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statistics
elementary statistics in social research

Questions and Answers of Elementary Statistics In Social Research

Assume that a survival time \(T\) has a constant hazard \(\lambda\) over time interval \(\left[\tau_{j}, \tau_{j+1}\right)\) with \(\log \lambda=\mathbf{x}^{\top} \boldsymbol{\beta}\). Prove
Fit the following models for genuinely discrete time to drop-out with age and gender as covariates for the DOS:(a) the proportional hazards models;(b) the proportional odds models.
Check that if \(p_{j}\left(\mathbf{x}_{i}\right)\) is small, then \(\frac{p_{j}\left(\mathbf{x}_{i}\right)}{1-p_{j}\left(\mathbf{x}_{i}\right)} \approx p_{j}\left(\mathbf{x}_{i}\right)\) and hence
For the Catheter Study, the patients were assessed bimonthly and the measurements about UTIs, catheter blockages, and replacements cover the previous two months. Thus, the patients were under
Prove (7.1) . y = (Y1, Y2,Yk) ~ MN (n,), i = 1, MN (n,), = (1, 2,..., k), (7.1) k. "
Show that the log-likelihood function in (7.3) can be maximized by maximizing the first term as a function of \(\beta\) and the second term as a function of \(\tau\) separately, and find the MLE of
Suppose that \(\left\{\mu_{i j}\right\}\) satisfy a multiplicative model\[\begin{equation*}\mu_{i j}=\mu \alpha_{i} \beta_{j}, \quad 1 \leq i \leq I, 1 \leq j \leq J \tag{7.30}\end{equation*}\]where
Use loglinear models to test the hypothesis concerning the distribution of depression diagnosis in Example2.2. Example 2.2 In the DOS, we are interested in testing the following hypothesis concerning
For the DOS, use the three-level depression diagnosis and variable MS for marital status as defined in Section 4.2.2 to test for uniform association, assuming that both the depression diagnosis and
In Example 7.3, we assessed the uniform association between the diagnosis of probands and informants. Perform the following alternative approaches for assessing uniform association and compare the
Prove that model (7.15) is indeed the model for the row-effect association. log (ij) = ++iy, 1iI,1 j J. (7.15)
For the DOS, use the three-level depression diagnosis and variable MS for marital status as defined in Section4.2.2. If MS is not ordered, we may not be able to talk about the uniform association.
Redo Example 2.12 using log-linear models. Example 2.12 Consider the association between gender and depression in the DOS. Although gender is nominal in nature, its binary representation in that
Each of the three random variables \(x, y\), and \(z\) has two levels: 0 and 1 . The joint distribution of these three variables can be determined from the facts \(\operatorname{Pr}(x=0, y=\) \(0,
Prove that under the mutual independence log-linear model (7.19), the three variables are indeed mutually independent. log ijk = logy + log i+++ log+j+ + log ++k = X + X + X + X. (7.19)
Prove that three variables being mutually independent implies that any two of them are marginally independent, conditionally independent, and any one of them is jointly independent with the others.
Prove that if \(x\) is jointly independent with \(y\) and \(z\), then \(x\) and \(y\) are marginally independent.
Prove that if \(x\) is jointly independent with \(y\) and \(z\), then \(x\) and \(y\) are conditionally independent.
To obtain the log-linear models for association homogeneity, we need the following two key facts:(a) Prove (7.25).(b) Prove that \(\lambda_{i^{\prime} j k}^{x y z}+\lambda_{i j^{\prime} k}^{x y
Verify that under the mutual independent log-linear model (7.23), the variable \(y\) is independent with the other two. log ijk = log +logi+k+log +j+ = X + X + x + X + X (7.23)
Verify the numbers of free parameters in the model (7.17). log ij+i+j+Aij, all i, j, (7.17)
Write down the log-linear model for quasi-symmetry, and count the number of free parameters in the model.
Prove under the paradigm of a multinomial distribution that if \(x\) and \(y\) are homogeneously associated, then \(y\) and \(z\) as well as \(x\) and \(z\) are also homogeneously associated.
Prove that \(\exp \left(\lambda_{111}^{x y z}\right)=\frac{\pi_{2,2,2} / \pi_{1,2,2}}{\pi_{2,1,2} / \pi_{1,1,2}} / \frac{\pi_{2,2,1} / \pi_{1,2,1}}{\pi_{2,1,1} / \pi_{1,1,1}}\) for a \(2 \times 2
For the DOS, use the three-level depression diagnosis.(a) Use the Poisson log-linear model to test whether depression and gender are independent.(b) Use methods for contingency tables studied in
For the DOS, use the three-level depression diagnosis and variable MS for marital status as defined(a) whether depression, gender, and MS are mutually independent;(b) whether depression is
