Questions and Answers of Categorical Data Analysis

In his autobiography A Sort of Life, British author Graham Greene described a period of severe mental depression during which he played Russian Roulette.This “game” consists of putting a bullet
When the 2000 General Social Survey asked subjects whether they would be willing to accept cuts in their standard of living to protect the environment, 344 of 1170 subjects said “yes.”a. Estimate
A sample of women suffering from excessive menstrual bleeding have been taking an analgesic designed to diminish the effects.Anewanalgesic is claimed to provide greater relief. After trying the new
Refer to the previous exercise. The researchers wanted a sufficiently large sample to be able to estimate the probability of preferring the new analgesic to within 0.08, with confidence 0.95. If the
When a recent General Social Survey asked 1158 American adults, “Do you believe in Heaven?”, the proportion who answered yes was 0.86. Treating this as a random sample, conduct statistical
To collect data in an introductory statistics course, recently I gave the students a questionnaire. One question asked whether the student was a vegetarian. Of 25 students, 0 answered “yes.” They
Refer to the previous exercise, with y = 0 in n = 25 trials.a. Showthat 0, the maximized likelihood underH0, equals (1 − π0)25, which is (0.50)25 for H0: π = 0.50.b. Show that 1, the maximum of
Sections 1.4.4 and 1.4.5 found binomial P-values for a clinical trial with y = 9 successes in 10 trials. Suppose instead y = 8. Using the binomial distribution shown in Table 1.2:a. Find the P-value
If Y is a variate and c is a positive constant, then the standard deviation of the distribution of cY equals cσ(Y ). Suppose Y is a binomial variate, and let p = Y/n.a. √Based on the binomial
Using calculus, it is easier to derive the maximum of the log of the likelihood function, L = log , than the likelihood function itself. Both functions have maximum at the same value, so it is
Suppose a researcher routinely conducts significance tests by rejecting H0 if the P-value satisfies P ≤ 0.05. Suppose a test using a test statistic T and righttail probability for the P-value has
For a given sample proportion p, showthat a value π0 for which the test statistic z = (p − π0)/√[π0(1 − π0)/n] takes some fixed value z0 (such as 1.96) is a solution to the equation (1 + z2
An article in the NewYork Times (February 17, 1999) about the PSA blood test for detecting prostate cancer stated that, of men who had this disease, the test fails to detect prostate cancer in 1 in 4
For diagnostic testing, let X = true status (1 = disease, 2 = no disease)and Y = diagnosis (1 = positive, 2 = negative). Let πi = P(Y = 1|X = i), i = 1, 2.a. Explain why sensitivity = π1 and
According to recent UN figures, the annual gun homicide rate is 62.4 per one million residents in the United States and 1.3 per one million residents in the UK.a. Compare the proportion of residents
Consider the following two studies reported in the New York Times:a. A British study reported (December 3, 1998) that, of smokers who get lung cancer, “women were 1.7 times more vulnerable than men
In the United States, the estimated annual probability that awoman over the age of 35 dies of lung cancer equals 0.001304 for current smokers and 0.000121 for nonsmokers [M. Pagano and K. Gauvreau,
For adults who sailed on the Titanic on its fateful voyage, the odds ratio between gender (female, male) and survival (yes, no) was 11.4. (For data, see R. Dawson, J. Statist. Educ. 3, no. 3,
A research study estimated that under a certain condition, the probability a subject would be referred for heart catheterization was 0.906 for whites and 0.847 for blacks.a. A press release about the
An estimated odds ratio for adult females between the presence of squamous cell carcinoma (yes, no) and smoking behavior (smoker, nonsmoker) equals 11.7 when the smoker category consists of subjects
Data posted at the FBI website (www.fbi.gov) stated that of all blacks slain in 2005, 91% were slain by blacks, and of all whites slain in 2005, 83% were slain by whites. Let Y denote race of victim
A20-year study of British male physicians (R. Doll and R. Peto, British Med. J., 2: 1525–1536, 1976) noted that the proportion who died from lung cancer was 0.00140 per year for cigarette smokers
A statistical analysis that combines information from several studies is called a meta analysis. A meta analysis compared aspirin with placebo on incidence of heart attack and of stroke, separately
Refer to Table 2.1 about belief in an afterlife.a. Construct a 90% confidence interval for the difference of proportions, and interpret.b. Construct a 90% confidence interval for the odds ratio, and
A poll by Louis Harris and Associates of 1249 adult Americans indicated that 36% believe in ghosts and 37% believe in astrology. Can you compare the proportions using inferential methods for
A large-sample confidence interval for the log of the relative risk isAntilogs of the endpoints yield an interval for the true relative risk. Verify the 95% confidence interval of (1.43, 2.30)
Table 2.12 comes from one of the first studies of the link between lung cancer and smoking, by Richard Doll and A. Bradford Hill. In 20 hospitals in London, UK, patients admitted with lung cancer in
Refer to Table 2.3. Find the P-value for testing that the incidence of heart attacks is independent of aspirin intake using (a) X2, (b) G2. Interpret results.
