Methods And Applications Of Linear Models Regression And The Analysis Of Variance 2nd Edition Ronald R. Hocking - Solutions

Discover comprehensive solutions and detailed answers for "Methods And Applications Of Linear Models Regression And The Analysis Of Variance 2nd Edition" by Ronald R. Hocking. Our online resource offers a complete answers key and solution manual, providing step-by-step answers to all solved problems. Access the solutions PDF for free download, covering all chapter solutions and the instructor manual. Enhance your learning with the test bank, questions and answers, and textbook solutions designed for in-depth understanding. Perfect for students and instructors seeking reliable and accessible educational materials.

Make a rough drawing of a hazard function from birth for a man who develops significant coronary artery disease at age 50 and undergoes coronary artery bypass surgery at age 55.
Use resampling to validate the Somers’ Dxy rank correlation between predicted logit and the ordinal outcome. Also validate the generalized R2, and slope shrinkage coefficient, all using a single R command. Comment on the quality (potential “export-ability”) of the model.
Display the shape of how each predictor relates to the log odds of exceeding any level of sfdm2 you choose, setting other predictors to typical values(one value per predictor). By default, Predict will make predictions for the second response category, which is a satisfactory choice here.
Compute partial tests of association for each predictor and a test of nonlinearity for continuous ones. Compute a global test of nonlinearity. Graphically display the ranking of importance of the predictors.
Fit a model to predict the ordinal response using all predictors. For continuous ones assume a smooth relationship but allow it to be nonlinear.Quantify the ability of the model to discriminate patients in the five outcomes. Do an overall likelihood ratio test for whether any variables are
Impute race using the most frequent category and pafi and alb using“normal” values.
Make two plots nonparametrically relating age to all of the cumulative proportions or their logits. You can use commands such as the following(to use the R Hmisc package).for(i in 1:4)plsmo (age , sfdm ≥ i, add=i>1, ylim=c(.2 ,.8), ylab = ' Proportion Y ≥ j ' )for(i in 1:4)plsmo (age , sfdm ≥
Use an R function such as the following to compute the logits of the cumulative proportions.sf ← function(y)c ( ' Y ≥ 1 ' =qlogis (mean(y ≥ 1)),' Y ≥ 2 ' =qlogis (mean(y ≥ 2)),' Y ≥ 3 ' =qlogis (mean(y ≥ 3)),' Y ≥ 4 ' =qlogis (mean(y ≥ 4)))As the Y = 3 category is rare, it may be
To allow for maximum adjustment of baseline functional status, treat this predictor as nominal (after rounding it to the nearest whole number; fractional values are the result of imputation) in remaining steps, so that all dummy variables will be generated. Make a single chart showing proportions
For the variables adlsc, sex, age, meanbp, and others if you like, make plots of means of predictors stratified by levels of the response, to check for ordinality. On the same plot, show estimates of means assuming the proportional odds relationship between predictors and response holds. Comment on
Show that for a two-sample problem, the numerator of the score test for comparing the two groups using a proportional odds model is exactly the numerator of the Wilcoxon-Mann-Whitney two-sample rank-sum test.
What is the advantage of a rank-based test over a parametric test based on log(cost)?
Why do all of the tests you did have the same number of degrees of freedom for the hypothesis of no association between dzgroup and totcst?
If you were selecting one of the rank-based tests for testing the association between disease and cost, which of any of the tests considered would you choose?
Instead of using only two cost groups, group cost into 3, 4, 5, 6, 8, 10, and 12 quantile groups. Describe the relationship between the number of intervals used to approximate the continuous response variable and the efficiency of the analysis. How many intervals of total cost, assuming that the
Use a binary logistic model to test for association between disease group and whether total cost exceeds the median of total cost. In other words, group total cost into two quantile groups and use this binary variable as the response. What is wrong with this approach?
