Confidence intervals for the ratio of two variances

In this paper we consider confidence intervals for the ratio of two population variances. We propose a confidence interval for the ratio of two variances based on the t-statistic by deriving its Edgeworth expansion and considering Hall's and Johnson's transformations. Then, we consider the coverage accuracy of suggested intervals and intervals based on the F-statistic for some distributions.     

Confidence intervals and tests on the variance ratio in random models with two variance components

The LM test is modified to test any value of the ratio of two variance components in a mixed effects linear model with two variance components. The test is exact, so it can be used to construct exact confidence intervals on this ratio.Exact Neyman-Pearson (NP) tests on the variance ratio are described.Their powers provide attainable upper bounds on powers of tests on the variance ratio.Efficiencies of LM tests, which include ANOVA tests, and NP tests are compared for unbalanced, random, one-way ANOVA models.Confidence intervals corresponding to LM tests and NP tests are described.     

Confidence intervals for the between group variance in the unbalanced one-way random effects model of anaylsis of variance

A confidence interval for the between group variance is proposed which is deduced from Wald'sexact confidence interval for the rtio of the two variance components in the one-way random effects model and the exact confidence interval for the error variance resp.an unbiased estimator of the error variance. In a simulation study the confidence coeffecients for these two intervals are compared with the confidence coefficients of two other commonly used confidence intervals. There the confidence interval derived here yields confidence coefficiends which are always greater than the prescriped level.     

Small samples confidencce intervals on common mean of two normal distributions variances

Let X₁,X₂,…,X_m be distributed normally with mean μ and variance σ² _X; Let Y₁,Y₂,…,Y_n be distributed normally with mean μ and variance σ² _Y; let X₁,X₂,…,X_m,Y₁,Y₂,…,Y_n be jointly independent. There have been several papers written concerning point estimation of μ for this problem, but very little is available in the literature concerning confidence intervals on the common mean μ. In this paper a method is proposed that results in a confidence interval with confidence coefficient essentially equal to a prescribed value 1 - α. The method is evaluated and compnred with other methods through the expected length of the confidence interval.     

Confidence intervals for variance components in unbalanced one-way random effects model using non-normal distributions

In scenarios where the variance of a response variable can be attributed to two sources of variation, a confidence interval for a ratio of variance components gives information about the relative importance of the two sources. For example, if measurements taken from different laboratories are nine times more variable than the measurements taken from within the laboratories, then 90% of the variance in the responses is due to the variability amongst the laboratories and 10% of the variance in the responses is due to the variability within the laboratories. Assuming normally distributed sources of variation, confidence intervals for variance components are readily available. In this paper, however, simulation studies are conducted to evaluate the performance of confidence intervals under non-normal distribution assumptions. Confidence intervals based on the pivotal quantity method, fiducial inference, and the large-sample properties of the restricted maximum likelihood (REML) estimator are considered. Simulation results and an empirical example suggest that the REML-based confidence interval is favored over the other two procedures in unbalanced one-way random effects model.     

Confidence intervals for the treatment effect on the treated

The average effect of the treatment on the treated is a quantity of interest in observational studies in which no definite parameter can be used to quantify the treatment effect, such as those where only a random subset of the data obtained by stratification can be used for analysis. Nonparametric confidence intervals for this quantity appear to be known only in the case where the responses to the treatment are binary and the data fall into a single stratum. We propose nonparametric confidence intervals for the average effect of the treatment on the treated in studies involving one or more strata and general numerical responses.     

Confidence intervals on the total variance in an unbalanced two-fold nested classification with equal subsampling

A problem of interest in a variance component analysis is the construction of a confidence interval on the variance of a single observation. This article considers an unbalanced two-fold nested classification with equal subsampling and compares two methods for constructing this interval . Computer simulations indicate that one of these methods in general will provide an interval that has an achieved confidence coefficient at least as great as the stated value.     

Approximate confidence intervals for a ratio of two variance components

New approximate confidence intervals for the ratio of two variance components in an unbalanced mixed .linear model with a single set of random effects are proposed. Contrary to the confidence intervals known in the literature the new intervals preserve the confidence coefficient and cover the exact confidence interval which, however, is not easy to establish as it requires the solution of complicated nonlinear equations.     

Confidence intervals for estimating the population signal-to-noise ratio: a simulation study

This paper considered several confidence intervals for estimating the population signal-to-noise ratio based on parametric, non-parametric and modified methods. A simulation study has been conducted to compare the performance of the interval estimators under both symmetric and skewed distributions. We reported coverage probability and average width of the interval estimators. Based on the simulation study, we observed that some of our proposed interval estimators are performing better in the sense of smaller width and coverage probability and have been recommended for the researchers.     

