This page shows an example of Poisson 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 Poisson 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 Poisson 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 Poisson model, as compared to other count models (i.e., negative binomial or zero-inflated models), is assumed the appropriate model. In other words, we assume that the dependent variable is not over-dispersed and does not have an excessive number of zeros. For more information on over-dispersion or excessive zeroes please review our Negative Binomial or Zero-Inflated webpages.
Also, note that 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 Poisson regression estimate 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.
The first half of this page interprets the coefficients in terms of Poisson regression coefficients and the second half interprets the coefficients in terms of incidence rate ratios.
We also run the estat ic command to calculate the likelihood ratio chi-square statistic.
use https://stats.idre.ucla.edu/stat/stata/notes/lahigh, clear
generate female = (gender == 1)
poisson daysabs mathnce langnce female
Iteration 0: log likelihood = -1547.9709
Iteration 1: log likelihood = -1547.9709
Poisson regression Number of obs = 316
LR chi2(3) = 175.27
Prob > chi2 = 0.0000
Log likelihood = -1547.9709 Pseudo R2 = 0.0536
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daysabs | Coef. Std. Err. z P>|z| [95% Conf. Interval]
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mathnce | -.0035232 .0018213 -1.93 0.053 -.007093 .0000466
langnce | -.0121521 .0018348 -6.62 0.000 -.0157483 -.0085559
female | .4009209 .0484122 8.28 0.000 .3060348 .495807
_cons | 2.286745 .0699539 32.69 0.000 2.149638 2.423852
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estat ic
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Model | Obs ll(null) ll(model) df AIC BIC
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. | 316 -1635.608 -1547.971 4 3103.942 3118.965
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Iteration Log, Model Summary and estat ic
Iteration 0: log likelihood = -1547.9709
Iteration 1: log likelihood = -1547.9709a
Poisson regression Number of obsc = 316
LR chi2(3)b = 175.27
Prob > chi2e = 0.0000
Log likelihood = -1547.9709b Pseudo R2f = 0.0536
estat ic
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Model | Obs ll(null)d ll(model)d df AIC BIC
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. | 316 -1635.608 -1547.971 4 3103.942 3118.965
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a. Iteration Log – This is a listing of the log likelihood at each iteration. Poisson regression uses maximum likelihood estimation, which is an iterative procedure to obtain parameter estimates. If you are familiar with other regression models that use maximum likelihood (e.g., logistic regression), you may notice this iteration log behaves differently. Specifically, the log likelihood at iteration 0 does not correspond to the likelihood for the empty (or null) model. This is evident when we look under ll(null) from the estat ic command, which provides the log likelihood for the empty model. The log likelihood for the fitted model is given in the last iteration of the iteration log and under ll(model) from estat ic; note that both values are equal (unlike ll(null) and the log likelihood from iteration 0). The log likelihood for the fitted model is then used with ll(null) to calculate the Likelihood ratio chi-square test statistic.
b. 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.
c. Number of obs – This is the number of observations used in the Poisson regression. It may be less than the number of cases in the dataset if there are missing values for some variables in the model. By default, Stata does a listwise deletion of incomplete cases.
d. LR chi2(3), ll(null) and ll(model) from estat ic – This is the LR test statistic for the omnibus test that at least one predictor variable regression coefficient is not equal to zero in the model. The degrees of freedom (the number in parenthesis) of the LR test statistic is defined by the number of predictor variables (3). LR chi2(3) is calculated as -2*[ll(null) – ll(model)] = -2*[-1635.608 – (-1547.971)] = 175.274.
e. Prob > chi2 – This is the probability of getting a LR test statistic as extreme as, or more so, than the one 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 test statistic (175.274) 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, p < 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).
f. Pseudo R2 – This is McFadden’s pseudo R-squared. It is calculated as 1 – ll(model)/ll(null) = 0.0536. Poisson regression does not have an equivalent to the R-squared found in OLS regression; however, many have tried to derive an equivalent measure. There are a variety of pseudo-R-square statistics. Because this statistic does not mean what R-square means in OLS regression (the proportion of variance of the response variable explained by the predictors), we suggest interpreting this statistic with caution.
