HW1:
Problems 1.1, 1.3, 1.7, 1.12, 1.19 in the textbook + 1.100 presented
below. (10 pts for 1.12, 8 pts for other problems). Due 1/19/2012
1.100: For testing certain hull hypothesis H_0 vs the alternative H_a,
we will use a test statistic T and greater value of T means more
extreme. The distribution of T under the null may be discrete.
Use the logic given the class, show that the expectation of
the mid p-value under the null is 0.5.
1.7(e) Exact binomial test. Let Y= # of observations where the new
drug is considered better. Then a sensible testing procedure for
testing H_0: p_i=0.5 vs H_a: p_i not= 0.5 is "reject H0 if |Y - n*0.5|
is large". Then the observed test stat = |20 - 20*0.5| = 10. By the
definition of a p-value, the p-value is
p-value = P[observing a test stat that is as extreme as the current
one or more extreme than the current one | H0]
= P[|Y- 20*0.5| >= 10|p_i=0.5] = P[Y= 0 or Y=20 | p_i=0.5] = ...
For exact CI for p_i, please read pages 108-117 of
1.7(f): To find the sample size such that the 2 limits of the 95% CI
of p_i are at most 0.05 away from its center when the true prob p_i = 0.9.
In 1.12, the statement that {Y_i} are independent applies only to (a) and
(d). In (c), we assume Y_i and Y_j are conditionally independent given
p_i for any i not= j. Implicitly, in (d) we assume (Y_i, pi_i) are
independent across i.
HW2:
Problems 1.9, 1.30, 2.1, 2.8, 2.10 in the textbook. Due: 1/26/2012
HINT for problem 1.9: Let Y = # of deaths in a corps-year. What we are
asked to test is H0: Y is from Poisson(mu). The original data is Y1,
Y2, .., Yn, where n=200 corps-years. What is given in Table 1.3 is the
number of Yi's that are 0, 1, 2, 3, 4 or >=5. Let (n0, n1, n2, n3, n4)
are the number of Yi's that are 0, 1, 2, 3, or >=4 (truncated at
4). Then (n0, n1, n2, n3, n4) has a multinomial distribution. Under
H0, the corresponding probabilities are given by the Poisson
distribution:
p0 = exp(-mu)
p1 = mu*exp(-mu)
p2 = mu^2 * exp(-mu)/2!
p3 = mu^3 * exp(-mu)/3!
p4 = 1-(p0+p1+p2+p3).
We can then use the Pearson Chi-squared test to test this null
hypothesis. In order to construct the test, we need to find out the MLE of
mu under H0. Try to find it out from the original un-truncated data.
HW3:
Problems 2.12, 2.18, 2.19, 2.30, 2.38 in the text book. Due 2/2/2012.
2.18 (a) In finding sample "odds" of lung cancer, we treat the data as
if from a prospective study or a cross-sectional study. Of course,
these "odds" are not the actual odds and therefore they themselves
alone are not very meaningful. Then we need to find 5 local odds-ratios using
the first row (non-smoking) as the reference level. The odds-ratios
will be well defined.
2.18 (d) What we need to do is to show the lung cancer probability
estimated with the given data increases or decreases when the
row number i increases. Equivalently, the "odds" of the lung cancer
probability increases or decreases when the row number i increases,
or the odds ratio defined in (a) increases or decreases when the row
number i increases.
HW4:
Problems 3.1, 3.5, 3.11 3.22 in the textbook and the following 3.100.
Due 2/9/2012
3.100: Using the multivariate delta method, show that the variance of log
odds-ratio estimate from a 2x2 table can be estimated by
1/n_11 + 1/n_12 + 1/n_21 + 1/n_22
where (n_11, n_12, n_21, n_22) is from a multinomial sampling.
For 3.11, please calculate the test statistics by hand and by using SAS.
HW5: Due 2/16/2012
Problem 3.4(a) + Test the association treating party identification as
an ordinal categorical varialbe (with scores 1, 2, 3). Run an ANOVA to
get the F-statistic and compare CMH2 to (I-1)*F. Are they close?
Problems 3.13 (also do d. conduct the exact Pearson chi-squared test by
hand); for dis-advantage of a mid p-value, please review your hw 1.19
(the actual error probability may be greater than the nominal level).
Problems 3.14, 3.17, 3.30 (note that z^2 is the Cochran-Armitage trend
test statistic and X^2 is the Pearson chi-squared statistic).