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Statistical Methods for the Social Sciences

Alan Agresti, Barbara Finlay

Chapter 15

Logistic Regression: Modeling Categorical Responses - all with Video Answers

Educators

Chapter Questions

01:26

Problem 1

A logistic regression model describes low the probablity $\pi$ of voting for the Republican candidate in a presidential election depends on $X$. the voter's total faunily income (un thousands of dollars) in the previous year. The prediction equation for a particular sainple is
$$
\log \left(\frac{\hat{\pi}}{1-\hat{\pi}}\right)=-1.00+.02 X
$$
a) For the region of $X$ values for which $\pi$ is nead .50 , give a linear approximation for the change in the probability for each thousand dollar mcrease in income
b) At which income level is the estimated probability of voting for the Republican candidate (i) equal to .50 ?, (ii) greater than .50 ?
c) Find the estimated probability of voting for the Republican candidate $u$ hen (i) income $=10$ thousand. (ii) income $=100$ thousand.
d) Explain how the odds of voting Republican depends on famuly income.
c) The estimated standard error of the coetficient of $X$ is 005 . Assumung the sample was randomly selected, test the hypothesis that vote for the Republican candidute is independent of family incone Report the $P$-value, and interpreL.

Nick Johnson
Nick Johnson
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Problem 2

Reler to the previous exercise. When the explanatory vanables are $X_1=$ family incume. $X_2=$ number ol jears of education, and $X_3=$ gendes $(1=$ male, $0=$ fomale $)$, the prediction equation is
$$
\operatorname{logit}(\hat{\pi})=-2.40+.02 X_1 \div .08 X_2+.20 X_3
$$

For this sample. $X_1$ ranges from 6 to 157 w ith a standard deviation of 25 , and $X_2$ ranges from 7 to 20 with a standard deviation of 3 .
a) Find the estimated probability of voting Republican fol (i) a man with 16 years of education and incomne 30 thousand dollars. (u) a woman with 16 years of education and income 30 thousand dollars.
b) Conver the probahilities in (a) to odds, and find the odds ratio. dividing the odds for men by the odds for females Interpret
c) Using the prediction equation, find the effect on the odds of changing $x_3$ from 0 to 1 . controlling for $X_1$ and $X_2$. Compare to (b), and interpret
d) Find the estimated effect on the odds of volung Republican of a one year increase in $X_2$, contiolling for $X_1$ and $X_3$. Interpret.
e) Holding the othes variables constant. find the estumated effect on the odds of voting Republican of

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02:22

Problem 3

A sample of 54 clderly men ane given a psychiatric examination to determine whether symptoms of senility are present. A sublest of the Wechsler Adult Intelligence Scale (WAIS) is the explanatory variable. Table 15.15 show's results. The WAlS scores range from 4 to 20 . with a mean of 11 . Higher values indicate more effective intellectual functioning.
a) Report the prediclion equation, and explain why uhis equation suggests that the probdbility of senility decreases at higher levels of the WAIS
b) Show that $\hat{\pi}=1 / 2$ at $X=7.2$ and that $\hat{\pi}<1 / 2$ for $X>7.2$
c) Find the predicted probability of senility at $X=20$.
d) Report the $\Sigma$ test statistic for $H_0 \beta=0$ and $H_a: \beta \neq 0$. What does the Wald statistic equal? What do you conclude about the association?
e) The least squares fit of the linear probability model is $\hat{\sim}=.847-051 X$ Find the predicted probability of senility at $X=20$ Does this make sense?

Sheryl Ezze
Sheryl Ezze
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00:54

Problem 4

For the 23 space shuttle flights that occurred before the Challenger mssion disaster in 1986. Table 15.16 shows the temperature (in degrees fahrenheit) at the tine of the flight and u hether ar leasi one primary O -ring suffered thernal distucss.

