Mplus Data Analysis Examples: Poisson Regression



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Mplus Data Analysis Examples
Poisson Regression

Examples of Poisson 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.

We have attendance data on 316 high school juniors from two urban high schools in the file poissonreg.dat 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.

NOTE: This example was done using Mplus version 4.21. The syntax may not work with earlier versions of Mplus.

Let's look at the data.

Data:
  File is d:\work\data\mplus\poissonreg.dat ;
Variable:
  Names are 
     id school male math langarts daysatt daysabs;
 Missing are all (-9999) ; 
 usevariables are male math langarts daysabs;
analysis:
  type = basic;
plot: type is plot1;

     SAMPLE STATISTICS


           Means
              MALE          MATH          LANGARTS      DAYSABS
              ________      ________      ________      ________
      1         0.487        48.751        50.064         5.810


           Covariances
              MALE          MATH          LANGARTS      DAYSABS
              ________      ________      ________      ________
 MALE           0.251
 MATH          -0.481       319.721
 LANGARTS      -1.507       220.942       321.815
 DAYSABS       -0.456       -21.672       -24.319        55.488


           Correlations
              MALE          MATH          LANGARTS      DAYSABS
              ________      ________      ________      ________
 MALE           1.000
 MATH          -0.054         1.000
 LANGARTS      -0.168         0.689         1.000
 DAYSABS       -0.122        -0.163        -0.182         1.000

Some Strategies You Might Be Tempted To Try

Before we show how you can analyze this with a Poisson regression analysis, 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.
Negative Binomial Regression - Negative binomial regression does better with over dispersed data, i.e. variance much larger than the mean.
Zero-inflated Regression Model - This might be a good if there are more zeros than would be expected by either a poisson or negative binomial model.

Mplus Poisson Regression Analysis

  Data:
  File is d:\work\data\mplus\poissonreg.dat ;
  Variable:
    Names are
       id school male math langarts daysatt daysabs;
   Missing are all (-9999) ;
   usevariables are male math langarts daysabs;
   count is daysabs;
  model:
    daysabs on male math langarts;

TESTS OF MODEL FIT

Loglikelihood

          H0 Value                       -1547.971
          H0 Scaling Correction Factor      10.455
            for MLR

Information Criteria

          Number of Free Parameters              4
          Akaike (AIC)                    3103.942
          Bayesian (BIC)                  3118.965
          Sample-Size Adjusted BIC        3106.278
            (n* = (n + 2) / 24)

MODEL RESULTS

                   Estimates     S.E.  Est./S.E.

 DAYSABS    ON
    MALE              -0.401    0.139     -2.877
    MATH              -0.004    0.008     -0.462
    LANGARTS          -0.012    0.005     -2.299

 Intercepts
    DAYSABS            2.688    0.218     12.340

The default estimation method is MLR - maximum likelihood parameter estimates with standard errors and a chi-square test statistic that are robust to non-normality and non-independence of observations when used with type = complex, according to the Mplus 4 manual. The MLR standard errors are computed using a sandwich estimator. This is what we generally call robust standard errors.

After the heading informing that "THE MODEL ESTIMATION TERMINATED NORMALLY" comes the information about the model. It begins with the information on log likelihood, AIC and BIC. They can be used in comparing models. Then we will find the Poisson regression coefficients for each of the variables along with standard errors, and the quantity that is labeled as Est./S.E. The last column is the quotient of the estimates divided by the standard errors and are basically z-scores if the sample size is reasonably large.

Now, we can also run the Poisson model without the robust standard error. This is done by specifying estimator = ML.

  Data:
  File is d:\work\data\mplus\poissonreg.dat ;
  Variable:
    Names are
       id school male math langarts daysatt daysabs;
   Missing are all (-9999) ;
   usevariables are male math langarts daysabs;
   count is daysabs;
  analysis: estimator = ml;
  model:
    daysabs on male math langarts;

TESTS OF MODEL FIT

Loglikelihood

          H0 Value                       -1547.971

Information Criteria

          Number of Free Parameters              4
          Akaike (AIC)                    3103.942
          Bayesian (BIC)                  3118.965
          Sample-Size Adjusted BIC        3106.278
            (n* = (n + 2) / 24)

MODEL RESULTS

                   Estimates     S.E.  Est./S.E.

 DAYSABS    ON
    MALE              -0.401    0.048     -8.281
    MATH              -0.004    0.002     -1.934
    LANGARTS          -0.012    0.002     -6.623

 Intercepts
    DAYSABS            2.688    0.073     36.994

Using the robust standard errors has resulted in a fairly large change in the standard error, which should be more appropriate. The z-tests still yield similar significant results, but give more realistic p-values.

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

Data:
  File is d:\work\data\mplus\poissonreg.dat ;
Variable:
  Names are 
     id school male math langarts daysatt daysabs;
 Missing are all (-9999) ; 
 usevariables are male  langarts daysabs;
 count is daysabs;
model:
  daysabs on male langarts;

TESTS OF MODEL FIT

Loglikelihood

          H0 Value                       -1549.857
          H0 Scaling Correction Factor       8.128
            for MLR

Information Criteria

          Number of Free Parameters              3
          Akaike (AIC)                    3105.713
          Bayesian (BIC)                  3116.980
          Sample-Size Adjusted BIC        3107.465
            (n* = (n + 2) / 24)



MODEL RESULTS

                   Estimates     S.E.  Est./S.E.

 DAYSABS    ON
    MALE              -0.409    0.135     -3.027
    LANGARTS          -0.015    0.003     -4.274

 Intercepts
    DAYSABS            2.647    0.182     14.516

One more thing that we might be interested in getting is the overall model fit comparing with the intercept-only model. We would want to know the overall effect of the predictors in the model. This can be done by comparing this model with the null model.

Data:
  File is d:\work\data\mplus\poissonreg.dat ;
Variable:
  Names are 
     id school male math langarts daysatt daysabs;
 Missing are all (-9999) ; 
 usevariables are  daysabs;
 count is daysabs;
model:
  [daysabs] ;

THE MODEL ESTIMATION TERMINATED NORMALLY

TESTS OF MODEL FIT

Loglikelihood

          H0 Value                       -1635.608
          H0 Scaling Correction Factor       9.520
            for MLR

Information Criteria

          Number of Free Parameters              1
          Akaike (AIC)                    3273.216
          Bayesian (BIC)                  3276.972
          Sample-Size Adjusted BIC        3273.800
            (n* = (n + 2) / 24)

MODEL RESULTS

                   Estimates     S.E.  Est./S.E.

 Means
    DAYSABS            1.760    0.072     24.436

The log likelihood for this model is -1635.608 with one degree of freedom and is -1549.857 for the full model with three degrees of freedom. We can do a chi-square test on the difference of the 2*loglikelihood. This leads to the omnibus test of chi-square of 171.502 with two degrees of freedom, yielding a p-value <.0001.

Sample Write-Up of the Analysis

The Poisson regression model predicting days absent from school stay 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 -0.0146. This translates to a decrease of about 1.46 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 Poisson models be applied to small samples. What constitutes a small sample does not seem to be clearly defined in the literature.

Mplus Data Analysis Examples Poisson Regression