Using margins to test for group differences in generalized

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					Using margins to test for group differences in
      generalized linear mixed models

                        Sarah Mustillo
                       Purdue University




          Sarah A. Mustillo, Ph.D   Stata Conference Chicago 2011
                                      Introduction   Problem
                                        Examples
                                       Application
                                       Conclusion




                               The problem

• Linear mixed models (LMM) are a standard model for estimating trajectories of
  change over time in longitudinal data.

• Theory, specification, estimation, and post-estimation evaluation techniques for
  LMMs are well-developed.

• Less so for generalized linear mixed models (GLMM).




                      Sarah A. Mustillo, Ph.D        Using Margins to test for group differences in GLMMs
                                       Introduction   Problem
                                         Examples
                                        Application
                                        Conclusion




             Testing for group differences

• In LMMs, researchers tend to include a group by time interaction term to test for
  group differences.

• Others have suggested that this same procedure can be used in nonlinear
  models. For example, Rabe-Hesketh and Skrondal (2005) note that the
  coefficient of the product term can be interpreted as indicating group differences
  in the rate of change over time in logistic models (pp.115-118) and ordinal
  models (155-161).

• But, interaction terms in nonlinear models are different than interaction terms in
  linear models.




                       Sarah A. Mustillo, Ph.D        Using Margins to test for group differences in GLMMs
                                         Introduction   Problem
                                           Examples
                                          Application
                                          Conclusion




Interpreting interactions in nonlinear models

 • For example, Ai and Norton (2004) argue that:
    • The coefficient of the interaction term in a linear model is the same as the
       first derivative or marginal effect and thus a group by time interaction term in
       a linear model can be interpreted as group differences in the effect of time
       on the DV.
    • In nonlinear models, the first derivative of the interaction term is not the
       interaction effect. For that, we need the cross-partial derivative of E(y) with
       respect to group and time.

 • -inteff- is one way to interpret interactions in logit and probit models, but it’s not a
   panacea for several reasons.
     • Only available for logit and probit.
     • Not available for longitudinal models.
     • Difficult to interpret and generalize.


                         Sarah A. Mustillo, Ph.D        Using Margins to test for group differences in GLMMs
                                       Introduction   Problem
                                         Examples
                                        Application
                                        Conclusion




                      Longitudinal models

• In the longitudinal, mixed model context, the interaction of a grouping variable
  and a time variable is a test for group differences in slope, but it’s a test on a
  ratio scale, which isn’t always what we want (or ever, in my case).

• The difference in the rate of change (rather than the ratio of change) can be
  measured by taking the derivative or partial derivative of the conditional
  expectation of Y with respect to time by group.

• When the ratio of change and the rate of change are close, both yield similar
  results. When they aren’t the same, they provide different results and answer
  different questions.




                       Sarah A. Mustillo, Ph.D        Using Margins to test for group differences in GLMMs
                                          Introduction   Motivating example
                                            Examples
                                           Application
                                           Conclusion




                                 Real example

• Using the Established Populations for Epidemiological Studies of the Elderly
  (EPESE) data, we were exploring the effects of baseline cognitive status on
  change in physical functioning over time. Physical functioning was measured as
  a count of instrumental tasks the subject could not perform. We used
  –xtmepoisson- with a cognitive impairment X time interaction term to test for the
  group difference in slope.

• Based on previous work, we expected baseline cognitive impairment to be
  associated with greater yearly increases in disability over time. Indeed,
  descriptive statistics showed an increase of .06 per year in the cognitively intact
  and .13 in the cognitively impaired.
              (2) (2)                                                    (2) (2)
      ln( | ζ ,ζ )  β  β Time  β Impairment  β Impariment *Time  ζ     ζ * Time
          it 1    2    0   1    it  2          i   3          i     it   1     2       it
               i   i                                                      i     i




                          Sarah A. Mustillo, Ph.D         Using Margins to test for group differences in GLMMs
                                       Introduction      Empirical example
                                         Examples
                                        Application
                                        Conclusion




 Results from –xtmepoisson-
Table 1. Estimated Mixed Poisson Model of Number of IADL's Regressed on Cognitive Impairment by Time,
EPESE Data.




