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statistics
elementary statistics in social research

Questions and Answers of Elementary Statistics In Social Research

Fit complementary log-log models to DOS data, using dichotomized depression as response and gender as the predictor. Comparing the results between modeling the probability of No depression and
Use the baseline information from the Catheter Study. We model the binary catheter blockage outcome with age, gender, and education as predictors. Assuming additive effect for all these covariates,
Show that the discrimination slope equals to the differences of the mean fitted probabilities between \(y=1\) and \(y=0\).
For the Catheter Study, apply the binomial regression model (4.49) with the UTI response replaced by the catheter blockage.(a) What conclusions you may have based on the model? Is there
In this problem, we perform a simulation study about clustered binary outcomes with sample size 1000 .(a) Generate random variable \(X\) from \(\mathrm{N}(0,1)\).(b) For each \(X\), generate five
Show that for a generalized logit model, \(T_{j k}(x)=\sum_{i=1}^{n} y_{i j} x_{i k}\) is a sufficient statistic for parameter \(\beta_{j k}(1 \leq j \leq J\) and \(1 \leq k \leq p)\).
Compute the fitted probabilities for females based on the generalized logistic model in Example 5.1. Example 5.1 Let us apply the generalized logit model to the DOS, using the three-level depression
Prove that the multinomial probit model defined in (5.8) does not depend on the selection of the reference level. = zij ij Yi = Hij + Eij EiJ, (5.8)
Compute the fitted probabilities for females based on the multinomial probit model in Example5.2 Example 5.2 Let us apply the multinomial probit model to Example 5.1. If DEP = 0 is selected as the
Prove that the generalized logit model (5.6) can be defined with utility functions \(y_{i j}^{*}=\alpha_{j}+\boldsymbol{\beta}_{j}^{\top} \mathbf{x}+\varepsilon_{i j}\), where \(\varepsilon_{i j}\)
For the DOS data set, treat the three-level depression diagnosis as a nominal outcome, and compare the results.(a) Fit a generalized logistic model with the three-level depression diagnosis as the
Verify (5.13) for the proportional odds model defined in (5.11). Yi (x1) / (1 Yi (x1)) Yi (x2)/(1-j (x2)) P (BT = exp (B (x1 -x2)), j=1,...,J 1. (5.13)
Find the Wald and LR test statistics for testing equality of slopes in Example5.5 Example 5.5 We change the link function in Example 5.4 to probit and fit the following cumulative probit model to the
Find the Wald and LR test statistics for testing equality of slopes in Example5.6 Example 5.6 We change the link function in Example 5.4 to complementary log-log and fit the following cumulative
For the DOS data set, fit a cumulative logit model with the three-level depression diagnosis as the ordinal response and Age, Gender, CIRS, and MS as covariates.(a) Repeat the analyses (a), (b), and
Repeat the analyses in Problem5.10 with continuation ratio models with logit, probit, and complementary log-log link functions. Problem5.10 5.10 For the DOS data set, fit a cumulative logit model
Compute the correlation matrix of the fitted probabilities for having major depression based on the models in Problems 5.6, 5.10, 5.11, 5.12, and 5.13.Problems 5.6:Problems 5.10:Problems
For an \(I \times J\) contingency table with ordinal column variable \(y(=1, \ldots, J)\) and ordinal row variable \(x(=1, \ldots, I)\), consider the model\[\operatorname{logit}[\operatorname{Pr}(y
For an \(I \times J\) contingency table with ordinal column variable \(y(=1, \ldots, J)\) and ordinal row variable \(x(=1, \ldots, I)\), consider the adjacent category model\[\log
Show that the log function in (6.2) is the canonical link for the Poisson model in (6.1). Yi Xi ~ Poisson (#), 1 in. (6.1)
Consider a Poisson regression model for a count response \(y\) with a single continuous covariate \(x, E(y \mid x)=\exp \left(\alpha_{0}+\alpha_{1} x\right)\). If \(x\) is measured on another
Show that for the Poisson regression model in (6.2), \(\sum_{i=1}^{n} y_{i} x_{i j}\) is a sufficient statistic for \(\beta_{j}(1 \leq j \leq p)\). log()=xB=Bxi1+...+pip (6.2)
Similar to logistic regression, give a definition of median unbiased estimate (MUE) of a parameter based on the exact conditional distribution.
