Questions and Answers of Linear State Space Systems

11.2 Suppose that (Yjk,xjk) are observations on the kth subject in cluster k (with j = 1,...,J;k = 1,...,K) and the goal is to fit a “regression through the origin” model E(Yjk) = βxjk,
11.1 The measurement of left ventricular volume of the heart is important for studies of cardiac physiology and clinical management of patients with heart disease. An indirect way of measuring the
10.6 The data in Table 10.5 are survival times, in months, of 44 patients with chronic active hepatitis. They participated in a randomized controlled trial of prednisolone compared with no treatment.
10.5 As the survivor function S(y) is the probability of surviving beyond time y, the odds of survival past time y are O(y) = S(y)1−S(y).For proportional odds models the explanatory variables for
10.4 For proportional hazards models the explanatory variables for subject i, ηi, act multiplicatively on the hazard function. If ηi = e xT iβ, then the hazard function for subject i is hi(y) = e
10.3 For accelerated failure time models the explanatory variables for subject
10.2 The log-logistic distribution with the probability density function f(y) = eθ λyλ−1(1+eθ yλ )2 is sometimes used for modelling survival times.(a) Find the survivor function S(y), the
10.1 The data in Table 10.4 are survival times, in weeks, for leukemia patients.There is no censoring. There are two covariates, white blood cell count(WBC) and the results of a test (AG positive and
9.7 Mittlbock and Heinzl (2001) compare Poisson and logistic regression models for data in which the event rate is small so that the Poisson distribution provides a reasonable approximation to the
9.6 Consider a 2×K contingency table (Table 9.14) in which the column totals y.k are fixed for k = 1,...,K.
9.5 Use log-linear models to examine the housing satisfaction data in Table 8.5. The numbers of people surveyed in each type of housing can be regarded as fixed.a. First, analyze the associations
9.4 For a 2×2 contingency table, the maximal log-linear model can be written asη11 = µ +α +β + (αβ), η12 = µ +α −β −(αβ),η21 = µ −α +β −(αβ), η22 = µ −α −β +
9.3 This question relates to the flu vaccine trial data in Table 9.6.a. Using a conventional chi-squared test and an appropriate log-linear model, test the hypothesis that the distribution of
9.2 The data in Table 9.13 are numbers of insurance policies, n, and numbers of claims, y, for cars in various insurance categories, CAR, tabulated by age of policy holder, AGE, and district where
9.1 Let Y1,...,YN be independent random variables with Yi ∼ Po(µi) and log µi = β1 +∑J j=2 xi jβj, i = 1,...,N.a. Show that the score statistic for β1 is U1 = ∑N i=1(Yi − µi).b. Hence,
8.4 Consider ordinal response categories which can be interpreted in terms of continuous latent variable as shown in Figure 8.2. Suppose the distribution of this underlying variable is Normal. Show
8.3 The data in Table 8.6 show tumor responses of male and female patients receiving treatment for small-cell lung cancer. There were two treatment regimes. For the sequential treatment, the same
8.2 The data in Table 8.5 are from an investigation into satisfaction with housing conditions in Copenhagen (derived from Example W in Cox and Snell, 1981, from original data from Madsen, 1971).
8.1 If there are only J = 2 response categories, show that models (8.4), (8.13),(8.15) and (8.16) all reduce to the logistic regression model for binary data.
7.5 Let Yi be the number of successes in ni trials with Yi ∼ Bin(ni,πi), where the probabilities πi have a Beta distributionπi ∼ Be(α,β).The probability density function for the Beta
7.4 Let l(bmin) denote the maximum value of the log-likelihood function for the minimal model with linear predictor x T β = β1, and let l(b) be the corresponding value for a more general model x T
7.3 Tables 7.16 and 7.17 show the survival 50 years after graduation of men and women who graduated each year from 1938 to 1947 from various faculties of the University of Adelaide (data compiled by
7.2 Odds ratios. Consider a 2×2 contingency table from a prospective study in which people who were or were not exposed to some pollutant are followed up and, after several years, categorized
7.1 The number of deaths from leukemia and other cancers among survivors of the Hiroshima atom bomb are shown in Table 7.14, classified by the radiation dose received. The data refer to deaths during
6.9 Examine if there is a non-linear association between age and cholesterol using the fractional polynomial approach for the data in Table 6.24.