Christopher Stanley
Christopher Stanley
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Problem 5

Refer to Table 15.13. Treal party affihation as the response variable, and use logistic regression to describe the effect of ideology by assigning scores $(1,2,3,4,5)$ to its levels. a) Report the predictuon equation, and obtain the predicted probability of Democrate affiliation at ideology level (i) very liberal. (ii) very conservative.
b) Usc the model to test whether the variables arc independent. Report the test statistic, $P$-value, and interpret.
c) Use the odds ratio wo describe the effect on party affiliation of a change in ideology from (i) very liberal to shghly liberal, (ii) slightly liberal to moderale.
d) Check the goodness of fil of the model Interpret.
e) II your software provides the option, oblain $95 \%$ confidence intervals for the probability of Democratic atfiliation at ideology level (i) ver liberal, (ii) very conservative. Compare.

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01:45

Problem 6

in the first mune decades of the twealieth century in baseball's National League, the percentage of times the starting pitcher pilched a complete game were 72.7 (1900-1909), 634,50 0,44.3, 41.6, 32.8, 27 2, 22.5, 13.3 (1980-1989) (George Will. New'sweek, April 10. 1989).
a) Let $X=1,2, \ldots, 9$ for the successive nine decades. Use a logistic regression model to describe the time trend in thcse data. (For simplicity, suppose the number of games was the same in each decade. For iustance, you can represent the peicentage 72.7 by 727 successes and 273 failures in 1000 trials.)
b) Use the firled model to predict the percentage of complete games in the decade $X=12$ (i.e.. 2010-2019).
c) Repeat the analysis using a lincar probability model. Is its future prediction realistic?

Carson Merrill
Carson Merrill
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01:03

Problem 7

Table 15.17 shows results of a study on the effects of AZT in slow ing the developinent of AIDS symploms. In the study, 338 veterans whose immune systems were beginning to falter after infection uith the AIDS virus were randomly assigned eithei to receive AZT inmediately or to wait until their T cells showed severe inmune weakness The table classifies the veterans' race. whether they received AZT immediatcly, and whether they developed AIDS symptoms during the three-year study. Fit a logit model with main effects to these data, and using it
a) Report and interprel the prediction equation.
b) For black vererans without immediate AZT use, find the predicted probability of showing AIDS symptoms.
c) Find the estimated conditional odds ratio between AZT use and the developinent of symptoms. Intelpret.
d) Test for the effect of AZT use. Inlerpret.
e) Test the goodness of fit of the inodel Inlerpret.

Dominador Tan
Dominador Tan
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Problem 8

A recent article used logistic regression with 1149 obsen ations from the National Election Studies to model the probability of voling for the Rcpublican candidate (George Bush) in the 1988 presidential election (K. Snvith, American Politics Quarterly. Vol. 22, 1994, p 354). The predictors dealt with attitudes abour abortion and also included a dummy variable that equaled 1 if the respondent was from a state in the old confederacy and 0 otherwise. The estimated southem effect was .503 , with a standard error of .215 .
a) Use the odds ratio to estimate the effect of southem residence on the vote for president. conırolling for abortion artitudes.
b) Test the significance of this effect, and interprel.
c) Construct and interpret a $95 \%$ confidence interval for the fruc odds 1 atio in (a).

Victor Salazar
Victor Salazar
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07:40

Problem 9

A recent study of molher's occupational stafus and childen's schooling (M Kalnin, American Sociological Review. Vol. 59, 1994, p 257) , cported the predierion equation
$$
\operatorname{logir}(\hat{\pi})=.75+.35 Y+.13 F+.09 M-30 F O+.21 M O-.92 M E-.16 S
$$
where $\pi$ is the probability the child obtains a high school degree, $Y=$ respondent's year of birth, $F=$ father's education. $M=$ mother's education (010 17). $F O=$ father's occupational level, $M O=$ mother 's occupational level ( 1 to 9 ). $M E=$ whether mother eniployed $(1=$ yes), $S=$ number of siblings All effects werc significant at the 01 lcvel
a) Interpiet the coefficient of mothe, 's education.
b) Interpret the cocfficient of whether mother employed
c) The author reported thar a one-point increase in mother's occupationdl level is associated with a $24 \%$ increase in the odds of a high school diplons. Explain hou he made this interprctation.

