A PC program for computing confidence bands for average and individual growth curves
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Growth curve (statistics)
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The increase in the squared multiple correlation coefficient (ΔR 2 ) associated with a variable in a regression equation is a commonly used measure of importance in regression analysis. Algina, Keselman, and Penfield found that intervals based on asymptotic principles were typically very inaccurate, even though the sample size was quite large (i.e., larger than 200). However, they also reported that probability coverage for the confidence intervals based on a bootstrap method was typically quite accurate, and moreover, this accuracy was obtained with relatively small sample sizes with six or fewer predictors. They further speculated that nonnormality would likely affect the accuracy of interval coverage. In the present study, the authors investigated the accuracy of coverage probability for confidence intervals obtained by using asymptotic and percentile bootstrap methodology when either predictors, residuals, or both are nonnormal. Coverage probability for asymptotic confidence intervals is poor, but adequate coverage probability can be obtained with reasonable sample sizes by using percentile bootstrap methodology. As well, the authors found that the width of these intervals was relatively precise (i.e., narrow) for the larger cases of sample size investigated.
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Confidence intervals for the between study variance are useful in random-effects meta-analyses because they quantify the uncertainty in the corresponding point estimates. Methods for calculating these confidence intervals have been developed that are based on inverting hypothesis tests using generalised heterogeneity statistics. Whilst, under the random effects model, these new methods furnish confidence intervals with the correct coverage, the resulting intervals are usually very wide, making them uninformative. We discuss a simple strategy for obtaining 95 % confidence intervals for the between-study variance with a markedly reduced width, whilst retaining the nominal coverage probability. Specifically, we consider the possibility of using methods based on generalised heterogeneity statistics with unequal tail probabilities, where the tail probability used to compute the upper bound is greater than 2.5 %. This idea is assessed using four real examples and a variety of simulation studies. Supporting analytical results are also obtained. Our results provide evidence that using unequal tail probabilities can result in shorter 95 % confidence intervals for the between-study variance. We also show some further results for a real example that illustrates how shorter confidence intervals for the between-study variance can be useful when performing sensitivity analyses for the average effect, which is usually the parameter of primary interest. We conclude that using unequal tail probabilities when computing 95 % confidence intervals for the between-study variance, when using methods based on generalised heterogeneity statistics, can result in shorter confidence intervals. We suggest that those who find the case for using unequal tail probabilities convincing should use the ‘1–4 % split’, where greater tail probability is allocated to the upper confidence bound. The ‘width-optimal’ interval that we present deserves further investigation.
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We construct bootstrap confidence intervals for reliability, R= P{X>Y}, where X and Y are independent normal random variables. One way ANOVA random effect models are assumed for the populations of X and Y, where standard deviations and are unequal. We investigate the accuracy of the proposed bootstrap confidence intervals and classical confidence intervals work better than classical confidence interval for small sample and/or large value of R.
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The Bitterlich Sampling (horizontal point sampling) is a common method in forest inventories. By this method, the Horvitz-Thompson estimator is used in a number of independent sampling points for the estimation of overall tree volume in a forest area/stand. In this paper, confidence intervals are constructed and evaluated using the normal approach and two bootstrap methods; the percentile method (Cα) and the bias-corrected and accelerated method (BCα). The simulation results show that the normal confidence interval has better coverage of true value at sample size 10. At sample sizes 20 and 30, it seems that there are no substantial differences in coverage between confidence intervals, although it could be noted a small superiority of BCα method. At sample size 40, the coverage of the three confidence intervals is higher than the nominal coverage (95%).
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This chapter contains sections titled: Confidence Intervals Sample Size Needed for a Desired Confidence Interval The t Distribution Confidence Interval for the Mean Using the t Distribution Estimating the Difference Between Two Means: Unpaired Data Estimating the Difference Between Two Means: Paired Comparison Problems
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Abstract Under the random effects model for meta-analysis, confidence intervals for the overall effect are typically constructed using quantiles of the standard normal distribution. We discuss confidence intervals based on both the standard normal distribution and the t-distribution, in conjunction with several methods of estimating the heterogeneity variance for a standardized mean difference, and we compare the empirical coverage probabilities of the intervals using simulation. The coverage probabilities of intervals based on an approximate t-statistic are higher than the coverage probabilities for the standard normal intervals, and are very close to the specified confidence level even for small meta-analysis sample size. Moreover, intervals based on the approximate t-statistic appear relatively robust to different methods of estimating the heterogeneity variance, unlike the normal intervals. Thus, we conclude that confidence intervals based on the t-statistic are superior to the standard normal confidence intervals for a standardized mean difference, and should be used by practitioners in place of the normal intervals. Keywords: Coverage probabilityConfidence limitsRandom effects modelSimulation study t-StatisticWeighted average estimation Acknowledgment The authors thank the referee for his helpful comments.
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Confidence intervals for the variance and difference of variances of Birnbaum-Saunders distributions
Herein, we present confidence intervals for the variance and difference of variances of Birnbaum-Saunders distributions constructed by using the bootstrap confidence interval (BCI), the generalized confidence interval (GCI), the Bayesian confidence interval (BayCI), and the highest posterior density interval (HPD). The performances of the proposed confidence intervals were investigated in terms of their coverage probabilities and average lengths by running a Monte Carlo simulation. The simulation results reveal that HPD performed the best, even for small sample sizes and/or different values of the shape parameter. To illustrate the efficacy of the proposed confidence intervals, we applied them to datasets of the PM 2.5 concentration in Chiang Mai, Thailand.
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The mean score of a sample deviates most probably from the mean score of the population, in which one is most interested. It is possible to calculate a kind of minimum and maximum value, the so called confidence interval, which indicates the position of the mean score of the population. Some worked examples elucidate the procedure of constructing such confidence intervals for the population mean, using the standard error of the (sample) mean (SEM).
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This study compares and analyzes the coverage probabilities and the average interval lengths of confidence interval for a population mean based on the dependent bootstrap procedure against those based on the independent bootstrap procedure. Both dependent and independent bootstrap confidence intervals for a population mean are computed by the Bootstrap-t, Percentile, and Modified Percentile methods. Simulations show that the coverage probabilities of the dependent bootstrap confidence intervals are similar to those of the independent bootstrap confidence intervals. The average interval lengths of the dependent bootstrap method are shorter for most situations. For both the independent and dependent bootstrap confidence intervals, the coverage probabilities increase and the average interval lengths decrease as the sample size n increase for Normal, Gamma, and Chi-square distributions, as well as three methods used in this work.
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In this study, we propose confidence intervals and their bootstrap versions for the difference of variances of two independent population using some robust variance estimators. The proposed confidence intervals are compared with Herbert confidence interval in terms of coverage probability and average width. A simulation study is conducted to evaluate performances of the proposed confidence intervals under different scenarios. The simulation results indicate that the coverage probabilities for the proposed confidence intervals are very close to nominal confidence levels when the difference of population variances is zero. Confidence interval based on binary distance produces the narrowest average widths. Herbert confidence interval have not perform well for skewed distribution populations. Confidence interval based on comedian is generally recommended when the difference of population variances for skewed distributions is not zero. Average widths of bootstrap percentile confidence intervals are smaller, and decreases as sample size and nominal size increases, as expected. Consequently, we recommend bootstrap percentile confidence interval based on binary distances for skewed distributions.
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Confidence distribution
Tolerance interval
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