Categorical Data Analysis 3rd Edition Alan Agresti - Solutions

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4. What is the employee lifecycle and how does it impact the data ecosystem at most organizations?
3. Why is descriptive analytics the most important form of analytics in HR?
2. What are the main differences between descriptive, predictive, and prescriptive analytics?
1. What is digital transformation and how is it different than HR Analytics? How does machine learning stand to impact both?
2. What are the two main types of professionals moving into the HR analytics space today? What skills do they bring and which do they need to develop?
1. What makes employee data different from other types of data used for data science?
What are the four parts of HR Analytics Ikigai? Why is each important?
What is the difference between deductive and inductive reasoning? What is the value of each?
1.1 Identify each variable as nominal, ordinal, or interval.a. UK political party preference (Labour, Liberal Democrat, Conservative)b. Anxiety rating (none, mild, moderate, severe, very severe)c. Patient survival (in number of months)d. Clinic location (London, Boston, Madison, Rochester,
1.2 Each of 100 multiple-choice questions on an exam has four possible answers, one of which is correct. For each question, a student guesses by selecting an answer randomly.a. Specify the distribution of the number of correct answers.b. Find the mean and standard deviation of that distribution.
1.4 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 in one of the six chambers of a pistol, spinning the chambers to select one at random, and then
1.5 When the 2010 General Social Survey asked, "Please tell me whether or not you think it should be possible for a pregnant woman to obtain a legal abortion if she is married and does not want any more children," 587 replied "yes" and 636 replied "no." Let n denote the population proportion who
1.6 Refer to the vegetarianism example in Section 1.4.3. For testing H0: π = 0.50 against Hα: π ≠ 0.50, show that:a. The likelihood-ratio statistic equals 2[25 log(25/12.5)] = 34.7.b. The chi-squared form of the score statistic equals 25.0.c. The Wald z or chi-squared statistic is infinite.
1.7 In a crossover trial comparing a new drug to a standard, π denotes the probability that the new one is judged better. It is desired to estimate π and test H0: π = 0.50 against Hα: π ≠ 0.50. In 20 independent observations, the new drug is better each time.a. Find and sketch the likelihood
1.8 Refer to the previous exercise. Suppose you wanted a large enough sample to estimate the probability of preferring the new drug to within 0.05, with confidence 0.95. If the true probability is 0.80, about how large a sample is needed?
1.9 In an experiment on chlorophyll inheritance in maize, for 1103 seedlings of self-fertilized heterozygous green plants, 854 seedlings were green and 249 were yellow.Theory predicts the ratio of green to yellow is 3:1. Test the hypothesis that 3:1 is the true ratio. Report the P-value, and
1.10 Table 1.3 contains Ladislaus von Bortkiewicz's data on deaths of soldiers in the Prussian army from kicks by army mules (Fisher 1934, Quine and Seneta 1987).The data refer to 10 army corps, each observed for 20 years. In 109 corps-years of exposure, there were no deaths, in 65 corps-years
1.11 A binomial experiment tests $$H_0: \pi = 0.50$$ against $$H_a: \pi eq 0.50$$ using significance level 0.05. Only n = 5 observations are available. Show that the true null probability of rejecting Ho is 0.00 for an exact binomial test and $$ \frac{1}{16}$$ using the large-sample score test.
1.12 A researcher routinely tests using a nominal P (type I error) = 0.05, rejecting Ho if the P-value < 0.05. An exact test using test statistic T has null distribution P(T =0) = 0.30, P(T = 1) = 0.62, and P(T = 2) = 0.08, where a higher T provides more evidence against the null.Table 1.3 Data on
1.13 The 2006 General Social Survey asked respondents how much government should spend on culture and the arts, with categories (much more, more, the same, less, much less). For 18-21 year-old females, the counts in these categories were (0, 8, 10, 9, 1). Find the Bayes estimates of the population
1.14 Refer to Example 1.6.4 on estimating the proportion of vegetarians. For the Jeffreys prior, find the posterior mean, the posterior 95% equal-tail interval, and the 95%highest posterior density interval.