Oluwadamilola Ameobi
Oluwadamilola Ameobi
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Problem 10

Refer to Table 8.29 in Poblein 8. [4. The sesponse vabiable is eves having sexual intercourse.
a) Fit a logit model with main effects for lace and gendel Report the prediction equation
b) Find the predicted probability of having experienced intercourse for (i) $u$ hite fernales.
(ii) black females.
c) Find the estimated conditional odds ratio between lace and intercourse. (i) using a model parameter estimate, (ii) using the estimated probahilities in (b). Interpret.
d) Test for the effect of race in this nodel. Intelpret
e) Find the estimated conditional odds ratio between gender and intercourse. Interpiet.
f) Test for the effect of gender in the model. Interpret.
g) Test the goodness of fit of the model. Interpret.
h) Summarize your main findings abour these data in a way that you could present to the geneial public, using as little technical jargon as poscible

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Problem 11

Reler to Table 15 8. treatiug marijuana use as the resjonse variable Table 1518 shows the SAS printout (PROC GENMOD) for a logit model with mann effects
a) Report the prediction equation, and interpret.
b) Find the predicted probability of having used marijuana (i) for those who have not used alcohol or cigarettes, (ii) for those who have used both alcohol and cigarettes.
c) Show how to convert the estimated cocfficients to estimated odds ratios. Interpret.
d) Test the goodness of fit of this logit model Intcrpret
e) Which loglinear model is equiv ajent to this logit nodel?

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Problem 12

Table 15.19 refers to passengers in sutos and light truchs involved in accidents in the state of Maine in 1991. The table classifies subjects by gender, location of accidem. seat-belt usc. and a resjonsc variable having categories (1) nol injured: (2) injured but not transported by emeigency medical sen ices. (3) injured and transpouted by emergency medical services but not hospitalized. (4) injured and hospitalized hut did not dic. (5) injurce and died. For this exercise. combine iesponse catcgories $2-5$, and consider models for $\pi=$ probability of injury.

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Problem 13

Refer to the previous exercise. Fit the model that also las the three tho-way interactions between predictors. Use a likelihood-ratio test to compare this model to the man effects model. InterpreL. (Hint The test statistuc is the difference between the $(-2 \log L$ ) values, or equivalently die difference between the $G^2$ test statistics for testing fit of the two models.)

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Problem 14

Let $\pi$ denote the probability that a randomly selecled respondent supports current laus legalizung abortion. predicted using gender of respondent ( $G=0$. male, $G=1$, fernale), religious affiliation $\left(R_1=1\right.$, Protestant, 0 otherwise: $R_2=1$, Catholic, 0 otherwise. $R_1=R_2=0$, Jewish). and political parl affiliation ( $P_1=1$. Democrat, 0 otherwise: $P_2=1$. Republican, 0 otherwise. $P_1=P_2=0$, Indcpendent). The logit model with main eflects has prediction equation
$$
\operatorname{logit}(\bar{\pi})=.11+.16 G-.57 R_1-.66 R_2+.47 P_1-1.67 P_2
$$
a) Give the effect of gender on the odds of supporting legalized abortion; that is, If the odds of support for females equal 0 times the odds of support for males, report $\hat{\theta}$.
b) Give the effect of being Democrat mstead of Independent on the estimated odds of support for legahized abortion.
c) Give the effect of being Democrat instead of Republican on the estimated odds of support for legalized abortion
d) Find the estimated probability of suppouting legalized abortion. for (i) female Jewish Democrats. (n) male Catholic Repubhcans.
e) Show that there ae 18 sample logits and that $d f=12$ for lesting the fit of this model. If $G^2=10.4$. test the model goodness of fit, and interjret.
f) Let $A$ denote the opinion on current laws lcgalizing abortion State the symbol for the loglinear model that is equivalent to this logit inodel.
g) State the logit model corresponding to the loglinear model (AR. AP,GRP).
h) The logit model in (g) has $G^2=116$ for testing fit. Determine whethel gender is a significant predictor of opinion on abortion. controlling for $R$ and $P$.

