A global goodness-of-fit statistic for Cox regression models

In this paper, a global goodness-of-fit test statistic for a Cox regression model, which has an approximate chi-squared distribution when the model has been correctly specified, is proposed. Our goodness-of-fit statistic is global and has power to detect if interactions or higher order powers of covariates in the model are needed. The proposed statistic is similar to the Hosmer and Lemeshow (1980, Communications in Statistics A10, 1043-1069) goodness-of-fit statistic for binary data as well as Schoenfeld's (1980, Biometrika 67, 145-153) statistic for the Cox model. The methods are illustrated using data from a Mayo Clinic trial in primary billiary cirrhosis of the liver (Fleming and Harrington, 1991, Counting Processes and Survival Analysis), in which the outcome is the time until liver transplantation or death. The are 17 possible covariates. Two Cox proportional hazards models are fit to the data, and the proposed goodness-of-fit statistic is applied to the fitted models.     

Likelihood methods for incomplete longitudinal binary responses with incomplete categorical covariates

We consider longitudinal studies in which the outcome observed over time is binary and the covariates of interest are categorical. With no missing responses or covariates, one specifies a multinomial model for the responses given the covariates and uses maximum likelihood to estimate the parameters. Unfortunately, incomplete data in the responses and covariates are a common occurrence in longitudinal studies. Here we assume the missing data are missing at random (Rubin, 1976, Biometrika 63, 581-592). Since all of the missing data (responses and covariates) are categorical, a useful technique for obtaining maximum likelihood parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). In using the EM algorithm with missing responses and covariates, one specifies the joint distribution of the responses and covariates. Here we consider the parameters of the covariate distribution as a nuisance. In data sets where the percentage of missing data is high, the estimates of the nuisance parameters can lead to highly unstable estimates of the parameters of interest. We propose a conditional model for the covariate distribution that has several modeling advantages for the EM algorithm and provides a reduction in the number of nuisance parameters, thus providing more stable estimates in finite samples.     

Specific non-native hydrophobic interactions in a hidden folding intermediate: implications for protein folding

Structures of intermediates and transition states in protein folding are usually characterized by amide hydrogen exchange and protein engineering methods and interpreted on the basis of the assumption that they have native-like conformations. We were able to stabilize and determine the high-resolution structure of a partially unfolded intermediate that exists after the rate-limiting step of a four-helix bundle protein, Rd-apocyt b(562), by multidimensional NMR methods. The intermediate has partial native-like secondary structure and backbone topology, consistent with our earlier native state hydrogen exchange results. However, non-native hydrophobic interactions exist throughout the structure. These and other results in the literature suggest that non-native hydrophobic interactions may occur generally in partially folded states. This can alter the interpretation of mutational protein engineering results in terms of native-like side chain interactions. In addition, since the intermediate exists after the rate-limiting step and Rd-apocyt b(562) folds very rapidly (k(f) approximately 10(4) s(-1)), these results suggest that non-native hydrophobic interactions, in the absence of topological misfolding, are repaired too rapidly to slow folding and cause the accumulation of folding intermediates. More generally, these results illustrate an approach for determining the high-resolution structure of folding intermediates.