1.15 You plan to use Bayesian methods to estimate binomial parameters in two cases, using n observations. In case (1) you want to estimate the probability that a new treatment for skin cancer is effective. In case (2) you want to estimate the probability of a head when you repeatedly flip a
1.16 It is easier to get a precise estimate of the binomial parameter when π is near 0 or 1 than when it is near. Explain why.
1.17 Suppose that P(Y₁ = 1) = 1 − P(Y₁ = 0) = π, i = 1,..., n, where {Y;} are inde-pendent. Let Y = ΣΥ.a. What is the distribution of Y? What are E(Y) and var(Y)?b. When {Y;} instead have pairwise correlation p > 0, show that var(Y) > ηπ(1π), overdispersion relative to the binomial.
1.18 For a sequence of independent Bernoulli trials, let Y be the number of successes before the kth failure. Explain why its probability mass function is the negative binomial, $$p(y) = \frac{(y + k - 1)!}{y!(k - 1)!} \pi^y (1 - \pi)^k$$, y = 0, 1, 2, ....[For it, E(Y) = kπ/(1 – π) and var(Y)
1.19 For the multinomial distribution, show that$$corr(n_j, n_k) = -\pi_j \pi_k/\sqrt{\pi_j (1 - \pi_j)\pi_k(1 - \pi_k)}.$$When c = 2, show that this simplifies to corr(n1, n2) = -1, and explain why this makes intuitive sense.
1.20 Show that the moment generating function (mgf) is (a) m(t) = (1 - π + πε')" for the binomial distribution, (b) m(t) = exp{u[exp(t) - 1]} for the Poisson distribution.For each distribution, use them to obtain the first two moments and to show a reproductive property.
1.21 A likelihood-ratio statistic equals to. At the ML estimates, show that the data are exp(to/2) times more likely under Ha than under Ho.
1.22 Suppose that y1, y2,..., yn are independent from a Poisson distribution.a. Obtain the likelihood function. Show that the ML estimator û = ӯ.b. Construct a large-sample test statistic for Ho: μ = μο using (i) the Wald method,(ii) the score method, and (iii) the likelihood-ratio method.c.
1.23 Inference for Poisson parameters can often be based on connections with binomial and multinomial distributions. Show how to test Ho: μ₁ = μ₂ for two populations based on independent Poisson counts (y1, y2), using a corresponding binomial test.[Hint: Condition on n = y1 + y2 and identify
1.24 Since the Wald confidence interval for a binomial parameter π is degenerate when t = 0 or 1, argue that the probability that the interval covers π cannot exceed[1 - π" – (1 – π)"]; hence, the infimum of the coverage probability over 0 < π < 1 equals 0, regardless of n.
1.25 We noted in Section 1.4.2 that the midpoint of the score confidence interval (1.14)for π is the sample proportion after adding 22/2 observations to the sample, half of each type. This motivates a simple confidence interval,$$Ñ± Ζα/2√元(1 – ñ)/n*$$, where n* = n + 2/2.Show that the
1.26 A binomial sample of size n has y = 0 successes.a. Show that the confidence interval for based on the likelihood function is[0.0, 1 exp(-22/2/2n)]. For a = 0.05, use the expansion of an exponential function to show that this is approximately [0, 1.92/n].b. For the score method, show that the
1.27 Suppose that P(T = tj) = πj, j = 1,.... Show that E(mid P-value) = 0.50.[Hint: Show that Σ; π;(π;/2 + πj+1 +۰۰۰) = (Σ; π;)²/2.]
1.28 For a statistic T with cdf F(t) and p(t) = P(T = t), the mid distribution func-tion is Fmid(t) = F(t) - 0.50p(t) (Parzen 1997). Given T = to, show that the mid P-value equals 1 - F(to). (It also satisfies E[Fmid(T)] = 0.50 and var [Fmid(T)] =(1/12){1 - E[p²(T)]}.)