Let \(y \sim \operatorname{Poisson}(\mu)\).(a) If \(\mu=n\) is an integer, show that the normalized variable \(\frac{y-\mu}{\sqrt{\mu}}\) has an asymptotic normal distribution \(N(0,1)\), i.e.,
Show that the asymptotic result in (6.7) still holds if \(\beta\) is replaced by the MLE \(\widehat{\beta}\). n P = i=1 (Yi - ) D~ Xn-p as for all 1
For the Sexual Health pilot study, consider modeling the number of unprotected vaginal sex behaviors during the three month period of the study as a function of three predictors, HIV knowledge,
Use the intake data for the Catheter Study to study the association between urinary tract infection (UTI) and demographic characteristics including age, gender, and marital status (ms). There are
Prove that the CMP distribution \(\operatorname{CMP}(\lambda, v)\) converges to(a) a Bernoulli distribution as \(v\) goes to infinite. Find the parameter for the limiting Bernoulli distribution;(b) a
Consider the Poisson log-linear model\[y_{i} \mid \mathbf{x}_{i} \sim \text { Poisson }\left(\mu_{i}\right), \quad \log \left(\mu_{i}\right)=\mathbf{x}_{i}^{\top} \boldsymbol{\beta}, \quad 1 \leq i
Show that inference about \(\beta\) based on \(\mathrm{EE}\) is valid even when the NB model does not describe the distribution of the count variable \(y_{i}\), provided that the systematic component
Let \(y\) follow the negative binomial distribution (6.20). Show that \(E(y)=\mu\) and \(\operatorname{Var}(y)=\mu(1+\alpha \mu)\), where \(\alpha\) is the dispersion parameter for the negative
Let \(y\) follow a mixture of structural zeros of probability \(p\) and a Poisson distribution with mean \(\mu\) of probability \(q=1-p\). Show that \(E(y)=q \mu\), and \(\operatorname{Var}(y)=q
Have you experienced Simpson's paradox in your professional and/or personal life? If so, please describe the context in which it occurred.
Suppose you test ten hypotheses and under the null hypothesis each hypothesis is to be rejected with type I error rate 0.05. Assume that the hypotheses (test statistics) are independent. Compute the
Show that the asymptotic distribution for the CMH test for a set of \(q 2 \times 2\) tables is valid as long as the total size is large. More precisely,
Let \(\mathbf{x}\) be a random vector and \(\boldsymbol{V}\) its variance matrix. Show that \(\mathbf{x}^{\top} \boldsymbol{V}^{-1} \mathbf{x}\) is invariant under linear transformation. More
Use the DOS data to test whether there is gender and depression (dichotomized according to no and minor/major depression) association by stratifying medical burden and education levels, where medical
Show that the odds ratio is a monotone function of \(p_{11}\) if marginal distributions are fixed.
Verify (3.6) . H1122+v #1221 +v (3.6)
In the Postpartum Depression Study (PPD), stratify the subjects according to the ages of the babies (0-6 months, 7-12 months, and 13-18 months) since it is known to affect postpartum depression.
Redo Problem 2.16 by stratifying the subjects according to baby ages as in Problem 3.11Problem 2.16Problem 3.11 2.16 In the PPD, each subject was diagnosed for depression using SCID along with
Use statistic software to verify the given estimates of (unweighted) kappa coefficients and their variances in Example3.6 for the two individual tables in Table3.5 Example 3.6 For the Detection of
A random sample of 16 subjects was taken from a target population to study the prevalence of a disease \(p\). It turned out that six of them were diseased.(a) Estimate the disease prevalence
Since the sample size in Problem 2.1 is not very large, it is better to use exact tests.(a) Apply exact tests to test the hypothesis in (2.48) for the data in Problem 2.1 and compare your results
Check that in the binary case \((k=2)\), the statistic in (2.7) is equivalent to the one in (2.1).
