SPSS Data Analysis Examples Negative Binomial Regression



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SPSS Data Analysis Examples
Negative Binomial Regression

Note: The examples on this page were done in SPSS 17. If you are using an earlier version of SPSS, you may need to use the genlog command.

Examples of Negative Binomial Regression

Example 1. School administrators study the attendance behavior of high school juniors at two schools. Predictors of the number of days of absence include gender of the student and standardized test scores in math and language arts.

Description of the Data

Let's pursue Example 1 from above. This is the same example that was used in the page on poisson regression.

We have attendance data on 316 high school juniors from two urban high schools in the file poissonreg.dta. The response variable of interest is days absent, daysabs. The variables math and langarts give the standardized test scores for math and language arts respectively. The variable male is a binary indicator of student gender.

Let's look at the data.

GET FILE='D:\work\data\spss\poissonreg.sav'.
DESCRIPTIVES
  VARIABLES=male math langarts daysabs
  /STATISTICS=MEAN STDDEV VAR MIN MAX .

GRAPH /HISTOGRAM=daysabs .

FREQUENCIES VARIABLES=male.

Some Strategies You Might Be Tempted To Try

Before we show how you can analyze this with a negative binomial regression model, let's consider some other methods that you might use.

OLS Regression - You could try to analyze these data using OLS regression. However, count data are highly non-normal and are not well estimated by OLS regression.
Poisson Regression - Ordinary poisson regression will have difficulty with over dispersed data, i.e. variance much larger than the mean.
Zero-inflated Regression Model - This might be a good option if there are more zeros than would be expected by either a poisson or negative binomial model.

SPSS Negative Binomial Regression Analysis

GENLIN
  daysabs WITH male langarts math
  /MODEL male langarts math
  INTERCEPT=YES
  DISTRIBUTION=NEGBIN(MLE)
  LINK=LOG.

Goodness of Fit^b
	Value	df	Value/df
Deviance	356.935	311	1.148
Scaled Deviance	356.935	311
Pearson Chi-Square	337.089	311	1.084
Scaled Pearson Chi-Square	337.089	311
Log Likelihood^a	-880.873
Akaike's Information Criterion (AIC)	1771.746
Finite Sample Corrected AIC (AICC)	1771.940
Bayesian Information Criterion (BIC)	1790.525
Consistent AIC (CAIC)	1795.525
Dependent Variable: days absent Model: (Intercept), male, langarts, math
a. The full log likelihood function is displayed and used in computing information criteria.
b. Information criteria are in small-is-better form.

Tests of Model Effects
Source	Type III
Source	Wald Chi-Square	df	Sig.
(Intercept)	136.380	1	.000
male	9.531	1	.002
langarts	6.608	1	.010
math	.109	1	.741
Dependent Variable: days absent Model: (Intercept), male, langarts, math

Parameter Estimates
Parameter			95% Wald Confidence Interval
Parameter	B	Std. Error	Lower	Upper
(Intercept)	2.716	.2326	2.260	3.172
male	-.431	.1397	-.705	-.157
langarts	-.014	.0056	-.025	-.003
math	-.002	.0048	-.011	.008
(Scale)	1^a
(Negative binomial)	1.288	.1231	1.068	1.554
Dependent Variable: days absent Model: (Intercept), male, langarts, math
a. Fixed at the displayed value.

Parameter Estimates
Parameter	Hypothesis Test
Parameter	Wald Chi-Square	df	Sig.
(Intercept)	136.380	1	.000
male	9.531	1	.002
langarts	6.608	1	.010
math	.109	1	.741
Dependent Variable: days absent Model: (Intercept), male, langarts, math

The output looks very much like the output from an OLS regression. The output begins the goodness of fit including log likelihood, AIC and BIC. These values can be used when comparing models.

Next comes the Tests of Model Effects. This section looks the same as the section of Parameter Estimates. This is because that we have entered all the variables as continuous variables. So each one of them has just one degree of freedom. With models where there are categorical predictor variables, this section will give the over effects of categorical variables and continuous variables as well.

The Parameter Estimates follows. You will find the negative binomial regression coefficients for each of the variables along with standard errors, Chi-Square values, p-values and 95% confidence intervals for the coefficients.

Now, just to be on the safe side, let's rerun the negbin command with the covb = robust option in order to obtain robust standard errors for the negative binomial regression coefficients.

GENLIN
  daysabs WITH male langarts math
 /MODEL male langarts math
  INTERCEPT=YES
  DISTRIBUTION=NEGBIN(MLE)
  LINK=LOG
 /CRITERIA METHOD=FISHER(1) SCALE=1 COVB=ROBUST.