Use the proportional odds ordinal logistic model generalization of the Wilcoxon–Mann–Whitney Kruskal–Wallis Spearman test. Group total cost into 20 quantile groups so that only 19 intercepts will need to be in the model, not one less than the number of subjects (this would have taken the
Use the Kruskal–Wallis rank test.
Consider a binary logistic regression model using the following predictors:age (years), sex, race (white, African-American, Hispanic, Oriental, other), blood pressure (mmHg). The fitted model is given by logit Prob[Y = 1|X] = Xβˆ = −1.36 + .03(race = African-American)− .04(race = hispanic) +
Consider the age–sex–response example in Section 10.1.3. This dataset is available from the text’s web site in the Datasets area.a. Duplicate the analyses done in Section 10.1.3.b. For the model containing both age and sex, test H0 : logit response is linear in age versus Ha : logit response
For a similar setup as the binary logistic model simulation in Section 9.7, do a Monte Carlo simulation to determine the coverage probabilities for ordinary Wald and for three types of bootstrap confidence intervals for the true x=5 to x=1 log odds ratio. In addition, consider the Wald-type
Simulate a binary logistic model for n = 300 with an average fraction of events somewhere between 0.15 and 0.3. Use 5 continuous covariates and assume the model is everywhere linear. Fit an unpenalized model, then solve for the optimum quadratic penalty λ. Relate the resulting effective d.f. to
Consider a univariate saturated model in 3 treatments (A, B, C) that is quadratic in age. Write out the model with all the βs, and write in detail the contrast for comparing treatment B with treatment C for 30 year olds.Sketch out the same contrast using the “difference in predictions”
For each problem below, rank Wald, score, and LR statistics by overall statistical properties and then by computational convenience.a. A forward stepwise variable selection (to be later accounted for with the bootstrap) is desired to determine a concise model that contains most of the independent
Several regression models were considered for predicting a response. LR χ2(corrected for the intercept) for models containing various combinations of variables are found in Table 9.4. Compute all possible meaningful LR χ2.For each, state the d.f. and an approximate P-value. State which LR χ2
A sample of size 100 from a normal distribution with unknown mean and standard deviation (μ and σ) yielded the following log likelihood values when computed at two values of μ.log L(μ = 10, σ = 5) = −800 log L(μ = 20, σ = 5) = −820.What do you know about μ? What do you know about Y ?
Test for additivity of age and blood pressure. Make a joint test for the overall absence of complexity in the model (linearity and additivity simultaneously).
Expand the model to not assume additivity of age and blood pressure.Use a tensor natural spline or an appropriate restricted tensor spline. If you run into any numerical difficulties, use 4 knots instead of 5. Plot in an interpretable fashion the estimated 3-D relationship between age, blood
Test linearity in age, linearity in blood pressure, and linearity in heart rate, and also do a joint test of linearity simultaneously in all three predictors.
Make appropriate graphical diagnostics for this model.
Fit a model, assuming the above predictors act additively, but do not assume linearity for the age and blood pressure effects. Use the truncated power basis for fitting restricted cubic spline functions with 5 knots. Estimate the shrinkage coefficient ˆγ.
Do an analysis that helps one determine how many d.f. to devote to each predictor.
Do a redundancy analysis of the predictors, using both a less stringent and a more stringent approach to assessing the redundancy of the multiple-level variable disease group.
Do an analysis to understand interrelationships among predictors, and find optimal scaling (transformations) that make the predictors better relate to each other (e.g., optimize the variation explained by the first principal component).
Using the multiple imputed values, develop an overall least squares model for total cost (using the log transformation) making optimal use of partial information, with variances computed so as to take imputation (except for cost) into account. The model should use the predictors in Problem 1 and
Use predictive mean matching to multiply impute cost 10 times per missing observation. Describe graphically the distributions of imputed values and briefly compare these to distributions of non-imputed values. State in a simple way what the sample variance of multiple imputations for a single
Use transcan to develop single imputations for total cost, commenting on the strength of the model fitted by transcan as well as how strongly each variable can be predicted from all the others.
