Questions and Answers of Linear State Space Systems

A question in a GSS asked subjects how many times they had sexual intercourse in the preceding month. The sample means were 5.9 for males and 4.3 for females; the sample variances were 54.8 and 34.4.
Table 7.6 is based on a study involving British doctors.Table 7.6 Data for Exercise 7.36 on Coronary Death Rates Person-Years Coronary Deaths Age Nonsmokers Smokers Nonsmokers Smokers 35–44 18,793
Does the inflated-variance QL approach make sense as a way to generalize the ordinary normal linear model with v(????i) = ????2? Why or why not?
Using E(y) = E[E(y|x)] and var(y) = E[var(y|x)] + var[E(y|x)], derive the mean and variance of the beta-binomial distribution.
Let y1 and y2 be independent negative binomial variates with common dispersion parameter ????.a. Show that y1 + y2 is negative binomial with dispersion parameter ????∕2.b. Conditional on y1 + y2,
Altham (1978) introduced the discrete distribution f(x; ????, ????) = c(????, ????)( n x)????x(1 − ????)n−x????x(n−x), x = 0, 1,…, n, where c(????, ????) is a normalizing constant. Show that
Sometimes sample proportions are continuous rather than of the binomial form (number of successes)/(number of trials). Each observation is any real number between 0 and 1, such as the proportion of a
Motivation for the quasi-score equations (8.2): suppose we replace v(????i) by known variance vi. Show that the equations result from the weighted least squares approach of minimizing ∑i[(yi −
Before R. A. Fisher introduced the method of maximum likelihood in 1922, Karl Pearson had proposed the method of moments as a general-purpose method for statistical estimation2. Explain how this
Ordinary linear models assume that v(????i) = ????2 is constant. Suppose instead that actually var(yi) = ????i. Using the QL approach for the null model ????i = ????, i = 1,…, n, show that u(????)
Suppose we assume v(????i) = ????i but actually var(yi) = ????2. For the null model????i = ????, find the model-based var(????̂), the actual var(????̂), and the robust estimate of that variance.
Suppose we assume v(????i) = ????i but actually var(yi) = v(????i) for some unspecified function v. For the null model ????i = ????, find the model-based var(????̂), the actual var(????̂), and the
Consider the null model ????i = ???? when the observations are independent counts.Of the Poisson-model-based and robust estimators of the variance of ????̂ = ȳpresented in Section 8.3.3, which
Let yij denote the response to a question about belief in life after death (1 =yes, 0 = no) for person j in household i, j = 1,…, ni, i = 1,…, n. In modeling P(yij = 1) with explanatory
Use QL methods to construct a model for the horseshoe crab satellite counts, using weight, color, and spine condition as explanatory variables. Compare results with those obtained with zero-inflated
Use QL methods to analyze Table 7.5 on counts of homicide victims. Interpret, and compare results with Poisson and negative binomial GLMs.
Refer to Exercise 7.35 on the frequency of sexual intercourse. Use QL methods to obtain a confidence interval for the (a) difference, (b) ratio of means for males and females.
For the teratology study analyzed in Section 8.2.4, analyze the data using only the group indicators as explanatory variables (i.e., ignoring hemoglobin).Interpret results. Is it sufficient to use
Table 8.3 shows the three-point shooting, by game, of Ray Allen of the Boston Celtics during the 2010 NBA (basketball) playoffs (e.g., he made 0 of 4 shots in game 1). Commentators remarked that his
Verify formula (9.1) for the effects of correlation on between-cluster and within-cluster effects.
How does positive correlation affect the SE for between-cluster effects with binary data? Let y11,…, y1d be Bernoulli trials with E(y1j) = ????1 and let y21,…, y2d be Bernoulli trials with E(y2j)
Suppose y1,…yn have E(yi) = ????, var(yi) = ????2, and corr(yi, yj) = ???? for i ≠ j.Show that E(s2) = ????2(1 − ????).
Formulate a normal random-effects model that generates the within-cluster and between-cluster effects described in Section 9.1.1.
In the analysis of covariance model, observation j in group i has ????ij = ????0 +????1xij + ????i for a quantitative variable xij and qualitative {????1,…, ????c}. Describe an application in which
A crossover study comparing d = 2 drugs observes a continuous response(yi1, yi2) for each subject for each drug. Let ????1 = E(yi1) and ????2 = E(yi2) and consider H0: ????1 = ????2.a. Construct the
For the normal linear mixed model (9.6), derive expression (9.7) for var(yi).
