Negative binomial regression is for modeling count variables, usually for over-dispersed count outcome variables.

**
Please note:** The purpose of this page is to show how to use various data
analysis commands. It does not cover all aspects of the research process which
researchers are expected to do. In particular, it does not cover data
cleaning and checking, verification of assumptions, model diagnostics or
potential follow-up analyses.

This page was updated using SAS 9.2.

## 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 the type of program in which the student is enrolled and a standardized test in math.

Example 2. A health-related researcher is studying the number of hospital visits in past 12 months by senior citizens in a community based on the characteristics of the individuals and the types of health plans under which each one is covered.

## Description of the data

Let’s pursue Example 1 from above.

We have attendance data on 314 high school juniors from two urban high
schools in the file https://stats.idre.ucla.edu/wp-content/uploads/2016/02/nb_data.sas7bdat. The
response variable of interest is days absent, **daysabs**. The variable **
math** gives the standardized math score for each student. The variable **
prog** is a three-level nominal variable indicating the type of instructional
program in which the student is enrolled.

Let’s look at the data. It is always a good idea to start with descriptive statistics and plots.

proc means data = nb_data; var daysabs math; run;

The MEANS Procedure Variable Label N Mean Std Dev Minimum Maximum ----------------------------------------------------------------------------------------------------- DAYSABS number days absent 314 5.9554140 7.0369576 0 35.0000000 MATH ctbs math pct rank 314 48.2675159 25.3623913 1.0000000 99.0000000 -----------------------------------------------------------------------------------------------------

proc univariate data = nb_data noprint; histogram daysabs / midpoints = 0 to 50 by 1 vscale = count ; run;

Each variable has 314 valid observations and their distributions seem quite reasonable. The mean of our outcome variable is much lower than its variance.

Let’s continue with our description of the variables in this dataset. The table below shows the average numbers of days absent by program type and seems to suggest that program type is a good candidate for predicting the number of days absent, our outcome variable, because the mean value of the outcome appears to vary by
**prog**. The variances within each level of **prog** are higher than the
means within each level. These are the conditional means and variances. These
differences suggest that over-dispersion is present and that a Negative Binomial
model would be appropriate.

proc sort data = nb_data; by prog; run; proc means mean var n data = nb_data; by prog; var daysabs; run;PROG=1 The MEANS Procedure Analysis Variable : DAYSABS number days absent Mean Variance N ----------------------------------- 10.6500000 67.2589744 40 ----------------------------------- PROG=2 Analysis Variable : DAYSABS number days absent Mean Variance N ----------------------------------- 6.9341317 55.4474425 167 ----------------------------------- PROG=3 Analysis Variable : DAYSABS number days absent Mean Variance N ----------------------------------- 2.6728972 13.9391642 107 -----------------------------------

## Analysis methods you might consider

Below is a list of some analysis methods you may have encountered. Some of the methods listed are quite reasonable, while others have either fallen out of favor or have limitations.

- Negative binomial regression – Negative binomial regression can be used for over-dispersed count data, that is when the conditional variance exceeds the conditional mean. It can be considered as a generalization of Poisson regression since it has the same mean structure as Poisson regression and it has an extra parameter to model the over-dispersion. If the conditional distribution of the outcome variable is over-dispersed, the confidence intervals for the Negative binomial regression are likely to be narrower as compared to those from a Poisson regression model.
- Poisson regression – Poisson regression is often used for modeling count data. Poisson regression has a number of extensions useful for count models.
- Zero-inflated regression model – Zero-inflated models attempt to account for excess zeros. In other words, two kinds of zeros are thought to exist in the data, "true zeros" and "excess zeros". Zero-inflated models estimate two equations simultaneously, one for the count model and one for the excess zeros.
- OLS regression – Count outcome variables are sometimes log-transformed and analyzed using OLS regression. Many issues arise with this approach, including loss of data due to undefined values generated by taking the log of zero (which is undefined), as well as the lack of capacity to model the dispersion.

## Negative binomial regression analysis

Negative binomial models can be estimated in SAS using **proc** **genmod**. On the **class** statement we list the variable **prog**.
After **prog**, we use two options, which are given in parentheses. The
**param=ref** option changes the coding of **prog** from effect coding,
which is the default, to reference coding. The **ref=first** option
changes the reference group to the first level of **prog**. We have
used two options on the **model** statement. The **type3** option is
used to get the multi-degree-of-freedom test of the categorical variables listed
on the **class** statement, and the **dist = negbin** option is used to
indicate that a negative binomial distribution should be used.

proc genmod data = nb_data; class prog (param=ref ref=first); model daysabs = math prog / type3 dist=negbin; run;