State briefly why single conditional medianh imputation is OK here.
Prepare for later development of a model to predict costs by developing reliable imputations for missing costs. Remove the observation having zero totcst.ga. The cost estimates are not available on 105 patients. Total hospital charges (bills) are available on all but 25 patients. Relate these two
Explore the variables and patterns of missing data in the SUPPORT dataset.a. Print univariable summaries of all variables. Make a plot (showing all variables on one page) that describes especially the continuous variables.b. Make a plot showing the extent of missing data and tendencies for some
Consider a two–stage procedure in which one tests for linearity of the effect of a predictor X on a property of the response C(Y |X) against a quadratic alternative. If the two–tailed test of linearity is significant at the α level, a two d.f. test of association between X and Y is done. If
Prove that each component variable is linear in X when X ≥ tk, the last knot, using general principles and not algebra or calculus. Derive an expression for the restricted spline regression function when X ≥ tk.
Derive the formulas for the restricted cubic spline component variables without cubing or squaring any terms.
In “The Class of 1988: A Statistical Portrait,” the College Board reported mean SAT scores for each state. Use an ordinary least squares multiple regression model to study the mean verbal SAT score as a function of the percentage of students taking the test in each state. Provide plots of
For a fitted model Y = a + bX + cX2 derive the estimate of the effect on Y of changing X from x1 to x2.
Consider models each with two predictors, temperature and white blood count (WBC), for which temperature is always assumed to be linearly related to the appropriate property of the response, and WBC may or may not be linear (depending on the particular model you formulate for each question).
Consider a linear additive model involving three treatments (control, drug Z, and drug Q) and one continuous adjustment variable, age. Test H0 :treatment group is not associated with response, adjusted for age. Also test H0 : response for drug Z has the same property as the response for drug Q,
A property of Y such as the mean is linear in age and blood pressure and there may be an interaction between the two predictors. Test H0 :there is no interaction between age and blood pressure. Also test H0 :blood pressure is not associated with Y (in any fashion). State the effect of blood
Conroy (2005) prefers to run ordinal logistic regressions in parallel with interval regressions for interval-censored data with fixed thresholds, as mentioned in Section 7.4.2. Analyze the grades data with interval regression and ordinal logistic regression and compare their results.
Estimate the Tobit and log-normal models for the euthanasia attitudes data. The dependent variable in the data is called “euth” and the predictor is “mident” You will need to add a positive number (e.g., 1) to euth to enable log-linear models to be estimated. Run the appropriate model
Prove that the Weibull model has the constant hazard ratio property.
Analyze the reading accuracy data using a log-normal model. Compare the results with the Tobit model described in Section 7.1. Is there evidence for one model’s superiority to the other?
Analyze the response time dataset. In R, the data are available via the command data("rtime", package = "smdata"). Stata users may access the file at https://dl.dropbox.com/u/1857674/SmiMerBook/SmiMerDoc.html via the link “ch7 rtime.dta”. Run the log-normal and log-logistic models, obtaining
(Advanced) Derive the expected rate of change for the predictor xk in the truncated dependent variable y in a regression model for an upper-censored Weibull random variable with a vector of predictors x.
(Advanced) Find the expected value of an upper-censored exponential random variable with censoring threshold τ.
Explain why it would be difficult to construct a procedure for correcting the estimate of mean income in a country whose census undercounted homeless people, if no other data than the census were available.
Consider the truncation example of the bank that does not record losses unless they exceed $1000. Suppose this bank has accumulated 25 years of loss data using this criterion. Describe a procedure for estimating the number of losses per month that would take this truncation into account.
Investigate the model diagnostics (residuals and influence measures) regarding the model presented in this chapter for Smithson et al. (2007), Study 1. Use a leave-one-out method. Are there any badly fitted and/or strongly influential observations? Are any of the model’s conclusions altered by
For the reading accuracy model, perform the Wald test of the hypothesis thatδ1 = −δ2 in the precision submodel.
