Repeated-measures t test for ratio level data

[Dr Graham D Smith]

I would like to start a discussion on a family of procedures that tend not to be emphasised in the literature. The procedures I have in mind are based upon the ratio between two sets of scores from the same sample.   To illustrate the discussion, I shall refer to the data from a simple psychological experiment I recently undertook (I have simplified the data, in fact the study involved several IVs). The study was designed to investigate an aspect of visual priming. In this kind of experiment there are two stimuli per trial; the prime and the target. The prime is presented before the target. The target is the stimulus to which the participant responds. The prime is said to be congruent if it similar to the target in some way or other. The prime is said to be incongruent if it dissimilar to the target.   In my experiment the target was either a blue or a green shape presented on a computer monitor. The participants responded by pressing one of two keys on the computer keyboard. Participants reaction times (RTs) in milliseconds were recorded. There were forty trials per condition. The DV was the medians of RTs by participant by condition (see below. You can ignore the following technical details if you wish. RTs typically have a right-skewed distribution. A standard procedure for dealing with RTs is to calculate the median RT by participant by condition. A further complication that can be overlooked is that only RTs from correctly identified targets are used to generate these medians. There are procedures designed to ensure that the median is not biased by them being calculated from different numbers observations that I shall not discuss here). The aim of the experiment was to determine what effect if any the prime had on response to the target. In my study the primes were either the same colour (i.e., congruent) or a different colour (i.e, incongruent) to the target.   Cong   Incong 553.50 637.25 563.50 591.00 656.88 682.00 533.25 537.13 719.63 799.75 632.25 599.75 516.88 538.38 765.00 741.00 445.50 453.38 593.38 606.00 478.25 517.00 539.25 554.75   My hypothesis was that reaction times will be shorter with congruent primes than with incongruent primes. One way of testing this hypothesis is to use a repeated-measures t test; i.e., calculate difference scores for each participant and perform a one-sample t test to determine whether the observed mean difference score is significantly different from zero. But I had recently heard about some researchers who had used the ratio of the lengths people's index and ring fingers as a variable. "Why a ratio and not a difference?", I wondered. If everyone had the same size hands then there would be no point. However, it is reasonable to expect that the relationship between hand size and the difference in two finger lengths is heteroscedastic. Bigger hands can have bigger finger length differences. Then I wondered whether the similar reasoning could be applied to RT data. RT is a ratio level measurement scale just like finger length. It is reasonable to expect that slower participants will have greater variability in their reaction times.   I wondered how I might test my hypothesis using ratios rather than differences. Here's "my" solution. I calculated ratio scores for each participant and perform a one-sample t test to determine whether the observed mean ratio score is significantly different from one. As ratios of interval level variables are meaningless I surmised that the t test on ratios should only be applied to ratio level data. For the above data the results are as follows;   one sample t test of ratios t (12) = -2.337, p = 0.039 Mean Ratio = 0.965 (i.e., participants in the congruent condition responded 3.5% quicker than participants in the incongruent condition) 95% Confidence Limits = 0.933 and 0.998   However, I was reinventing the wheel. Later I learned that Howell (1997) describes the procedure on pages 180-181 and was previously used by Kaufman & Rock (1962).   My feeling is that the t test for ratios should have a similar status and profile as the repeated measures t test (on differences). I suspect that the t test for differences is often used when the t test for ratios would be more suitable. So why is the procedure not more widely used? Perhaps this is only a problem within psychology where ratio level data is not commonly used.   Also I wonder whether the t test of ratios is a more powerful test than the t test of differences. If so then the ratio t test should be used in preference to the difference t test. By the way for the above data there is a significant difference for the t test of ratios but not for the t test of differences;   one sample t test of differences t (12) = 2.163, p = 0.053 Mean difference = 21.68     95% Confidence Limits = -0.38 and 43.73   Thinking about these issues has caused me to reassess the assumptions underpinning the use of the repeated measures t test (for differences). For a long time, I have thought that the homogeneity of variance assumption is meaningless for the RM t test. In other words there is no point in comparing the variability of scores from one condition with the variability of scores in the other condition prior to using the test. I thought this because, once the difference scores are calculated homogeneity of variance is meaningless. The t test is performed on the differences not the scores themselves whose variances may differ (so what?). However, I now wonder whether in fact one should look at homoscedasticity of the relationship between the difference of the scores in the two conditions and the sum of the scores in the two conditions; for example, for my data the relationship between Incong-Cong and Incong+Cong. (Actually the data from my study were not clearly heteroscedastic).   Finally, surely an independent measures t test for ratios must be possible.   Graham   ************************************************* Dr Graham D. Smith Psychology Division Park Campus University College Northampton Boughton Green Rd. Northampton NN2 7AL   Tel: +44 (0) 1604 735500 Ext 2393 E-mail: [EMAIL PROTECTED] *************************************************