Table 2.13 shows data from the 2002 General Social Survey cross classifying a person’s perceived happiness with their family income. The table displays the observed and expected cell counts and the
Table 2.14 was taken from the 2002 General Social Survey.a. Test the null hypothesis of independence between party identification and race. Interpretb. Use standardized residuals to describe the
In an investigation of the relationship between stage of breast cancer at diagnosis(local or advanced) and a woman’s living arrangement (D. J. Moritz and W. A. Satariano, J. Clin. Epidemiol., 46:
Each subject in a sample of 100 men and 100 women is asked to indicate which of the following factors (one or more) are responsible for increases in teenage crime: A, the increasing gap in income
Table 2.15 classifies a sample of psychiatric patients by their diagnosis and by whether their treatment prescribed drugs.a. Conduct a test of independence, and interpret the P-value.b. Obtain
Table 2.16, from a recent General Social Survey, cross-classifies the degree of fundamentalism of subjects’ religious beliefs by their highest degree of education. The table also shows standardized
Formula (2.8) has alternative formula X2 = n(pij − pi+p+j )2/pi+p+j .Hence, for given {pij }, X2 is large when n is sufficiently large, regardless of whether the association is practically
For tests of H0: independence, { ˆ μij = ni+n+j/n}.a. Show that { ˆ μij } have the same row and column totals as {nij }.b. For 2 × 2 tables, show that ˆμ11ˆμ22/ˆμ12 ˆμ21 = 1.0. Hence, {
A chi-squared variate with degrees of freedom equal to df has representation Z2 1+· · ·+Z2 df , where Z1, . . . , Zdf are independent standard normal variates.a. If Z has a standard normal
A study on educational aspirations of high school students (S. Crysdale, Int. J.Comp. Sociol., 16: 19–36, 1975) measured aspirations using the scale (some high school, high school graduate, some
By trial and error, find a 3 × 3 table of counts for which the P-value is greater than 0.05 for the X2 test but less than 0.05 for the M2 ordinal test. Explain why this happens.
A study (B. Kristensen et al., J. Intern. Med., 232: 237–245, 1992) considered the effect of prednisolone on severe hypercalcaemia in women with metastatic breast cancer. Of 30 patients, 15 were
Table 2.17 contains results of a study comparing radiation therapy with surgery in treating cancer of the larynx. Use Fisher’s exact test to testH0: θ = 1 against Ha: θ > 1. Interpret results.
Refer to the previous exercise.a. Obtain and interpret a two-sided exact P-value.b. Obtain and interpret the one-sided mid P-value. Give advantages of this type of P-value, compared with the ordinary
Of the six candidates for three managerial positions, three are female and three are male. Denote the females by F1, F2, F3 and the males by M1, M2, M3.The result of choosing the managers is (F2, M1,
In murder trials in 20 Florida counties during 1976 and 1977, the death penalty was given in 19 out of 151 cases in which a white killed a white, in 0 out of 9 cases in which a white killed a black,
Smith and Jones are baseball players. Smith had a higher batting average than Jones in 2005 and 2006. Is it possible that, for the combined data for these two years, Jones had the higher batting
At each age level, the death rate is higher in South Carolina than in Maine, but overall the death rate is higher in Maine. Explain how this could be possible.(For data, see H.Wainer, Chance, 12: 44,
Give a “real world” example of three variables X, Y , and Z, for which you expect X and Y to be marginally associated but conditionally independent, controlling for Z.
Based on murder rates in the United States, the Associated Press reported that the probability a newborn child has of eventually being a murder victim is 0.0263 for nonwhite males, 0.0049 for white
For three-way contingency tables:a. When any pair of variables is conditionally independent, explain why there is homogeneous association.b. When there is not homogeneous association, explain why no
True, or false?a. In 2 × 2 tables, statistical independence is equivalent to a population odds ratio value of θ = 1.0.b. We found that a 95% confidence interval for the odds ratio relating having a
Describe the purpose of the link function of a GLM. Define the identity link, and explain why it is not often used with the binomial parameter.
