# Crawford-Howell (1998) t-test for case-control comparisons

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Cognitive neuropsychologists (like me) often need to compare a single case to a small control group, but the standard two-sample t-test does not work for this because the case is only one observation. Several different approaches have been proposed and in a new paper just published in

Crawford, J.R., & Howell, D.C. (1998). Comparing an Individual’s Test Score Against Norms Derived from Small Samples. The Clinical Neuropsychologist, 12 (4), 482-486 DOI: 10.1076/clin.12.4.482.7241

Crawford, J. R., & Garthwaite, P. H. (2012). Single-case research in neuropsychology: A comparison of five forms of t-test for comparing a case to controls. Cortex, 48 (8), 1009-1016 DOI: 10.1016/j.cortex.2011.06.021

*Cortex*, Crawford and Garthwaite (2012) demonstrate that the Crawford-Howell (1998) t-test is a better approach (in terms of controlling Type I error rate) than other commonly-used alternatives. As I understand it, the core issue is that with a typical t-test, you’re testing whether two means are different (or, for a one-sample t-test, whether one mean is different from some value), so the more observations you have, the better your estimate of the mean(s). In a case-control comparison you want to know how likely it is that the case value came from the distribution of the control data, so even if your control group is very large, the variability is still important — knowing that your case is below the control mean is not enough, you want to know that it is below 95% (for example) of the controls. That is why, as Crawford and Garthwaite show, Type I error increases with control sample size for the other tests, but not for the Crawford-Howell test.It is nice to have this method validated by Monte Carlo simulation and I intend to use it next time the need arises. I’ve put together a simple R implementation of it (it takes a single value as case and a vector of values for control and returns a data frame containing the t-value, degrees of freedom, and p-value):

CrawfordHowell <- function(case, control){ tval <- (case - mean(control)) / (sd(control)*sqrt((length(control)+1) / length(control))) degfree <- length(control)-1 pval <- 2*(1-pt(abs(tval), df=degfree)) #two-tailed p-value result <- data.frame(t = tval, df = degfree, p=pval) return(result) }

Crawford, J.R., & Howell, D.C. (1998). Comparing an Individual’s Test Score Against Norms Derived from Small Samples. The Clinical Neuropsychologist, 12 (4), 482-486 DOI: 10.1076/clin.12.4.482.7241

Crawford, J. R., & Garthwaite, P. H. (2012). Single-case research in neuropsychology: A comparison of five forms of t-test for comparing a case to controls. Cortex, 48 (8), 1009-1016 DOI: 10.1016/j.cortex.2011.06.021

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