Refer to Problem 3.27. The sample size may need to be quite large for the sampling distribution of γ̂ to be approximately normal, especially if |γ| is large. The Fisher-type transform ξ̂ = 1/2 log[(1 + γ̂)/(1 − γ̂)] converges more quickly to normality.

a. Show that the asymptotic variance of ξ̂ equals the asymptotic variance of γ̂ multiplied by (1 − γ²)^?2.

b. Explain how to construct a confidence interval for ξ and use it to obtain one for γ.

c. Show that ξ̂  = 1/2 log(C/D). For 2 × 2 tables, show that this is half the log odds ratio.

Data from Problem 3.27:

For ordinal variables, consider gamma (2.14). Let

Where i and j are fixed in the summations. Show that II_c = ∑_i ∑_j π_ij π^(c)_ij and II_d = ∑_i ∑_j π_ij π^(d)_ij. Use the delta method to show that the large-sample normality (3.9) applies for γ̂ , with (Goodman and Kruskal 1963)

Transcribed Image Text:

Τ Σ Σ Τ ai b>j π Σ ΣΤ + ΣΣπ aj a>i b