impact of the distributional assumptions of random effects on model fitting xueying (sherri) zhang...

Impact of the Distributional Assumptions of Random Effects on

Model Fitting

Xueying (Sherri) Zhang

Research Scientist

California Department of Health Services

CDIC/ Tobacco control section

2

Overview

1. Introduction

2. Data Source and Study Population3. Methods: Model Description and Simulations

4. Results

5. Discussion and Conclusions

3

Introduction

Distribution of the random effects

-0.31969 0.08031 0.48031 0.88031 1.28031 1.68031 2.08031 2.48031 2.88031

bi

0

1

2

3

4

5

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Random Effect model: Logit [E (Yij=1| bi)] = ß0 + ß1Xij1+ ß2Xij2 + ß2Xij3 +bi

4

Introduction

Research Questions:

1. What is the impact of this normality assumption on the

estimates of fixed effects?

2. When a cluster- level confounder is omitted from the model and

the random effects are associated with the covariates in the

model, are the estimates of RE model correct?

5

Data Source and Study Population

1. Regional Rural Injury Study-II(RRIS-II) was a population-based, prospective cohort study, which was designed to investigate the incidence and consequences of agricultural injury in the five state region of Minnesota, Wisconsin, North Dakota, South Dakota, and Nebraska in 1999.

2. 3,765 household, including 16,538 persons, participated in the study.

3. We modeled the probability of agricultural activity-related injury (Yes/No). Gender, prior injury, and working hours per week on the agricultural operation were chosen as the covariates.

4. Clustered binary data---the same operations and the potential similarity of behaviors between parents and children.

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Model Description

Generalized linear mixed models (GzLMM) with a random intercept is expressed as:

Logit [E (Yij=1| bi)] = ß0 + ß1Xij1+ ß2Xij2 + ß3Xij3+bi

Yij indicates whether the agricultural activity-related injury happened or not in 1999 for the jth person in the ith family

bi is the random effect for the ith family

Xijk indicates gender, age, education, marital status, prior injury, working hours on the farm and the percentage of prior injury within the family (PPI).

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Methods

Random effect model :The marginal likelihood for Bernoulli data is as follows:

i

m

i

i

n

j

dbbfijy

ijy

YDL pp ijij

1

)1(

1

)()/,( )1(

Where Pij =E(Yij | bi) and with a logit link:

bZXP

PLog iijij

ij

ij

)

1( bi ~ N(0, σ2)

ebib

if 2

0

2

2

1

22

1)(

8

Methods

Conditional model:

Logit [E (Yij=1| bi)] = ß’0+ ß’

1Xij1+ ß’2Xij2 + ß’

3Xij3+bi

Given the sufficient statistics for bi, Sij, the conditional likelihood can

be expressed as:

i j iiiij

j iiiij

bxxSbxxy

f

f

),,...,|(

),,...,|(

31

31L=

9

Methods

Marginal model only takes the fixed effects into account, the model is:

Logit [E (Yij=1| bi)] = ß*0+ ß*

1Xij1+ ß*2Xij2 + ß*

3Xij3

Var(Yij)=φVar(E(Yij))

It estimates β* by solving a quasi score function:

0))((1

1

'

xy iii

m

ii f

Marginal model:

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Simulations---(1) True model for the first simulation:

Logit [E (Yij | bi)] = ß0 + ß1*gender + ß2*workhour +ß3 *priorinj +bi

Random effects:

nbi=sqrt (σ2/v2)*(bi-µ) nbi~ ( 0, σ2 )

σ2 the estimated variance for random effects from the true model.

v2 the variance of the predicted bi from the true model.

µ the mean of estimated bi from the true model.

bi the predicted random effects from the true model.

Pij= Exp (Xijß+nbi)/(1+ Exp(Xijß+nbi))

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Simulations---(1)

1000 different seeds Random numbers (R) from U(0,1) for each individual

If R≤ Pij, SimY=1; else, SimY=0

Pr (simY) = Pij

Covariates remain the same with the real data

a marginal model, RE model and conditional model was fit for each data set.

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Simulations---(2) True model for the second simulation:

Logit [E (Yij | bi)] = ß0 + ß1*gender + ß2*workhour +ß3 *priorinj + ß4 *PPI + bi

PPI-Percentage of prior injury within family.

Random effects:nbi=sqrt (σ2/v2)*(bi-µ) nbi~ ( 0, σ2 )

σ2 the estimated variance for random effects from the true model. v2 the variance of the predicted bi from the true model. µ the mean of estimated bi from the true model. bi the predicted random effects from the true model.

Pij= Exp (Xijß+nbi)/(1+ Exp(Xijß+nbi))

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Simulations---(2)

1000 different seeds Random numbers (R) from U(0,1)

If R≤ Pij, SimY=1; else, SimY=0

Pr (simY) = Pij

Covariates remain the same with the real data

A marginal model, RE model and conditional model was fit for each data set.

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Results - Results from the model using real data:

Marginal model RE model Conditional model

Working hours Estimate 95% C.I. Estimate 95% C.I. Estimate 95% C.I.

