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 https://stats.idre.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 Log^a

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 log of expected counts of the
response variable changes 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. (z_α/2)*(Std.Err.), where z_α/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( μ_x0+1) – log( μ_x0 ), 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( μ_x0+1) – log( μ_x0 ) = log( μ_x0+1
/  μ_x0 ), 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.