Several characteristics of confidence intervals for a binomial parameter are considered. In particular confidence intervals when zero success is observed are of our interest. There have been proposed two constructing methods for such cases, which seems to be troublesome. In order to resolve the confict we examine the methods from the viewpoint of coverage probabilities of the central and shortest confidence intervals. After a brief review of several constructing methods of confidence intervals, a simple iterative procedure to find the shortest interval is introduced. Then, in terms of actual coverage probabilities, we examine the properties of several intervals. Some guidelines for selecting appropriate interval are provided.
An exponential-type nonlinear mixed-effect model was fitted to estimate and predict bone mineral density (BMD) at three sites -the lumbar spine, femoral neck, and radius -in 89 postmenopausal women. Years since menopause (YSM) and age at menopause (AAM) were used as explanatory variables. The model was assumed as (BMD)ij=α+ai+(β+bi)exp(γ· (YSM)ij) + δ ((AAM)i -sigma) + eij (i: subject; j: time point; α,β,γ and δ are the population means; ai and bi are the random effects among subjects; eij are random error). Because BMD decreases rapidly after menopause and changes slowly thereafter, an exponential model was assumed. The fitted models were as follows: lumbar spine: (BMD)ij=0.792+ai+(0.179+bi)exp(-0.185 x (YSM)ij)-0.00251((AAM)i -50.6), sigma = (0.106)2, sigma =(0.0442)2, sigmaab = 0.0000720, sigma = (0.0185)2 , femoral neck: (BMD)ij =0.594+ai+(0.161+bi)exp(-0.108 x (YSM)ij)-0.00605((AAM)i -50.6), sigma=(0.108)2, sigma = (0.108)2, sigmaab=-0.00619, sigma = (0.0165)2 , radius: (BMD)ij = 0.458 +ai+ 0.152+bi)exp(-0.0885 x (YSM)ij)-0.00248((AAM)i -50.6) , sigma = (0.108)2 , sigma = (0.106)2, sigmaab=-0.00987, sigma =(0.0115)2. The maximum likelihood model was done with a specific algorithm which took account of highly unbalanced data. These formulae were also used to derive a prediction formula for each individual using a Bayesian approach. This prediction formula with individual longitudinal data makes it possible to predict the future BMD and to estimate the risk of osteoporosis more accurately in individual subjects than the prediction model in cross sectional data.
Artificial selection results in an inflation of inbreeding through reduced effective population size. In recent years, various selection and mating procedures have been proposed to reduce rates of inbreeding in breeding populations. Since these procedures are generally based on optimal solutions of mathematical programming with a huge number of variables, they could not be implemented in most of the practical situations. To overcome this problem, the use of quasi-optimal solution via simulated annealing was considered in this study. The simulated annealing algorithm is an analogy to the fact that if a metal is cooled slowly, the atoms find their optimal positions to achieve a state of minimum energy. The algorithm was illustrated with the problem for finding mating allocation to give minimum inbreeding (minimum coancestry mating). Effect of parameters to control the imaginary temperature was examined in simulated populations, and a reasonable set of parameters in the practical applications was suggested. Using the simulated annealing with the suggested parameters, the effectiveness of the minimum coancestry mating and selection for maximizing genetic diversity was evaluated in simulated pig and broiler populations.
The α-spending function approach is a flexible interim analysis method for clinical trials, which can control the type I error probability without specifying the times and the number of analyses before the trial starts. However validity of the method can be lost when the times or the number of analyses depend on the results of the previous interim analysis because it may cause the inflation of type I error probability. Although data dependent interim analysis plans have not been recommended for this reason, it is possible to calculate the boundaries which can preserve the type I error probability for a certain simple data dependent plan. In this article, we show the way to calculate the adjusted boundaries, and evaluate the power and expected stopping time of the trial for the adjusted and no adjusted boundaries. Then we will show that type I error adjustment for a simple data dependent analysis plan is admissible although the plan does not have significant advantages for the power and the expected stopping time of the trial.