Fixed parameters
                                                          B               SE          IRR
Cognitive impairment                                  2.817***         (0.161)       16.73
Time                                                  0.541***         (0.060)       1.72
Cog impairment X Time                                 -0.188***        (0.046)       0.83
Intercept                                             -4.555***        (0.144)

Random components

Slope variance                                          0.102          (0.0187)
Intercept variance                                      7.562           (0.679)
Covariance                                             -0.487           (0.115)


Summary Statistics

N                                                   15016
Chi square                                         434.050
Log likelihood                                    -8346.699

Note: Standard errors in parentheses
* p< .05 **p<.01 *** p<.001




                     Sarah A. Mustillo, Ph.D              Using Margins to test for group differences in GLMMs
                                                                                                                 Introduction              Fake example
                                                                                                                   Examples
                                                                                                                  Application
                                                                                                                  Conclusion



    Fake example - Graphs of generated count variables with gender differences in slope.
Graph of gender interaction in simulated Poisson variable with a mean of 4.                         Graph of gender interaction in simulated Poisson variable with mean = 5.                          Graph of gender interaction in simulated Poisson variable with a mean of 6.




                                                                                                                                                                                                 10
                                                                                                9
8




                                                         Simulated DV, mean=5




                                                                                                                                                            Simulated DV, mean=6
                                                                                                7




                                                                                                                                                                                                  8
6




                                                                                                                                                                                                  6
                                                                                                5
4




                                                                                                                                                                                                  4
                                                                                                3
2




                                                                                                                                                                                                  2
                                                                                                1
0




     0                     1                      2                              3                   0                      1                       2                              3                      0                    1                       2                     3
                                       time                                                                                             Time                                                                                               time

                                Male            Female                                                                          Males             Females                                                                           Male             Female




             Histogram of simulated Poisson variable with a mean of 4.                                        Histogram of simulated Poisson variable with a mean of 5.                                       Histogram of simulated Poisson variable with a mean of 6.
.6




                                                                                                .4




                                                                                                                                                                                                 .4
                                                                                                .3




                                                                                                                                                                                                 .3
.4




                                                                                      Density




                                                                                                                                                                                       Density
                                                                                                .2




                                                                                                                                                                                                 .2
.2




                                                                                                .1




                                                                                                                                                                                                 .1
    0




                                                                                                    0




                                                                                                                                                                                                      0
         0                    5                      10                              15                  0               5                10                15                     20                     0               5                10                15             20
                        Simulated Poisson variable, mean = 4                                                             Simulated Poisson variable, mean = 5                                                             Simulated Poisson variable, mean = 6




                                                                                Sarah A. Mustillo, Ph.D                                        Using Margins to test for group differences in GLMMs
                                                                                                      Introduction              Fake example
                                                                                                        Examples
                                                                                                       Application
                                                                                                       Conclusion



Fake example - Graphs of generated count variables with gender differences in slope.
Graph of gender interaction in simulated Poisson variable with a mean of 4.              Graph of gender interaction in simulated Poisson variable with mean = 5.                  Graph of gender interaction in simulated Poisson variable with a mean of 6.




                                                                                                                                                                              10
                                                                                     9
8




                                                       Simulated DV, mean=5




                                                                                                                                                   Simulated DV, mean=6
                                                                                     7
6




                                                                                                                                                                               8
                                                                                                                                                                                                   1.22
            1.32                                                                                   1.25
4




                                                                                                                                                                               6
                                                                                     5
                                            1.40                                                                                            1.24                                                                                    1.17
2




                                                                                                                                                                               4
                                                                                     3
0




                                                                                                                                                                               2
                                                                                     1
    0                    1                      2                               3         0                      1                      2                                 3           0                     1                      2                      3
                                     time                                                                                    Time                                                                                       time

                              Male            Female                                                                 Males             Females                                                                   Male            Female




        Ratio Female/Male =                                                                     Ratio Female/Male =                                                                    Ratio Female/Male =
        1.32/1.40=.93                                                                           1.25/1.24=1.01                                                                         1.22/1.17=1.03




                                                                              Sarah A. Mustillo, Ph.D                               Using Margins to test for group differences in GLMMs
                                                          Introduction    Fake example
                                                            Examples
                                                           Application
                                                           Conclusion




Table 2. Mixed Poisson Regression Models Estimated for Generated Count Variables in EPESE Data (n=16,648).


                                        Model 1_______                     Model 2______                       Model 3_____
   Mean Outcome=                              4                                 5                                6

                             B            (S.E)     IRR         b            (S.E)       IRR          B         (S.E)     IRR
   Time                     .340***       (0.008)   1.406***   .222***       (0.006)     1.250***   .165***     (0.059)   1.180***


   Female                   1.037***      (0.020)   2.822***   .671***       (0.016)     1.957***   .498***     (0.013)   1.646***


   Female*Time              -0.065***     (0.009)   0.937***   .005          (0.007)     1.006      .029***     (0.006)   1.030***


   Intercept                0.080***      (0.019)              0.741***      (0.014)                1.397***    (0.011)




   Chi square               13824.89                           11435.03                             9775.27


   Log likelihood           28298.13                           31527.75                             34031.66


   Note: Standard errors in parentheses
   * p< .05 **p<.01 *** p<.001




                                          Sarah A. Mustillo, Ph.D         Using Margins to test for group differences in GLMMs
                                       Introduction   Margins
                                         Examples
                                        Application
                                        Conclusion




     Using –margins- to assess the group difference

• The interaction term does not test what we want to test here.

