Correcting for Regression to the Mean in Behavior and Ecology Colleen Kelly and Trevor D. Price, 2005 The American Naturalist Presented by: Sifat Tasnim Casey Engstrom

What is regression to the mean (RTM)? Statistical phenomenon Also known as regression effect If two successive trait measurements have less than perfect correlation, individuals on average tend to be closer toward the mean on the second measurement There is a negative correlation between individuals state at time 1 and time 2 Test 1 Test 2 http://www.socialresearchmethods.net/kb/regrmean.php

Problem in Observational studies - no comparison available Problem of Regression effect (in Behavior and Ecology) Problem in Observational studies - lack of control group - no comparison available 2. Can draw erroneous conclusions - statistical artifacts

Examples: Illustrate the problem of Regression effect Example1: Mass Loss depends on initial mass Example2: Mating success of hybrid drosophila in two different generations

Example1: Mass Loss depends on initial mass Amount of mass lost = Initial mass – final mass r = 0.55, P = .0096 Need further statistical analysis

Example2: Mating success of hybrid drosophila in two different generations Need further statistical analysis

Taking RTM into account With better experiment/study methods: Controls (manipulative experiments) Multiple baseline measurements over several days Increase N, replicates In analysis: Pitman-Morgan test Can adjust data to “exclude” RTM, make a new linear model, etc

Pitman-Morgan test: Is there a change in values beyond that expected by RTM? Uses variance to measure RTM T-value is a function of sample size, correlation, and variance Interpreting the t test

1. Pitman-Morgan underlying concept: Variance1 = Variance2, -> suggests RTM Data scenario 1: RTM Day 1 Day 2 mean mean frequency a b b a light heavy

1. Pitman-Morgan underlying concept: Variance1 = Variance2, -> suggests RTM Data scenario 1: RTM Day 1 Day 2 Overall actual population distribution frequency No change in overall distribution, our samples experienced RTM a b b a light heavy

1. Pitman-Morgan underlying concept: Variance1 = Variance2, -> suggests RTM Data scenario 1: RTM Day 1 Day 2 Change in variance Interpretation No change Overall the population stays the same, the extreme values have simply “regressed to the norm” frequency a b b a light heavy Data scenario 2: real effect “a” and “b” same as scenario 1, but all data values are converging, variance decreases heavy birds are getting lighter, light birds getting heavier a b a b

2. Pitman-Morgan equation sd1 Sample size sd2 Inputs: correlation Output: T-value: “ratio of signal to noise”

3. Interpreting t-value T= Generally, greater T values -> worse fit with model (in this case, the null hypothesis) Source: http://blog.minitab.com/blog/adventures-in-statistics-2/understanding-t-tests-t-values-and-t-distributions

Taking into account RTM After Correction Before Correction

When doing your own study… Conclusions When reading… Assume RTM, unless proven otherwise! When in doubt, run Pitman-Morgan yourself on the data When doing your own study… Control or multiple baseline measurements (if experiment or clinical trial possible) Placate skeptics with Pitman-Morgan t test or similar