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In settings with independent observations, sample size is one way to quickly characterize the precision of an estimate. But what if your estimate is based on weighted data, where each observation doesn’t necessarily contribute to equally to the estimate? Here, one useful way to gauge the precision of an estimate is the effective sample size or ESS. Suppose that we have \(N\) independent observations \(Y_1,...,Y_N\) drawn from a population with standard deviation \(\sigma\), and that observation \(i\) receives weight \(w_i\).


Earlier this month, I taught at the Summer Research Training Institute on Single-Case Intervention Design and Analysis workshop, sponsored by the Institute of Education Sciences’ National Center for Special Education Research. While I was there, I shared a web-app for simulating data from a single-case design. This is a tool that I put together a couple of years ago as part of my ARPobservation R package, but haven’t ever really publicized or done anything formal with.


I’m very happy to share a new paper, co-authored with my student Danny Swan, “A gradual effects model for single-case designs,” which is now available online at Multivariate Behavioral Research. You can access the published version at the journal website (click here for free access while supplies last) or the pre-print on PsyArxiv (always free!). Here’s the abstract and the supplementary materials. Danny wrote R functions for fitting the model, (available as part of the SingleCaseES package) as well as a slick web interface, if you prefer to point-and-click.


Last night I attended a joint meetup between the Austin R User Group and R Ladies Austin, which was great fun. The evening featured several lightning talks on a range of topics, from breaking into data science to network visualization to starting your own blog. I gave a talk about sandwich standard errors and my clubSandwich R package. Here are links to some of the talks: Caitlin Hudon: Getting Plugged into Data Science Claire McWhite: A quick intro to networks Nathaniel Woodward: Blogdown Demo!


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 \[ \text{Var}(r) \approx \frac{1}{N}\left(1 - \rho^2\right)^2. \] But what if the observations are drawn from a multi-stage sample? If one uses the raw correlation between the observations (ignoring the multi-level structure), then the \(r\) will actually be a weighted average of within-cluster and between-cluster correlations (see Snijders & Bosker, 2012).


Recent Publications

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(2019). Current practices in meta-regression in psychology, education, and medicine. Research Synthesis Methods, forthcoming.

Preprint Dataset Journal

(2018). A history of meta-regression: Technical, conceptual, and practical developments between 1974 and 2018. Research Synthesis Methods, forthcoming.

Preprint Journal

(2018). Testing for funnel plot asymmetry of standardized mean differences. Research Synthesis Methods, forthcoming.

Preprint Code Journal

(2018). Small sample methods for cluster-robust variance estimation and hypothesis testing in fixed effects models. Journal of Business and Economic Statistics, 36(4), 672–683.

Preprint Journal R package

(2018). Between-case standardized effect size analysis of single case design: Examination of the two methods. Research in Developmental Disabilities, 79, 88-96.


Recent Presentations

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Combining robust variance estimation with models for dependent effect sizes
Combining robust variance estimation with models for dependent effect sizes
Meta-analysis of dependent effects: A review and consolidation of methods
Meta-analysis of single-case research: A brief and breezy tour
A gradual effects model for single case designs



Simulate systematic direct observation data.


Single-case design effect size calculator.


cluster-robust variance estimation.


Between-case SMD for single-case designs.