02185
 
 Means and their standard errors
   23.0075   33.2210   40.8063   51.1718   61.8852
     .6122     .4183    1.0436    1.4953    4.7820
 
 Sigmas and their standard errors
    2.2952    3.1821    4.1105    5.4059    5.7198
     .4834     .3473     .9976    1.0565    3.3099
 
 Degrees of freedom = 25 - 1 +  35 -  7 - 14 -  11 =  27
 
 Chi-squared =  17.5757            (P =  .9160)
 * WARNING *  GOODNESS-OF-FIT TEST MAY BE INVALID;  14 EXPECTED COUNTS ARE < 1
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