Indexes of contrast and quantitative significance for comparisons of two groups

Abstract

Although boundaries for 'large' and 'small' differences are needed to plan research and interpret results, the diverse indexes of descriptive contrast for the central indexes, A and B, of two groups, have not received intensive attention. For two means, the increment of mid R:A-Bmid R: reflects the slope of a line showing the 'effect', but is altered by different units of measurement. Division of mid R:A-Bmid R: by the common standard deviation produces the standardized increment (SI), which is sometimes called the 'effect size'. Despite many advant ages, it does not contrast the relative magnitudes of A and B. For the latter contrast, the relative change or proportionate increment (mid R:A-Bmid R:/B) is particularly easy to understand, and the relative translocation (mid R:A-Bmid R:/[A+B]) produces a bounded range from -1 to +1. Nevertheless, all indexes of relative magnitude ablate the scales of measurement, thereby increasing difficulty in interpretation. Although seldom applied, proportionate reduction in overall system variance can be highly useful. Its square root leads to eta, the analogue of a correlation coefficient, which corresponds to a standardized slope for the direct increment. The values of eta usually approximate (SI)/2. Although arbitrary levels have been proposed for 'quantitative significance' of the SI, the proportionate reduction in overall system variance is often regarded as ineffectual unless >/=10 per cent. With this belief, minimum boundaries for quantitative significance can often be set at eta>/=0.3 and SI>/=0.6. In indexes of relative magnitude for two proportions (or rates), p(A) and p(B), confusion is produced if q(A) and q(B) are alternatively chosen for the denominators. The odds ratio, (p(A) q(B)/p(B) q(A)), avoids these choices, but is often difficult to interpret. For easy understanding and communication, the preferred index is NNE, the number needed to produce one excess effect, calculated as the inverse of the direct increment, that is, 1/mid R:p(A)-p(B)mid R: The standardized increment, mid R:p(A)-p(B)mid R:/ radical(PQ), (where P is the average of p(A) and p(B) and Q=1-P) could offer a single index applicable to both dimensional and binary data, but when P becomes quite small, that is, <0.01, radical(PQ) requires special calculations and also approaches the value of radicalP. Boundaries of 'quantitative significance' are particularly difficult to establish for comparisons of two rates, because of additional consequences in populational extrapolations and clinical implications. Nevertheless, the principles of quantitative significance can aid the ad hoc construction of boundaries that must be set for medical importance when sample sizes are calculated and when results are interpreted for studies of either efficacy or equivalence.