This site is a compendium of R code meant to highlight the various uses of simulation to aid in the understanding of probability, statistics, and study design. I frequently draw on examples using my R package `simstudy`

. Occasionally, I opine on other topics related to causal inference, evidence, and research more generally.

Recently I wrote about the challenges of trying to learn too much from a small pilot study, even if it is a randomized controlled trial. There are limitations on how much you can learn about a treatment effect given the small sample size and relatively high variability of the estimate. However, the temptation for researchers is usually just too great; it is only natural to want to see if there is any kind of signal of an intervention effect, even though the pilot study is focused on questions of feasibility and acceptability.
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## When proportional odds is a poor assumption, collapsing categories is probably not going to save you

Continuing the discussion on cumulative odds models I started last time, I want to investigate a solution I always assumed would help mitigate a failure to meet the proportional odds assumption. I’ve believed if there is a large number of categories and the relative cumulative odds between two groups don’t appear proportional across all categorical levels, then a reasonable approach is to reduce the number of categories. In other words, fewer categories translates to proportional odds.
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## Considering the number of categories in an ordinal outcome

In two Covid-19-related trials I’m involved with, the primary or key secondary outcome is the status of a patient at 14 days based on a World Health Organization ordered rating scale. In this particular ordinal scale, there are 11 categories ranging from 0 (uninfected) to 10 (death). In between, a patient can be infected but well enough to remain at home, hospitalized with milder symptoms, or hospitalized with severe disease.
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## To stratify or not? It might not actually matter...

Continuing with the theme of exploring small issues that come up in trial design, I recently used simulation to assess the impact of stratifying (or not) in the context of a multi-site Covid-19 trial with a binary outcome. The investigators are concerned that baseline health status will affect the probability of an outcome event, and are interested in randomizing by health status. The goal is to ensure balance across the two treatment arms with respect to this important variable.
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## Simulation for power in designing cluster randomized trials

As a biostatistician, I like to be involved in the design of a study as early as possible. I always like to say that I hope one of the first conversations an investigator has is with me, so that I can help clarify the research questions before getting into the design questions related to measurement, unit of randomization, and sample size. In the worst case scenario - and this actually doesn’t happen to me any more - a researcher would approach me after everything is done except the analysis.
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## Yes, unbalanced randomization can improve power, in some situations

Last time I provided some simulations that suggested that there might not be any efficiency-related benefits to using unbalanced randomization when the outcome is binary. This is a quick follow-up to provide a counter-example where the outcome in a two-group comparison is continuous. If the groups have different amounts of variability, intuitively it makes sense to allocate more patients to the more variable group. Doing this should reduce the variability in the estimate of the mean for that group, which in turn could improve the power of the test.
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## Can unbalanced randomization improve power?

Of course, we’re all thinking about one thing these days, so it seems particularly inconsequential to be writing about anything that doesn’t contribute to solving or addressing in some meaningful way this pandemic crisis. But, I find that working provides a balm from reading and hearing all day about the events swirling around us, both here and afar. (I am in NYC, where things are definitely swirling.) And for me, working means blogging, at least for a few hours every couple of weeks.
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## When you want more than a chi-squared test, consider a measure of association

In my last post, I made the point that p-values should not necessarily be considered sufficient evidence (or evidence at all) in drawing conclusions about associations we are interested in exploring. When it comes to contingency tables that represent the outcomes for two categorical variables, it isn’t so obvious what measure of association should augment (or replace) the \(\chi^2\) statistic.
I described a model-based measure of effect to quantify the strength of an association in the particular case where one of the categorical variables is ordinal.
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## Alternatives to reporting a p-value: the case of a contingency table

I frequently find myself in discussions with collaborators about the merits of reporting p-values, particularly in the context of pilot studies or exploratory analysis. Over the past several years, the American Statistical Association has made several strong statements about the need to consider approaches that measure the strength of evidence or uncertainty that don’t necessarily rely on p-values. In 2016, the ASA attempted to clarify the proper use and interpretation of the p-value by highlighting key principles “that could improve the conduct or interpretation of quantitative science, according to widespread consensus in the statistical community.
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## Clustered randomized trials and the design effect

I am always saying that simulation can help illuminate interesting statistical concepts or ideas. The design effect that underlies much of clustered analysis is could benefit from a little exploration through simulation. I’ve written about clustered-related methods so much on this blog that I won’t provide links - just peruse the list of entries on the home page and you are sure to spot a few. But, I haven’t written explicitly about the design effect.
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