Confidence intervals on intraclass correlation coefficients in a balanced two-factor random design

A modified large-sample (MLS) approach and a generalized confidence interval (GCI) approach are proposed for constructing confidence intervals for intraclass correlation coefficients. Two particular intraclass correlation coefficients are considered in a reliability study. Both subjects and raters are assumed to be random effects in a balanced two-factor design, which includes subject-by-rater interaction. Computer simulation is used to compare the coverage probabilities of the proposed MLS approach (GiTTCH) and GCI approaches with the Leiva and Graybill [1986. Confidence intervals for variance components in the balanced two-way model with interaction. Comm. Statist. Simulation Comput. 15, 301–322] method. The competing approaches are illustrated with data from a gauge repeatability and reproducibility study. The GiTTCH method maintains at least the stated confidence level for interrater reliability. For intrarater reliability, the coverage is accurate in several circumstances but can be liberal in some circumstances. The GCI approach provides reasonable coverage for lower confidence bounds on interrater reliability, but its corresponding upper bounds are too liberal. Regarding intrarater reliability, the GCI approach is not recommended because the lower bound coverage is liberal. Comparing the overall performance of the three methods across a wide array of scenarios, the proposed modified large-sample approach (GiTTCH) provides the most accurate coverage for both interrater and intrarater reliability.     

Approximate confidence bounds for ratios of variance components in two-way unbalanced crossed models with interactions

Consider the general unbalanced two-factor crossed components-of-variance model with interaction given by Y_ijk: = μ+A_i: +B_j: + C_ij: +E_ijk: (i = 1,2, … a; j = 1,…,b; k = 1,…,.n_ij:=0) A_i:,B_j:, C_ij: and E_ijk: are independent unobservable random variables. Also A_i:sim; N(0,σ² _A),B_j: ～ N(0,σ² _B), C_ij:～N(0,s² _C:) and E_ijk:～N(0,s² _E:). In this paper approximate confidence bounds are obtained for ρ_A: = ρ² _A/² and ρ_B: = ρ² _B:/ρ² (where σ² = σ² _A:+ σ² _B+σ² _Cσ² _E) for special cases of the above model. The balanced incomplete block model is studied as a special case.     

Bias in linear model power and sample size due to estimating variance

Planning a study using the General Linear Univariate Model often involves sample size calculation based on a variance estimated in an earlier study. Noncentrality, power, and sample size inherit the randomness. Additional complexity arises if the estimate has been censored. Left censoring occurs when only significant tests lead to a power calculation, while right censoring occurs when only non-significant tests lead to a power calculation. We provide simple expressions for straightforward computation of the distribution function, moments, and quantiles of the censored variance estimate, estimated noncentrality, power, and sample size. We also provide convenient approximations and evaluate their accuracy. The results allow demonstrating that ignoring right censoring falsely widens confidence intervals for noncentrality and power, while ignoring left censoring falsely narrows the confidence intervals. The new results allow assessing and avoiding the potentially substantial bias that censoring may create.     

Confidence intervals for the length of a vector mean

Suppose we have a random sample of size n from a multivariate distribution with finite moments, for which a parametric form is not available. We wish to obtain a confidence interval (CI) for the length of its mean. The usual method is to Studentize. The resulting CIs are not exact. The error in their nominal levels is ～n ^?1/2 and ～n ^?1 in the one-sided and two-sided cases. We show how to reduce these errors to ～n ^?3/2 and ～n ^?2.     

An upper bound on a ratio of variances

Several processes may be monitored in terms of their variances relative to each other by the maximum ratio of variances. Based on the observed value an upper confidence bound for the population value is derived and tables are supplied for its implementation. Such bounds may be used when the experimenter wishes to establish that the variances are "not too different". The usual testing of variances for equality is noticed to be inappropriate in such cases.     

Confidence intervals for survival quantiles in the Cox regression model

Median survival times and their associated confidence intervals are often used to summarize the survival outcome of a group of patients in clinical trials with failure-time endpoints. Although there is an extensive literature on this topic for the case in which the patients come from a homogeneous population, few papers have dealt with the case in which covariates are present as in the proportional hazards model. In this paper we propose a new approach to this problem and demonstrate its advantages over existing methods, not only for the proportional hazards model but also for the widely studied cases where covariates are absent and where there is no censoring. As an illustration, we apply it to the Stanford Heart Transplant data. Asymptotic theory and simulation studies show that the proposed method indeed yields confidence intervals and bands with accurate coverage errors.