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Problem 15

Consider a four-way cross-classification of variables $W, X . Y$, and $Z$.
a) State the sy mbol for the loglinear model in which
i) All pairs of variables are independent.
ii) $X$ and $Y$ are associaled but other pairs of vanables are independent.
iu) All pairs of vanables are associated, but there is no interaction.
b) Suppose all vanables are bunary, and $Y$ is the response variable Whte down the logit model that is equivalent to the loglinear model symbolized by (WXZ.YX.YZ)

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Problem 16

The hypothetical population cell proportions in Table 15.20 describe the relationship among income, gender, and college of employnient for faculty at Nornal State U.
a) Calculate the conditional odds ratios for each pair of variables, and interpret.
b) Treating incoune as the response. which logit model do these dala satisfy? Why?
c) Which loghnear model do these cell proportions satisly? Why?
d) Construct the two-way income-gender table, collapsing over college Cakculate the odds ralo. Why is it so different from the conditional odds ratios for income and gender?

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Problem 17

Refer to the loglinear model analyses reported in Section 15.4.
a) Use software to conduct the analyses.
b) Based on the table of parameter estimates (Table 15.9), construci a $95 \%$ confidence interval for the conditional ouds ratio betwcen cigaretle use and ınarijuana use. Intcrpret.

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Problem 17

Table 15.21 refers to individuals who applied for admussion into graduate school at the University of California in Berkeley. for the fall 1973 session. Data arc presented for five of the six largest graduate departunents at the university. The vanables for the $2 \times 2 \times 5$ table are denoted by
A: Whether adnitted (Jes, no)
$G$ : Gender of applican (male, femalc)
D. Department to which apphcation was sent $\left(D_1, D_2, D_2, D_4 . D_5\right)$

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Problem 19

Refer to Table 8.17 Treating ob sdlisfaction as the response variable. Table 1522 shows resulls of a SAS printout for the cumulative logit inodel, using scores $(3,15,30)$ for income, and chi-squared test of independence and ordinal measure (gatnma. Kendan's tau-b) analyses of Chapter 8 .
d) Show that, for testing independence, one would get similar results to the model using a test based on a measure of association that also uses the ordering infonnation.

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Problem 20

Refer to Table 8.32 and Probleın 8.27. Treat detection of breast cancer as the response variable.
a) Pit the cumulative logit model. using scores (3, 2. 3) for mamnography expericnce. Report and interpret thc effect.
b) Test the hypotlesis of independence. by lesting that a paraineter in this model equals
0 . Report the test statistic and $P$-value, and interpiet.
c) The choice of scores for these rou $s$ is unclear. To check sensifivity of the resules to this choice, repeat the anal ses in (a) and (b) using scores (i) (1.2.5, 3), (ii) (1, 1.5.3). Which of the three sets seems most sensible to you? Why?

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Problem 21

Refer to Problem 15 12. Now. using all five response categories. fit a cumulative logat model with main effects
a) Report the estimates of the effects of the threc predictors, and indicate which catcgory of each predictor tends to be associated with nore serious injunes.
b) Test the hypothesis of no sear belt effect. Interpret
c) Interptct the seat belt extimate. using an odds ratuo.
d) Constuct a $95 \%$ confidence interval for the true odds ratio describing the scat belt e1fect

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Problem 22

Refer to the WWW dara set (Prublem 1.7). Using compuler software. conduct and interpret a logistic regression analysis using $Y=$ opinion about abortion and predictors a) political ideology:
b) gender and whether support affinnative action
c) gender and political ideology:

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Problem 23

Refer to the WWW data set (Problern 1.7). Build a mode! for predicting whether one supports affirmative action. Prepare a shor repor, explaining bow you built the model and hou you interpiel results from that model.

