### Stata Annotated Output Negative Binomial Regression

This page shows an example of negative binomial regression analysis with footnotes explaining the output. The data collected were academic information on 316 students. The response variable is days absent during the school year (daysabs), from which we explore its relationship with math standardized tests score (mathnce), language standardized tests score  (langnce) and gender (female).

As assumed for a negative binomial model our response variable is a count variable, and each subject has the same length of observation time. Had the observation time for subjects varied, the model would need to be adjusted to account for the varying length of observation time per subject. This point is discussed later in the page. Also, the negative binomial model, as compared to other count models (i.e., poisson or zero-inflated models), is assumed the appropriate model. In other words, we assume that the dependent variable is over-dispersed and does not have an excessive number of zeros. The first half of this page interprets the coefficients in terms of negative binomial regression coefficients, and the second half interprets the coefficients in terms of incidence rate ratios.

use http://www.ats.ucla.edu/stat/stata/notes/lahigh, clear

generate female = (gender == 1)

nbreg daysabs mathnce langnce female

Fitting Poisson model:

Iteration 0:   log likelihood = -1547.9709
Iteration 1:   log likelihood = -1547.9709

Fitting constant-only model:

Iteration 0:   log likelihood = -897.78991
Iteration 1:   log likelihood = -891.24455
Iteration 2:   log likelihood = -891.24271
Iteration 3:   log likelihood = -891.24271

Fitting full model:

Iteration 0:   log likelihood = -881.57337
Iteration 1:   log likelihood = -880.87788
Iteration 2:   log likelihood = -880.87312
Iteration 3:   log likelihood = -880.87312

Negative binomial regression                      Number of obs   =        316
LR chi2(3)      =      20.74
Dispersion     = mean                             Prob > chi2     =     0.0001
Log likelihood = -880.87312                       Pseudo R2       =     0.0116

------------------------------------------------------------------------------
daysabs |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
mathnce |   -.001601     .00485    -0.33   0.741    -.0111067    .0079048
langnce |  -.0143475   .0055815    -2.57   0.010    -.0252871    -.003408
female |   .4311844   .1396656     3.09   0.002     .1574448     .704924
_cons |   2.284885   .2098761    10.89   0.000     1.873535    2.696234
-------------+----------------------------------------------------------------
/lnalpha |   .2533877   .0955362                      .0661402    .4406351
-------------+----------------------------------------------------------------
alpha |   1.288383   .1230871                      1.068377    1.553694
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0:  chibar2(01) = 1334.20 Prob>=chibar2 = 0.000

### Iteration Loga

Fitting Poisson model:

Iteration 0:   log likelihood = -1547.9709
Iteration 1:   log likelihood = -1547.9709

Fitting constant-only model:

Iteration 0:   log likelihood = -897.78991
Iteration 1:   log likelihood = -891.24455
Iteration 2:   log likelihood = -891.24271
Iteration 3:   log likelihood = -891.24271

Fitting full model:

Iteration 0:   log likelihood = -881.57337
Iteration 1:   log likelihood = -880.87788
Iteration 2:   log likelihood = -880.87312
Iteration 3:   log likelihood = -880.87312

a. Iteration Log - This is the iteration log for the negative binomial model. Note there are three sections; Fitting Poisson model, Fitting constant-only model and Fitting full model. Negative binomial regression is a maximum likelihood procedure and good initial estimates are required for convergence; the first two sections provide good starting values for the negative binomial model estimated in the third section.

The first section, Fitting Poisson model, fits a poisson model to the data. Estimates from the last iteration serve as the starting values for the parameter estimates in the final section. The second section, Fitting constant-only model, finds the maximum likelihood estimate for the mean and dispersion parameter of the response variable. The dispersion parameter is plugged in as the starting value for the dispersion parameter. Once starting values are obtained, the negative binomial model iterates until the algorithm converges. The trace option can be specified to see how parts from the first two iteration components are used for the final iteration component.

### Model Summary

Negative binomial regression                      Number of obs   =        316d
LR chi2(3)      =      20.74e
Dispersion     = meanb                            Prob > chi2     =     0.0001f
Log likelihood = -880.87312c                      Pseudo R2       =     0.0116g

b. Dispersion - This refers how the over-dispersion is modeled. The default method is mean dispersion.

c. Log Likelihood - This is the log likelihood of the fitted model. It is used in the calculation of the Likelihood Ratio (LR) chi-square test of whether all predictor variables' regression coefficients are simultaneously zero and in tests of nested models.

d. Number of obs - This is the number of observations used in the regression model. It may be less than the number of cases in the dataset if there are missing values for some variables in the equation. By default, Stata does a listwise deletion of incomplete cases.

