# distribution theory

## Variance component estimates in meta-analysis with mis-specified sampling correlation

$\def\Pr{{\text{Pr}}} \def\E{{\text{E}}} \def\Var{{\text{Var}}} \def\Cov{{\text{Cov}}}$ In a recent paper with Beth Tipton, we proposed new working models for meta-analyses involving dependent effect sizes. The central idea of our approach is to use a working model that captures the main features of the effect size data, such as by allowing for both between- and within-study heterogeneity in the true effect sizes (rather than only between-study heterogeneity).

## Implications of mean-variance relationships for standardized mean differences

I spend more time than I probably should discussing meta-analysis problems on the R-SIG-meta-analysis listserv. The questions that folks pose there are often quite interesting—especially when they’re motivated by issues that they’re wrestling with while trying to complete meta-analysis projects in their diverse fields.

## Standardized mean differences in single-group, repeated measures designs

I received a question from a colleague about computing variances and covariances for standardized mean difference effect sizes from a design involving a single group, measured repeatedly over time.

## Finding the distribution of significant effect sizes

In basic meta-analysis, where each study contributes just a single effect size estimate, there has been a lot of work devoted to developing models for selective reporting. Most of these models formulate the selection process as a function of the statistical significance of the effect size estimate; some also allow for the possibility that the precision of the study’s effect influences the probability of selection (i.

## Simulating correlated standardized mean differences for meta-analysis

As I’ve discussed in previous posts, meta-analyses in psychology, education, and other areas often include studies that contribute multiple, statistically dependent effect size estimates. I’m interested in methods for meta-analyzing and meta-regressing effect sizes from data structures like this, and studying this sort of thing often entails conducting Monte Carlo simulations.

## Sampling variance of Pearson r in a two-level design

Consider Pearson’s correlation coefficient, $$r$$, calculated from two variables $$X$$ and $$Y$$ with population correlation $$\rho$$. If one calculates $$r$$ from a simple random sample of $$N$$ observations, then its sampling variance will be approximately

## The multivariate delta method

The delta method is surely one of the most useful techniques in classical statistical theory. It’s perhaps a bit odd to put it this way, but I would say that the delta method is something like the precursor to the bootstrap, in terms of its utility and broad range of applications—both are “first-line” tools for solving statistical problems.

## 2SLS standard errors and the delta-method

I just covered instrumental variables in my course on causal inference, and so I have two-stage least squares (2SLS) estimation on the brain. In this post I’ll share something I realized in the course of prepping for class: that standard errors from 2SLS estimation are equivalent to delta method standard errors based on the Wald IV estimator.

## Alternative formulas for the standardized mean difference

The standardized mean difference (SMD) is surely one of the best known and most widely used effect size metrics used in meta-analysis. In generic terms, the SMD parameter is defined as the difference in population means between two groups (often this difference represents the effect of some intervention), scaled by the population standard deviation of the outcome metric.

## The sampling distribution of sample variances

A colleague and her students asked me the other day whether I knew of a citation that gives the covariance between the sample variances of two outcomes from a common sample.