1.29 Genotypes AA, Aa, and aa occur with probabilities [02, 20(1 – 0), (1–0)2].A multinomial sample of size n has frequencies (n1, n2, n3) of these three genotypes.a. Form the log likelihood. Show that = (2n1 + n2)/(2n1 + 2n2 + 2n3).b. Show that -2L(0)/102 = [(2n1 + n2)/02] + [(n2 + 2n3)/(1 -
1.30 Refer to Section 1.5.6 and the model for pneumonia infections in calves. Using the likelihood function to obtain the information, show that the approximate standard error of t is √π(1 – π)/n(1+ π).
1.31 Refer to Section 1.5.6. Let a denote the number of calves that got a primary, sec-ondary, and tertiary infection, b the number that received a primary and secondary but not a tertiary infection, c the number that received a primary but not a secondary infection, and d the number that did not
1.32 Refer to quadratic form (1.18) that leads to the Pearson chi-squared.a. Verify that the matrix quoted in the text for ¹ is the inverse of 20.b. Show that (1.18) simplifies to Pearson's statistic (1.16).2c. For the zs statistic (1.11), show that z X2 for c = 2.=
1.33 For testing Ho: π; = πjo, j = 1,...,c, using sample multinomial proportions {;}, the likelihood-ratio statistic (1.17) is 2G² = −2η Σπ; log(njo/nj).j jShow that G² ≥ 0, with equality if and only if ₁ = πjo for all j. [Hint: Apply Jensen's inequality to E(-2n log X), where X
1.34 For counts {n,}, the power divergence statistic for testing goodness of fit (Cressie and Read 1984, Read and Cressie 1988) is 2λ(λ + 1)λΣη; [(π/μ₁) – 1]a. For λ = 1, show that this equals X2.for −00 < λb. As a → 0, show that it converges to G2. [Hint: log t = limno(th-1)/h.]c.
1.35 The chi-squared mgf with df = v is m(t) = (1 - 2t)/2, for |t|
1.36 For the multinomial (n, {π;}) distribution with c > 2, a possible set of score-type simultaneous confidence limits for π; are the solutions of(π; – π;)2/[π;(1 – π;)/n] = (2a/2c)², j =1,..., c.a. Using the Bonferroni inequality, argue that for large n these c intervals
1.37 Consider the Bayesian equal-tail posterior interval for a binomial parameter π, using a beta or logit-normal prior. When y = 0, explain why the lower limit for a can never be 0, unlike the frequentist approach based on inverting a score or likelihood-ratio test.
1.38 Consider estimating the ratio πί/π; of two multinomial parameters. Should the estimate depend at all on the counts in other categories?a. With a frequentist approach, explain why the ML estimate of πί/π; is n₁/nj.b. For a Dirichlet prior, show that using the Bayes estimates of π; and
1.39 Given π, Y has a bin(n, π) distribution, and a has a uniform prior distribution. Show that the marginal distribution of Y is uniform over 0, 1,...,n.
1.40 Consider the Bayes estimator of the binomial parameter π using a beta prior distri-bution.a. Show that the ML estimator is a limit of Bayes estimators, for a certain sequence of beta prior parameter values.b. Find an improper prior density such that the Bayes estimator coincides with the ML
1.41 For the Dirichlet prior for multinomial probabilities, show the posterior expected value of π¡ is formula (1.19). Derive the expression for this Bayes estimator as a weighted average of pi and E(π₁).
2.1 According to the FBI website (www.fbi.gov), in 2008, of female murder victims, 1710 were slain by males and 200 by females, whereas of male murder victims, 4351 were slain by males and 455 by females. Let Y denote sex of victim and X denote sex of offender. Report the sample (a) joint
2.2 According to the FBI website, of all blacks slain in 2008, 92% were slain by blacks, and of all whites slain in 2005, 85% were slain by whites. Let Y denote race of victim and X denote race of offender.a. Which conditional distribution do these statistics refer to, Y given X, or X given Y?b.
