A generalized bivariate Bernoulli model withcovariate dependence

Fan Zhang

Outline

• Introduction• Model proposed• Simulation• Remarks

Introduction

• Dependence in outcome variables may arise in various fields such as epidemiology, time series, environment, public health, economics, anthropology, etc.

• Examples: 1) pre-post tests; 2) proposed diagnostic tests vs. standard procedure on selected individuals; 3) any twin studies.

• Dependence in outcome variables may pose formidable difficulty in analyzing data in longitudinal studies.

Previous Methods

Most common approach: Marginal way• In the past, most of the studies made attempts

to address this problem using the marginal models.

• Example: 1) marginal odds ratios by Lipsitz et al.; 2) marginal model based on the binary Markov Chain by Azzalini.

Previous Methods

Less common approach: conditional way• Example: 1) Markov models for covariate

dependence of binary sequence by Muenz et al.; 2) logistic model by Bonney et al.

Other attemps:• quadratic exponential form model• multivariate Plackett distribution

Previous Methods

• Limitations: using the marginal models alone, it is difficult to specify the measures of dependence in outcomes due to association between outcomes as well as between outcomes and explanatory variables.

• Neither conditional approach alone can resolve the problems

Model proposed

• Bivariate Bernoulli distribution, a joint model.• Model setting:

• Joint probability:

Model proposed

• The bivariate probabilities as a function of covariates X are as follows:

• In terms of the exponential family for the generalized linear model:

Model proposed

• Log likelihood for size n:

• Link function:

where η0 is the baseline link function, η2 is the link function for Y1, η1 is the link function for Y2 and η3 is the link function for dependence between Y1 and Y2.

Model proposed• Express the conditional probabilities in terms

of the logit link function as:

Model proposed

• The marginal probabilities are as:

• Assume:

Model proposed

• Now write:

Model proposed

• Thus,

Model proposed

• Hence, if there is no association between Y1 and Y2 then P00(x)*P11(x)/P01(x)*P10(x) = 1 and Ln(1)=0. This indicates β11= β01.

• This is a new formulation to measure the dependence in terms of the parameters of the conditional models obtained from the joint mass function.

Model proposed

• In case of no dependence, it is expected that η3 = 0 which is evident if, alternatively, β11= β01. We can test the equality of two sets of regression parameters, β11 and β01 using the statistic:

which is distributed asymptotically as chi-square with p+1 degree of freedom.

Model proposed• Comparison with regressive model, another

widely used technique. maybe a typo

Regressive model:

It is noteworthy that γ is the parameter associated with the outcome variable Y1 such that, H0 : γ = 0 indicates a lack of dependence between Y1 and Y2

Model proposed• Comparison with regressive model

Regressive model:

However, one of the major limitations arises from the fact that dependence in Y1 and Y2 depends on the dependence between the outcome variables and the covariates as well. Hence, in many instances, the regressive model may fail to recognize the true nature of relationship between Y1 and Y2 in the presence of covariates X1, X2, . . . , Xp in the model.

Simulation

df=2H0: independence

Simulation

“It is clearly evident that the true correlations between Y1 and Y2 are zero and the average conditional correlations between Y2 and X for given Y1 = 0 and Y1 = 1 are similar or closely indicating a lack of dependence in the outcome variables as revealed by the proposed test (17). However, the regressive model (18) fails to reveal that due to the non-zero correlation between the previous outcome variable (Y1) and explanatory variable (X). This is indicative of the fact that the proposed test can reveal the nature of dependence in a wider range of situations in reality.”

Conclusion

“The problem of dependence in the repeated measures outcomes is one of the formidable challenges to the researchers. In the past, the problem had been resolved on the basis of marginal models with very strict assumptions. The models based on GEE with various correlation structures have been employed in most of the cases. Another widely used technique is the regressive logistic regression model. However, both these approaches provide either inadequate or, in some instances, misleading results due to use of only marginal or conditional approaches, instead of joint models. We need to specify the bivariate or multivariate outcomes specifying the underlying correlations for a more detailed and more meaningful models. This paper shows the model for bivariate binary data using the conditional and marginal models to specify the joint bivariate probability functions. A test procedure is suggested for testing the dependence.”

A heuristic point of this paper for me is that it kind of parameterizing something that is hard to measure and transforming the problem into a parameter testing problem!