Reference no: EM132232346

Question 1. For a two-way contingency table, let Π_ij be the cell probability for the (i, j)th cell for i=1,...,I; j=1,...,J.

a) For n= Σ_ij n_ij independent observations, write down the multinomial probability that n₁₁ fall in the (1, 1 )th cell,..., n_ij fall in the (i, j )th cell, ..., n_IJ fall in the (I, J )th cell.

b) Let e_ij be the expected counts for the cell (i,j) under H0 hypothesis. Show that the likelihood- ratio test for two way contingency table is given by

G² = 2 ∑_ij n_ijln(n_ij/e_ij) = -2 ∑_ij n_ijln(e_ij/n_ij).

c) Based on Taylor's expansion of natural log at x= 1 below
Ln(x) ≈ (x-1) - (x-1)²/2,

show that likelihood-ratio test can be approximated by

∑_ij (n_ij - e_ij)²/n_ij .

[Hint: Apply Taylor expansion of Ln(x) at x = e_ij/n_ij in the second expression of G².]

d) Let X be row random variable with categories indexed by i=1,..,I. Let Y be column random variable with categories indexed by j=1,..,J. Show that the degree of freedom of the likelihood- ratio test for independence between X and Y is (I-1)(J-1).

Question 2. The conditional distribution of Y given X for a 2 by 2 table is given below

	Y
X	Π1	1-Π1
	Π2	1-Π2

Show that X and Y are independent if and only if Π₁ = Π₂.

Question 3. Suppose we have 2 random categorical variables X and Y whose underlying probability structure for the joint distribution is given by

		Y
		1	2
X	1	Π11	Π12
	2	Π21	Π22

and we are interested in testing the hypothesis: H₀ : Π₁₁ = η (1-η), Π₁₂ = (1-η)², Π₂₁= η², Π₂₂ = η (1- η) where 0 < η < 1. Do the following:

a) Find the marginal distributions of X and Y under H₀ in terms of η.
b) Find the conditional distribution of Y at X = x under H₀ in terms of η for x=1,2, respectively.
c) Show that X and Y are independent under H₀.
d) Suppose we have following data n_ij's from a multinomial sampling

		Y
		1	2
X	1	25	15
	2	27	33

Find the MLE of η under H₀.

e) For the data in (d), find the (estimated) expected value of n_ij for cell (i,j) under H₀ for i=1,2;,j=1,2.
f) For the data in (d), conduct the Pearson χ² for testing H₀ and test H₀ at level α = 0.05 using this test statistic.
g) For the data in (d), conduct the likelihood ratio test for testing H₀ and test H₀ at level α = 0.05 using this test statistic.

Question 4. In the United States, the estimated annual probability that a woman over the age of 35 dies of lung cancer equals 0.001304 for current smokers and 0:000121 for nonsmokers (M.Pagano and K. Gauvreau, Principles of Biostatistics, 1993, p. 134).

(a) Calculate and interpret the difference of proportions and the relative risk. Which of these measures is more informative for these data? Why?

(b) Calculate and interpret the odds ratio. Explain why the relative risk and odds ratio take similar values.

Question 5. The following table was taken from the 1991 General Social Survey.

	Belief in Afterlife
Race	Yes	No or Undecided
White	621	239
Black	89	42

a) Identify each classification as a response or explanatory variable.

b) Calculate sample i) odds ratio ii) relative risk iii) difference in proportions. Interpret the direction and strength of association based on odds ratio.

c) Obtain a 95% confidence interval for i) odds ratio, ii) relative risk iii) difference in proportions. Test the independence between belief in afterlife and race.

Note that

A large-sample confidence interval for the log of the relative risk is 

log(p₁/p₂)±z_α/2√((1-p₁)/(n₁p₁) + (1-p₂)/(n₂p₂))

where n₁ = n₁+ and n₂ = n₂₊; and p₁ = n₁₁/n₁₊ and p₂ = n₂₁/n₂₊.

d) Are the conclusions in i), ii) and iii) consistent?

Question 6. In an article about crime in the United States, Newsweek magazine (Jan. 10, 1994) quoted FBI statistics stating that of all blacks slain in 1992, 94% were slain by blacks, and of all whites slain in 1992, 83% were slain by whites. Let Y denote the race of victim and X denote race of murderer.

(a) Which conditional distribution do these statistics refer to, Y given X, or X given Y ?

(b) Calculate and interpret the odds ratio between X and Y.

(c) Given that a murderer was white, can you estimate the probability that the victim was white? What additional information would you need to do this?