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Problem 24

Reler to the data file you cieated in Problem 1.7. For variables chosen by you instructor. fit a logistic regression model and conduct descnptive and infeıential statistical analyses. Interpret and summarize your findings.

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Problem 25

Reier to the previous exercise For variables chosen by your instructol, conduct a loghnear model analysis. Interpret and summarize your findings.

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08:43

Problem 26

The data given in Problem 10.8 in Chapter 10 came from an early study on the application of the death penalty in Florida. Analyze those data using methods of this chapter.

Heather Duong
Heather Duong
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03:31

Problem 27

The U'S National Collegiate Athletic Association (NCAA) conducted a srudy of gradtittion rates for student athletes who were freshmen dusing the 1984-85 academic year Table 15.23 shows the data Analyze. In youn repout. interpret the model, explain the results ol statistical inference and checks of the adequacy of the model. and summalize your conclusions

Lucas Finney
Lucas Finney
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11:28

Problem 28

According to the Independent newspaper (London, March 8, 1994), the Metropolitan Police in London 1eported 30.475 people as missing in the year ending March 1993. For those of age 13 or less, 33 of 3271 mussing males and 38 of 2486 missing females were still missing a yeat later For ages 14-18, the values were 63 of 7256 males and 108 of 8877 females; for ages 19 and above, the values were 157 of 5065 males and 159 of 3520 femules. Analyze and interpret these date. (Thanks to Dr PM.E. Altham for showing ine these data.)

Trent Speier
Trent Speier
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Problem 29

In a study designed to evaluate whether an educationd program makes sexually active adolescents mole likely to obtain condoms, adolescents were randonly assigned to two expenmental groups. The educational program, involving a lecture and vidcotape about transmssion of the HIV vilus. was prot ided to one group but not the other In logistic regression modcls. factors observed to influence a teenager to obtain condoms were gender, socjoeconomic status. lifetime number of partuers, and the experimental group. Table 1524 summanzes the study results.
a) lnterpret the odds ratio and the related confidence interval for the effect of group.
b) Find the parameter estimates for the fitted model, using (1.0) dummy variables for the first three piedictors.
c) Explain why etther the estimate of 1.38 for the odds ratio for gender or the corresponding confidence interval seems incorrect. The confidence interval is hased on taking antilogs ol endpoints of a confidence interial for the log odds ratio. Show that if the reported couffidence intcrval is correct. then 1.38 is actually the og odds ratio. and the estimated odds ratio equals 3.98

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02:17

Problem 30

L'sing samples taken1 in the late 1980s by the Insutute of Social Research in Zagreb, a recent study used logistic regression to model the probability that a subject identified themselves as Yugoslav, when given the option of choosing that category or Cioat, Serb. Moslem, or some other nationality Table 15.25 shous estimates for the predictors: urbanism of residence $(1=$ village. $2=$ town. $3=$ city$)$, memberslip at some time in the Communist Party $(1=$ yes. $0=$ no), age. nationall mixed puentagc $(1=$ yes, $0=$ no). Explain how to interpret results in this table
3

Victor Salazar
Victor Salazar
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01:54

Problem 31

Table 15.26 displays data finm the $1987-88$ National Survey of Families and Households. The sample consists of currently mamed couples. married for 20 or fewer years, among

Anna Jones
Anna Jones
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Problem 32

Table 15.27 is a contingency table summanzng observations on the degrec of participation in school sports for random samples of 30 boys and 30 girls in middle schools.
a) Use ch-squared (Section 8.2) to test whether participation is independent of gender. Report the $P$-value.
b) Now, conduct a test using an ordinal analysis, and report the $P$-value
c) Compare the results of parts (a) and (b), and discuss why it is generally inappropnate to use the chn-squared test if one or both variables are ordinal.

