Distribution of the Sample Version of the Measure of Association, Gamma

Abstract Random sampling experiments were performed to determine whether Goodman and Kruskal's asymptotic theory (2) for the sample version (G) of the measure of association Gamma (γ) is applicable to small samples. First, the asymptotic theory was checked against the results obtained with a series of 5 × 5 population cross classifications with values of γ covering the full range. For samples of size fifty, the tails of the distribution of (G — γ)/σ corresponded closely to the tails of the unit normal distribution when σ2 was the appropriate asymptotic variance derived in (2). Asymptotic normality still worked fairly well in the tails when σ2 was replaced by its maximum likelihood estimator, providing that |γ| <.50. There was, however, a clear tendency to have more observations in the tails than predicted from the asymptotic theory and this became very marked for large γ's. When σ was replaced by an upper bound derived in (2), (G — γ)/σ was of course more tightly concentrated. For the same series of population cross classifications, the power of a test that γ = 0, using a test statistic based on G and a specially applicable variance formula, was investigated. For sample sizes twenty-five and fifty, the power of this test was high except when γ approached closely to zero. For the case γ = 0 (independence of the two classifications), and 5 × 5 population cross classifications, an investigation was made of the effect of varying the marginal totals. The tails of the obtained distribution corresponded fairly closely to the tails of the unit normal distribution for samples of size twenty-five and fifty unless the population cross classifications had a high percentage of the population concentrated in one category.