e. LR chi2(3)  - This is the test statistic that all regression coefficients in the model are simultaneous equal to zero. It is calculated as negative two times the difference of the likelihood for the null model and the fitted model. The null model corresponds to the last iteration from Fitting constant-only model. Piecing parts from the iteration log together, the LR chi2(3) value is -2[-891.24 - (-880.87)] = 20.74.

f. Prob > chi2 - This is the probability of getting a LR test statistic as extreme as, or more so, than the observed under the null hypothesis; the null hypothesis is that all of the regression coefficients are simultaneously equal to zero. In other words, this is the probability of obtaining this chi-square statistic (20.74) if there is in fact no effect of the predictor variables. This p-value is compared to a specified alpha level, our willingness to accept a Type I error, which is typically set at 0.05 or 0.01. The small p-value from the LR test,  <0.00001, would lead us to conclude that at least one of the regression coefficients in the model is not equal to zero. The parameter of the chi-square distribution used to test the null hypothesis is defined by the degrees of freedom in the prior line, chi2(3).

g. Pseudo R2 - This is McFadden's pseudo R-squared. It is calculated as 1 - ll(model)/ll(null) = 0.0116. Negative binomial regression does not have an equivalent to the R-squared measure found in OLS regression; however, many people have attempted to create one. Because this statistic does not mean what R-square means in OLS regression (the proportion of variance for the response variable explained by the predictors), we suggest interpreting this statistic with caution.

### Parameter Estimates

------------------------------------------------------------------------------
daysabsf|      Coef.g  Std. Err.h      zi   P>|z|i    [95% Conf. Interval]j
-------------+----------------------------------------------------------------
mathnce |   -.001601     .00485    -0.33   0.741    -.0111067    .0079048
langnce |  -.0143475   .0055815    -2.57   0.010    -.0252871    -.003408
female |   .4311844   .1396656     3.09   0.002     .1574448     .704924
_cons |   2.284885   .2098761    10.89   0.000     1.873535    2.696234
-------------+----------------------------------------------------------------
/lnalpha |   .2533877   .0955362                      .0661402    .4406351
-------------+----------------------------------------------------------------
alpha |   1.288383   .1230871                      1.068377    1.553694
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0:  chibar2(01) = 1334.20 Prob>=chibar2 = 0.000k

f. daysabs - This is the response variable in the negative binomial regression. Underneath are the predictor variables, the intercept and the dispersion parameter.

g. Coef. - These are the estimated negative binomial regression coefficients for the model. Recall that the dependent variable is a count variable that is either over- or under-dispersed, and the model models the log of the expected count as a function of the predictor variables. We can interpret the negative binomial regression coefficient as follows: for a one unit change in the predictor variable, the difference in the logs of expected counts of the response variable is expected to change by the respective regression coefficient, given the other predictor variables in the model are held constant.

mathnce - This is the negative binomial regression estimate for a one unit increase in math standardized test score, given the other variables are held constant in the model. If a student were to increase her mathnce test score by one point, the difference in the logs of expected counts would be expected to decrease by 0.0016 unit, while holding the other variables in the model constant.

langnce - This is the negative binomial regression estimate for a one unit increase in language standardized test score, given the other variables are held constant in the model. If a student were to increase her langnce test score by one point, the difference in the logs of expected counts would be expected to decrease by 0.0143 unit, while holding the other variables in the model constant.

female -  This is the estimated negative binomial regression coefficient comparing females to males, given the other variables are held constant in the model. The difference in the logs of expected counts is expected to be 0.4312 unit higher for females compared to males, while holding the other variables constant in the model.

_cons - This is the negative binomial regression estimate when all variables in the model are evaluated at zero. For males (the variable female evaluated at zero) with zero mathnce and langnce test scores, the log of the expected count for daysabs is 2.2849 units. Note that evaluating mathnce and langnce at zero is out of the range of plausible test scores. If the test scores were mean-centered, the intercept would have a natural interpretation: the log of the expected count for males with average mathnce and langnce test scores.

/lnalpha - This is the estimate of the log of the dispersion parameter, alpha, given on the next line..

alpha - This is the estimate of the dispersion parameter. The dispersion parameter alpha can be obtained by exponentiating /lnalpha. If the dispersion parameter equals zero, the model reduces to the simpler poisson model. If the dispersion parameter, alpha, is significantly greater than zero than the data are over dispersed and are better estimated using a negative binomial model than a poisson model.

h. Std. Err. - These are the standard errors for the regression coefficients and dispersion parameter for the model. They are used in both the calculation of the z test statistic, superscript i, and confidence intervals, superscript j.

i. z and P>|z| - These are the test statistic and p-value, respectively, that the null hypothesis that an individual predictor's regression coefficient is zero, given that the rest of the predictors are in the model. The test statistic z is the ratio of the Coef. to the Std. Err. of the respective predictor. The z value follows a standard normal distribution which is used to test against a two-sided alternative hypothesis that the Coef. is not equal to zero. The probability that a particular z test statistic is as extreme as, or more so, than what has been observed under the null hypothesis is defined by P>|z|.

