Fitting the negative binomial to frequency data

Obs y obsfreq yexp yobs
1 0 24 -0.1 0.1
2 1 16 0.9 1.1
3 2 16 1.9 2.1
4 3 18 2.9 3.1
5 4 15 3.9 4.1
6 5 9 4.9 5.1
7 6 6 5.9 6.1
8 7 5 6.9 7.1
9 8 3 7.9 8.1
10 9 4 8.9 9.1
11 10 3 9.9 10.1
12 11 0 10.9 11.1
13 12 1 11.9 12.1

Fitting the negative binomial to frequency data

The UNIVARIATE Procedure

Variable: y

 

Freq: obsfreq

Moments
N 120 Sum Weights 120
Mean 3.16666667 Sum Observations 380
Std Deviation 2.78752724 Variance 7.77030812
Skewness 0.91183392 Kurtosis 0.32893349
Uncorrected SS 2128 Corrected SS 924.666667
Coeff Variation 88.0271761 Std Error Mean 0.25446526
Basic Statistical Measures
Location Variability
Mean 3.166667 Std Deviation 2.78753
Median 3.000000 Variance 7.77031
Mode 0.000000 Range 12.00000
    Interquartile Range 4.00000
Tests for Location: Mu0=0
Test Statistic p Value
Student's t t 12.4444 Pr > |t| <.0001
Sign M 48 Pr >= |M| <.0001
Signed Rank S 2328 Pr >= |S| <.0001
Quantiles (Definition 5)
Level Quantile
100% Max 12
99% 10
95% 9
90% 7
75% Q3 5
50% Median 3
25% Q1 1
10% 0
5% 0
1% 0
0% Min 0
Extreme Observations
Lowest Highest
Value Freq Obs Value Freq Obs
0 24 1 7 5 8
1 16 2 8 3 9
2 16 3 9 4 10
3 18 4 10 3 11
4 15 5 12 1 13

Fitting the negative binomial to frequency data

The UNIVARIATE Procedure

Fitting the negative binomial to frequency data; Histogram for y 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 y 0 10 20 30 40 Count Distribution of y

Fitting the negative binomial to frequency data

Obs n ybar var
1 120 3.16667 7.77031

Fitting the negative binomial to frequency data

The GENMOD Procedure

Model Information
Data Set WORK.NEGBIN
Distribution Negative Binomial
Link Function Log
Dependent Variable y
Frequency Weight Variable obsfreq
Number of Observations Read 13
Number of Observations Used 12
Sum of Frequencies Read 120
Sum of Frequencies Used 120
Missing Values 1
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 119 140.6185 1.1817
Scaled Deviance 119 140.6185 1.1817
Pearson Chi-Square 119 104.3325 0.8767
Scaled Pearson X2 119 104.3325 0.8767
Log Likelihood   92.3889  
Full Log Likelihood   -272.1343  
AIC (smaller is better)   548.2685  
AICC (smaller is better)   548.3711  
BIC (smaller is better)   553.8435  
Algorithm converged.
Analysis Of Maximum Likelihood Parameter Estimates
Parameter DF Estimate Standard
Error		 Wald 95% Confidence Limits Wald Chi-Square Pr > ChiSq
Intercept 1 1.1527 0.0858 0.9845 1.3209 180.40 <.0001
Dispersion 1 0.5680 0.1298 0.3629 0.8890    

Note:The negative binomial dispersion parameter was estimated by maximum likelihood.

Fitting the negative binomial to frequency data

Obs n ybar var m k y obsfreq yexp yobs nbprob expfreq
1 120 3.16667 7.77031 3.16667 1.76049 0 24 -0.1 0.1 0.16335 19.6024
2 120 3.16667 7.77031 3.16667 1.76049 1 16 0.9 1.1 0.18483 22.1793
3 120 3.16667 7.77031 3.16667 1.76049 2 16 1.9 2.1 0.16396 19.6747
4 120 3.16667 7.77031 3.16667 1.76049 3 18 2.9 3.1 0.13209 15.8503
5 120 3.16667 7.77031 3.16667 1.76049 4 15 3.9 4.1 0.10103 12.1237
6 120 3.16667 7.77031 3.16667 1.76049 5 9 4.9 5.1 0.07481 8.9770
7 120 3.16667 7.77031 3.16667 1.76049 6 6 5.9 6.1 0.05417 6.5008
8 120 3.16667 7.77031 3.16667 1.76049 7 5 6.9 7.1 0.03860 4.6319
9 120 3.16667 7.77031 3.16667 1.76049 8 3 7.9 8.1 0.02717 3.2599
10 120 3.16667 7.77031 3.16667 1.76049 9 4 8.9 9.1 0.01893 2.2722
11 120 3.16667 7.77031 3.16667 1.76049 10 3 9.9 10.1 0.01309 1.5714
12 120 3.16667 7.77031 3.16667 1.76049 11 0 10.9 11.1 0.00900 1.0797
13 120 3.16667 7.77031 3.16667 1.76049 12 1 11.9 12.1 0.00615 0.7379
Fitting the negative binomial to frequency data; Plot of expfreq by yexp expfreq 0 10 20 30 yexp -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Fitting the negative binomial to frequency data PLOT expfreq obsfreq