Binary logistic regression repeated measures
Analysis of data with repeated measures is often accomplished through the use of Generalized Estimating Equations GEE methodology. Although methods exist for assessing the adequacy of the fitted models for uncorrelated data with likelihood methods, it is not appropriate to use these methods for models fitted with GEE methodology.
Barnhart and Williamson proposed model-based and robust empirically corrected goodness-of-fit tests for GEE modeling with binary responses based on partitioning the space of covariates into distinct regions and forming score statistics that are asymptotically distributed as chi-square random variables with the appropriate degrees of freedom.
In their suggested GEE approach the correlation between two responses was not considered. We here proposed an alternative binary logistic regression repeated measures based on GEE where the correlation between two responses was considered. We extended their work using different correlation structures exchangeable, autoregressive and pairwise correlation along with their suggested identity correlation structure.
The use of Generalized Estimating Equations GEE to binary logistic regression repeated measures repeated binary data has become increasingly common in the health sciences. The analysis of correlated binary responses is often accomplished through the use of GEE methodology for parameter estimation.
Assessment of the adequacy of the fitted GEE model is problematic since no likelihood exists and the residuals are correlated within a cluster.
Tsiatis  proposed a goodness-of-fit test for the logistic regression model which is asymptotically chi-squared and is computed binary logistic regression repeated measures a quadratic form of observed counts minus the expected counts.
Stuart  proposed a goodness-of-fit test statistic for regression with heterogeneous variance, which is asymptotically chi-square if the given model is correct. The test statistic is computed as a quadratic form of observed minus predicted responses. Cessie  discussed a new global test statistic for models with continuous covariates and binary response is introduced. The test statistic is based on nonparametric binary logistic regression repeated measures methods.
Explicit expressions are given the mean and variance of the test statistic. Asymptotic properties are considered and approximate corrections due to parameter estimation are presented. Also Cessie  considered testing the goodness-of-fit of regression models. Emphasis is on a goodness-of-fit test for generalized linear models with canonical link function and known dispersion parameter.
The test based on the score test for extra variation in a random effect model. By choosing a suitable form for the dispersion matrix, a goodness-of-fit test statistic is obtained which is quite similar to test statistics based on non-parametric kernel methods.
The aim of present study was to utilize the BIRDEM data to parameter estimate in binary logistic regression repeated measures main effect model and another model binary logistic regression repeated measures includes the same main effects, the regions, time effects and interaction effects and then to test the goodness-of-fit by using various correlation structures.
Let us consider that each individual is observed for T occasions. Thus we have a Y x 1 random vector of responses for the ith individual where the response variable is binary. Here the response variable binary logistic regression repeated measures dichotomous. We took k independent variables, so for ith individual we have a T x k matrix of covariates. So the variance of y ij is. By first partitioning the covariate space into M distinct region in P-dimensional space.
Let be an be an M x 1 vector, where, I itm is the indicator variable that equals one if the ith subject is in the mth region at the tth occasion and zero otherwise. They define the T x M matrix I i as:. A goodness-of-fit statistic consists of testing H 0: Denote U be the L binary logistic regression repeated measures 1 vector with lth component:.
Then binary logistic regression repeated measures H 0: Where, is a T x T matrix. Note that cov Y i can be consistently estimated by If the correlation matrix R i is correctly specified, then the asymptotic covariance matrix U reduces to. Let H 1 and H 2 be the design matrices in models 1 and 4respectively.
Then intuitively, the degrees-of-freedom of the above chi-square random variables is equal to rank H 2 - H 1. Let design matrix for the ith subject in model 4.
It is easily shown that the tj th element of is equal to Therefore, the goodness-of-fit binary logistic regression repeated measures statistics Q and Q R can be readily calculated once is obtained from the estimating Eq. Data set and covariates: In our study we have used the repeated measures data diabetes mellitus to carry out the analysis. Here the follow up data on patients registered at BIRDEM Bangladesh Institute of Research and Rehabilitation in Diabetes, Endocrine and Metabolic disorders in is used to identify the risk factors responsible for the transitions from controlled diabetic to confirmed diabetic state as well as confirm diabetic to controlled stage of diabetes.