The GENMOD Procedure Model Information Data Set WORK.NB_DATA Distribution Negative Binomial Link Function Log Dependent Variable DAYSABS number days absent Number of Observations Read 314 Number of Observations Used 314 Class Level Information Design Class Value Variables PROG 1 0 0 2 1 0 3 0 1 Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 310 358.5193 1.1565 Scaled Deviance 310 358.5193 1.1565 Pearson Chi-Square 310 339.8771 1.0964 Scaled Pearson X2 310 339.8771 1.0964 Log Likelihood 2151.5227 Full Log Likelihood -865.6289 AIC (smaller is better) 1741.2578 AICC (smaller is better) 1741.4526 BIC (smaller is better) 1760.0048 Algorithm converged. Analysis Of Maximum Likelihood Parameter Estimates Standard Wald 95% Confidence Wald Parameter DF Estimate Error Limits Chi-Square Pr > ChiSq Intercept 1 2.6153 0.1964 2.2304 3.0001 177.40 <.0001 MATH 1 -0.0060 0.0025 -0.0109 -0.0011 5.71 0.0168 PROG 2 1 -0.4408 0.1826 -0.7986 -0.0829 5.83 0.0158 PROG 3 1 -1.2787 0.2020 -1.6745 -0.8828 40.08 <.0001 Dispersion 1 0.9683 0.0995 0.7916 1.1844 NOTE: The negative binomial dispersion parameter was estimated by maximum likelihood. LR Statistics For Type 3 Analysis Chi- Source DF Square Pr > ChiSq MATH 1 5.61 0.0179 PROG 2 45.05 <.0001

this test. The non-significant p-value suggests that the negative binomial model is a good fit for the data.

data test; pval = 1 - probchi(339.8771, 310); run; proc print data = test; run;Obs pval 1 0.11703

**math**has a coefficient of -0.006, which is statistically significant. This means that for each one-unit increase in

**math**, the expected log count of the days absent decreases by .0006. The indicator for

**prog=2**is the expected difference in log count between group 2 and the reference group (

**prog**=1). The expected log count for level 2 of

**prog**is 0.44 lower than the expected log count for level 1. The indicator variable

**prog=3**is the expected difference in log count between group 3 and the reference group. The expected log count for level 3 of

**prog**is 1.28 lower than the expected log count for level 1. To determine if

**prog**itself, overall, is statistically significant, we can look at the LR Statistics for Type 3 Analysis table that includes the two degrees-of-freedom test of this variable. The two degree-of-freedom chi-square test indicates that

**prog**is a statistically significant predictor of

**daysabs**. The chi-square value for this test is 45.05 with a p-value of .0001. This indicates that the variable

**prog**is a statistically significant predictor of

**daysabs**.

We can also see the results as incident rate ratios by using **estimate** statements with the **exp** option.

proc genmod data = nb_data; class prog (param=ref ref=first); model daysabs = math prog / type3 dist=negbin; estimate 'prog 2' prog 1 0 / exp; estimate 'prog 3' prog 0 1 / exp; estimate 'math' math 1 / exp; run;< - some output omitted - >Contrast Estimate Results Mean Mean L'Beta Standard L'Beta Chi- Label Estimate Confidence Limits Estimate Error Alpha Confidence Limits Square prog 2 0.6435 0.4500 0.9204 -0.4408 0.1826 0.05 -0.7986 -0.0829 5.83 Exp(prog 2) 0.6435 0.1175 0.05 0.4500 0.9204 prog 3 0.2784 0.1874 0.4136 -1.2787 0.2020 0.05 -1.6745 -0.8828 40.08 Exp(prog 3) 0.2784 0.0562 0.05 0.1874 0.4136 math 0.9940 0.9892 0.9989 -0.0060 0.0025 0.05 -0.0109 -0.0011 5.71 Exp(math) 0.9940 0.0025 0.05 0.9892 0.9989

The output above indicates that the incident rate for **prog=2** is 0.64
times the incident rate for the reference group (**prog=1**). Likewise, the
incident rate for **prog=3** is 0.28 times the incident rate for the
reference group holding the other variables constant. The percent change in the
incident rate of **daysabs** is a 1% decrease (1 - .99) for every unit
increase in **math**.

The form of the model equation for negative binomial regression is the same as that for Poisson regression. The log of the outcome is predicted with a linear combination of the predictors:

log(daysabs) = Intercept + b

_{1}(prog=2) + b_{2}(prog=3) + b_{3}math.

This implies:

daysabs = exp(Intercept + b

_{1}(prog=2) + b_{2}(prog=3)+ b_{3}math) = exp(Intercept) * exp(b_{1}(prog=2)) * exp(b_{2}(prog=3)) * exp(b_{3}math)

The coefficients have an *additive* effect in the log(y) scale and the
IRR have a *multiplicative* effect in the y scale. The dispersion
parameter in negative binomial regression does not effect the expected counts,
but it does effect the estimated variance of the expected counts.

For additional information on the various metrics in which the results can be
presented, and the interpretation of such, please see *Regression Models for
Categorical Dependent Variables Using Stata, Second Edition* by J. Scott Long
and Jeremy Freese (2006).