• We want to calculate the partial derivative of E(Y) with respect to time by group
  and then test for a significant difference using a Wald test.

• Hmmm…does Stata have a command that can do that?




                       Sarah A. Mustillo, Ph.D         Using Margins to test for group differences in GLMMs
                                       Introduction   Margins
                                         Examples
                                        Application
                                        Conclusion




     Using –margins- to assess the group difference

• The interaction term does not test what we want to test here.

• We want to calculate the partial derivative of E(Y) with respect to time by group
  and then test for a significant difference using a Wald test.

• Hmmm…does Stata have a command that can do that?

• xtmepoisson yvar i.female##c.time || person:time,
  cov(unstr) var mle

• margins , dydx(time) over(female) predict(fixedonly) post

• lincom _b[0.female] - _b[1.female]



                       Sarah A. Mustillo, Ph.D         Using Margins to test for group differences in GLMMs
                                                 Introduction   Margins
                                                   Examples
                                                  Application
                                                  Conclusion




  Table 3. Using –margins- following –xtmepoisson- to test for group differences in slope in the fake examples




                            Model 1_______                       Model 2______                          Model 3_____
Mean Outcome=                     4                                    5                                  6

Fem ratio/                      0.93                                 1.01                                 1.03
 Male ratio




dy/dt




 Male                      0.693*** (0.018)                     0.659***(0.021)                      0.687***(0.027)



 Female                    1.359***(0.021)                      1.325*** (0.022)                     1.329*** (0.025)



 Difference                0.667***(0.028)                      0.665***(0.031)                      0.642***(0.037)




                                 Sarah A. Mustillo, Ph.D         Using Margins to test for group differences in GLMMs
                                                        Introduction      Empirical example
                                                          Examples
                                                         Application
                                                         Conclusion



Table 4. Using –margins- following –xtmepoisson- to test for group differences in slope in the original example

                                                                                        Disability

                                                                 b                         SE             IRR
                                                                                                       1.716***

              Time                                            .541***                    (0.102)
                                                                                                       16.727***

              Cognitive impairment                            2.817***                   (2.686)
                                                                                                       0.830***

              Cog impairment X Time                           -0.187***                  (0.038)


              Intercept                                       -4.555***

              dy/dt


               No cog impairment                   0.015***                   (0.002)


               Cog impairment                      0.108***                   (0.018)


               Difference                          0.093***                   (0.017)


              Note: Random coefficients omitted, * p<0.05, ** p<0.01, *** p<0.001




                                      Sarah A. Mustillo, Ph.D              Using Margins to test for group differences in GLMMs
                                        Introduction   Empirical example
                                          Examples
                                         Application
                                         Conclusion


Table 5. Using –margins- following –xtmepoisson- to test for group differences in slope
    in the original example with additional covariates and an additional interaction
                                                                                   Disability
                                                               B                      SE                       IRR
Time                                                        .564***                 (0.082)               1.758***
Cognitive impairment                                        2.130***                (1.299)               8.413***
Cog impairment X Time                                       -0.186***               (0.035)               0.830***
Age                                                         0.114***                (0.008)               1.121***
Female                                                       -0.032                 (0.113)                0.969
Black                                                        0.225*                 (0.131)                1.253*
Income                                                      -0.029***               (0.006)               0.972***
Married                                                      0.005                  (0.152)                1.005
Married X Time                                               -0.025                 (0.040)                0.976
Intercept                                                  -12.701***


dy/dt
 No cog impairment                                          0.022***                (0.003)
 Cog impairment                                             0.163***                (0.024)
 Difference                                                 0.141***                (0.023)
 Married                                                   0.019***                 (0.003)
 Unmarried                                                 0.052***                 (0.006)
 Difference                                                0.033***                 (0.005)




                        Sarah A. Mustillo, Ph.D         Using Margins to test for group differences in GLMMs
                                       Introduction
                                         Examples
                                        Application
                                        Conclusion




                                    Summary

• In the generalized linear mixed model, the group by time interaction term is
  measuring differences in the ratio of change, e.g., change on a multiplicative
  scale.

• This isn’t wrong – it just wasn’t what we wanted.

• -margins- provides an easy way to test group difference in rate of change over
  time on an additive scale by allowing us to calculate the partial derivative of the
  response with respect to time separately by group and then run a significance
  test between the two.




                       Sarah A. Mustillo, Ph.D        Using Margins to test for group differences in GLMMs

				
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