Find the method of moments estimates of the precisions in the reading accuracy data.
(Advanced) Prove the identity in Equation (6.8).
Analyze the data from Smithson et al. (2007), Study 3. In R, the data file is available via the command data("guilt3", package = "smdata"). Stata users may access the file via the link “ch6 guilt3.dta” at https://dl.dropbox.com/u/1857674/SmiMerBook/SmiMerDoc.html. It contains 96 observations
Prove the identity between Equations (6.3) and (6.4).
In your own words, explain why “A score of 0% or 100% on an exam is a true score if the exam includes all relevant test items, but not if it is considered to include only a sample of relevant items.”
Obtain the workdays data from the smdata package or from https://dl.dropbox.com/u/1857674/SmiMerBook/SmiMerDoc.html, which contains extra variables not described in this chapter’s examples. In particular, service provides information about whether each individual has been in the U.S. Armed Forces
(Advanced) The gamma distribution can be defined as f(λ|α,β) = 1(α −1)!βαλα−1 exp(−λ/β) (5.40)for λ ≥ 0, α a positive integer, and β > 0. (α is not required to be an integer, but we will assume this for simplicity here.)Assuming the hierarchical distribution for y given by
Write a hurdle model (in the manner of Equation (5.34)) that uses a rightcensored Poisson model for the hurdle component and a truncated Poisson model for the count component. Show that, when x = z and γ = β , the hurdle model is equivalent to a regular Poisson regression model.
Construct a Pearson residual-based test of overdispersion for the negative binomial model.
Show that the hurdle model in Equation (5.34) satisfies the following conditions to be considered a probability mass function. (The conditioning on the left side of Equation (5.34) is omitted here but should be implicitly assumed.)(a) P(Y = y) ≥ 0 for all y.(b)∞∑i=0 P(Y = i) = 1.
Compute the probability of observing each possible value of y, when y arises from a Poisson(λ = 3) distribution that is right-truncated at 6.
Your roommate has written up the results of a Poisson regression in which she is studying the relationship between number of speeding tickets (response) and college GPA (predictor). She writes that “with every point increase in GPA, there is a 110% decrease in number of speeding tickets.” How
Obtain the Hulsey data from the R package smdata or from https://dl.dropbox.com/u/1857674/SmiMerBook/SmiMerDoc.html. This file includes two other predictors that were not described in the text. The first predictor, effectiveness, contains subjects’ ratings of the importance of treatment
Modify Equation (4.33) to obtain Pearson residuals for a stage model.
Latent variable formulation of cumulative models: Show that, if we use the normal distribution instead of the logistic distribution to model e in (4.9), we obtain a cumulative ordinal model with probit link.
Show that, when J = 2, models described under all three approaches (cumulative, stage, and adjacent) reduce to a logistic regression model as described in Chapter 2.
Use cumulative proportional odds model predictions for P(y ≤ j), j = 1,2,..., J, to show why the constraints α1 < α2 < ... < αJ−1 are necessary.
The proportional odds assumption: Using a cumulative model, define the odds of being at a particular point on the scale or lower as P(y ≤ j|x)1−P(y ≤ j|x).The “proportional odds” assumption derives from the fact that, when the assumption is employed, the odds ratio associated with two
For an ordinal response variable with J categories, show that the predicted P(y = j) (j = 1,..., J) from a continuation ratio model equal the expressions in Equation (4.26).
(R users) Estimate the MPT model in the WPWSE example. The datafile is available in the smdata package via the command data("phono", package="smdata").
Explain why, in Exercise 3.4, it would be impossible to include an interaction effect between the public versus private transportation variable and transport alternatives.
Estimate a transportation choice model as a CL model, with sex as a subjectlevel covariate and the distinction between public and private transport as an alternative-level variable. The data file for R users is available in the smdata package via the command data("trlong", package="smdata")and for
Verify that the MPT model in Equation (3.11) results in category probabilities that sum to 1.
