Approximate Methods For Analyzing Semi-Parametric Longitudinal Models With Non-Ignorable Missing Responses

In this thesis, we suggest and explore semi-parametric generalized partially linear mixed models for longitudinal data with non-ignorable and non-monotone missing responses.The key subject of our attention is the estimation of mean response parameters and variance components using a semi-parametric Monte Carlo EM method, where the conditional mean response is semiparametric.We first discuss the penalized regression spline method, which is often referred to as P-splines, for linear mixed model.We investigate the methods for estimating mean response parameters and variance components with complete data.We also investigate the connection between P-splines and linear mixed model through incorporating the non-parametric mean functions into longitudinal linear mixed model.An extensive simulation study using different semiparametric mean response functions are presented.Our simulation study reports that when the true underlying model is partially linear, the penalized spline method provides unbiased and efficient estimators.On the other hand, when the mean response is a correctly specified linear model, the P-spline still provides reliable estimates of the model parameters.Next, we present semi-parametric generalized partially linear mixed models for longitudinal data with non-ignorable missing responses.In this situation, we introduce a parametric model for non-ignorable missing data and incorporate it into the likelihood function.We obtain the asymptotic variances of the proposed estimators by the method of Louis (cf.[2], [7]).In addition, we propose and explore a semi-parametric Monte Carlo EM (MCEM) algorithm for simultaneous estimation of the regression parameters and variance components in partially and in-laws (Ibrahim, AbduAllah, Hesham, Saleh, Yousf, AbduAlaziz, Khawlah and Norah) for their unbelievable love and supports no matter what.For most of all, I would like to thank my loving, supportive, encouraging, and patient husband Waleed Alrajhi whose true support during my Ph.D. studies is very appreciated.The past years have not been an easy journey for both of us in many aspects.I deeply thank him for standing by my side, always and at any cost.Last but not least, my three amazing children, Ali, Mohammed and Almas, who have been an everlasting source of love and enthusiasm.v List of Tables 2.1 Empirical biases, mean squared errors (MSEs), coverage probabilities (CPs), and average lengths of 95% confidence intervals for m = 100 clusters and n = 4 measurements, S = 500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 2.2 Empirical biases, mean squared errors (MSEs), coverage probabilities (CPs), and average lengths of 95% confidence intervals for m = 200 clusters and n = 4 measurements, S = 500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 2.3 Empirical biases, mean squared errors (MSEs), coverage probabilities (CPs), and average lengths of 95% confidence intervals for m = 300 clusters and n = 4 measurements, S = 500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 2.4 Summary statistics of smoking status for youth group 12 -19 years old. . . . . .29 2.5 ML estimates and standard errors of parameters for CCHS. . . . . . . . . . . . .30 3.1 Empirical biases, mean squared errors (MSEs), coverage probabilities (CPs), and average lengths of 95% confidence intervals of maximum likelihood estimators (MLEs) of regression parameters and variance components, under different sample sizes m = 100, 200 and 300 and n = 2.The simulation is under NMAR model with roughly 23% missing response values, the missing data parameters φ φ φ = (-2.5,0.2, 0.3, 1) t , and m i = M = 2000 Gibbs sampling size, under S = 1000 simulation size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 xii 3.2 Empirical biases, mean squared errors (MSEs), coverage probabilities (CPs), and average lengths of 95% confidence intervals of maximum likelihood estimators (MLEs) of regression parameters and variance components, under different sample sizes m = 100, 200 and 300, and n = 2.The simulation is under NMAR model with roughly 30% missing response values, the missing data parameters φ φ φ = (-3, 0.2, 0.3, 1) t , and m i = M = 2000 Gibbs sampling size, under S = 1000 simulation size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 4.1 Comparison of Henderson's method (H) for mixed P-spline model with EM method for LMM.The simulated biases and mean squared errors (MSEs) are for the estimators of mean responses at five different values of x, and variance components, under 500 simulation size.True mean response m 0 (x) is linear. . .78 4.2 Comparison of Henderson's method (H) for mixed P-spline model with EM method for LMM.The simulated biases and mean squared errors (MSEs) are for the estimators of mean responses at five different values of x, and variance components, under 500 simulation size.True mean response m 0 (x) is quadratic.79 4.3 Comparison of Henderson's method (H) for mixed P-spline model with EM method for LMM.The simulated biases and mean squared errors (MSEs) are for the estimators of mean responses at five different values of x, and variance components, under 500 simulation size.True mean response m 0 (x) is exponential.80 5.1 Comparison of our proposed semi-parametric MCEM method (Method 1) with the MCEM method of Ibrahim et al. (2001) (Method 2).The simulated biases and mean squared errors (MSEs) are for the estimators of mean responses at five different values of x and variance components under 1000 simulation size.Missing data parameters φ φ φ = (-2.5,0.2, 0.3, 1) t leads to NMAR with 23% missing responses.True mean response m 0 (x) is linear. . . . . . . . . . . . . . . . . . .110 xiii 5.2 Comparison of our proposed semi-parametric MCEM method (Method 1) with the MCEM method of Ibrahim et al. (2001) (Method 2).The simulated biases and mean squared errors (MSEs) are for the estimators of mean responses at five different values of x and variance components under 1000 simulation size.Missing data parameters φ φ φ = (-2.5,0.2, 0.3, 1) t leads to NMAR with 23% missing responses.True mean response m 0 (x) is nonlinear (quadratic) . . . . . . . . . .111 5.3 Simulated biases and mean squared errors (MSEs) of the semi-parametric MCEM estimators of mean responses at five different values of x and variance components under 1000 simulation size.Missing data parameters φ φ φ = (-3, 0.2, 0.3, 1) t leads to NMAR with 30% missing responses.True mean response m 0 (x) is nonlinear (quadratic). . . . . . . . . . . . . .

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