Parameter Estimates
Parameter			95% Wald Confidence Interval
Parameter	B	Std. Error	Lower	Upper
(Intercept)	2.716	.2135	2.298	3.135
male	-.431	.1401	-.706	-.157
langarts	-.014	.0054	-.025	-.004
math	-.002	.0063	-.014	.011
(Scale)	1^a
(Negative binomial)	1.288	.1231	1.068	1.554
Dependent Variable: days absent Model: (Intercept), male, langarts, math
a. Fixed at the displayed value.

Parameter Estimates
Parameter	Hypothesis Test
Parameter	Wald Chi-Square	df	Sig.
(Intercept)	161.818	1	.000
male	9.472	1	.002
langarts	7.124	1	.008
math	.066	1	.798
Dependent Variable: days absent Model: (Intercept), male, langarts, math

Using the covb = robust option has resulted in a fairly large change in the model chi-square, which is now a Wald chi-square, based on log pseudo likelihoods, instead of a likelihood ratio chi-square. The robust standard errors attempt to adjust for heterogeneity in the model. The variable math was not significant without the covb = robust option and is even less so with it.

Since math is not significant in the model with robust standard errors, we will rerun the model dropping that variable.

GENLIN
  daysabs WITH male langarts
 /MODEL male langarts
  INTERCEPT=YES
  DISTRIBUTION=NEGBIN(MLE)
  LINK=LOG
 /CRITERIA METHOD=FISHER(1) SCALE=1 COVB=ROBUST.

Goodness of Fit^b
	Value	df	Value/df
Deviance	2238.318	313	7.151
Scaled Deviance	2238.318	313
Pearson Chi-Square	2752.913	313	8.795
Scaled Pearson Chi-Square	2752.913	313
Log Likelihood^a	-1549.857
Akaike's Information Criterion (AIC)	3105.713
Finite Sample Corrected AIC (AICC)	3105.790
Bayesian Information Criterion (BIC)	3116.981
Consistent AIC (CAIC)	3119.981
Dependent Variable: days absent Model: (Intercept), male, langarts
a. The full log likelihood function is displayed and used in computing information criteria.
b. Information criteria are in small-is-better form.

Omnibus Test^a
Likelihood Ratio Chi-Square	df	Sig.
171.503	2	.000
Dependent Variable: days absent Model: (Intercept), male, langarts
a. Compares the fitted model against the intercept-only model.

Parameter Estimates
Parameter			95% Wald Confidence Interval
Parameter	B	Std. Error	Lower	Upper
(Intercept)	2.647	.1823	2.290	3.004
male	-.409	.1352	-.674	-.144
langarts	-.015	.0034	-.021	-.008
(Scale)	1^a
Dependent Variable: days absent Model: (Intercept), male, langarts
a. Fixed at the displayed value.

Parameter Estimates
Parameter	Hypothesis Test
Parameter	Wald Chi-Square	df	Sig.
(Intercept)	210.718	1	.000
male	9.164	1	.002
langarts	18.263	1	.000
Dependent Variable: days absent Model: (Intercept), male, langarts

Finally, we will use the emmeans option to get the predicted value in days absent for male and female. In order to use the emmeans option, we will have to specify variable male to be a categorical variable. The model specified this way is the same as the one above since male is a binary variable, except the reference group for male is now switched to male = 1. That is why the sign for the parameter coefficients are reversed.

GENLIN
  daysabs
  BY male
  WITH langarts
 /MODEL
  male langarts
  INTERCEPT=YES
  DISTRIBUTION=NEGBIN(MLE)
  LINK=LOG
 /CRITERIA METHOD=FISHER(1) SCALE=1 COVB=ROBUST
 /EMMEANS TABLES=male SCALE=ORIGINAL.

Estimates
male			95% Wald Confidence Interval
male	Mean	Std. Error	Lower	Upper
0	6.82	.690	5.47	8.17
1	4.43	.432	3.59	5.28
Covariates appearing in the model are fixed at the following values: langarts=50.0638

Sample Write-Up of the Analysis

The negative binomial regression model predicting days absent from school from language arts and gender was statistically significant with likelihood ratio chi-square = 171.503, df=2 yielding p-value <.0001. The predictors langarts and male were each statically significant. For these data, the expected log count for a one-unit increase in language arts was -0.015. This translates to a decrease of about 1.5 days absent for a one standard deviation increase in language arts when gender is held constant. Male students had an expected log count -0.41 less than female students which amounts to about 2.27 fewer days absent than females while holding language arts constant.

Cautions, Flies in the Ointment

It is not recommended that negative binomial models be applied to small samples. What constitutes a small sample does not seem to be clearly defined in the literature.