2.3 An article in The New York Times (Feb. 17, 1999) about the PSA blood test for detecting prostate cancer stated: "The test fails to detect prostate cancer in 1 in 4 men who have the disease (false-negative results), and as many as two-thirds of the men tested receive false-positive results." Let
2.4 Table 2.10 shows fatality results for drivers and passengers in auto accidents in Florida in 2008, according to whether the person was wearing a seat belt.a. Estimate the probability of fatality, conditional on seat-belt use in category (i) no and (ii) yes.b. Estimate the probability of wearing
2.5 Consider the following two studies reported in The New York Times.a. A British study reported (Dec. 3, 1998) that of smokers who get lung cancer,"women were 1.7 times more vulnerable than men to get small-cell lung cancer."Is 1.7 the odds ratio or the relative risk?b. A National Cancer
2.6 According to a report by the United Nations Office on Drugs and Crime, the number of homicides involving firearms per million people is about 62.4 in the United States, 6.0 in Canada, 5.6 in Australia, and 1.3 in the UK. Use the relative risk to compare the United States with the other
2.7 An article in The Economist (July 3, 2010) stated that the number of people in prison is 154 per 100,000 in England and Wales, 96 per 100,000 in France, 87 per 100,000 in Germany, and 753 per 100,000 in the United States Explain how to use the relative risk to compare the U.S. rate to the
2.8 At the start of the 2010 World Cup, the betting exchange Betfair stated that the odds against being the winning team were 9/2 for Spain, 11/2 for Brazil, 6/1 for England, and 90/1 for the United States. Find the corresponding prior probabilities of winning for these four teams.
2.9 In a recent survey of people aged 50-71 in the United States summarized by N.Freedman et al. (Lancet Oncol. 9: 649-656, 2008), during a follow-up period the annual probability of lung cancer occurrence was about 0.00023 for people who had never smoked and about 0.01284 for current smokers who
2.10 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. J. M.Dawson, J. Statist. Ed. 3, 1995.)a. What is wrong with the interpretation, "The probability of survival for females was 11.4 times
2.11 A research study estimated that under a certain condition, the probability that a subject would be referred for heart catheterization was 0.906 for whites and 0.847 for blacks.a. A press release about the study stated that the odds of referral for cardiac catheter-ization for blacks are 60% of
2.13 For the Women's Health Study, heart attacks were reported for 198 of 19,934 taking aspirin and for 193 of 19,942 taking placebo (J. Am. Med. Assoc. 295: 306-313, 2006). Construct the 2 x 2 table that cross-classifies the treatment with whether a heart attack was reported. Estimate the odds
2.14 According to poll results released by the Pew Research Center (www.people-press.org) in 2010, when adults in the United States were asked whether there is solid evidence that the average temperature on earth has been getting warmer over the past few decades, the estimated odds of a yes
2.15 Table 2.11 refers to applicants to graduate school at the University of California at Berkeley, for fall 1973. It presents admissions decisions by gender of applicant for Table 2.11 Data for Exercise 2.15 on Graduate Admissions Whether Admitted Male Female Department Yes No Yes No A512 313 89
2.16 State three "real-world" variables X, Y, and Z for which you expect a marginal association between X and Y but conditional independence controlling for Z.
2.17 Based on murder rates in the United States, an Associated Press story reported that the probability that a newborn child has of eventually being a murder victim is 0.0263 for nonwhite males, 0.0049 for white males, 0.0072 for nonwhite females, and 0.0023 for white females.a. Find the
2.18 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(2): 44, 1999.]
2.20 Table 2.12 is from an early study on the death penalty in Florida. Analyze these data and show that Simpson's paradox occurs.Table 2.12 Data for Exercise 2.20 on the Death Penalty Death Penalty Victim's Race Defendant's Race Yes No White White 19 132 Black 11 52 Black White 09 Black 697
2.21 Smith and Jones are baseball players. Smith has a higher batting average than Jones in each of K years. Is is possible that for the combined data from the K years, Jones has the higher batting average? Explain, creating some data with K = 2 to illustrate.