For the extension of the random-intercept linear mixed model (9.8) that assumes cov(????ij, ????ik) = ????2???? ????|j−k|, show that corr(yij, yik) = (????2 u + ????|j−k|????2????)∕(????2 u +
Consider the model discussed in Section 9.2.4 having a random intercept and a random slope. Is the fit for subject i any different than using least squares to fit a line using only the data for
For the linear mixed model, show that ????̃ (for known {Vi}) is unbiased, and derive its variance. Show how ????̃ and var(????̃) simplify when the model does not contain random effects.
When Xi and Vi in the linear mixed model are the same for each subject, show that the generalized least squares solution (9.10) can be expressed in terms of ȳ = (1∕n)∑i yi.
For the balanced random-intercept linear mixed model (9.8) based on conditional independence given the random effect (Section 9.2.1), show that????̃ = (XTV−1X)−1 XTV−1y simplifies to the
Using the joint normal density of y and u, derive Henderson’s mixed-model equations.
When R = ????2???? I, show that as ????−1 u tends to the zero matrix, the mixed model equations for a linear mixed model tend to the ordinary normal equations that treat both ???? and u as fixed
For the random-effects one-way layout model (Section 9.3.2), show that ????̂0 and ũi are as stated there (i.e., with the known variances). Show that ũi is a weighted average of 0 and the least
BLUPs are unbiased. Explain why this does not imply that E(ũ ∣ u) = u. Illustrate by a simulation using a simple linear mixed model. For your simulation, show how ũ tends to shrink toward 0
Show how fitting and prediction results of Section 9.3 for linear mixed models simplify when var(????i) = ????2???? I instead of R.
For the REML approach for the normal null model described in Section 9.3.3, find L, derive the distribution of Ly, and find the REML estimator of ????2.
For the binary matched-pairs model (9.5), consider a strictly fixed effects approach, replacing ????0 + ui in the model by ????0i. Assume independence of responses between and within subjects.a. Show
Refer to the previous exercise. Unlike the conditional ML estimator of ????1, the unconditional ML estimator is inconsistent (Andersen 1980, pp. 244–245).a. Averaging over the population, explain
A binary response yij = 1 or 0 for observation j on subject i, i = 1,…, n, j = 1,…,d. Let ȳ.j = ∑i yij∕n, ȳi. = ∑j yij∕d, and ȳ = ∑i∑j yij∕nd. Regard{yi+} as fixed, and
For the GLMM for binary data using probit link function,Φ−1[P(yij = 1 ∣ ui)] = xij???? + zijui show that the corresponding marginal model is Φ−1[P(yij = 1] = xij????[1 +zij????uzT
From Exercise 7.25, with any link and a factor predictor in the one-way layout, the negative binomial ML fitted means equal the sample means. Show this is not true for the Poisson GLMM.
For the Poisson GLMM (9.12) with random intercept, use the normal mgf to show that for j ≠ k, cov(yij, yik) = exp[(xij + xik)????][exp (????2 u) (exp (????2 u)− 1)]Find corr(yij, yik). Explain
For recent US Presidential elections, in each state wealthier voters tend to be more likely to vote Republican, yet states that are wealthier in an aggregate sense are more likely to have more
Construct a marginal model that is a multivariate analog of the normal linear model for a balanced two-way layout, assuming an absence of interaction.Show how to express the two main effect
For subject i, yi = (yi1,…, yid)T are d repeated measures on a response variable. Consider a model by which yi ∼ N(????,????????) with ????1 = ⋯ = ????d and???? = corr(yij, yik) for all j ≠
When y ∼ N(0, ????????), in terms of elements kij from the concentration matrix K = ????−1???? , show that the pdf f for y has the form log f(y) = constant − 1 2∑i kiiy2 i −∑i∑j
For the GEE (9.15) with R(????) = I, show that the equations simplify to(1∕????)∑n i=1 XT i ????i(yi − ????i) = 0, where ????i is the diagonal matrix with elements ????????ij∕????????ij on
Generalizing the heuristic argument in Section 8.3.2, justify why formula(9.16) is valid for the sandwich covariance matrix (Liang and Zeger 1986, Appendix).
