Divergence from, and Convergence to, Uniformity of Probability Density Quantiles.

We demonstrate that questions of convergence and divergence regarding shapes of distributions can be carried out in a location- and scale-free environment. This environment is the class of probability density quantiles (pdQs), obtained by normalizing the composition of the density with the associated quantile function. It has earlier been shown that the pdQ is representative of a location-scale family and carries essential information regarding shape and tail behavior of the family. The class of pdQs are densities of continuous distributions with common domain, the unit interval, facilitating metric and semi-metric comparisons. The Kullback-Leibler divergences from uniformity of these pdQs are mapped to illustrate their relative positions with respect to uniformity. To gain more insight into the information that is conserved under the pdQ mapping, we repeatedly apply the pdQ mapping and find that further applications of it are quite generally entropy increasing so convergence to the uniform distribution is investigated. New fixed point theorems are established with elementary probabilistic arguments and illustrated by examples.

Meta-analysis of ratios of sample variances.

When conducting a meta-analysis of standardized mean differences (SMDs), it is common to use Cohen's d, or its variants, that require equal variances in the two arms of each study. While interpretation of these SMDs is simple, this alone should not be used as a justification for assuming equal variances. Until now, researchers have either used an F-test for each individual study or perhaps even conveniently ignored such tools altogether. In this paper, we propose a meta-analysis of ratios of sample variances to assess whether the equality of variances assumptions is justified prior to a meta-analysis of SMDs. Quantile-quantile plots, an omnibus test for equal variances or an overall meta-estimate of the ratio of variances can all be used to formally justify the use of less common methods when evidence of unequal variances is found. The methods in this paper are simple to implement and the validity of the approaches are reinforced by simulation studies and an application to a real data set.

Transforming the Model T: random effects meta-analysis with stable weights.

Standard meta-analytic theory assumes that study outcomes are normally distributed with known variances. However, methods derived from this theory are often applied to effect sizes having skewed distributions with estimated variances. Both shortcomings can be largely overcome by first applying a variance stabilizing transformation. Here we concentrate on study outcomes with Student t-distributions and show that we can better estimate parameters of fixed or random effects models with confidence intervals using stable weights or with profile approximate likelihood intervals following stabilization. We achieve even better coverage with a finite sample bias correction. Further, a simple t-interval provides very good coverage of an overall effect size without estimation of the inter-study variance. We illustrate the methodology on two meta-analytic studies from the medical literature, the effect of salt reduction on systolic blood pressure and the effect of opioids for the relief of breathlessness. Substantial simulation studies compare traditional methods with those newly proposed. We can apply the theoretical results to other study outcomes for which an effective variance stabilizer is available.

Robust analogs to the coefficient of variation.

The coefficient of variation (CV) is commonly used to measure relative dispersion. However, since it is based on the sample mean and standard deviation, outliers can adversely affect it. Additionally, for skewed distributions the mean and standard deviation may be difficult to interpret and, consequently, that may also be the case for the CV . Here we investigate the extent to which quantile-based measures of relative dispersion can provide appropriate summary information as an alternative to the CV. In particular, we investigate two measures, the first being the interquartile range (in lieu of the standard deviation), divided by the median (in lieu of the mean), and the second being the median absolute deviation, divided by the median, as robust estimators of relative dispersion. In addition to comparing the influence functions of the competing estimators and their asymptotic biases and variances, we compare interval estimators using simulation studies to assess coverage.

Combining the evidence using stable weights.

In a meta-analysis one seeks to combine the results of several studies in order to improve the accuracy of decisions. Here we compare by simulation four methods for combining estimates of the risk difference, namely the Cochran and Mantel-Haenszel (MH) methods, the inverse-variance weights approach and a recent variance-stabilized weights approach. Both the level and power of corresponding test statistics, as well as the coverage of related confidence intervals are compared over a wide range of sample size and parameter configurations. We found that the inverse-variance weights methodology is unreliable and is not recommended, while for equal risks, the Cochran test and the associated confidence intervals are the most reliable. Under alternatives of unequal risks, the coverage probabilities of the variance-stabilized confidence intervals are almost uniformly more reliable than those based on other methods except when the average risk is small in which case the MH confidence intervals are preferable. Copyright © 2011 John Wiley & Sons, Ltd.

Confidence intervals for the standardized effect arising in the comparison of two normal populations.

Confidence intervals for a standardized effect are derived after stabilizing the variance of the Welch t-statistic. Simulation studies demonstrate the viability of the resulting intervals for a wide range of parameter values and sample sizes as small as five. The methodology is extended to the combination of results from several studies, so as to obtain a confidence interval for a representative standardized effect for all the studies. The methods are illustrated on a recent meta-analytic study of systolic blood pressure reduction during a weight reducing regime, as well as the classical Mumford data on psychological intervention and hospital length of stay.