6.8 Table 6.27 shows the data from a fictitious two-factor experiment.a. Test the hypothesis that there are no interaction effects.b. Test the hypothesis that there is no effect due to Factor A(i) by
6.7 For the balanced data in Table 6.12, the analyses in Section 6.4.2 showed that the hypothesis tests were independent. An alternative specification of the design matrix for the saturated Model
6.6 The weights (in grams) of machine components of a standard size made by four different workers on two different days are shown in Table 6.26;five components were chosen randomly from the output
6.5 Table 6.25 shows plasma inorganic phosphate levels (mg/dl) one hour after a standard glucose tolerance test for obese subjects, with or without hyperinsulinemia, and controls (data from Jones,
6.4 It is well known that the concentration of cholesterol in blood serum increases with age, but it is less clear whether cholesterol level is also associated with body weight. Table 6.24 shows for
6.3 Analyze the carbohydrate data in Table 6.3 using appropriate software (or, preferably, repeat the analyses using several different regression programs and compare the results).a. Plot the
6.2 Table 6.23 shows response of a grass and legume pasture system to various quantities of phosphorus fertilizer (data from D. F. Sinclair; the results were reported in Sinclair and Probert, 1986).
6.1 Table 6.22 shows the average apparent per capita consumption of sugar(in kg per year) in Australia, as refined sugar and in manufactured foods(from Australian Bureau of Statistics, 1998).Table
5.4 For the leukemia survival data in Exercise 4.2:a. Use the Wald statistic to obtain an approximate 95% confidence interval for the parameter β1.b. By comparing the deviances for two appropriate
5.3 Suppose Y1,...,YN are independent identically distributed random variables with the Pareto distribution with parameter θ.a. Find the maximum likelihood estimator θb of θ.b. Find the Wald
5.2 Consider a random sample Y1,...,YN with the exponential distribution f(yi;θi) = θi exp(−yiθi).Derive the deviance by comparing the maximal model with different values of θi for each Yi and
5.1 Consider the single response variable Y with Y ∼ Bin(n,π).a. Find the Wald statistic (πb − π)T I(πb − π), where πb is the maximum likelihood estimator of π and I is the
4.3 Let Y1,...,YN be a random sample from the Normal distribution Yi ∼N(logβ,σ2) where σ2 is known. Find the maximum likelihood estimator of β from first principles. Also verify Equations
4.2 The data in Table 4.6 are times to death, yi, in weeks from diagnosis and log10 (initial white blood cell count), xi, for seventeen patients suffering from leukemia. (This is Example U from Cox
4.1 The data in Table 4.5 show the numbers of cases of AIDS in Australia by date of diagnosis for successive 3-month periods from 1984 to 1988.(Data from National Centre for HIV Epidemiology and
3.12 See some more relationships between distributions in Figure 3.3.
3.11 For the Pareto distribution, find the score statistics U and the information I = var(U). Verify that E(U) = 0.
3.10 Let Y1,...,YN be independent random variables with E(Yi) = µi = β0 +log(β1 +β2xi); Yi ∼ N(µ,σ2)for all i = 1,...,N. Is this a generalized linear model? Give reasons for your answer.
3.9 Suppose Y1,...,YN are independent random variables each with the Pareto distribution and E(Yi) = (β0 +β1xi)2.Is this a generalized linear model? Give reasons for your answer.
3.8 Is the extreme value (Gumbel) distribution, with probability density function f(y;θ) = 1φexp(y−θ)φ−exp(y−θ)φ(where φ > 0 is regarded as a nuisance parameter) a member of the
3.7 Consider N independent binary random variables Y1,...,YN with P(Yi = 1) = πi and P(Yi = 0) = 1−πi.The probability function of Yi, the Bernoulli distribution B(π), can be written asπyi
3.6 Do you consider the model suggested in Example 3.5.3 to be adequate for the data shown in Figure 3.2? Justify your answer. Use simple linear regression (with suitable transformations of the
3.5a. For a Negative Binomial distribution Y ∼ NBin(r,θ), find E(Y) and var(Y ).b. Notice that for the Poisson distribution E(Y) = var(Y ), for the Binomial distribution E(Y) > var(Y) and for the
3.4 Use results (3.9) and (3.12) to verify the following results:a. For Y ∼ Po(θ), E(Y) = var(Y ) = θ.b. For Y ∼ N(µ,σ2), E(Y) = µ and var(Y ) = σ2.c. For Y ∼ Bin(n,π), E(Y) = nπ and
3.3 Show that the following probability density functions belong to the exponential family:a. Pareto distribution f(y;θ) = θy−θ−1.b. Exponential distribution f(y;θ) = θe−yθ
3.2 If the random variable Y has the Gamma distribution with a scale parameter β, which is the parameter of interest, and a known shape parameterα, then its probability density function is f(y;β)
3.1 The following associations can be described by generalized linear models.For each one, identify the response variable and the explanatory variables, select a probability distribution for the
2.5 The model for two-factor analysis of variance with two levels of one factor, three levels of the other and no replication is E(Yjk) = µjk = µ +αj +βk; Yjk ∼ N(µjk,σ2), where j = 1,2; k =
2.4 Suppose you have the following data x: 1.0 1.2 1.4 1.6 1.8 2.0 y: 3.15 4.85 6.50 7.20 8.25 16.50 and you want to fit a model with E(Y) = ln(β0 +β1x+β2x 2).Write this model in the form of
2.3 For Model (2.7) for the data on birthweight and gestational age, using methods similar to those for Exercise 1.4, show Sb1 =J∑j=1 K∑k=1(Yjk −aj −bjxjk)2=J∑j=1 K∑k=1(Yjk −(αj
2.2 The weights, in kilograms, of twenty men before and after participation in a “waist loss” program are shown in Table 2.8 (Egger et al. 1999). We want to know if, on average, they retain a
2.1 Genetically similar seeds are randomly assigned to be raised in either a nutritionally enriched environment (treatment group) or standard conditions (control group) using a completely randomized
1.6 The data in Table 1.4 are the numbers of females and males in the progeny of 16 female light brown apple moths in Muswellbrook, New South Wales, Australia (from Lewis, 1987).a. Calculate the
1.5 This exercise is a continuation of the example in Section 1.6.2 in which Y1,...,Yn are independent Poisson random variables with the parameter θ.a. Show that E(Yi) = θ for i = 1,...,n.b.