In the DOS, we are interested in testing the following hypothesis concerning the distribution of depression diagnosis for the entire sample:\[\begin{aligned}\operatorname{Pr}(\text { No depression })
Suppose \(x \sim B I(n, p)\) follows a binomial distribution of size \(n\) and probability \(p\). Let \(k\) be an integer between 0 and \(n\). Show that \(\operatorname{Pr}(x \geq k)\), looking as a
Prove that(a) If \(y \sim \operatorname{Poisson}(\lambda)\), then both the mean and variance of \(y\) are \(\lambda\).(b) If \(y_{1}\) and \(y_{2}\) are independent and \(y_{j} \sim
Following the MLE method, the information matrix is closely related with the asymptotic variance of MLE. For the MLE of Poisson distribution,(a) First compute the Fisher information matrix then plug
Derive the negative binomial (NB) distribution.(a) Suppose \(y\) follows a Poisson \((\lambda)\), where the parameter \(\lambda\) itself is a random variable following a gamma distribution
Prove the equation below (2.11). - P11 P1+P+1 ~a N (P11 P1+P+1, [P+P+1 (1 P1+) (1 P+1)]). (2.11) n
Consider the statistic in (2.14).(a) Show that this statistic is asymptotically normal with the asymptotic variance given
For the DOS, test whether education is associated with depression. To simplify the problem, we dichotomize both variables; use no and major/minor for depression diagnosis and at most and more than 12
Derive the relationships among the eight versions of odds ratios of Section 2.2.2.When twovariables(orrowandcolumn)areactuallyassociated,wemaywanttoknowthenature
Let \(p_{1}=\operatorname{Pr}(y=1 \mid x=1)=0.8\) and \(p_{2}=\operatorname{Pr}(y=1 \mid x=0)=0.4\).(a) Compute the relative risk of response \(y=1\) of population \(x=1\) to population \(x=0\), and
Show that the hypergeometric distribution \(H G\left(k ; n, n_{1+}, n_{+1}\right)\) has mean \(\frac{n_{1+} n_{+1}}{n}\) and variance \(\frac{n_{1+} n_{+1} n_{+2} n_{2+}}{n^{2}(n-1)}\).
In the PPD, each subject was diagnosed for depression using SCID along with several screening tests including EPDS. By repeatedly dichotomizing the EPDS outcome, answer the following questions:(a)
The data set "intake" contains baseline information of the Catheter Study. Use the two binary outcomes on whether urinary tract infections (UTIs) and catheter blockages occurred during the last two
Group the count responses on UTIs and catheter blockages in the data set "intake" into three levels: no occurrence, only once, and more than once. Use these three-level outcomes to assess(a) whether
For the DOS, use the three-level depression diagnosis and dichotomized education (more than 12 years education or not) to check the association between education and depression.(a) Test whether
The data set "DosPrepost" contains depression diagnosis of patients at baseline (pretreatment) and one year after treatment (posttreatment) in the DOS. We are interested in whether there is any
Let \(p\) denote the prevalence of a disease of interest. Express \(P P V\) and \(N P V\) as a function of \(p, S e\), and \(S p\).
Prove that the weighted kappa for \(2 \times 2\) tables will reduce to the simple kappa, no matter which weights are assigned to the two levels.
Verify the variance formula for the MWW statistics (2.29). nin2 (N+1) 12 r 12N (N-1) (n+j - 1) n+j (n+j+1) (2.29) j=1
Use the three-level depression diagnosis and dichotomized education (more than 12 years education or not) in the DOS data to test the association between education and depression.(a) Use the Pearson
For the \(2 \times r\) table with scores as in Section 2.3.1,(a) verify that the MLE of \(\beta\) in the linear regression model in (2.26) is \(\widehat{\beta}=\frac{\overline{x y}-\bar{x}
For the DOS, compute the indices, Pearson correlation, Spearman correlation, Goodman-Kruskal \(\gamma\), Kendall's \(\tau_{b}\), Stuart's \(\tau_{c}\), Somers' D, lambda coefficients, and uncertainty
Many measures of association for two-way frequency tables consisting of two ordinal variables are based on the numbers of concordant and discordant pairs. To compute such indices, it is important to
Suppose \(x\) is a random variable with \(m\) levels such that \(\operatorname{Pr}(x=i)=p_{i}\) for \(i=\) \(1,2, \ldots, m\) with \(\sum_{i=1}^{m} p_{i}=1\). In other words, \(x \sim \mathrm{MN}(1,
Let \(x\) be a binary variable with outcomes 0 and 1 . Let \(p=\operatorname{Pr}(x=1)\). Show that entropy has the maximum at \(p=0.5\).
For an \(r \times s\) table, the probability of concordant (discordant) pair \(p_{s}\left(p_{d}\right) \leq \frac{m-1}{m}\), where \(m=\min (r, s)\).
EPDS is an instrument (questionnaire) for depression in postpartum women. This instrument is designed so that a person with a higher EPDS score has a higher chance to be depressed. Use the PPD data
Suppose \(x\) is a random variable with at least two levels, with \(\operatorname{Pr}\left(x=x_{i}\right)=p_{i}\), for \(i=1,2\). Let \(x^{\prime}\) be the new random variable based on \(x\) with the
If a fair die is thrown, then each number from 1 to 6 has the same chance of being the outcome. Let \(X\) be the random variable to indicate whether the outcome is 5,
For random variables \(X\) and \(Y\), show that \(E[E(X \mid Y)]=E(X)\) and \(\operatorname{Var}(X)=\operatorname{Var}(E(X \mid Y))+E(\operatorname{Var}(X \mid Y))\).