In the baby quartet example it was shown that the model coefficients can be used to derive the predicted probabilities for each baby in those cases where gaze has been initiated by another baby. Describe the remaining cases for which choice probabilities can be derived from the model coefficients.
Perform the simple-effects tests for the sex×race effect on the inhale+inject and freebase+inject subsets of the cocaine usage data, as described in this chapter. Compare the sex×race model with the sex+race model, and evaluate the sex+race model against the corresponding null model.The data for
The model using age to predict cocaine usage method yielded very similar coefficients for the effect of age on the odds ratios involving inhale and freebase(see Table 3.3 and compare β21 with β31). Determine whether those coefficients differ significantly from one another.
Obtain the skipping data from the smdata package or from https://dl.dropbox.com/u/1857674/SmiMerBook/SmiMerDoc.html, which includes two other potential predictors that were not used in the text. The first is income, which reflects household income on an ordinal scale. The levels are “less
Demonstrate that the ROC curve associated with a binary regression model will always include the points (0,0) and (1,1).
In the “skipping school” example, we fit two models to the data: a binary logistic regression model and a binomial logistic regression model with n = 30. We found that the estimated slopes were very similar across the two models, but the intercepts were considerably different (see Table 2.2).
For a simple logistic regression model with one predictor variable, show that the odds ratio associated with a one-unit increase in the predictor variable can indeed be obtained by exponentiating the regression weight associated with the predictor.
Show that the inverse of the logit transformation corresponds to Equation (2.4).
Use the fitted logistic regression model from Equation (2.13) to calculate the model-predicted probabilities of passing calculus for individuals with GPAs of 1.5, 2.75, and 3.6.
Verify the Bonferroni inequality for the case k=2. Recall that ENE EUE. That is, the compliment of the intersection of E, and E is the union of the compliments of E, and E. The general result follows by induction.
Consider the cell means model, where the errors are independent, N(0,). For testing the hypothesis of equal means, we may use the (p-1) xp matrix H = (I|- J).a. Determine the matrix H(XX)HT and its inverse (see Appendix A.1.10). Show that the last (p-1) rows of the Helmert matrix defined in
a. In the extension of the Tukey simultaneous testing method, verify the relation shown in (18.34).b. Verify the generalization in (18.37).e. Write the acceptance region in (18.37) if the single-degree-of-freedom hypotheses are aph.
Show that the result in (18.18) can be viewed as an application of Cochran's theorem.
To establish the desirable feature of using the eigenvectors to define a set of orthogonal comparisons, determine the circumscribed rectangle for the ellipse Mr I with minimum volume. Hint: From Exercise 18.1 we describe the rectangle by the pairs of planes Show that the square of the volume of
Establish the Scheff inequality as follows: A tangent plane with direction numbers a has the form a(z-za)=b. Applying the general expression for the linear approximation to a function, given in Appendix A.II.2, show that the tangent plane at a specified point a has the form (2*-20) M(z - 20) = 1.
Consider the linear model y z +. The data have been standardized so that 0, SS, 1 and normal equations are given as [0.842 [0.9640] [8] - [0.793] 0.94][]a. Determine the parameter estimates.b. Develop the test statistic for the hypothesis, H: BB = 0 and perform the test with a 0.05.c. Develop the
Consider the cell means model from Exercise 17.18 for the special case p=4. The hypothesis of equal means can be described by the hypothesis matrix -1 H-1 0-1 0 0 0 0a. Determine the matrix M as in (17.51), where G H and G satisfies the conditions in (17.561), and define the reparameterization =
Consider the linear model with design matrix and parameter vector given by B J. 0 x JB1 J B B 0 J. B: B J. -Ju Ba. Determine the estimates of B anda. b. Determine the numerator sum of squares for testing the hypothesis H: B=0.

Showing 200 - 300 of 1264

Methods And Applications Of Linear Models Regression And The Analysis Of Variance 2nd Edition Ronald R. Hocking - Solutions

Step by Step Answers