2.22 Table 2.13 summarizes responses from a General Social Survey about homosexual sex and premarital sex. Find and interpret a measure of association.
2.23 For the data in Table 2.13, the two marginal distributions are dependent rather than independent samples, but the measure Δ can still compare those distributions. Find it, and interpret.
2.24 Table 2.14 cross-classifies job satisfaction by race. Determine whether the groups are stochastically ordered, and estimate the difference between the probability that job satisfaction is higher for blacks than whites and the probability that job satisfaction is higher for whites than
2.25 For a diagnostic test of a certain disease, let π₁ denote the probability that the diagnosis is positive given that a subject has the disease, and let π₂ denote the probability that the diagnosis is positive given that a subject does not have it. Let ρdenote the probability that a
2.26 Show that the odds ratio and relative risk need not be similar when π; is close to 1.0 for both groups.
2.27 Let D denote having a certain disease and E denote having exposure to a certain risk factor. The attributable risk (AR) is the proportion of disease cases attributable to that exposure (see Benichou 2005).a. Let P(E) = 1 - P(E). Explain why AR = [P(D) – P(D|E)]/P(D).b. Show that AR relates
2.28 In comparing new and standard treatments with success probabilities π₁ and π₂, the number needed to treat (NNT) is the number of patients that would need to be treated with the new treatment instead of the standard in order for one patient to benefit.Explain why a natural estimate of
2.29 For a 2 × 2 table of counts {nij}, show that the odds ratio is invariant to (a) inter-changing rows with columns, and (b) multiplication of cell counts within rows or within columns by c≠ 0. Show that the difference of proportions and the relative risk do not have these properties.
2.30 For given π₁ and π₂, show that the relative risk cannot be farther than the odds ratio from their independence value of 1.0.
2.31 Let πijk = P(X = i, Y = j|Z = k). Explain why XY conditional independence isπijk = πi+|k π+j|k for all i and j and k.
2.32 For a 2 x 2 x 2 table, show that homogeneous association is a symmetric property, by showing that equal XY conditional odds ratios is equivalent to equal YZ conditional odds ratios.
2.33 For a 2 x 2 x 2 table, suppose θXY(1) = θXY(2) = θ. For a possibly confounding vari-able Z, let θc denote the common value of θ(i)YZ. Let π1 = P(Z = 1|X = 1, Y = 2)and π2 = P(Z = 1|X = 2, Y = 2).a. Show (Breslow and Day 1980, p. 96) thatθXY = θθcπ1 + (1 – π1)θcπ2 + (1 –
2.34 When X and Y are conditionally dependent at each level of Z yet marginally inde-pendent, Z is called a suppressor variable. Specify joint probabilities for a 2 x 2x2 table to show that this can happen (a) when there is homogeneous association, and(b) when the association has opposite direction
2.35 Show that the {aij} in (2.11) determine all odds ratios formed from pairs of rows and pairs of columns.
2.36 For I × J contingency tables, explain why the variables are independent when the (I - 1)(J-1) differences πji - πji = 0, i = 1, ..., I – 1, j = 1, ..., J-1.
2.38 For 2 x 2 tables, Yule (1900, 1912) introduced Q =π11 π22 - π12 π21π11 π22 + π12 π21, which he labeled Q in honor of the Belgian statistician Quetelet. It is now called Yule's Q.a. Show that for 2 x 2 tables, Goodman and Kruskal's y = Q.b. Show that Q relates to the odds ratio by Q =
2.39 Goodman and Kruskal (1954) proposed an association measure (tau) for nominal variables based on variation measure V (Y) = Σπ+j(1 − π+j) = 1 − Σπ2+j.a. Show that V(Y) is the probability that two independent observations on Y fall in different categories. Show that V (Y) = 0 when π+j =
2.40 The measure of association lambda for nominal variables (Goodman and Kruskal 1954) has V (Y) = 1 - max{π+j} and V(Y\i) = 1 - maxj{πj|i }. Interpret lambda as a proportional reduction in error for predictions which select the response category that is most likely. Show that independence
2.41 Show that Δ in (2.15) relates to a = P(Y1 > Y2) + (+)P(Y1 = Y2) byα = (Δ + 1)/2, Δ = 2α – 1, with α having range [0, 1] and null value 1⁄2.