In the 2000 US Presidential election, Palm Beach County in Florida was the focus of unusual voting patterns apparently caused by a confusing “butterfly ballot.” Many voters claimed they voted
Refer to Table 2.7 on x = mother’s alcohol consumption and Y = whether a baby has sex organ malformation. WIth scores (0, 0.5, 1.5, 4.0, 7.0) for alcohol consumption, ML fitting of the linear
Refer to the previous exercise and the solution to (b).a. The sample proportion of malformations is much higher in the highest alcohol category than the others because, although it has only one
For Table 3.1 on snoring and heart disease, re-fit the linear probability model or the logistic regression model using the scores (i) (0, 2, 4, 6), (ii) (0, 1, 2, 3),(iii) (1, 2, 3, 4). Compare the
In Section 3.2.2 on the snoring and heart disease data, refer to the linear probability model. Would the least squares fit differ from the ML fit for the 2484 binary observations? (Hint: The least
Access the horseshoe crab data in Table 3.2 at www.stat.ufl.edu/∼aa/introcda/appendix.html. Let Y = 1 if a crab has at least one satellite, and let Y = 0 otherwise. Using weight as the predictor,
Refer to the previous exercise for the horseshoe crab data.a. Report the fit for the probit model, with weight predictor.b. Find ˆπ at the highest observed weight, 5.20 kg.c. Describe the weight
Table 3.6 refers to a sample of subjects randomly selected for an Italian study on the relation between income and whether one possesses a travel credit card(such as American Express or Diners Club).
Refer to Problem 4.1 on cancer remission. Table 3.7 shows output for fitting a probit model. Interpret the parameter estimates (a) finding the remission value at which the estimated probability of
An experiment analyzes imperfection rates for two processes used to fabricate silicon wafers for computer chips. For treatment A applied to 10 wafers, the numbers of imperfections are 8, 7, 6, 6, 3,
Refer to Problem 3.11. The wafers are also classified by thickness of silicon coating (z = 0, low; z = 1, high). The first five imperfection counts reported for each treatment refer to z = 0 and the
Access the horseshoe crab data of Table 3.2 at www.stat.ufl.edu/∼aa/introcda/appendix.html.a. Using x = weight and Y = number of satellites, fit a Poisson loglinear model. Report the prediction
Refer to the previous exercise. Allow overdispersion by fitting the negative binomial loglinear model.a. Report the prediction equation and the estimate of the dispersion parameter and its SE. Is
A recent General Social Survey asked subjects, “Within the past 12 months, how many people have you known personally that were victims of homicide”? The sample mean for the 159 blacks was 0.522,
One question in a recent General Social Survey asked subjects howmany times they had had sexual intercourse in the previous month.a. The sample means were 5.9 for males and 4.3 for females; the
A study dealing with motor vehicle accident rates for elderly drivers (W. Ray et al., Am. J. Epidemiol., 132: 873–884, 1992) indicated that the entire cohort of elderly drivers had 495 injurious
Table 3.8 lists total attendance (in thousands) and the total number of arrests in a season for soccer teams in the Second Division of the British football league.a. Let Y denote the number of
Table 3.4 showed data on accidents involving trains.a. Is it plausible that the collision counts are independent Poisson variates with constant rate over the 29 years? Respond by comparing a Poisson
Table 3.9, based on a study with British doctors conducted by R. Doll and A. Bradford Hill, was analyzed by N. R. Breslow in A Celebration of Statistics, Berlin: Springer, 1985.a. For each age,
For rate data, a GLM with identity link isμ/t = α + βx Explain why you could fit this model using t and tx as explanatory variables and with no intercept or offset terms.
True, or false?a. An ordinary regression (or ANOVA) model that treats the response Y as normally distributed is a special case of a GLM, with normal random component and identity link function.b.