Use the log-linear model to test whether SCID (two levels: no depression and depressed including major and minor depression) and dichotomized EPDS (EPDS \(\leq 9\) and EPDS \(>9)\) are homogeneously
Check that for Example 7.7, you may obtain different (incorrect) results if the random zero is removed from the data set for data analysis. Example 7.7 Let us check the quasi-independence between the
Complete the forward model selection in Example7.9 and compare it with the models selected in Examples7.10 Example 7.9 Consider the relationship among gender (g), three-level marital status (m),
Check that [edgm \(][\) cdgm \(]\) is a graphical model. Education Gender MS Dep CIRS
Check that there are at least \(2\left(\begin{array}{c}n \\ 3\end{array}\right)\) different hierarchical models which contain all twoway interaction terms for an \(n\)-way contingency table.
Consider a random variable \(x\) following the standard logistic distribution with the \(\mathrm{CDF}\) and PDF given in (4.3) and (4.4).(a) Show that the PDF in (4.4) is symmetric about 0.(b) Show
Prove that if\[\operatorname{logit}\left(\operatorname{Pr}\left(y_{i}=1 \mid \mathbf{x}_{i}\right)\right)=\beta_{0}+\mathbf{x}^{\top}
If \(\Sigma\) is an \(n \times n\) invertible matrix and \(K\) is a \(k \times n\) matrix with \(\operatorname{rank} k(k \leq n)\), show that \(K \Sigma K^{\top}\) is invertible.
Show that the Wald statistic in (4.15) does not depend on the specific equations used. Specifically, suppose that \(K\) and \(K^{\prime}\) are two equivalent systems of equations for a linear
Use a logistic model to assess the relationship between CIRSD and Depd, with Depd as the outcome variable.(a) Write down the logistic model.(b) Write down the null hypothesis that CIRSD has no
Based on a logistic regression of Depd on some covariates, we obtained the following prediction equation:\[\begin{align*}\operatorname{logit}[\widehat{\operatorname{Pr}}(D e p d=1)] & =1.13-0.02
In suicide studies, alcohol use is found to be an important predictor of suicide ideation. Suppose the following logistic model is used to model the
Consider the logistic regression in (4.25). Show that for each \(j(1 \leq j \leq p), T_{j}(x)=\) \(\sum_{i=1}^{n} y_{i} x_{i j}\) is a sufficient statistic for \(\beta_{j}\). logit (T) = log 1 i ) ==
Use the fact that \(x \log x=(x-1)+(x-1)^{2} / 2+o\left((x-1)^{2}\right)\) to show that the deviance test statistic \(D^{2}\) in (4.43) has the same asymptotic distribution as the general Pearson
For the DOS, we are interested in how MS and gender are related with the depression outcome. Based on the logistic model: Dep \(\sim\) MS | Gender (MS, Gender, and their interaction), answer the
We add age as a continuous covariate to the model in the last problem and consider the logistic model: Dep \(\sim\) MS+ Gender+Age.(a) Test the goodness of fit of the model.(b) Use Box-Tidwell test
For the exponential family of distributions defined in (4.41), show(a) \(E(y)=\frac{d}{d \theta} b(\theta)\).(b) \(\operatorname{Var}(y)=a(\phi) \frac{d^{2}}{d \theta^{2}} b(\theta)\).(c) Assume that
Prove that a sufficient statistic for the parameter \(\beta_{j}\) in the model (4.25) is given by \(T_{j}(x)=\sum_{i=1}^{n} y_{i} x_{i j}(1 \leq j \leq p)\). logit (T) = log = Box10 + + BpTip = Tx.
This problem illustrates why exact inference may not behave well when conditional on continuous covariates.(a) Consider the following equation where \(a_{1}, a_{2}, \ldots, a_{n}\) are some known
Verify the conditional likelihood (4.29). k II i=1 1 1+ exp( d;)' (4.29)
Suppose \(y_{i}=\sum_{j=1}^{n_{i}} y_{i j}\), where \(y_{i j} \sim \operatorname{Bernoulli}\left(p_{i}\right)\) and are positively correlated with \(\operatorname{cor}\left(y_{i j}, y_{i
Prove that the deviance and Pearson chi-square test statistics are asymptotically equivalent.
Plot and compare the CDFs of logistic, probit, and complementary log-log variables after they are centered at their medians and scaled to unit variances.
Let \(y_{i}^{*}=\beta_{0}+\boldsymbol{\beta}^{\top} \mathbf{x}_{i}+\varepsilon_{i}\), where \(\varepsilon_{i} \sim N(0,1)\) (a standard normal with mean 0 and variance 1 ) and \(y_{i}\) is determined
Prove that if\[\operatorname{Pr}\left(y_{i}=1 \mid \mathbf{x}_{i}\right)=\Phi\left(\beta_{0}+\mathbf{x}^{\top} \boldsymbol{\beta}\right), \quad \operatorname{Pr}\left(y_{i}=0 \mid
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

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