Scale in Poisson/Neg Bin from Genmod

[Miracle]

Dear Sir or Madam,

Can I please ask which result is more appropriate? I tried both Poisson and Negative Binomial with noscale, pscale and dscale but I am not sure which one is more appropriate. It seems NegBin with scale=pearson is better than Poisson because of lower Log likelihood, AIC, AICC and BIC. Am I on the right track?

Any insight is much appreciated. Thank you very much.

ods select modelfit ParameterEstimates;
proc genmod data=tmp order=formatted;
	title1 "Poisson-noscale";
	where occ>0;
	format raps raps.;
	class raps;
	model Inv=raps / type3 dist=poisson link=log noscale; 
run;

Poisson-noscale

             Criteria For Assessing Goodness Of Fit

Criterion                     DF           Value        Value/DF

Deviance                    1978        727.6564          0.3679
Scaled Deviance             1978        727.6564          0.3679
Pearson Chi-Square          1978       4544.5249          2.2975
Scaled Pearson X2           1978       4544.5249          2.2975
Log Likelihood                         -383.7010
Full Log Likelihood                    -444.8931
AIC (smaller is better)                 899.7862
AICC (smaller is better)                899.8166
BIC (smaller is better)                 927.7480


                     Analysis Of Maximum Likelihood Parameter Estimates

                                    Standard   Wald 95% Confidence         Wald
Parameter           DF   Estimate      Error          Limits         Chi-Square   Pr > ChiSq

Intercept            1    -3.6990     0.1644    -4.0212    -3.3767       506.25       <.0001
raps        Four     1     2.9105     0.4765     1.9766     3.8444        37.31       <.0001
raps        One      1     1.5888     0.2341     1.1299     2.0476        46.06       <.0001
raps        Three    1     3.0466     0.2589     2.5392     3.5541       138.48       <.0001
raps        Two      1     1.5242     0.3061     0.9243     2.1241        24.80       <.0001
raps        Zero     0     0.0000     0.0000     0.0000     0.0000          .          .
Scale                0     1.0000     0.0000     1.0000     1.0000

NOTE: The scale parameter was held fixed.

===============================================================================================================

ods select modelfit ParameterEstimates;
proc genmod data=tmp order=formatted;
	title1 "Poisson:scale=Pearson";
	where occ>0;
	format raps raps.;
	class raps;
	model Inv=raps / type3 dist=poisson link=log scale=p; 
run;

Poisson:scale=Pearson

             Criteria For Assessing Goodness Of Fit

Criterion                     DF           Value        Value/DF

Deviance                    1978        727.6564          0.3679
Scaled Deviance             1978        316.7118          0.1601
Pearson Chi-Square          1978       4544.5249          2.2975
Scaled Pearson X2           1978       1978.0000          1.0000
Log Likelihood                         -167.0055
Full Log Likelihood                    -444.8931
AIC (smaller is better)                 899.7862
AICC (smaller is better)                899.8166
BIC (smaller is better)                 927.7480


                     Analysis Of Maximum Likelihood Parameter Estimates

                                    Standard   Wald 95% Confidence         Wald
Parameter           DF   Estimate      Error          Limits         Chi-Square   Pr > ChiSq

Intercept            1    -3.6990     0.2492    -4.1874    -3.2106       220.34       <.0001
raps        Four     1     2.9105     0.7222     1.4950     4.3260        16.24       <.0001
raps        One      1     1.5888     0.3548     0.8933     2.2842        20.05       <.0001
raps        Three    1     3.0466     0.3924     2.2775     3.8158        60.27       <.0001
raps        Two      1     1.5242     0.4640     0.6149     2.4336        10.79       0.0010
raps        Zero     0     0.0000     0.0000     0.0000     0.0000          .          .
Scale                0     1.5158     0.0000     1.5158     1.5158

NOTE: The scale parameter was estimated by the square root of Pearson's Chi-Square/DOF.

===============================================================================================================

ods select modelfit ParameterEstimates;
proc genmod data=tmp order=formatted;
	title1 "Poisson:scale=Deviance";
	where occ>0;
	format raps raps.;
	class raps;
	model Inv=raps / type3 dist=poisson link=log scale=d; 
run;

Poisson:scale=Deviance

             Criteria For Assessing Goodness Of Fit

Criterion                     DF           Value        Value/DF

Deviance                    1978        727.6564          0.3679
Scaled Deviance             1978       1978.0000          1.0000
Pearson Chi-Square          1978       4544.5249          2.2975
Scaled Pearson X2           1978      12353.4536          6.2454
Log Likelihood                        -1043.0205
Full Log Likelihood                    -444.8931
AIC (smaller is better)                 899.7862
AICC (smaller is better)                899.8166
BIC (smaller is better)                 927.7480


                     Analysis Of Maximum Likelihood Parameter Estimates

                                    Standard   Wald 95% Confidence         Wald
Parameter           DF   Estimate      Error          Limits         Chi-Square   Pr > ChiSq

