Interval Estimation of Simple Difference under Independent Negative Binomial Sampling

When the sample size is not large or when the underlying disease is rare, to assure collection of an appropriate number of cases and to control the relative error of estimation, one may employ inverse sampling, in which one continues sampling subjects until one obtains exactly the desired number of cases. This paper focuses discussion on interval estimation of the simple difference between two proportions under independent inverse sampling. This paper develops three asymptotic interval estimators on the basis of the maximum likelihood estimator (MLE), the uniformly minimum variance unbiased estimator (UMVUE), and the asymptotic likelihood ratio test (ALRT). To compare the performance of these three estimators, this paper calculates the coverage probability and the expected length of the resulting confidence intervals on the basis of the exact distribution. This paper finds that when the underlying proportions of cases in both two comparison populations are small or moderate (≤0.20), all three asymptotic interval estimators developed here perform reasonably well even for the pre-determined number of cases as small as 5. When the pre-determined number of cases is moderate or large (≥50), all three estimators are essentially equivalent in all the situations considered here. Because application of the two interval estimators derived from the MLE and the UMVUE does not involve any numerical iterative procedure needed in the ALRT, for simplicity we may use these two estimators without losing efficiency.     

A Note on Interval Estimation of the Simple Difference in Data with Correlated Matched Pairs

When we employ cluster sampling to collect data with matched pairs, the assumption of independence between all matched pairs is not likely true. This paper notes that applying interval estimators, that do not account for the intraclass correlation between matched pairs, to estimate the simple difference between two proportions of response can be quite misleading, especially when both the number of matched pairs per cluster and the intraclass correlation between matched pairs within clusters are large. This paper develops two asymptotic interval estimators of the simple difference, that accommodate the data of cluster sampling with correlated matched pairs. This paper further applies Monte Carlo simulation to compare the finite sample performance of these estimators and demonstrates that the interval estimator, derived from a quadratic equation proposed here, can actually perform quite well in a variety of situations.     

Multiple Comparisons in Nonlinear Repeated Measurements

It is interest to compare functions of parameters in nonlinear models for repeated measurements. In a pharmacokinetic model, the maximum value of the model would be a nonlinear function of some unknown parameters. In this paper, simultaneous confidence intervals of functions of parameters in a nonlinear model for repeated mesurement data are considered to compare the populations.     

Sample Size Calculation for Intraclass Correlation in the Presence of Random Loss of Sampled Patients

When a trial involves an invasive laboratory test procedure or requires patients to make a commitment to follow a restrictive test schedule, we can often lose a great proportion of our sampled patients due to refusal of participation into our study. Therefore, incorporating the possible loss of patients into sample size calculation is certainly important in the planning stage of a study. In this paper, we have generalized the sample size calculation procedure for intraclass correlation by accounting for the random loss of patients in the beginning of a trial. We have demonstrated that the simple ad hoc procedure, that raises the estimated sample size in the absence of loss of patients by the factor 1/p_o, where p_o is the retention probability for a randomly selected patient, is adequate when p_o is large (=0.80). When p_o is small (i.e., a high refusal rate), however, use of this simple ad hoc procedure tends to underestimate the required sample size. Furthermore, we have found that if the individual retention probability varied substantially among patients, then the magnitude of the above underestimation could even be critical and therefore, the application of the simple direct adjustment procedure in this situation should be avoided.     

Interval estimation for a difference between intraclass kappa statistics

Model-based inference procedures for the kappa statistic have developed rapidly over the last decade. However, no method has yet been developed for constructing a confidence interval about a difference between independent kappa statistics that is valid in samples of small to moderate size. In this article, we propose and evaluate two such methods based on an idea proposed by Newcombe (1998, Statistics in Medicine, 17, 873-890) for constructing a confidence interval for a difference between independent proportions. The methods are shown to provide very satisfactory results in sample sizes as small as 25 subjects per group. Sample size requirements that achieve a prespecified expected width for a confidence interval about a difference of kappa statistic are also presented.     

Analysis of Repeated Measurements with Ante-Dependence Covariance Models

This paper considers the use of ante-dependence models in problems with repeated measures through time. These are conditional regression models which reflect the dependence of a measure on some of the previous observations from the same subject. We present maximum likelihood estimators of the covariance matrix and procedures for selecting the order of ante-degendence based on penalized like-lihoods. Extensions to missing data situations are discussed. We propose Wald-type test statistics and apply them in two situations common in experiments with repeated measures: one with pre-study observations and another one with small sample size relative to the number of time periods. In these examples, tests assuming ante-dependence find effects which are not detected using competing procedures.     