0 -2.78 -3.38 -2.18 -2.90 -3.53 -2.26 -2.50 -3.28 -1.73

0-20 -1.24 -1.55 -0.92 -1.33 -1.66 -0.99 -1.15 -1.65 -0.66

21-40 -0.63 -0.91 -0.34 -0.69 -1.00 -0.38 -0.55 -1.00 -0.09

41-60 -0.31 -0.58 -0.04 -0.33 -0.63 -0.04 -0.13 -0.58 0.32

61-80 -0.20 -0.46 0.07 -0.22 -0.51 0.08 -0.05 -0.49 0.40

81+ * 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

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Results - Results from the model using real data:

-4.00

-3.50

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

0 0-20 21-40 41-60 61-80 81+ *

Farm working hours

Es

tim

ate

s

Marginal Model RE Model Conditional Model

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Results--Simulation (1) of the model without PPI:

The true model for the simulation is the RE model:

Logit [E (Yij | bi)] = ß0 + ß1*gender + ß2*workhour +ß3 *priorinj +bi

1. The average estimates of the RE model are closer to those of the conditional model.

2. The bias for prior injury in the RE model is 0.1130, much larger than the estimates from the marginal model and conditional model: 0.0573 and -0.0036.

3. The MSE for prior injury from the RE model is 0.0182, which is much bigger than the MSE from the marginal model and conditional models: 0.0073 and 0.0089.


VariablesWorkinghours Gender

Prior injury

Working hours Gender

Prior injury

Working hours Gender

Prior injury

Bt

(True values) 0.9448 0.5716 1.1263 1.0392 0.6287 1.2388 1.0392 0.6287 1.2388

Bs (Avg

estimates) 0.9048 0.5172 1.1836 1.0613 0.5765 1.3518 1.0516 0.6249 1.2352

Bias (Bs-Bt) -0.0400 -0.0544 0.0573 0.0221 -0.0522 0.1130 0.0124 -0.0038 -0.0036

MSE mean((Bi-Bt)^2)) 0.0073 0.0081 0.0073 0.0083 0.0092 0.0182 0.0124 0.0083 0.0089


Working GenderPrior injury Working Gender

Prior injury Working Gender

Prior injury

Percentage of 80% C.I. coverage 0.7770 0.6810 0.6950 0.8080 0.7230 0.4830 0.7990 0.8000 0.8620





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Results -Simulation of the Model with PPI:

Hypotheses for incorrect estimates for prior injury in the RE model: an important cluster level confounder related to prior injury was omitted from the model.

The random effects were significantly associated with prior injury (ß =0.0162, p=0.0021).

After PPI was included in the model, the random effects were independent of prior injury (ß=0.0027, p=0.6101) and PPI (ß=0.0084, p=0.3668).

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PPI is a confounder for the effect of prior injury because it is significant in the model (ß=0.5205, p<0.001) and also associated with prior injury.

True model for the second simulation:

Logit [E (Yij | bi)] = ß0 + ß1*gender + ß2*workhour +ß3 *priorinj + ß4 *PPI + bi


Variables Working GenderPrior injury Working Gender

Prior injury Working Gender

Prior injury

Bt

(True values) 0.9688 0.5940 0.9827 1.0653 0.6531 1.0805 1.0653 0.6531 1.0805

Bs (Avg estimates) 0.9145 0.5375 1.0935 1.0698 0.6285 1.2517 1.0429 0.6249 1.2389

Bias (Bs-Bt) -0.0543 -0.0565 0.1109 0.0045 -0.0246 0.1712 -0.0224 -0.0282 0.1584

MSE mean((Bi-Bt)^2)) 0.0057 0.0049 0.0067 0.0080 0.0072 0.0088 0.0141 0.0077 0.0097

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The average estimates of the RE model are almost equal to those of the conditional model.

The biases of the estimates for these three models are very similar to each other.

The mean squared errors (MSE) of the estimates of these three models are also similar.


VariablesWorkinghours Gender

Prior injury

Workinghours Gender

Prior injury

Workinghours Gender

Prior injury






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The percentage of C.I. coverage is higher than the corresponding confidence interval.

For instance, the percentage of 80% C.I. coverage for working hours in the marginal model is 83.4%, higher than 80%.

The percentages of C.I. coverage are greatly improved, especially for the prior injury in the RE model.

For instance, only 48.3% of the 80% confidence intervals of the prior injury estimates cover the true value, but in Table 7, 83.5% of the 80% confidence intervals cover the true value.

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Discussion and Conclusions

The fixed effects from the RE model are correct even in cases where the distribution of random effects does not follow the normal distribution.

The random effects should be independent of the covariates in the model.

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In this project, after we include PPI, the random effects were independent of all the covariates, the random effects still did not follow a normal distribution

--- unknown or unmeasured variables may exist which affects the probability of injury within the family.

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One limitation of the project is that the true values for marginal model were not available.

For further study, the impacts of the normality assumption of random effects on the estimates of random effects are of interest.

Thank You!

impact of the distributional assumptions of random effects on model fitting xueying (sherri) zhang...

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