Intercept            1    -3.6990     0.0997    -3.8944    -3.5035      1376.14       <.0001
raps        Four     1     2.9105     0.2890     2.3441     3.4769       101.43       <.0001
raps        One      1     1.5888     0.1420     1.3105     1.8670       125.20       <.0001
raps        Three    1     3.0466     0.1570     2.7389     3.3544       376.44       <.0001
raps        Two      1     1.5242     0.1857     1.1603     1.8881        67.40       <.0001
raps        Zero     0     0.0000     0.0000     0.0000     0.0000          .          .
Scale                0     0.6065     0.0000     0.6065     0.6065

NOTE: The scale parameter was estimated by the square root of DEVIANCE/DOF.

===============================================================================================================
ods select modelfit ParameterEstimates;
proc genmod data=tmp order=formatted;
	title1 "NegBin:noscale";
	where occ>0;
	format raps raps.;
	class raps;
	model Inv=raps / type3 dist=negbin noscale; 
run;

NegBin:noscale

             Criteria For Assessing Goodness Of Fit

Criterion                     DF           Value        Value/DF

Deviance                    1978        727.6564          0.3679
Scaled Deviance             1978        727.6564          0.3679
Pearson Chi-Square          1978       4544.5249          2.2975
Scaled Pearson X2           1978       4544.5249          2.2975
Log Likelihood                         -383.7010
Full Log Likelihood                    -444.8931
AIC (smaller is better)                 899.7862
AICC (smaller is better)                899.8166
BIC (smaller is better)                 927.7480


                     Analysis Of Maximum Likelihood Parameter Estimates

                                     Standard   Wald 95% Confidence         Wald
Parameter            DF   Estimate      Error          Limits         Chi-Square   Pr > ChiSq

Intercept             1    -3.6990     0.1644    -4.0212    -3.3767       506.25       <.0001
raps         Four     1     2.9105     0.4765     1.9766     3.8444        37.31       <.0001
raps         One      1     1.5888     0.2341     1.1299     2.0476        46.06       <.0001
raps         Three    1     3.0466     0.2589     2.5392     3.5541       138.48       <.0001
raps         Two      1     1.5242     0.3061     0.9243     2.1241        24.80       <.0001
raps         Zero     0     0.0000     0.0000     0.0000     0.0000          .          .
Dispersion            0     0.0000     0.0000      .          .

NOTE: The negative binomial dispersion parameter was held fixed.

===============================================================================================================

ods select modelfit ParameterEstimates;
proc genmod data=tmp order=formatted;
	title1 "NegBin:scale=Pearson";
	where occ>0;
	format raps raps.;
	class raps;
	model Inv=raps / type3 dist=negbin scale=p; 
run;


NegBin:scale=Pearson

             Criteria For Assessing Goodness Of Fit

Criterion                     DF           Value        Value/DF

Deviance                    1978        281.1836          0.1422
Scaled Deviance             1978        216.5476          0.1095
Pearson Chi-Square          1978       2568.4019          1.2985
Scaled Pearson X2           1978       1978.0000          1.0000
Log Likelihood                         -227.8614
Full Log Likelihood                    -357.0665
AIC (smaller is better)                 726.1330
AICC (smaller is better)                726.1755
BIC (smaller is better)                 759.6872


                     Analysis Of Maximum Likelihood Parameter Estimates

                                     Standard   Wald 95% Confidence         Wald
Parameter            DF   Estimate      Error          Limits         Chi-Square   Pr > ChiSq

Intercept             1    -3.6990     0.2119    -4.1142    -3.2837       304.85       <.0001
raps         Four     1     2.9105     1.2786     0.4045     5.4165         5.18       0.0228
raps         One      1     1.5888     0.3609     0.8815     2.2960        19.38       <.0001
raps         Three    1     3.0466     0.6338     1.8045     4.2888        23.11       <.0001
raps         Two      1     1.5242     0.4922     0.5594     2.4890         9.59       0.0020
raps         Zero     0     0.0000     0.0000     0.0000     0.0000          .          .
Dispersion            1    11.2685     3.0312     6.6511    19.0914

NOTE: The covariance matrix was multiplied by a factor of Pearson's Chi-Square/DOF.

