Abstract
We introduce linear contrasts between treatment group means as a principled way for constructing t-tests and confidence intervals for treatment comparisons. We consider a variety of contrasts, including contrasts for estimating time trends and for finding minimal effective doses. Multiple comparison procedures control the family-wise error rate, and we introduce four commonly used methods by Bonferroni, Tukey, Dunnett, and Scheffé. Finally, we discuss a larger real-life example to demonstrate the use of linear contrasts and highlight the need for careful definition of contrasts to correctly reflect the desired comparisons.
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Notes
- 1.
We can do this using a ‘contrast’ \(\mathbf {w}=(1/k,1/k,\dots ,1/k)\), even though its weights do not sum to zero.
- 2.
A small subtlety arises from estimating the contrasts: since all \(t\)-tests are based on the same estimate of the residual variance, the tests are still statistically dependent. The effect is usually so small that we ignore this subtlety in practice.
- 3.
It is plausible that measurements on the same mouse are more similar between timepoints close together than between timepoints further apart, a fact that ANOVA cannot properly capture.
- 4.
If 200 hypotheses seem excessive, consider a simple microarray experiment: here, the difference in expression level is simultaneously tested for thousands of genes.
- 5.
The authors of this study kindly granted permission to use their data. Purely for illustration, we provide some alternative analyses to those in the publication.
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Kaltenbach, HM. (2021). Comparing Treatment Groups with Linear Contrasts. In: Statistical Design and Analysis of Biological Experiments. Statistics for Biology and Health. Springer, Cham. https://doi.org/10.1007/978-3-030-69641-2_5
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DOI: https://doi.org/10.1007/978-3-030-69641-2_5
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