The sequence \(\left\{\frac{1}{n}\right\}_{n=1}^{\infty}\) converges to 0 . If we treat each constant, \(\frac{1}{n}\), as a constant random variable, then the corresponding \(\mathrm{CDF}\)
Suppose \(X_{n} \sim \chi_{n}^{2}\), the chi-square with \(n\) degrees of freedom. Show that\[\frac{1}{\sqrt{2 n}}\left(X_{n}-n\right) \rightarrow_{d} N(0,1)\]
Let \(X_{1}, \ldots, X_{n}\), be a sequence of i.i.d. random variables that follow the Poisson distribution with mean \(\mu\). Then the sample average \(\bar{X}_{n}=\frac{X_{1}+\cdots+X_{n}}{n}\) is
Prove Slutsky's theorem.
Prove that under some regularity conditions such as the exchangeability of the integral and differentiation, we have(a) \(E\left[\frac{1}{f\left(X_{i}, \boldsymbol{\theta}\right)}
A random variable \(X\) follows an exponential distribution with parameter \(\lambda\) that takes positive values and \(\operatorname{Pr}(X
For an independent sample of \(Y_{i}\) and \(X_{i}(1 \leq i \leq n)\), suppose that\[\begin{equation*}Y_{i}=\mathbf{X}_{i}^{\top} \beta+\epsilon_{i}, \quad \epsilon_{i} \sim\left(0,
Let \(\mathbf{f}(\boldsymbol{\theta})\) be a \(n \times 1\) and \(\mathbf{g}(\boldsymbol{\theta})\) a \(1 \times m\) vector-valued function of \(\boldsymbol{\theta}=\left(\theta_{1}, \ldots,
Use the properties of differentiation in Problem1.11 to prove (1.13) and (1.14).Problem1.11 To prove (1.13) and (1.14). 1.11 Let f (0) be an x1 and g (0) a 1 xm vector-valued function of 0 =
Prove (1.21). ZEEE = E- (XX) E [X; (Y; X] B) X] E (X;X). - (1.21) n
Let \(\mathbf{X}_{i}(1 \leq i \leq n)\) be an i.i.d. sample of random vectors and let \(\mathbf{h}\) be a vector-valued symmetric function \(m\) arguments.
Show that the U-statistic \(\widehat{\sigma}^{2}\) in (1.24) is the sample variance of \(\sigma^{2}\), i.e., \(\widehat{\sigma}^{2}\) can be expressed as \(\widehat{\sigma}^{2}=\frac{1}{n-1}
Consider the function \(\mathbf{h}\left(\mathbf{Z}_{1}, \mathbf{Z}_{2}\right)\) for the U-statistic in (1.22).(a)
Let \(\mathbf{X}_{i}(1 \leq i \leq n)\) and \(\mathbf{Y}_{i}(1 \leq i \leq n)\) be two independent i.i.d. samples of random vectors from two different population. Let \(\mathbf{h}\) be a
Install the statistical software packages that you will use for the book in your computer. Read the DOS data set using your statistical software and find out the number of observations and the number
Referring back to Problem 19, the same doctor then used the three variables—pain level (pain), age, and gender—to predict whether or not patients were admitted to the hospital 1admit = 12 by the
When the requirements of Pearson’s r cannot be met, we might still be able to employa. A t ratio.b. A parametric measure of correlation.c. A nonparametric measure of correlation.d. None of the above
Using SPSS, analyze the General Social Survey to find out if respondents’ attitudes toward assistance with healthcare costs for the sick (HELPSICK) are related to their attitudes about the
A researcher wonders whether people will give similar answers to questions about abortion depending upon the circumstances. In particular, she wants to know whether people will differ (or not) when
The phi and contingency coefficients are extensions of which test?a. Pearson’s rb. Spearman’s rank orderc. Chi-squared. Gamma
A researcher suspects that there may be a relationship between political views (POLVIEWS) and attitudes toward national support on issues related to foreign aid (NATAID), military, armaments and
To use a Spearman’s correlation coefficient, the data must bea. Measured at the interval level.b. Measured at the categorical level.c. Normally distributed.d. Rank-ordered with respect to the
The General Social Survey includes a variety of questions on activities that Americans may do at night. Apply Gamma to test the null hypothesis that there is no relationship between each of the
For Spearman’s correlation coefficient, if two (or more) positions are tied (assigned the same rank), you must adjust the data bya. Calculating the “difference” using the variance of the
Are males more likely than females to have ever injected drugs? Find phi for testing the null hypothesis that there is no relationship between ever injecting drugs (EVIDU) and sex (SEX).

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