For both linear mixed models and marginal models, the generalized least squares estimator ????̂ = (XTV−1X)−1 XTV−1y is useful. Suppose V is not the true var(y) but we can consistently estimate
For the smoking prevention and cessation study (Section 9.2.3), fit multilevel models to analyze whether it helps to add any interaction terms. Interpret fixed and random effects for the model that
Using the R output shown for the simple analyses of the FEV data in Section 9.2.5, show that the estimated values of corr(yi1, yi2) and corr(yi1, yi8) are 0.74 for the random intercept model and 0.86
Refer to Exercise 1.21 and the longitudinal analysis in Section 9.2.5. Analyze the data in file FEV2.dat at www.stat.ufl.edu/~aa/glm/data , investigating the correlation structure for the eight FEV
A field study11 analyzed associations between a food web response measure and plant invasion and tidal restriction in salt marsh habitats. The response, observed in a species of small marsh fish
For Table 7.5 on counts of victims of homicide, specify and fit a Poisson GLMM. Interpret estimates. Show that the deviance decreases by 116.6 compared with the Poisson GLM, and interpret.
The data file Maculatum.dat at the text website is from a study13 of salamander embryo development. These data refer to the spotted salamander(Ambystoma maculatum). One purpose of the study was to
Refer to the previous exercise. For all the embryos, use a logistic GLMM to model the probability of hatching in terms of the treatment. Interpret results, and compare to those obtained with an
Download the file Rats.dat at the text website for the teratology study in Table 8.1.a. Use the GEE approach to fit the logistic model, assuming an exchangeable working correlation structure for
Refer to Exercise 9.34. Analyze the FEV data with marginal models, and compare results to those obtained with linear mixed models.
A crossover study analyzed by B. Jones and M. Kenward (1987, Stat. Med. 6:555–564) compared three drugs on a binary outcome (success = 1, failure =0). Counts for the eight possible response
Suppose y1,…, yc are independent from a Poisson distribution with mean ????.Conditional on ∑ yi = n, are y1,…, yc exchangeable? Independent? Explain.
Independent observations y = (y1,…, yn) come from the N(????, ????2) distribution, with ????2 known, and ???? has a N(????, ????2) prior. Show that the posterior predictive distribution for a
Find the posterior mean and variance for ???? in the null model with a N(????, ????2) response for unknown ????2, using the improper-priors approach of Section 10.2.5.
Suppose y1,…, yn are independent from a N(????,????) distribution, and ???? has a N(????0,????0) prior distribution. With ???? known, derive the posterior distribution.Explain how the posterior
For the Bayesian ordinary normal linear model, using a flat improper prior for ???? but a proper inverse-gamma distribution (10.5) for ????2, find the posterior Bayes estimate of ????2 and express it
Suppose we assume the Bayesian normal linear model for a one-way layout, but the actual conditional distribution of y is highly skewed to the right (e.g., y = annual income). For large {ni}, would
You regard m potential models, M1,…, Mm, to be (a priori) equally likely.Use Bayes’ theorem to conduct Bayesian model averaging by finding an expression for the posterior P(Mi ∣ y) in terms of
With a beta(????1, ????2) prior for the binomial parameter ???? and sample proportion y, if n is large relative to ????1 + ????2, show that the posterior distribution of ????has approximate mean y
With a beta prior for the binomial parameter ???? having ???? = ????1∕(????1 + ????2)and letting n∗ = ????1 + ????2, find E( ̃???? − ????)2 and express it as a weighted average of [????(1 −
A beta prior for the binomial parameter ???? has ????1 = ????2 = ????.a. When y = 0, for what values of ???? is the posterior density of ???? monotone decreasing and hence the HPD posterior interval
In Exercise 4.11 for the binomial probability ???? of being a vegetarian, the proportion y = 0 of n = 25 students were vegetarians.a. Report the ML estimate. Find the 95% confidence interval based on
This exercise is based on an example in the keynote lecture by Carl Morris(see www.youtube.com/watch?v=JOovvj_SKOg) at a symposium held in his honor in October 2012. Before a Presidential election,
For a binomial distribution with beta prior, show that the marginal distribution of s = ny is the beta-binomial. State its mean and variance.
