Using margins to test for group differences in generalized - PowerPoint Presentation

Slide1

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

Sarah MustilloPurdue University

Stata Conference Chicago 2011Slide2

The problemIntroductionExamples

ApplicationConclusion

Problem

Using Margins to test for group differences in GLMMs

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).Slide3

Testing for group differencesIntroductionExamples

ApplicationConclusion

Problem

Using Margins to test for group differences in GLMMs

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.Slide4

Interpreting interactions in nonlinear modelsIntroductionExamples

ApplicationConclusion

Problem

Using Margins to test for group differences in GLMMs

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.Slide5

Longitudinal modelsIntroductionExamples

ApplicationConclusion

Problem

Using Margins to test for group differences in GLMMs

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.Slide6

Real exampleIntroductionExamples

ApplicationConclusion

Motivating example

Using Margins to test for group differences in GLMMs

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.Slide7

Results from –xtmepoisson-Introduction

ExamplesApplication

Conclusion

Empirical example

Using Margins to test for group differences in GLMMs

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 impairment2.817***(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        Slide8

Fake example - Graphs of generated count variables with gender differences in slope.

IntroductionExamplesApplication

Conclusion

Fake example

Using Margins to test for group differences in GLMMsSlide9

Fake example - Graphs of generated count variables with gender differences in slope.

IntroductionExamplesApplication

Conclusion

Fake example

Using Margins to test for group differences in GLMMs

1.32

1.40

1.25

1.24

1.22

1.17

Ratio Female/Male = 1.32/1.40=.93

Ratio Female/Male = 1.25/1.24=1.01

Ratio Female/Male = 1.22/1.17=1.03Slide10

IntroductionExamplesApplication

ConclusionFake example

Using Margins to test for group differences in GLMMs

Mean Outcome=

Model 1

_______

4

Model 2______

5

Model 3_____

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***Intercept0.080***(0.019) 0.741***(0.014) 1.397***(0.011)           Chi square13824.89  11435.03   9775.27  Log likelihood28298.13  31527.75  34031.66  Note: Standard errors in parentheses * p< .05 **p<.01 *** p<.001Table 2. Mixed Poisson Regression Models Estimated for Generated Count Variables in EPESE Data (n=16,648).Slide11

Using –margins- to assess the group differenceIntroductionExamples

ApplicationConclusion

Margins

Using Margins to test for group differences in GLMMs

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?Slide12

Using –margins- to assess the group differenceIntroductionExamples

ApplicationConclusion

Margins

Using Margins to test for group differences in GLMMs

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]Slide13

IntroductionExamplesApplication

ConclusionMargins

Using Margins to test for group differences in GLMMs

Table 3. Using –margins- following –

xtmepoisson

- to test for group differences in slope in the fake examples

Fem ratio/

Male ratio

0.93

1.01

1.03

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)

Difference0.667***(0.028)0.665***(0.031)0.642***(0.037) Mean Outcome= Model 1_______4 Model 2______5 Model 3_____6 Slide14

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

ExamplesApplicationConclusion

Empirical example

Using Margins to test for group differences in GLMMs

Disability

b

SE

IRR

Time

.541***

(0.102)

1.716***

Cognitive impairment

2.817***

(2.686)

16.727***

Cog impairment X Time

-0.187***

(0.038)

0.830***

Intercept

-4.555***

dy

/

dt

No cog impairment

0.015***

(0.002)

Cog impairment

0.108***

(0.018) Difference0.093***(0.017) Note: Random coefficients omitted, * p<0.05, ** p<0.01, *** p<0.001  Slide15

Table 5. Using –margins- following –xtmepoisson- to test for group differences in slope in the original example with additional covariates and an additional interaction

IntroductionExamplesApplicationConclusion

Empirical example

Using Margins to test for group differences in GLMMs

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)0.969Black    0.225*(0.131)1.253*Income    -0.029***(0.006)0.972***Married    0.005(0.152)1.005Married X Time   -0.025(0.040)0.976Intercept    -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) Slide16

SummaryIntroductionExamples

ApplicationConclusion

Using Margins to test for group differences in GLMMs

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.