Abstract
Various effect sizes have been proposed. However, different effect size measures are suitable for different types of data, and the interpretations of effect sizes are generally arbitrary and remain problematic. In this article, the concepts of contrast variable, its standardized mean (SMCV) and c + -probability are explored to link together the commonly used effect sizes including the probabilistic index and ratios of mean difference to variability. A contrast variable can provide both a probabilistic meaning and an index of signal-to-noise ratio to interpret the strength of a comparison, which offers us a strong base to classify the strength of a comparison. Contrast variable, SMCV and c + -probability not only give interpretations to both Cohen’s and McLean’s criteria but also work effectively and consistently for either relationship or group comparison in either independent or correlated situations and in either two or more than 2 groups. Treatment effect, main effect, interaction effect, linear relationship, quadratic relationship and any other contrasts in factorial experiments can all be addressed consistently using contrast variables and the SMCV-based classifying rule, as demonstrated using examples in this article. Therefore, contrast variable, SMCV, c + -probability and associated classifying rules may have the potential to offer a consistent interpretation to effect sizes.
Highlights
Traditional contrast analysis tackles the question of whether a linear combination of group means is exactly zero using significance testing
standardized mean of contrast variable (SMCV) and the coefficients in (11)–(14), we can use the classifying rule in Table 1 to provide a consistent interpretation to any effects that are commonly used in two-way Analysis of Variance (ANOVA)
When the concept of contrast variable is applied to 2-group comparison, the contrast variable represents the difference between two groups
Summary
Traditional contrast analysis tackles the question of whether a linear combination of group means is exactly zero using significance testing. The p-value from classical t-test of testing a traditional contrast can capture data variability; it is affected by both sample size and the strength of a comparison This issues with significance testing of mean difference are reflected in Tukey’s comments [33] “It is foolish to ask ‘Are the effects of A and B different?’ They are always different --- for some decimal place.”. A contrast variable is defined as a linear combination of random variables representing the values in the groups (or treatment levels) themselves, not just a linear combination of their means, where the coefficients sum to zero.
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