Some Comments about Chapter 1 of Hollander & Wolfe
Section 1.1
As I imply in my
comments about the preface, H&W
tend to glorify nonparametric procedures and downplay the fact that they
do have restrictions even though a parametric model doesn't have to be
assumed. But nonparametric methods do have their good points, and p. 1
of the text describes some of these good points.
- Nonparametric tests are generally easier to do with pencil and paper, and
sometimes it is desirable to literally do a "back of the envelope"
calculation.
- Nonparametric tests are generally not very sensitive to outliers.
If a somewhat strange observation may be a mistake, but one isn't
sure, it's nice that it typically won't matter a whole lot if one
deletes it or keeps it in for a nonparametric procedure. Whereas, with
a normal theory procedure, removing a single data value may make an
appreciable difference in the result, and the decision to keep it or
remove it can be an important one. Of course a strange observation may
be legitimate, giving us no good reason to remove it. If it is kept in
the data set and a normal theory procedure is applied, it may
tremendously influence and distort the results. But with a
nonparametric procedure, a single unusual observation can be dealt with
in a "fair" way, and need not distort the results (because a
nonparametric procedure is not based on an assumption of normality).
- In some settings, there is a nonparametric test which is nearly as
good (powerful) as a normal theory procedure if indeed the assumptions of the
normal theory procedure are met, and the nonparametric test is also
accurate if the assumption of normality isn't a good one.
- In many situations, it'll be possible to obtain an exact p-value
from a nonparametric procedure, as opposed to using an approximate
p-value (perhaps from a normal or chi-square approximation). (H&W
contains a lot of tables, and the StatXact software can do lots of
different nonparametric tests exactly.) Unfortunately, some routinely
use approximations with nonparametric procedures (which may be okay for
large sample sizes, but not okay for small sample sizes), and thus do
not take advantage of one of the strengths of nonparametric methods.
Section 1.2
This section introduces the notion of a distribution-free test
statistic using the Wilcoxon rank sum test from Ch. 4 as an example.
During the first lecture, I'll do something similar using the sign test
from Ch. 3 as an example: the sign test is a distribution-free test about the
median of a continuous distribution. Then I'll mention some other
nonparametric tests for one sample problems (or matched-pairs
observations, which some refer to as paired samples), and point out that
an additional assumption of symmetry has to be made if one wants to view
them as being distribution-free tests about the distribution
median/mean.
While H&W describe what is meant by a distribution-free procedure, they
don't make a point of describing how nonparametric is not the same as
distribution-free. Basically, a nonparametric procedure is
one which is not based on a particular parametric model, and can be
applied if one of a large number of parametric models happens to be the
appropriate one. A variety of density estimation methods are
referred to as being nonparametric because they can be applied the same
way whatever may be the continuous distribution underlying the data. We
can say that the simple sample median is a nonparametric estimate of the
distribution median because it can provide a sensible estimate of the
median whatever the underlying distribution is. Since the sampling
distribution of the sample median depends on what the underlying
distribution is, I wouldn't say that the sample median is a distribution-free
estimator, but I would refer to it as being nonparametric. So you can
think of distribution-free procedures as being a subset of nonparametric
procedures. Interestingly, although the null sampling distribution of a
distribution-free test statistic doesn't depend on the distribution
underlying the data, the sampling distribution of the test statistic if
the alternative hypothesis is true does depend on the distribution(s)
underlying the data (and so the power function of a distribution-free
test is not distribution-free).
Section 1.3
While the examples given in this section are okay, without looking at
the data it isn't real clear why nonparametric methods should be used
as opposed to some other types of methods.
Note that Example 1.5 seems to deal with a categorical data
analysis problem, as opposed to what is usually thought of as
nonparametric statistics. In Ch. 2 and Ch. 10, H&W cover some settings
that could be covered in a course in categorical data analysis. I plan
to cover those chapters somewhat briefly, since they are a bit out of
the mainstream of nonparametric statistics. But it certainly won't hurt
for you to get a bit more comfortable with such categorical data
analysis settings, and we'll see some connections with
nonparametric statistics. (Plus, since StatXact does both
categorical and nonparametric procedures, you'll get some experience in
using the software for categorical data analysis problems.)
Section 1.4
This section of Ch. 1 describes what is covered in each of the
subsequent chapters. Since the topics in the
course syllabus closely match the
chapters in the book, by reading this section you should get a fairly
good idea about the types of things we'll cover this semester. But I do
intend to deviate from the text a bit, mainly by adding some material.
For example, in Chapters 4 and 5, H&W seem to assume that a treatment
has a constant effect if it has any effect. That is, that it will cause
a shift in the distribution of values if it does anything. Since this
assumption may not be true all of the time, I plan to discuss the use of
the procedures in Chapters 4 and 5 in more general settings in addition
to the simple shift model described in H&W.
Section 1.5
Since I intend to follow the text rather closely in this course, it'll
be good for you to read this section in order to gain a better
understanding of the format, organization, and philosophy of the text.
It can be noted that I, like H&W, at times like to compare approximate
results with exact ones, based on the same data, in order to learn
something about the accuracy of various approximations.
In the Ties paragraph, H&W indicate that after adjusting for ties,
the "adjusted procedure should then be viewed as an approximation." But
with StatXact, sometimes ties can be handled in such as way so
that one is
able to use an exact null sampling distribution for a test
statistic even though there are tied values.
Section 1.6
Like H&W, I tend to use StatXact and Minitab to do most of my
nonparametric statistics computing. While I think that StatXact is the best
software package for standard nonparametric statistical methods, there
is nothing special about Minitab except that it is easy to use,
and it seems to be an adequate complement to StatXact. It should
be noted that Minitab typically uses normal and chi-square
approximations, with continuity corrections and adjustments for ties in
some cases, for it's nonparametric computations, with the exception of
the sign test which is done exactly. Some software packages for
statistics, such as S-Plus and SAS, give some exact
p-values for nonparametric procedures, but none of the others compare
well to StatXact. Note that PROC-StatXact is not a
SAS product, but rather it's a
Cytel product that can be used with SAS (and since it's
not a SAS product, a lot of places that use SAS won't have
PROC-StatXact installed).
Section 1.7
This section gets into the history of nonparametric statistics, which I
find to be interesting.
Although I think it's safe to say that the use of nonparametric procedures
really took off in the mid 1940s with the publication of Wilcoxon's
important paper, H&W point out that several nonparametric procedures
were already in place by the 1930s. But to indicate that the true
beginning of nonparametric statistics was 1936, as Savage did, seems a
bit questionable. One can note that Spearman had a paper on rank
correlation published in 1904, and the 1936 paper by Hotelling and Pabst
suggests that prior to Spearman, Galton did some similar things. Also,
Karl Pearson treated rank correlation in a 1907 book.
It can be noted that H&W include jackknifing and bootstrapping as
nonparametric procedures (but some nonparametric books do not cover
these important techniques). I covered these topics to a limited extent
in my summer course (in Summer Session 2002), and I know that it takes a
while to present even an elementary description of jackknifing and
bootstrapping. My guess is that we won't cover these topics in general in
STAT 657, but I will cover them to the extent that they are dealt with in
H&W.
Section 1.8
This section is just a large table that is a nice complement to Section
1.4.