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we refer to the article “Television Viewing and Physical Fitness in Adults”
(Res. Quart. Exercise Sport, 1990: 315-320). Subjects were asked about their
television-viewing habits and were classified as physically fit if they scored in
the excellent or very good category on a step test. Table 13.11 shows the results in
the form of a 4 x 2 table. The TV column gives the hours per day
Table 13.11 TV versus fitness results
TV Time Unfit Fit
i oe
i
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13.3 Two-Way Contingency Tables 751
The rows need to be given specific numeric values for computational pur-
poses, and it is convenient to make these just 1, 2, 3, 4, because consecutive integers
correspond to the assumption of a common odds ratio from row to row. The
columns may need to be labeled as 0 and 1 for input to a program. The logistic
regression results from MINITAB are shown in Figure 13.5, where the estimated
coefficient f, for TV is given as -.29 and the odds ratio is given as .75 = e~°. This
means that, for each increase in TV watching category, the odds of being fit decline
to about 3/4 of the previous value. There is a loss of 25% for each increment in TV.
The output shows two tests for f), a z based on the ratio of the coefficient to
its estimated standard error and G, which is based on a likelihood ratio test and
gives the chi-squared approximation for the difference of log likelihoods. The two
tests usually give very similar results, with G being approximately the square of z.
In this case they agree that the P-value is around .02, which means that we should
reject at the .05 level the hypothesis that ,; = 0, and we can conclude that there is a
relationship between TV watching and fitness. Of course, the existence of a
relationship does not imply anything about one causing the other. By the way, a
chi-squared test yields xr = 6.161 with 3 df, P = .104, so with this test we would
not conclude that there is a relationship, even at the 10% level. There is an
advantage in using logistic regression for this kind of data.
Logistic Regression Table
odds 95% CI
Predictor Coef SE Coef Zz P Ratio Lower Upper
Constant -1.21316 0.267486 -4.54 0.000
TV -0.290693 0.125588 -2.31 0.021 0.75 0.58 0.96
Log-Likelihood = -483.205
Test that all slopes are zero: G = 5.501, DF = 1, P-Value = 0.019
Figure 13.5 Logistic regression for TV versus fitness =
Suppose there are two ordinal factors, each with more than two levels. This
too can be handled with logistic regression, but it requires a procedure called
ordinal logistic regression that allows an ordinal dependent variable. When one
factor is ordinal and the other is not, the analysis can be done with multinomial
(also called nominal or polytomous) logistic regression, which allows a non-ordinal
dependent variable.
Models and methods for analyzing data in which each individual is cate-
gorized with respect to three or more factors (multidimensional contingency tables)
are discussed in several of the references in the chapter bibliography.
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752 = cuarrer 13 Goodness-of-Fit Tests and Categorical Data Analysis
Exercises | Section 13.3 (23-35)
23. Reconsider the Cubs data of Exercise 56 women the number of individuals whose feet
in Chapter 10. Form a 2 x 2 table for the data were the same size, had a bigger left than right
and use a 7 statistic to test the hypothesis of foot (a difference of half a shoe size or more), or
equal population proportions. The 7° statistic had a bigger right than left foot.
should be the square of the z statistic in Exer- Sample
cise 56 of Chapter 10. How are the P-values L>R L=R L<R_ Size
related?
24, The accompanying data refers to leaf marks
both long-grass areas and short-grass areas
(The Biology ob the: Leal ark Polymnorpiusen Does the data indicate that gender has a strong
wy iatouunn werens Lio* Hereding 1916: ffect on the development of foot asymmetry?
306-325). Use a 77 test to decide whether the Sista thes Hoon endaltensth Wet
true proportions of different marks aré identical te RE AROIANS TOES oma YEE
. eses, compute the value of y°, and obtain infor-
Gee OWES ORIER IONE mation about the P-value.
Type of Mark Sample 27, The article “Susceptibility of Mice to Audio-
dy BEYSGNT 0 LOMIEES FSize genic Seizure Is Increased by Handling Their
Long: Dams During Gestation” (Science, 1976:
Grass] 409 22 | 7 | 277] 726 427-428) reports on research into the effect of
Areas different injection treatments on the frequencies
Short- foo fs] | | of audiogenic seizures.
Gras | 512) 4 | 14 220") 76L No Wild Clonic Tonic
Areas Treatment Response Running Seizure Seizure
25. The following data resulted from an experiment “Thlenyialaning “4
to study the effects of leaf removal on the ability Solvent 34
of fruit of a certain type to mature (“Fruit Set,
Herbivory, Fruit Reproduction, and the Fruiting Sham 48
Strategy of Catalpa speciosa,” Ecology, 1980: Unhandled 32
57-64). Does the data suggest that the chance
of a fruit maturing is affected by the number of.
leaves removed? State and test the appropriate bes thie aca gest thiat thevinie: percentages
hypotheses at level 01. in the different Tesponse categories depend on
the nature of the injection treatment? State and
Number Nuraber test the appropriate hypotheses using x = .005.
of Fruits of Fruits 28. The accompanying data on sex combinations of
Treatment Matured Aborted two recombinants resulting from six different
TAT TN male genotypes appears in the article “A New
Control 141 206 Method for Distinguishing Between Meiotic and
Two leaves removed 28 69 Premeiotic Recombinational Events in Drosoph-
Four leaves removed 25 73 ila melanogaster” (Genetics, 1979: 543-554).
Six leaves removed 24 78 Does the data support the hypothesis that the