Lainey Roebuck
Lainey Roebuck
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Problem 33

Refer to Table 8.39 and Problem 8 39. Using a cumulauve logit model, analyze and interpret these data.

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02:02

Problem 34

Reter to Problem 8.28. Using a model, analyze these data, compare results to the Pearson chi-squared test of independence, and explain the discrepancy.

Sheryl Ezze
Sheryl Ezze
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03:13

Problem 35

Refe, to Table 15.4. Show that the association between the defendant's race and the death penalty verdict satisfies Simpson's paradox What causes this?

Hossam Mohamed
Hossam Mohamed
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02:17

Problem 36

One reason logistic regression is usually preferıed over the linear probability nodel is that a fixed change in $X$ often bas a stnaller impact on $\pi$ when $\pi$ is near 0 or neat 1 than when $\pi$ is ncar the middle of its range. Let $Y$ refer to the decision to rent on to buy a home. with $\pi=$ the probability of buymg and let $X=$ weekl; family income. In which case do you think an increase of $$\$ 100$$ in $X$ has greater effect: when $$X=\$ 50,000$$ (for which $\pi$ is near 1) or when $X=\$ 500^{\circ}$ Explain how your answer relates to the choice of a linear versus logistic regression model

Victor Salazar
Victor Salazar
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Problem 37

' For the logistic regression model, fiom the linear approximation $\beta / 4$ for the rate of change in the probability at $\pi=5$, show that $1 /|\beta|$ is the appioximate distance between the $X$-values at which $\pi=1 / 4$ (or $\pi=3 / 4$ ) and at which $\pi=1 / 2$. Thus. the larger the value of $|\beta|$. the less the $X$-distance over which this change in probability occurs

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Problem 38

State the symbols for the loglinear models for categorical variables that are implied by the causal diagrans in Figure 15.5.
โa!
(b)
$$
X \longrightarrow Y
$$
(c)
fd 1
Figure 15.5

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Problem 39

* Fol a two-wdy contingency tablc, let $r_t$ denote the $i$ tht row total, let $c_i$ denote the $j$ th colujnn totdl. and let $n$ denote the total sample size Section 82 showed that the cell in row $i$ and column $j$ bas $f_i=r_i c_j / n$ for the independence model. Show that the log of the expected frequency has an additive fornula with terms representing the influence of the $i$ th 1 ou total. the $j$ th column total, and the sample size. This formula is the loglnear model for independence in (wo-way contingency tables

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03:05

Problem 40

For a $2 \times 2$ table with cell counts $a . b, c . d$. the sanipic $\log$ odds ratio $\log \hat{\theta}$ has approximatel) a nurnal sampling distribution with standard errol
$$ \hat{\sigma}(\log \hat{\theta})=\sqrt{\frac{1}{a+.5}+\frac{1}{b+.5}+\frac{1}{c+.5}+\frac{1}{d+5}} $$
The antilogs of the endpoints of the confidence interval for $\log (\theta)$ are endpoints of the confidence interval for $\theta$ Construct a $95 \%$ confidence interval for the odds ratio for Table 8.15 in Section 8.4 Interpret.

Willis James
Willis James
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Problem 41

* Explain what is meant by the absence of statistical inleraction in modeling the relationship between a response variable $Y$ and two explanatory vanables $X_1$ and $X_2$ in each of the following cases. Use graphs or tables to illustrate.
a) $Y, X_1$. and $X_2$ are guantitative
b) $Y$ and $X_1$ are quantitative; $X_2$ is qualutative.
c) $Y$ is guantilative. $X_1$ and $X_2$ are qualitative.
d) $Y . X_1$. and $X_2$ are binary.
e) $Y$ is binary: $X_1$ and $X_2$ are quantitative.

Shu Naito
Shu Naito
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