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4.0% 12.1% 10.1% 20.2% 53..5%
10.5% 16.2% 18.5% 41.7% 68.8%
Column 38 74 54 48 77 291
Total 13.1% 25.4% 18.6% 16.5% 26.5% 100.0%
Chi-Square D.F. Significance Min E.F. Cells with E.F. <5
70.64156 8 -0000 12.405 None
in the sample space is quite large). The paper Impaired Neurocognitive Performance in Colle-
“Probability Models for the NCAA Regional giate Soccer Players” (Amer. J. Sports Med.
Basketball Tournaments”(Amer. Statist., 1991: 2002: 157-162) investigated this issue from
35-38) proposed several different models for several perspectives.
the P;;’s. a. The paper reported that 45 of the 91 soccer
a. One model postulated P;; = .5 — 2 — j) with players in their sample had suffered at least one
2=4, (from which Pig =4, Pro =2, concussion, 28 of 96 nonsoccer athletes had suf-
etc.). Based on this, P(seed #1 wins) = .27477, fered at least one concussion, and only 8 of 53
P(seed #2 wins) = .20834, and P(seed #3 student controls had suffered at least one con-
wins) = .15429. Does this model appear to cussion. Analyze this data and draw appropriate
provide a good fit to the data? conclusions.
b. A more sophisticated model has P;; = .5 + b. For the soccer players, the sample correlation
.2813625(z; — z;), where the z’s are measures coefficient calculated from the values of
of relative strengths related to standard normal X = soccer exposure (total number of
percentiles [percentiles for successive highly competitive seasons played prior to enrollment
seeded teams are closer together than is the in the study) and y = score on an immediate
case for teams seeded lower, and .2813625 memory recall test was r = -.220. Interpret this
ensures that the range of probabilities is the result.
same as for the model in part (a). The resulting ¢. Here is summary information on scores on a
probabilities of seeds 1, 2, or 3 winning their controlled oral word-association test for the
regional tournaments are .45883, .18813, and soccer and nonsoccer athletes:
82 respectively. Assess the fit of this my = 26, % = 37.50,5; = 9.13,
money iy = 56; % = "39.63, 59= 10,19
46. Have you ever wondered whether soccer players , . .
suffer adverse effects from hitting “headers”? ee i ius datasand draw appropnetticonclus
The authors of the article “No Evidence of ° .
--- Trang 770 ---
Bibliography 757
d. Considering the number of prior nonsoccer Consider nonoverlapping groups of two digits,
concussions, the values of mean + SD for the and let pj denote the long-run proportion of
three groups were soccer players, .30 + .67; groups for which the first digit is i and the
nonsoccer athletes, .49 + .87; and student con- second digit is j. What hypotheses about these
trols, .19 + .48. Analyze this data and draw proportions should be tested, and what is df for
appropriate conclusions. the chi-squared test?
47. Do the successive digits in the decimal expansion © Consider nonoverlapping groups of 5 digits.
2 Could a chi-squared test of appropriate hypoth-
of 7 behave as though they were selected from a ©.
- ‘ aes . eses about the pijum’s be based on the first
random number table (or came from a computer’s the Pijkim §
100,000 digits? Explain.
random number generator)? 5 ef a
7 d. The paper “Are the Digits of an Independent
a. Let po denote the long-run proportion of digits : ee pe
nore and Identically Distributed Sequence?” (Amer.
in the expansion that equal 0, and define p,,...,
. Statist., 2000: 12-16) considered the first
po analogously. What hypotheses about these
‘ae ; 1,254,540 digits of x, and reported the follow-
proportions should be tested, and what is df for ! \ so
the cai equaiea test? ing P-values for group sizes of 1, ..., 5 digits:
“ . -572, .078, .529, .691, .298. What would you
b. Ho of part (a) would not be rejected for the conclude?
nonrandom sequence 012 ...901... 901... Soneiaes
Agresti, Alan, An Introduction to Categorical Data but informative survey of methods for analyzing
Analysis (2nd ed.), Wiley, New York, 2007. An categorical data, exposited with a minimum of
excellent treatment of various aspects of categori- mathematics.
cal data analysis by one of the most prominent Mosteller, Frederick, and Richard Rourke, Sturdy Sta-
researchers in this area. tistics, Addison-Wesley, Reading, MA, 1973. Con-
Everitt, B. S., The Analysis of Contingency Tables (2nd tains several very readable chapters on the varied
ed.), Halsted Press, New York, 1992. A compact uses of chi-square.
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°
Alternative
Approaches
to Inference
Introduction
In this final chapter we consider some inferential methods that are different in
important ways from those considered earlier. Recall that many of the confidence
intervals and test procedures developed in Chapters 9-12 were based on some sort
of a normality assumption. As long as such an assumption is at least approximately
satisfied, the actual confidence and significance levels will be at least approxi-
mately equal to the “nominal” levels, those prescribed by the experimenter through
the choice of particular t or F critical values. However, if there is a substantial
violation of the normality assumption, the actual levels may differ considerably
from the nominal levels (e.g., the use of tos in a confidence interval formula may
actually result in a confidence level of only 88% rather than the nominal 95%).
In the first three sections of this chapter, we develop distribution-free or non-
parametric procedures that are valid for a wide variety of underlying distributions
rather than being tied to normality. We have actually already introduced several
such methods: the bootstrap intervals and permutation tests are valid without
restrictive assumptions on the underlying distribution(s).