Some Comments about Chapter 2 of Hollander & Wolfe
Since I believe that STAT 554 covers this material more thoroughly than
H&W do, hopefully you're quite comfortable with the material in this
section. Still, I'll add some comments below.
- p. 21, line 1
- While the test as described is a
alpha test, it is
alpha test, and indicating that the size is
alpha gives more information (since a level alpha test can
have size alpha or a size less than alpha).
- pp. 21-22
- Note that they fail to use a continuity correction for their large
sample approximation. I've found that the use of a continuity
correction tends to make the normal approximation better.
- p. 22, Example 2.1
- I think that this is an odd choice for an initial example for Ch.
2. I would guess that p is not constant for all nutrient-poor
and dry soil sites, and I would also worry about the lack of
independence (thinking that there are dependencies related to the
spatial orientation of the gaps in the clumps). One way to firm things
up would be to define a fixed population of finite size, and take a
simple random sample from the population, and make an inference about
the population proportion using the hypergeometric distribution.
- pp. 23-24, Comment 1
- The sign test is covered in Ch. 3. When I cover this test about
the distribution median when we get to it in Ch. 3, I'll extend the
scheme to handle tests about quantiles other than the median.
- p. 24, Comment 2
- I find it to be extremely odd to refer to a test about the
parameter of a specified parametric model as being a distribution-free
test! By the same logic we could say that the t test is a
distribution-free test about the mean of a normal distribution.
- p. 25, between (2.15) and (2.16)
- I refer to the symbol under consideration as a binomial
coefficient, and a lot of people read it as "n choose
- p. 26, Comment 9
- Hopefully you're very comfortable with p-values. Still, I suggest
that you read Comment 9 from the text very carefully as a review.
- pp. 26-27, Comment 10
- Note that the powers are rather low because the sample size is very
small and the alternative under consideration isn't very different from
the null hypothesis value. Also note that H&W use beta fot the
probability of a type II error, like a lot of elementary books do, while
I use beta for power, like advanced books sometimes do.
- p. 28, Problem 1
- I think that a two-sample test (of the kind that will be covered in
Ch. 10) may be more appropriate, since one
has two samples of binary observations. Still, there would be concerns about
whether the assumptions are satisfied for such a test.
- p. 29, Problem 9
- I tend to get annoyed with silly situations for problems. What is
the significance of the value 0.6 for this problem? It seems to me that
an interval estimate of p would be more meaningful than a test
with 0.6 as the null hypothesis value. Besides, since the voles were of
different sizes, I think that it would be better to study the
relationship between vole body mass and the probability of success, and
the sizes of the adders in the model (since it can be noted that the
title of the article on which the problem was based is The advantage
of a big head: swallowing performance in adders).
Most of the material in this section is presented in STAT 554. I'll
make a few comments below.
- p. 29, (2.20) & (2.21)
- The standard deviation of an estimator is often referred to as the
standard error of the estimator. So (2.20) gives the standard
error. (2.21) gives the estimated standard error. (Note: Some
statistical software packages sometimes label an estimated standard
error as a standard error. My guess is that this is done to make the
output less cluttered.)
- p. 30, Comment 17
- H&W don't do a good job of stating the good points of the
alternative estimator reported on by Hodges and Lehmann in their 1950
paper. Although, asymptotically, the alternative estimator wins for no
values of p when the MSE is used as the measure of goodness, for
any finite value of n, the alternative estimator is better for
values of p in an interval that is symmetric about 0.5, with the
width of the interval decreasing with increasing n. For smallish
n, the interval can be rather wide. For example, for n =
16, the alternative estimator has a smaller MSE than the standard
estimator if p belongs to (0.2, 0.8).
I'm very disappointed that H&W don't provide much information about the
Clopper-Pearson confidence interval. During the last hour or so of the
first lecture, I'll try to explain it to you and provide some more
details. (I don't have a lot of comments here since copies of my
overhead projector presentation pertaining to this section will be /
were distributed in class.)