Parameter Estimates
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daysabsg | Coef.h Std. Err.i zj P>|z|j [95% Conf. Interval]k
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mathnce | -.0035232 .0018213 -1.93 0.053 -.007093 .0000466
langnce | -.0121521 .0018348 -6.62 0.000 -.0157483 -.0085559
female | .4009209 .0484122 8.28 0.000 .3060348 .495807
_cons | 2.286745 .0699539 32.69 0.000 2.149638 2.423852
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g. daysabs – This is the response variable in the Poisson regression. Underneath daysabs are the predictor variables and the intercept (_cons).
h. Coef. – These are the estimated Poisson regression coefficients for the model. Recall that the dependent variable is a count variable, and Poisson regression models the log of the expected count as a function of the predictor variables. We can interpret the Poisson regression coefficient as follows: for a one unit change in the predictor variable, the difference in the logs of expected counts is expected to change by the respective regression coefficient, given the other predictor variables in the model are held constant.
mathnce – This is the Poisson 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 0.0035 unit, while holding the other variables in the model constant.
langnce – This is the Poisson 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 0.0122 unit while holding the other variables in the model constant.
female – This is the estimated Poisson 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 0.4010 unit higher for females compared to males, while holding the other variables constant in the model.
_cons – This is the Poisson 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 2.2867 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.
i. Std. Err. – These are the standard errors of the individual regression coefficients. They are used both in the calculation of the z test statistic, superscript j, and the confidence interval of the regression coefficient, superscript k.
j. 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|.
mathnce – The z test statistic testing the slope for mathnce on daysabs is zero, given the other variables are in the model, is (-0.0035/0.0018) -1.93, with an associated p-value of 0.053. If we set our alpha level at 0.05, we would fail to reject the null hypothesis and conclude the Poisson regression coefficient for mathnce is not statistically different from zero given langnce and female are in the model.
langnce – The z test statistic testing the slope for langnce on daysabs is zero, given the other variables are in the model, is (-0.0122/0.0018) -6.62, with an associated p-value of <0.0001. If we set our alpha level at 0.05, we would reject the null hypothesis and conclude the Poisson regression coefficient for langnce is statistically different from zero given mathnce and female are in the model.
female – The z test statistic testing the difference between the log of expected counts between males and females on daysabs is zero, given the other variables are in the model, is (0.4009/0.04841) -8.28, with an associated p-value of <0.0001. If we set our alpha level at 0.05, we would reject the null hypothesis and conclude that the coefficient for female is statistically different from zero given mathnce and langnce are in the model.
_cons – The z test statistic testing _cons is zero, given the other variables are in the model and evaluated at zero, is (2.2867/0.0670) -32.69, with an associated p-value of <0.0001. If we set our alpha level at 0.05, we would reject the null hypothesis and conclude that _cons on daysabs has been found to be statistically different from zero given mathnce, langnce and female are in the model and evaluated at zero.
k. [95% Conf. Interval] – This is the confidence interval (CI) of an individual poisson 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 Poisson 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 where the “true” parameter may lie and the precision of the point estimate.
Incidence Rate Ratio Interpretation
The following is the interpretation of the Poisson regression in terms of incidence rate ratios, which can be obtained by poisson, irr after running the Poisson 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 Poisson regression coefficients as a difference between the logs of expected counts to incidence rate ratios. In the discussion above, Poisson regression coefficients were interpreted as the difference between the log of expected counts, where formally, this can be written as β = log( μx+1) – log( μx ), where β is the regression coefficient, μ is the expected count and the subscripts represent where the predictor variable, say x, is evaluated at x and x+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( μx+1) – log( μx ) = log( μx+1 / μx ), 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 can also be called a rate. By definition a rate is the number of events per time (or space), which our response variable qualifies as. Hence, we could also interpret the Poisson 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.
poisson daysabs mathnce langnce female, irr
Iteration 0: log likelihood = -1547.9709
Iteration 1: log likelihood = -1547.9709
Poisson regression Number of obs = 316
LR chi2(3) = 175.27
Prob > chi2 = 0.0000
Log likelihood = -1547.9709 Pseudo R2 = 0.0536
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daysabs | IRRa Std. Err. z P>|z| [95% Conf. Interval]b
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mathnce | .996483 .0018149 -1.93 0.053 .9929321 1.000047
langnce | .9879214 .0018127 -6.62 0.000 .984375 .9914806
female | 1.493199 .072289 8.28 0.000 1.35803 1.641823
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a. IRR – These are the incidence rate ratios for the Poisson model shown earlier. We obtain at the incidence rate ratio by exponentiating the Poisson regression coefficient
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 ratio for daysabs would be expected to decrease by a factor of 0.9965, 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 ratio for daysabs would be expected to decrease by a factor 0.9880, 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 1.493 times greater for daysabs.
b. [95% Conf. Interval] – This is the CI for the rate ratio given the other predictors are in the model. For a given predictor 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 incidence rate ratio, given the other variables are in the model.
References: 1. Regression Analysis of Count Data, Second Edition by A. Colin Cameron and Pravin K. Trivedi 2. Modeling Count Data by Joseph M. Hilbe