===============================================================================================================

ods select modelfit ParameterEstimates;
proc genmod data=tmp order=formatted;
	title1 "NegBin:scale=Deviance";
	where occ>0;
	format raps raps.;
	class raps;
	model Inv=raps / type3 dist=negbin scale=d; 
run;

NegBin:scale=Deviance

             Criteria For Assessing Goodness Of Fit

Criterion                     DF           Value        Value/DF

Deviance                    1978        281.1836          0.1422
Scaled Deviance             1978       1978.0000          1.0000
Pearson Chi-Square          1978       2568.4019          1.2985
Scaled Pearson X2           1978      18067.5485          9.1343
Log Likelihood                        -2081.3431
Full Log Likelihood                    -357.0665
AIC (smaller is better)                 726.1330
AICC (smaller is better)                726.1755
BIC (smaller is better)                 759.6872


                     Analysis Of Maximum Likelihood Parameter Estimates

                                     Standard   Wald 95% Confidence         Wald
Parameter            DF   Estimate      Error          Limits         Chi-Square   Pr > ChiSq

Intercept             1    -3.6990     0.0701    -3.8364    -3.5616      2784.62       <.0001
raps         Four     1     2.9105     0.4230     2.0813     3.7397        47.33       <.0001
raps         One      1     1.5888     0.1194     1.3547     1.8228       177.06       <.0001
raps         Three    1     3.0466     0.2097     2.6356     3.4576       211.09       <.0001
raps         Two      1     1.5242     0.1629     1.2050     1.8434        87.58       <.0001
raps         Zero     0     0.0000     0.0000     0.0000     0.0000          .          .
Dispersion            1    11.2685     1.0030     9.4647    13.4161

NOTE: The covariance matrix was multiplied by a factor of DEVIANCE/DOF.

[Ksharp]

Looks like both mode are not good ,both have sparse data problem. Maybe you should check other model :

Usage Note 56549: Poisson regression using a generalized Poisson distribution for overdispersed data

http://support.sas.com/kb/56/549.html

[Miracle]

Hi @Ksharp,

Thank you for your quick reply.

May I know what do you mean by sparse data? The number of observation in each level of RAPS? 

Thank you.

[Ksharp]

Sorry. It is known as overdispersion.  It is what you are talking about.  Both model 's Deviance/PearsonChi­Square  are not equal to 1 .

Try to use a generalization of the Poisson distribution that allows for more variability. 

[Miracle]

Hi @Ksharp.

Thank you for your reply.

I tried proc fmm from the link you gave. The result from proc fmm indicates it is better due to lower Fit Statistics.

Will this better Fit Statistics suffice to report the result?

Thank you very much.

proc fmm data=tmp;
	where occ>0;
	format raps raps.;
	class raps;
	model Inv=raps / dist=genpoisson;
	output out=fout pred;
run;
                 Model Information

Data Set             WORK.TMP
Response Variable    INV
Type of Model        Homogeneous Regression Mixture
Distribution         Generalized Poisson
Components           1
Link Function        Log
Estimation Method    Maximum Likelihood


          Class Level Information

Class    Levels    Values

raps          5    Four One Three Two Zero


Number of Observations Read        1987
Number of Observations Used        1983


           Optimization Information

Optimization Technique        Dual Quasi-Newton
Parameters in Optimization    6
Mean Function Parameters      5
Scale Parameters              1
Lower Boundaries              1
Upper Boundaries              0
Number of Threads             4


                          Iteration History

                               Objective                         Max
Iteration    Evaluations        Function          Change    Gradient

        0              5    409.36479106       .            98.97307
        1              2    376.19104009     33.17375098    39.94237
        2              3     371.9062368      4.28480328     12.1904
        3              3    370.07577813      1.83045867    11.37286
        4              2    368.76519848      1.31057965    49.32246
        5              2    366.71073524      2.05446324    31.20835
        6              7    359.51367257      7.19706267    34.19023
        7              3    356.74474967      2.76892290    19.68732
        8              3    355.58963948      1.15511019    7.101384
        9              3    354.99227853      0.59736095    3.494614
       10              3    354.95037557      0.04190296    1.079578
       11              3    354.94476031      0.00561526    0.188769
       12              3     354.9446361      0.00012421     0.04672
       13              3    354.94462919      0.00000691    0.006375
       14              3    354.94462905      0.00000014    0.001309
 
         Convergence criterion (GCONV=1E-8) satisfied.


           Fit Statistics

-2 Log Likelihood              709.9
AIC  (smaller is better)       721.9
AICC (smaller is better)       721.9
BIC  (smaller is better)       755.4
Pearson Statistic             2280.3


          Parameter Estimates for 'Generalized Poisson' Model

                                        Standard
Effect             raps     Estimate       Error    z Value    Pr > |z|

Intercept                    -3.4037      0.1972     -17.26      <.0001
raps               Four       2.6827      0.5947       4.51      <.0001
raps               One        0.8117      0.3204       2.53      0.0113
raps               Three      2.5387      0.3460       7.34      <.0001
raps               Two        1.5412      0.3286       4.69      <.0001
raps               Zero            0           .        .         .
Scale Parameter               0.4364     0.08798

[Ksharp]

Yes. I think this mode is better than negitive binomial model, since all these 

AIC (smaller is better)                 
AICC (smaller is better)               
BIC (smaller is better)               

are smaller . 

[Babloo]

I think for Time series also smaller AIC and BIC is better. Could you please tell me why we need to choose the model with  smaller AIC and BIC?