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categorical data analysis

Questions and Answers of Categorical Data Analysis

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 =
Table 4.20 appeared in a national study of 15- and 16-year-old adolescents.The event of interest is ever having sexual intercourse. Analyze these data and summarize in a one-page report, including
The US National Collegiate Athletic Association (NCAA) conducted a study of graduation rates for student athletes who were freshmen during the 1984–1985 academic year. Table 4.21 shows the data.
Refer to Table 7.3, treating marijuana use as the response variable. Analyze these data. Prepare a one-page report summarizing your descriptive and inferential results.
See http://bmj.com/cgi/content/full/317/7153/235 for a meta analysis of studies about whether administering albumin to critically ill patients increases or decreases mortality. Analyze the data for
Fowlkes et al. (J. Am. Statist. Assoc., 83: 611–622, 1988) reported results of a survey of employees of a large national corporation to determine how satisfaction depends on race, gender, age, and
For the model, logit[π(x)] = α + βx, show that eα equals the odds of success when x = 0. Construct the odds of success when x = 1, x = 2, and x = 3. Use this to provide an interpretation of β.
The slope of the line drawn tangent to the probit regression curve at a particular x value equals (0.40)β exp[−(α + βx)2/2].a. Show this is highest when x = −α/β, where it equals 0.40β. At
When β > 0, the logistic regression curve (4.1) has the shape of the cdf of a logistic distribution with mean μ = −α/β and standard deviation σ =1.814/β. Section 4.1.3 showed that the
For data from Florida on Y = whether someone convicted of multiple murders receives the death penalty (1 = yes, 0 = no), the prediction equation is logit(πˆ ) = −2.06 + .87d − 2.40v, where d

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