Below we use **estimate** statements to calculate the predicted number of events at each level of
**prog**, holding all other variables (in this example, **math**) in the
model at their means.

proc genmod data = nb_data; class prog (param=ref ref=first); model daysabs = math prog / type3 dist=negbin; estimate 'prog 1' intercept 1 prog 0 0 math 48.2675 / exp; estimate 'prog 2' intercept 1 prog 1 0 math 48.2675 / exp; estimate 'prog 3' intercept 1 prog 0 1 math 48.2675 / exp; run;< - some output omitted - >Contrast Estimate Results Mean Mean L'Beta Standard L'Beta Chi- Label Estimate Confidence Limits Estimate Error Alpha Confidence Limits Square prog 1 10.2369 7.4291 14.1058 2.3260 0.1636 0.05 2.0054 2.6466 202.22 Exp(prog 1) 10.2369 1.6744 0.05 7.4291 14.1058 prog 2 6.5879 5.5916 7.7618 1.8852 0.0837 0.05 1.7213 2.0492 507.76 Exp(prog 2) 6.5879 0.5512 0.05 5.5916 7.7618 prog 3 2.8501 2.2720 3.5753 1.0473 0.1157 0.05 0.8207 1.2740 82.00 Exp(prog 3) 2.8501 0.3296 0.05 2.2720 3.5753

In the output above, we see that the predicted number of
events for level 1 of **prog** is about 10.24, holding **math** at its
mean. The predicted number of events for level 2 of **prog** is lower at
6.59, and the predicted number of events for level 3 of **prog** is about
2.85. Note that the predicted count of level 2 of **prog** is
(6.5879/10.2369) = 0.64 times the predicted count for level 1 of **prog**.
This matches what we saw in the after in the incident rate ratio output table.

We can similarly obtain the predicted number of events for values of **math**
while holding **prog** constant.

proc genmod data = nb_data; class prog (param=ref ref=first); model daysabs = math prog / type3 dist=negbin; estimate 'math 20' intercept 1 prog 0 0 math 20 / exp; estimate 'math 40' intercept 1 prog 0 0 math 40 / exp; run;Contrast Estimate Results Mean Mean L'Beta Standard L'Beta Chi- Label Estimate Confidence Limits Estimate Error Alpha Confidence Limits Square math 20 12.1267 8.6305 17.0391 2.4954 0.1735 0.05 2.1553 2.8355 206.80 Exp(math 20) 12.1267 2.1043 0.05 8.6305 17.0391 math 40 10.7569 7.8092 14.8172 2.3755 0.1634 0.05 2.0553 2.6958 211.38 Exp(math 40) 10.7569 1.7576 0.05 7.8092 14.8172

The table above shows that when **prog** held at its reference level and
**math** at 20, the predicted count (or average number of days absent) is about
12.13; when **prog** held at its reference level and **math** at 40, the
predicted count is about 10.76. If we compare the predicted counts at these two
levels of **math**, we can see that the ratio is (10.7569/12.1267) = 0.887. This
matches the IRR of 0.994 for a 20 unit change: 0.994^20 = 0.887.

You can
graph the predicted number of events using the commands below. **
Proc genmod** must be run with the **output** statement to obtain the
predicted values in a dataset we called **pred1**. We then sorted our
data by the predicted values and created a graph with **proc sgplot**.

The graph indicates that the most days absent are predicted for those in program 1. The lowest number of predicted days absent is for those students in program 3.

proc genmod data = nb_data; class prog (param=ref ref=first); model daysabs = math prog / type3 dist=negbin; output out = nb_pred predicted = pred1; run; proc sort data = nb_pred; by pred1; run; proc sgplot data = nb_pred; series x=math y=pred1 / group = prog; run;

## Things to consider

- It is not recommended that negative binomial models be applied to small samples.
- Negative binomial models assume that only one process generates the data. If more than one process generates the data, then it is possible to have more 0s than expected by the negative binomial model; in this case, a zero-inflated model (either zero-inflated Poisson or zero-inflated negative binomial) may be more appropriate.
- If the data generating process does not allow for any 0s (such as the
number of days spent in the hospital), then a zero-truncated model may be
more appropriate. Such models can be estimated with
**proc countreg**. - Count data often have an exposure variable, which indicates the number
of times the event could have happened. This variable should be
incorporated into your negative binomial model with the use of the
**offset**option on the**model**statement. - The outcome variable in a negative binomial regression cannot have negative numbers.

## References

- Long, J. S. 1997.
*Regression Models for Categorical and Limited Dependent Variables.*Thousand Oaks, CA: Sage Publications. - Long, J. S. and Freese, J. 2006.
*Regression Models for Categorical Dependent Variables Using Stata, Second Edition*. College Station, TX: Stata Press. - Cameron, A. C. and Trivedi, P. K. 2009.
*Microeconometrics Using Stata*. College Station, TX: Stata Press. - Cameron, A. C. and Trivedi, P. K. 1998.
*Regression Analysis of Count Data*. New York: Cambridge Press. - Cameron, A. C. Advances in Count Data Regression Talk for the Applied Statistics Workshop, March 28, 2009. http://cameron.econ.ucdavis.edu/racd/count.html .
- Dupont, W. D. 2002.
*Statistical Modeling for Biomedical Researchers: A Simple Introduction to the Analysis of Complex Data.*New York: Cambridge Press.

## See also

- Annotated output for negative binomial regression
- SAS Online Manual