The response variable is defined in terms of the observed glucose level two hours of 75 g-glucose load binary logistic regression repeated measures visit. The cut-off point for the blood binary logistic regression repeated measures level is If the observed response is less than We included two independent variables in the study.
They are age and sex. Out of these variables, age represents the age responds at each visit. The variable is a continuous variable and used directly in the analysis. Sex is categorical variables. Here sex is a dichotomous variable with two categories 0 and 1, 0 stands for female and 1 stands for male. In order to assess the performance of the proposed goodness-of-fit tests, we used data simulated with known distributions from models in the alternative hypothesis to test the goodness-of-fit.
To conduct the proposed goodness-of-fit tests, the following regions were partitioned binary logistic regression repeated measures region1 if age greater than or equal to 50 and male, region 2 if age greater than or equal to 50 and female, region 3 if age less than 50 and male and region 4 if age less than 50 and female.
If any individual occurs any of the four regions then binary logistic regression repeated measures 1 otherwise 0. Time effect represents the two consecutive visits.
Time effect is a dichotomous variable with two categories 0 and 1, 0 stands for first visit and 1 stands for second visit. Interaction 1, interaction 2, interaction 3, interaction 4 are component wise multiplication of region 1, region 2, region 3, region 4 and time effect. The logistic regression model is considered as one of the most important and widely applicable techniques in analyzing repeated outcome variables.
To assess the fit of a model, it is necessary to identify the influential elements. In the logistic regression analysis for repeated binary measures we adjust for setting and the covariates. We assumed independence, exchangeable, autoregressive and pairwise working correlation structures and we obtained standard errors. Table 1 lists the parameter estimates and standard errors for the initial model having only main effects.
According to likelihood test the null hypothesis is rejected under all correlation structures in GEE. In this case has an interpretation that at binary logistic regression repeated measures one of the coefficients binary logistic regression repeated measures different from zero. There exits positive association between the response variable and sex.
The estimated coefficient of the variable age is found to be insignificant in all cases. Hence it may be conclude that these variables has no significant effect on the transition from confirmed diabetes state to controlled diabetes state.
In terms of odds ratio, we may comment that, male patients are 1. We considered additions to this main effects model to provide a better fit to the data. Table 2 displays the results from a model that includes regions, time effects and interactions. In this case we see that several of the effects are significant, indicating their importance in modeling.
Reject the null binary logistic regression repeated measures by likelihood test under independence, exchangeable autoregressive and pairwise correlation structures. So rejection of null hypotheses in this case has an interpretation that at least binary logistic regression repeated measures of the coefficients is different from zero. We also found that under all assumptions region 1 and time effect show positive association and interaction1 shows negative association.
The other coefficients of the variables are found to be insignificant in all cases. From the Table 3the model suggested by Barnhart and Williamson  is highly significant by model based test. Also we see that the null hypothesis is rejected by the empirically corrected test and the model 4 is highly significant.
In this case has an interpretation that the covariates have significant effect. The both goodness-of-fit test provided no evidence for lack of fit by adding regions, time effect and interaction effects. We fit two models to the data. The first model only includes the main effects of age and sex and the second model includes the same main effects and the treatment and time interaction.
Because all the covariates are discrete, the covariate categories were used to form four regions with frequencies. Both the goodness-of-fit tests suggest that the model with only main effects did not fit the data well.
There is a significant time and treatment interaction effect indicating that patients with new treatment improved significantly faster than the patients with the standard treatment.
The model with this interaction term included has a good fit to the data. The parameter estimates and the goodness-of-fit tests obtained here are very similar to the results obtained by using a weighted least squares approach. We are indebted to the Binary logistic regression repeated measures, Department of Statistics, University of Dhaka, Bangladesh for his kind cooperation through this research.
Similar Articles in this Journal. Search in Google Scholar. How to cite this article: Journal of Applied Sciences, 5: Estimates obtained by GEE assuming various correlation structures within repeated outcomes with associated Wald test. Goodness-of-fit test for GEE modeling with binary responses. A goodness-of-fit test for binary regression binary logistic regression repeated measures based on smoothing models. Testing the fit of a regression model via score tests in random effects models.
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