Data with surrogate outcomes frequently arise in medical research when true outcomes of interest are financially or technologically unable to ascertain, but surrogate outcomes, some correlates of the true outcomes, are often measured instead. To solve the problem with surrogate outcomes in multiple clusters, we proposed a semi-parametric maximum likelihood estimation method for random effects logistic regression model in this thesis. A kernel smoother is used to impute the probability of obtaining surrogate outcomes from validation data. This method is an extension of the method developed by Pepe in 1992. Pepe's method can only be used for fixed models while our proposed method takes the random effects into account and is for mixed models. We compared our proposed method with other methods using logistic regression model, Pepe's model and generalized linear mixed model by extensive simulation studies. The results show that the proposed method can gain the precision and accuracy for regression parameters estimation only if the surrogate is a good correlate of the true outcome or if the amount of missing data is moderate or low. The possible working region in terms of validation fraction and association between true and surrogate outcomes is given when surrogate and true outcomes are dichotomous variables.