A study used logistic regression to determine characteristics associated with Y = whether a cancer patient achieved remission (1 = yes). The most important explanatory variable was a labeling index
Refer to the previous exercise. Using information from Table 4.9:a. Conduct aWald test for the LI effect. Interpret.b. Construct aWald confidence interval for the odds ratio corresponding to a 1-unit
In the first nine decades of the twentieth century in baseball’s National League, the percentage of times the starting pitcher pitched a complete game were: 72.7(1900–1909), 63.4, 50.0, 44.3,
Consider the snoring and heart disease data of Table 3.1 in Section 3.2.2.With scores {0, 2, 4, 5} for snoring levels, the logistic regression ML fit is logit(πˆ ) = −3.866 + 0.397x.a. Interpret
For the 23 space shuttle flights before the Challenger mission disaster in 1986, Table 4.10 shows the temperature (◦F) at the time of the flight and whether at least one primary O-ring suffered
Refer to Exercise 3.9. Use the logistic regression output reported there to (a)interpret the effect of income on the odds of possessing a travel credit card, and conduct a (b) significance test and
Hastie and Tibshirani (1990, p. 282) described a study to determine risk factors for kyphosis, which is severe forward flexion of the spine following corrective spinal surgery. The age in months at
For the horseshoe crab data (Table 3.2, available at www.stat.ufl.edu/∼aa/intro-cda/appendix.html), fit the logistic regression model for π = probability of a satellite, using weight as the
For the horseshoe crab data, fit a logistic regression model for the probability of a satellite, using color alone as the predictor.a. Treat color as nominal scale (qualitative). Report the
An international poll quoted in an Associated Press story (December 14, 2004)reported low approval ratings for President GeorgeW. Bush among traditional allies of the United States, such as 32% in
Moritz and Satariano (J. Clin. Epidemiol., 46: 443–454, 1993) used logistic regression to predict whether the stage of breast cancer at diagnosis was advanced or local for a sample of 444
Exercise 2.33 mentioned a study in Florida that stated that the death penalty was given in 19 out of 151 cases in which a white killed a white, in 0 out of 9 cases in which a white killed a black, in
Refer to (d) in the previous exercise. The Cochran–Mantel–Haenszel test statistic for this hypothesis equals 7.00.a. Report the null sampling distribution of the statistic and the P-value.b.
Refer to the results that Table 4.5 shows for model (4.5) fitted to the data from the AZT and AIDS study in Table 4.4.a. For black veterans without immediateAZT use, use the prediction equation to
Table 4.12 refers to ratings of agricultural extension agents in North Carolina.In each of five districts, agents were classified by their race and by whether they qualified for a merit pay
Table 4.13 shows the result of cross classifying a sample of people from the MBTI Step II National Sample (collected and compiled by CPP, Inc.) on whether they report drinking alcohol frequently (1 =
Refer to the previous exercise. Table 4.14 shows the fit of the model with only E/I and T/F as predictors.a. Find ˆπ for someone of personality type introverted and feeling.b. Report and interpret
A study used the 1998 Behavioral Risk Factors Social Survey to consider factors associated with American women’s use of oral contraceptives. Table 4.15 summarizes effects for a logistic regression
A sample of subjects were asked their opinion about current laws legalizing abortion (support, oppose). For the explanatory variables gender (female, male), religious affiliation (Protestant,
Table 4.16 shows results of an eight-center clinical trial to compare a drug to placebo for curing an infection. At each center, subjects were randomly assigned to groups.a. Analyze these data,
In a study designed to evaluate whether an educational program makes sexually active adolescents more likely to obtain condoms, adolescents were randomly assigned to two experimental groups. The
Refer to model (4.11) with width and color effects for the horseshoe crab data.Using the data at www.stat.ufl.edu/∼aa/intro-cda/appendix.html:a. Fit the model, treating color as nominal-scale but
Table 4.18 shows estimated effects for a fitted logistic regression model with squamous cell esophageal cancer (1 = yes, 0 = no) as the response variable Y . Smoking status (S) equals 1 for at least
Table 4.19 shows results of a study about Y = whether a patient having surgery with general anesthesia experienced a sore throat on waking (1 = yes) as a function of D = duration of the surgery (in
For model (4.11) for the horseshoe crabs with color and width predictors, add three terms to permit interaction between color and width.a. Report the prediction equations relating width to the
Model (4.11) for the probability π of a satellite for horseshoe crabs with color and width predictors has fit logit(πˆ ) = −12.715 + 1.330c1 + 1.402c2 + 1.106c3 + 0.468x Consider this fit for
The prediction equation for the horseshoe crab data using width and quantitative color (scores 1, 2, 3, 4) is logit(πˆ ) = −10.071 − 0.509c + 0.458x. Color has mean = 2.44 and standard
For recent General Social Survey data, a prediction equation relating Y =whether attended college (1 = yes) to x = family income (thousands of dollars, using scores for grouped categories), m =

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