Discrete-time Markov chains have been used to analyze the transition of subjects from intact cognition to dementia with mild cognitive impairment and global impairment as intervening transient states, and death as competing risk. A multinomial logistic regression model is used to estimate the probability distribution in each row of the one-step transition matrix that correspond to the transient states. We investigate some goodness of fit tests for a multinomial distribution with covariates to assess the fit of this model to the data. We propose a modified chi-square test statistic and a score test statistic for the multinomial assumption in each row of the transition probability matrix. Multinomial logistic regression with categorical covariates can be analyzed by contingency tables. Exact p-value of goodness of fit test can be calculated based on MCMC samples. We show a hybrid scheme of the sequential importance sampling (SIS) procedure and the MCMC procedure for two-way contingency tables. We apply the SIS-MCMC procedure to the Nun Study data, a cohort of 461 participants on aging disease. The presence of the APOE-4 allele, levels of education are included as covariates in the application. Different grouping methods on age are also discussed. Separating data into four groups based on quantiles of age is recommended in the Nun Study. The traditional logistic regression model restricts the analysis on observations with complete covariate data, and ignores the incomplete observations due to missing or censored covariates. However, much information is lost in this approach. We introduce a maximum likelihood estimation based on the joint distribution of binary response variable, complete covariate and a right censored covariate. Simulation results show that the estimates with the new method are more accurate than those with the traditional complete case method when the sample size is relatively small or medium, across different censoring pattern. The proposed method is also applied to a model to analyze the relationship between the presence of arteriolosclerosis and the stay time in mild cognitive impairment of patients from SMART Study.