For a binomial distribution with beta prior, show how to conduct Bayesian estimation of {????i} for c groups in the one-way layout.
In a diagnostic test for a disease, let D denote the event of having the disease, and let + (−) denote a positive (negative) diagnosis by the test. Let????1 = P(+ ∣ D) (the sensitivity), ????2 =
Consider independent binary observations from two groups with ????i =P(yi = 1) = 1 − P(yi = 0), and a binary predictor x. For the 2×2 contingency table summarizing the two binomials, let
In the previous exercise, suppose ????̂ = 0 and the HPD interval for ???? is (0, U).Is (1∕U, ∞) the HPD interval or the equal-tail posterior interval for ????∗?Do you think it is a sensible
Re-do the analysis with prior ???? = 1 for the example in Section 10.3.2, but use (0, 1) coding for the indicator variable x1 instead of −0.5 and 0.5. Why is the posterior mean for ????1 so
For a Poisson random variable y with mean ????, show that the Jeffreys prior distribution for ???? is improper. Using it, find the posterior distribution and indicate whether it is improper.
For iid Poisson variates y1,…, yn with parameter ????, suppose ???? ∼ gamma(????, k) (Recall Section 4.7.2).a. Show that the posterior distribution of ???? is gamma (i.e., the prior is
Show how the results in (a) in the previous exercise generalize to estimating Poisson parameters for c groups in the one-way layout. For fixed gamma hyperparameters, show that the estimate of ????i
Show how results in Section 10.1.3 generalize for n independent multinomial trials and a Dirichlet prior for the multinomial probabilities. (Good (1965)gave one of the first Bayesian analyses with
Two independent multinomial variates have ordered response categories.Using independent Dirichlet priors, explain how to simulate to find the posterior probability that those two distributions are
Refer to the Presidential election data in Section 10.4.3.a. Use a Bayesian approach to fit model (10.7), with independent N(0, 100)priors for {????i}. Find corresponding estimates of {????i}, and
In the previous exercise, repeat (b), but using an empirical Bayes approach for a N(logit(qi), ????2) prior distribution for ????i.
Conduct Bayesian analogs of the frequentist modeling for the FEV clinical trial data from Exercise 3.31. Compare Bayesian and frequentist results.
Conduct Bayesian modeling for the smoking prevention data of Section 9.2.3.Compare results to the frequentist results presented there.
Refer to Exercise 5.34 on horseshoe crabs. Repeat this exercise using Bayesian methods.
For the endometrial cancer example in Section 10.3.2, fit the logistic model using a hierarchical Bayesian approach with a diffuse inverse chi-squared distribution for the ????2 hyperparameter.
Show that M-estimation with ????(ei) = |ei| gives the ML solution assuming a Laplace distribution for the response.
The breakdown point of an estimator is the proportion of observations that must be moved toward infinity in order for the estimator to also become infinite. The higher the breakdown point, the more
Refer to the equations solved to obtain ????̂ for M-estimation and the expression for var(????̂). Show how they simplify for least squares.
In M-estimation, let ????(x) = 2(√1 + x2∕2 − 1). Find the influence function, and explain why this gives a compromise between least squares and ????(x) =|x|, having a bounded influence function
Since the Gauss-Markov theorem says that least squares estimates are “best,”are not estimates obtained using M-estimation necessarily poorer? Explain.
For the saturated model, E(yi) = ????i, i = 1,…, n, find the ridge-regression estimate of ????i and interpret the impact of ????.
For the normal linear model, explain how (a) the ridge-regression estimates relate to Bayesian posterior means when {????j} have independent N(0, ????2)distributions, (b) the lasso estimates relate
Consider the linear model yi = ????1xi1 + ????2xi2 + ⋯ + ????40xi,40 + ????i with ????1 = 1 and ????2 = ⋯ = ????40 = 0, where xij = ui + vj with {ui}, {vj}, and{????i} being iid N(0, 1) random
Refer to the Dirichlet prior distribution introduced for multinomial parameters in Section 11.2.2. Explain why a multivariate normal prior for multinomial logits provides greater flexibility.
Refer to Copas’s kernel smoother (11.1) for binary regression, with ????(u) =exp(−u2∕2).a. To describe how close this estimator falls at a particular x value to a corresponding smoothing in the
When p > n, why is backward elimination not a potential method for selecting a subset of explanatory variables?

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