j. [95% Conf. Interval] - This is the confidence interval (CI) of an individual negative binomial regression coefficient, given the other predictors are in the model. For a given predictor variable with a level of 95% confidence, we'd say that we are 95% confident that upon repeated trials 95% of the CI's would include the "true" population regression coefficient. It is calculated as Coef. ± (zα/2)*(Std.Err.), where zα/2 is a critical value on the standard normal distribution. The CI is equivalent to the z test statistic: if the CI includes zero, we'd fail to reject the null hypothesis that a particular regression coefficient is zero, given the other predictors are in the model. An advantage of a CI is that it is illustrative; it provides information on the precision of the point estimate.

k. Likelihood-ratio test of alpha=0 - This is the likelihood-ratio chi-square test that the dispersion parameter alpha is equal to zero. The test statistic is negative two times the difference of the log-likelihood from the poisson model and the negative binomial model, -2[-1547.9709 -(-880.87312)] = 1334.1956 with an associated p-value of <0.0001. The large test statistic would suggest that the response variable is over-dispersed and is not sufficiently described by the simpler poisson distribution.

### Incidence Rate Ratio Interpretation

The following is the interpretation of the negative binomial regression in terms of incidence rate ratios, which can be obtained by nbreg, irr after running the negative binomial model or by specifying the irr option when the full model is specified. This part of the interpretation applies to the output below.

Before we interpret the coefficients in terms of incidence rate ratios, we must address how we can go from interpreting the regression coefficients as a difference between the logs of expected counts to incidence rate ratios. In the discussion above, regression coefficients were interpreted as the difference between the log of expected counts, where formally, this can be written as β = log( μx0+1) - log( μx0 ), where β is the regression coefficient, μ is the expected count and the subscripts represent where the predictor variable, say x, is evaluated at x0 and x0+1 (implying a one unit change in the predictor variable x). Recall that the difference of two logs is equal to the log of their quotient, log( μx0+1) - log( μx0 ) = log( μx0+1 /  μx0 ), and therefore, we could have also interpreted the parameter estimate as the log of the ratio of expected counts: This explains the "ratio" in incidence rate ratios. In addition, what we referred to as a count is technically a rate. Our response variable is the number of days absent over the school year, which by definition, is a rate. A rate is defined as the number of events per time (or space). Hence, we could also interpret the regression coefficients as the log of the rate ratio: This explains the "rate" in incidence rate ratio. Finally, the rate at which events occur is called the incidence rate; thus we arrive at being able to interpret the coefficients in terms of incidence rate ratios from our interpretation above.

Also, each subject in our sample was followed for one school year. If this was not the case (i.e., some subjects were followed for  half a year, some for a year and the rest for two years) and we were to neglect the exposure time, our regression estimates would be biased, since our model assumes all subjects had the same follow up time. If this was an issue, we would use the exposure option, exposure(varname), where varname corresponds to the length of time an individual was followed to adjust the poisson regression estimates.

nbreg, irr

Negative binomial regression                      Number of obs   =        316
LR chi2(3)      =      20.74
Dispersion     = mean                             Prob > chi2     =     0.0001
Log likelihood = -880.87312                       Pseudo R2       =     0.0116

------------------------------------------------------------------------------
daysabs |        IRRa   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
mathnce |   .9984003   .0048422    -0.33   0.741     .9889547    1.007936
langnce |   .9857549    .005502    -2.57   0.010     .9750299    .9965978
female |   1.539079   .2149564     3.09   0.002     1.170516    2.0236933
-------------+----------------------------------------------------------------
/lnalpha |   .2533877   .0955362                      .0661402    .4406351
-------------+----------------------------------------------------------------
alpha |   1.288383   .1230871                      1.068377    1.553694
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0:  chibar2(01) = 1334.20 Prob>=chibar2 = 0.000

a. IRR - These are the incidence rate ratios for the negative binomial regression model shown earlier.

mathnce - This is the estimated rate ratio for a one unit increase in math standardized test score, given the other variables are held constant in the model. If a student were to increase his mathnce test score by one point, his rate for daysabs would be expected to decrease by a factor of 0.9984, while holding all other variables in the model constant.

langnce - This is the estimated rate ratio for a one unit increase in language standardized test score, given the other variables are held constant in the model. If a student were to increase his langnce test score by one point, his rate for daysabs would be expected to decrease by a factor 0.9857, while holding all other variables in the model constant.

female -  This is the estimated rate ratio comparing females to males, given the other variables are held constant in the model. Females compared to males, while holding the other variable constant in the model, are expected to have a rate 1.539 times greater for daysabs.

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