Estimation of Sample Size for Reference Interval Studies

The currently used criterion for sample size calculation in a reference interval study is not well stated and leads to imprecise control of the ratio in question. We propose a generalization of the criterion used to determine sufficient sample size in reference interval studies. The generalization allows better estimation of the required sample size when the reference interval estimation will be using a power transformation or is nonparametric. Bootstrap methods are presented to estimate sample sizes required by the generalized criterion. Simulation of several distributions both symmetric and positively skewed is presented to compare the sample size estimators. The new method is illustrated on a data set of plasma glucose values from a 50‐g oral glucose tolerance test. It is seen that the sample sizes calculated from the generalized criterion leads to more reliable control of the desired ratio.     

Estimation of Treatment Difference and Standard Deviation with Blinded Data in Clinical Trials

In an ongoing clinical trial and while the randomized treatment codes remain blinded, it is often desirable to estimate the treatment difference and standard deviation for a normally‐distributed response variable. This is particularly useful for estimating the sample size for future trials or for adjusting the sample size for the ongoing trial. We describe the limitations of an available EM algorithm‐based procedure to reestimate the standard deviation without unblinding the codes for the two treatments. We introduce a new procedure and propose a clinical trial design for estimating both the treatment difference and standard deviation without unblinding. The performance of the proposed procedure is evaluated in a simulation study.     

Interval Estimation of the Effective Population Size from Heterozygote‐Excess in SNP Markers

The effective population size N_e is an important parameter in population genetics and conservation biology. In recent years, there has been great interest in the use of molecular markers to estimate N_e. Although the point estimates from molecular markers in general suffer from a low reliability, the use of single nucleotide polymorphism (SNP) markers over a wide range of genome is expected to remarkably improve the reliability. In this study, expressions were derived for interval estimates of N_e from one published method, the heterozygote‐excess method, when it is applied to SNP markers. The conditional variance theory is applied to the derivation of a confidence interval for N_e under random union of gametes, monogamy and polygyny. Stochastic simulation shows that the obtained confidence interval is slightly conservative, but fairly useful for practical applications. The result is illustrated with real data on SNP markers in a pig strain.     

The Link between Dynamics and Statistics in Analysis of Repeated Measurements

A method of analysis of repeated measurements is proposed. When a drug is administered orally or by intramuscular injection, the concentration of the drug in the blood-time curve can often be modeled using a one-compartment model assuming first-order absorption in the field of pharmacokinetics. As to incorporation of individual variation, which is important statistically in modeling of repeated measurements, variations within an individual are taken into account in the model by a Wiener process and variations between individuals are expressed as differences in parameters. Accordingly, in this study, repeated measurements are modeled by a continuous time series statistically which was obtained from a one-compartment model assuming first-order absorption representing the dynamics affecting the data. The proposed model is not a random or mixed effect model but a fixed effect model; however, the covariance matrix includes values which differ between individuals. This is a new type of statistical modeling for repeated measurements, with flexibility, due to use of the link between statistics and dynamics. On the basis of the above, hypotheses of equality of treatment effects and equality of parameters between individuals are tested.     

Testing Equality of Means and Simultaneous Confidence Intervals in Repeated Measures with Missing Data

In this paper, repeated measures with intraclass correlation model is considered when the observations are missing at random. An exact test for the equality of the mean components and simultaneous confidence intervals (Scheffé and Bonferroni inequality types) are given for linear contrasts of the mean components when the missing observations are of a monotone type. When the missing observations are not of the monotone type, the maximum likelihood estimates are obtained numerically by iterative methods given in Srivastava and Carter (1986). These estimators are then used to obtain asymptotic tests and confidence intervals for the equality of mean components and linear contrasts, respectively. An example is given to illustrate the method.     

A Further Problem of Interval Estimation of an Unknown Regressor in Model I of Linear Regression

For the model y = α + βx + ? (model I) of linear regression we dealt with in KUHNERT and HORN (1980) the determination of a confidence interval for that x₀ where the expectation Ey reaches a given value y₀. Here we start with realizations of random variables y (i = 1,…, m) being independent of x which are given in addition to the realizations of-y. Now y₀ denotes the unknown value of \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \sum \limits_{i = 1}^m $\end{document} ciEy and x₀ the x-value where the expectation Ey reaches that value y₀. For this x₀ we give a confidence interval. Applications stem from dose response assays.     

