The objective of this work is to introduce a new method called the Survivorship Instantaneous Log-odds Ratios (SILOR); to illustrate the creation of SILOR from empirical bivariate survival functions; to also derive standard errors of estimation; to compare results with those derived from logistic regression. Hip fracture, AGE and BMI from the Third National Health and Nutritional Examination Survey (NHANES III) were used to calculate empirical survival functions for the adverse health outcome (AHO) and non-AHO. A stable copula was used to create a parametric bivariate survival function, that was fitted to the empirical bivariate survival function. The bivariate survival function had SILOR contours which are not constant. The proposed method has better advantages than logistic regression by following two reasons. The comparison deals with (i) the shapes of the survival surfaces, S(X1, X2), and (ii) the isobols of the log-odds ratios. When using logistic regression the survival surface is either a hyper plane or at most a conic section. Our approach preserves the shape of the survival surface in two dimensions, and the isobols are observed in every detail instead of being overly smoothed by a regression with no more than a second degree polynomial. The present method is straightforward, and it captures all but random variability of the data.

<p>This thesis proposes a simplification of the model for dependent Bernoulli variables presented in Cox and Snell (1989). The simplified model, referred to as the simplified Cox model, is developed for identically distributed and dependent Bernoulli variables.</p><p>Properties of the model are presented, including expressions for the loglikelihood function and the Fisher information. The special case of a bivariate symmetric model is studied in detail. For this particular model, it is found that the number of design points in a locally D-optimal design is determined by the log-odds ratio between the variables. Under mutual independence, both a general expression for the restrictions of the parameters and an analytical expression for locally D-optimal designs are derived.</p><p>Focusing on the bivariate case, score tests and likelihood ratio tests are derived to test for independence. Numerical illustrations of these test statistics are presented in three examples. In connection to testing for independence, an E-optimal design for maximizing the local asymptotic power of the score test is proposed.</p><p>The simplified Cox model is applied to a dental data. Based on the estimates of the model, optimal designs are derived. The analysis shows that these optimal designs yield considerably more precise parameter estimates compared to the original design. The original design is also compared against the E-optimal design with respect to the power of the score test. For most alternative hypotheses the E-optimal design provides a larger power compared to the original design.</p>

This thesis proposes a simplification of the model for dependent Bernoulli variables presented in Cox and Snell (1989). The simplified model, referred to as the simplified Cox model, is developed for identically distributed and dependent Bernoulli variables. Properties of the model are presented, including expressions for the loglikelihood function and the Fisher information. The special case of a bivariate symmetric model is studied in detail. For this particular model, it is found that the number of design points in a locally D-optimal design is determined by the log-odds ratio between the variables. Under mutual independence, both a general expression for the restrictions of the parameters and an analytical expression for locally D-optimal designs are derived. Focusing on the bivariate case, score tests and likelihood ratio tests are derived to test for independence. Numerical illustrations of these test statistics are presented in three examples. In connection to testing for independence, an E-optimal design for maximizing the local asymptotic power of the score test is proposed. The simplified Cox model is applied to a dental data. Based on the estimates of the model, optimal designs are derived. The analysis shows that these optimal designs yield considerably more precise parameter estimates compared to the original design. The original design is also compared against the E-optimal design with respect to the power of the score test. For most alternative hypotheses the E-optimal design provides a larger power compared to the original design.

This paper develops a new approach to the analysis of longitudinal paired comparison data, where comparisons of the same objects by the same judges are made on more than one occasion. As an alternative to other recent approaches to such data, which are based on Kalman filter- ing, our approach treats the problem as one of multivariate multinomial data, allowing dependence terms between comparisons over time to be incorporated. The resulting model can be fitted as a Poisson log-linear model and has parallels with the quadratic binary exponential distribution of Cox. An example from the British Household Panel Survey illustrates the approach. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics