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Omega Squared

Omega squared (ω2) is a descriptive statistic used to quantify the strength of the relationship between a qualitative explanatory (independent or grouping) variable and a quantitative response (dependent or outcome) variable. The relationship is interpreted in terms of the proportion of variation in the response variable that is associated with the explanatory variable. As a proportion, it can have values between 0 and 1, with 0 indicating no relationship and 1 indicating that all of the variation in the response variable is attributed to the explanatory variable. Omega squared is used as an effect-size index to judge the meaningfulness of the observed relationship identified using the analysis of variance F test. It can supplement the results of hypothesis tests comparing two or more population means. The research design may be either experimental, involving the random assignment of units to levels of the explanatory variable (e.g., different drug treatments), or nonexperimental, involving a comparison of several groups representing existing populations (e.g., underweight, normal weight, overweight, obese).

A Data Example

Consider an experimental study designed to evaluate the merits of three drug treatments to reduce the number of cigarettes smoked (i.e., reduce smoking behavior). From a volunteer group of 150 moderate to heavy smokers, 50 individuals are randomly assigned to each of the three identified drug treatments (n = 50). After a 6-week treatment period, participants are asked to record the number of cigarettes smoked during Week 7. Hypothetical means and standard deviations are reported in Table 1.

To test the hypothesis that there is no difference in the average number of cigarettes smoked by individuals exposed to the three drug treatments (i.e., H0: μ1 = μ2 = μ2 = μ3), the analysis of variance (ANOVA) F test could be used. The sample means estimate the population means, but they are subject to sampling error. The statistical test provides information on whether the observed difference among sample means provides sufficient evidence to conclude that population means differ, or whether it is just a reflection of sampling error. More importantly, the ANOVA F test does not provide any indication of the amount of difference or the strength of the relationship between the treatments and smoking behavior. The statistical test provides only the probability of the observed F statistic if the null hypothesis (i.e., the drugs are equally effective) is true. Differences among population means may be very small, even trivial, and these differences can be detected by the statistical test if the number of individuals in the samples is sufficiently large. Statistical significance does not imply meaningful significance. For the data in Table 1, the results might be judged to be statistically significant [F(2,147) = 4.573, p = .011], and it might be concluded that the drug treatments are differentially effective with respect to smoking behavior. Or, it might be concluded that a relationship does exist between the drug treatments and smoking behavior.

Table 1 Mean Number of Cigarettes Smoked and Standard Deviations for Each of Three Drug Treatments
TreatmentMeanStandard Deviation
Drug 1159.8
Drug 21810.7
Drug 3129.2

Effect Size

Omega squared provides a measure of association between the grouping factor (e.g., drug type) and the outcome variable (e.g., number of cigarettes smoked). That is, it estimates the proportion of variation in the outcome variable that is associated with the grouping variable. Theoretically, the variance of the population means, σ2G, plus the variance of individual scores within a population, σ21/G (which is assumed to be equal for all populations), determines the total variance in the response variable, σ2Y, (i.e., σ2Y = σ2G + σ21/G). In a balanced design (equal number of units per group), ω2

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