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Exercise 16.7.5. Refer back to the snoring and heart disease data in Exercise 14.9.7. The data consists of (Y1,X1),.... (Ym,x,) independent observations, where each Y, ~ Bernoulli(p) indicates whether person i has heart disease. The py's follow a logistic regression model, logit(p;) = a + Bx; where x, is the extent to which the person Snores. Here are the data again: Heart disease? + Yes No Frequency of snoring 4 Xi Never -3 24 1355 Occasionally -1 35 603 (16.112) Nearly every night 1 21 192 Every night 3 30 224 The MLEs are @ = -2.79558, 8 = 0.32726. Consider testing Ho : 8 = 0 versus HA : 6 + 0. We could perform an approximate z-test, but here find the LRT. The loglikelihood is Et(wi log(p:) + (1 - y) log(1 - p:)). (a) Find the value of the log- likelihood under the alternative, I,(a, B). (b) The null hypothesis implies that the pa's are all equal. What is their common MLE under Ho? What is the value of the loglikelihood? (c) Find the 2log(A(y, x)) statistic. (d) What are the dimensions of the two parameter spaces? What are the degrees of freedom in the asymptotic x ? (e) Test the null hypothesis with a = 0.05. What do you conclude