3.1 A meta-analysis (Moore et al., Lancet 370: 319-328, 2007) of studies on the asso-ciation between cannabis use (yes, no) and presence of psychosis (yes, no) reported a pooled odds ratio estimate of 1.41, with 95% confidence interval of (1.20, 1.65).Explain how to interpret this interval.
3.2 For 239 golf tournaments on the PGA tour between 2004 and 2009, the economists D. Pope and M. Schweitzer evaluated risk aversion by comparing percentages of putts made when putting for a par versus putting for a birdie (Am. Econ. Rev. 101:129-157, 2011). For 2828 pairs of putts taken from
3.3 Table 3.10 uses the GSS to cross-classify a subject's political party ID with their opinion about whether homosexuals should have the right to marry, for subjects hav-ing strong identification with a particular party and strong agreement or disagreement with homosexual marriage. Show that (a)
3.4 For Table 2.10 on seat-belt use and results of auto accidents, find and interpret 95%confidence intervals for the conceptual population (a) odds ratio, (b) difference of proportions, and (c) relative risk.
3.5 Refer to Table 2.5 on lung cancer and smoking. Conduct an inferential analysis, and interpret results.
3.6 A study considered the effect of prednisolone on severe hypercalcemia in women with metastatic breast cancer (B. Kristensen et al., J. Intern. Med. 232: 237-245, 1992). Of 30 patients, 15 were randomly selected to receive prednisolone. The other 15 formed a control group. Normalization in their
3.7 In professional basketball games during 2009-2010, when Kobe Bryant of the Los Angeles Lakers shot a pair of free throws, 8 times he missed both, 152 times he made both, 33 times he made only the first, and 37 times he made only the second. Is it plausible that the successive free throws are
3.8 Refer to Exercise 3.3 and Table 3.10.a. Find the z statistic (3.12) and explain how it relates to a chi-squared test.b. Find a score or profile likelihood confidence interval for the odds ratio, and compare it to the Wald interval.
3.9 Go to sda.berkeley.edu/GSS and download a contingency table relating attained education and the fundamentalism of one's religious beliefs, for the most recent survey. The GSS variable names are EDUCATION and FUND, and you can enter the year in the Selection Filter, such as YEAR(2010). Using the
3.10 As in the previous exercise, download recent GSS data and perform analyses to answer the questions asked.a. Are people happier who believe in life after death? Analyze using the GSS variables HAPPY and POSTLIFE.b. Is belief in the existence of God associated with party ID? Analyze the 3 x 6
3.11 Refer to Table 3.11, GSS data on party ID and race.a. Using X2 and G2, test the hypothesis of independence between party identifica-tion and race. Report the P-values and interpret.b. Use standardized residuals to describe the evidence of association.c. Partition chi-squared into components
3.12 Using the 2008 GSS, we cross-classified party ID with gender. Table 3.12 shows some results. Explain how to interpret all the results on this printout. (Reschi denotes the Pearson residual and StReschi denotes the standardized residual.)
3.13 A recent study (by R. Armenio et al., J. Am. Dent. Assoc. 139: 592-597, 2008) re-ported results of a double-blind randomized clinical trial comparing tooth sensitivity for 14 patients using a fluoride gel to 15 patients using placebo. Each patient had weekly visits for responses, between 3 and
3.14 Table 3.13 classifies a sample of psychiatric patients by their diagnosis and by whether their treatment prescribed drugs. Partition chi-squared into three compo-nents to describe differences and similarities among the diagnoses, by comparing(i) the first two rows, (ii) the third and fourth

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Categorical Data Analysis 3rd Edition Alan Agresti - Solutions

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