Defining and Detecting Second Order Correlations in Repeated Measurements Designs

It is shown that 3-point response curves from N individuals imply a second order correlation (triplet correlation) between the 3 series of repeated measurements if the curves are clustered as to their shapes. Coefficients are defined and tests are suggested for triplet correlations. Nonparametric alternatives to ANOVA are discussed for repeated measurement designs involving shapeclustered response curves.     

Confidence Interval of a Proportion with Over‐dispersion

A new approach that extends the classical Clopper‐Pearson procedure is proposed for the estimation of the (1–α)% confidence interval of a proportion with over‐dispersion. Over‐dispersion occurs when a proportion of interest shows more variation (variance inflation) than predicted by the binomial distribution. There are two steps in the approach. The first step consists of the estimation of the variance inflation factor. In the second step, an extended Clopper‐Pearson procedure is applied to calculate the confidence interval after the effective sample size is obtained by adjusting with the estimated variance inflation factor. The performance of the extended Clopper‐Pearson procedure is evaluated via a Monte Carlo study under the setup motivated from head lice studies. It is demonstrated that the 95% confidence intervals constructed from the new approach generally have the closest coverage rate to target (95%) when compared with those constructed from competing procedures.     

Interval estimation of the kappa coefficient with binary classification and an equal marginal probability model

We derive a likelihood score method for interval estimation of the intraclass version of the kappa coefficient of agreement with binary classification using a general theory of Bartlett (1953, Biometrika 40, 306-317). By exact evaluation, we investigate statistical properties of the score method, the chi-square goodness-of-fit procedure (Donner and Eliasziw, 1992, Statistics in Medicine 11, 1511-1519; Hale and Fleiss, 1993, Biometrics 49, 523-534), and a crude confidence interval for small and medium sample sizes. Actual coverage percentages of the score and chi-square methods are satisfactorily close to the nominal confidence coefficient, while that of the crude method is quite unsatisfactory. The expected length of the score method is shorter than that of the chi-square procedure when the response rate is very small or very large.     

Diminished Confidence Levels in the Interval Estimation of an Unknown Regressor in Model I of Linear Regression

For the model y=β₀ + β₁ x + e (model I of linear regression) in the literature confidence estimators of an unknown position x₀ are given at which either the expectation of y is given (see FIELLER, 1944; FINNEY, 1952), or realizations of y are given (see GRAYBILL, 1961). These confidence regions with level 1—α need not be intervals. The occurrence of interval shape is a random event. Its probability is equal to the power of the t test for the examination of the hypothesis H: β₁ = 0. The papers mentioned above claim to provide confidence intervals with level 1 ? α. But because of the restriction of (1 —α)—confidence regions to intervals the true confidence probability is the conditional probability W_c: W_c = P (the confidence region covers x₀| the region has interval shape). Here this conditional probability is shown to be less than 1 —α. Evidence on the possible deviations from 1 —α has been obtained by simulations.     

Approximate Confidence Interval on Ratio of Two Variances in a Two-way Crossed Model

The model considered is a two-factor cross-classification variance components model with one observation per cell. Let the two factors be A and B, the problem is to obtain an approximate confidence interval for the ratio of variance component A over variance component B. In this paper, a method of solving this problem is established and simulations are performed to check the method.     

Highest Density Difference Region Estimation with Application to Flow Cytometric Data

Motivated by the needs of scientists using flow cytometry, we study the problem of estimating the region where two multivariate samples differ in density. We call this problem highest density difference region estimation and recognise it as a two‐sample analogue of highest density region or excess set estimation. Flow cytometry samples are typically in the order of 10 000 and 100 000 and with dimension ranging from about 3 to 20. The industry standard for the problem being studied is called Frequency Difference Gating, due to Roederer and Hardy ( 2001 ). After couching the problem in a formal statistical framework we devise an alternative estimator that draws upon recent statistical developments such as patient rule induction methods. Improved performance is illustrated in simulations. While motivated by flow cytometry, the methodology is suitable for general multivariate random samples where density difference regions are of interest.     

Sample Size Estimation in Cluster Randomized Studies with Varying Cluster Size

Cluster randomized studies are common in community trials. The standard method for estimating sample size for cluster randomized studies assumes a common cluster size. However often in cluster randomized studies, size of the clusters vary. In this paper, we derive sample size estimation for continuous outcomes for cluster randomized studies while accounting for the variability due to cluster size. It is shown that the proposed formula for estimating total cluster size can be obtained by adding a correction term to the traditional formula which uses the average cluster size. Application of these results to the design of a health promotion educational intervention study is discussed.