1.4 Let Y1,...,Yn be independent random variables each with the distribution N(µ,σ2). Let Y =1 nn∑i=1 Yi and S 2 =1 n−1 n∑i=1(Yi −Y)2.a. What is the distribution of Y?b. Show that S 2 =1
1.3 Let the joint distribution of Y1 and Y2 be MVN(µ,V) withµ =2 3and V =4 1 1 9 .a. Obtain an expression for (y−µ)TV−1(y−µ). What is its distribution?b. Obtain an expression for y
1.2 Let Y1 and Y2 be independent random variables with Y1 ∼ N(0,1) and Y2 ∼N(3,4).a. What is the distribution of Y 21?b. If y =Y1(Y2 −3)/2, obtain an expression for y T y. What is its
1.1 Let Y1 and Y2 be independent random variables with Y1 ∼ N(1,3) and Y2 ∼ N(2,5). If W1 = Y1 +2Y2 and W2 = 4Y1 −Y2, what is the joint distribution of W1 and W2?
For the anorexia study of Exercise 1.24, write a report in which you pose a research question and then summarize your analyses, including graphical description, interpretation of a model fit and its
For the Student survey.dat data file at the text website, model how political ideology relates to number of times per week of newspaper reading and religiosity. Prepare a report, posing a research
Refer to the study for comparing instruction methods mentioned in Exercise 2.45. Write a short report summarizing inference for the model fitted there, interpreting results and attaching edited
In a study11 at Iowa State University, a large field was partitioned into 20 equal-size plots. Each plot was planted with the same amount of seed corn, using a fixed spacing pattern between the
For the house selling price data, fit the model with size of home as the sole explanatory variable. Find a 95% confidence interval for E( y) and a 95%prediction interval for y, at the sample mean
Using the house selling price data at the text website, describe the predictive power of various models by finding adjusted R2 when (i) size is the sole predictor, (ii) size and new are main-effect
For the house selling price data of Section 3.4, when we include size, new, and taxes as explanatory variables, we obtain--------------------------------------------------------------->
Section 3.4.1 used x1 = size of house and x2 = whether new to predict y = selling price. Suppose we instead use a GLM, log(????i) = ????0 + ????1 log(xi1) +????2xi2.a. For this GLM, interpret ????1
Refer to Exercise 2.47 on carapace width of attached male horseshoe crabs.Extend your analysis of that exercise by conducting statistical inference, and interpret.
For the horseshoe crab dataset Crabs.dat at the text website, analyze inferentially the effect of color on the mean number of satellites, treating the data as a random sample from a conceptual
Refer to Exercise 1.21 on a study comparing forced expiratory volume (y =fev1 in the data file) for three drugs (x2), adjusting for a baseline measurement(x1).a. Fit the normal linear model using
Suppose the relationship between y = college GPA and x = high school GPA satisfies yi ∼ N(1.80 + 0.40xi, 0.302). Simulate and construct a scatterplot for n = 1000 independent observations taken
Construct a Q–Q plot for the model for the house selling prices that uses size, new, and their interaction as the predictors, and interpret. To get a sense of how such a plot with a finite sample
In the previous exercise, suppose truncation instead occurs on x. Would you expect this to affect (a) E(????̂1)? (b) inference about ????1? Why?
Selection bias: Suppose the normal linear model ????i = ????0 + ????1xi holds with????1 > 0, but the responses are truncated and we observe yi only when yi > L(or perhaps only when yi < L) for some
In the one-way layout with c groups and a fixed common sample size n, consider simultaneous confidence intervals for pairwise comparisons of means, using family-wise error probability ???? = 0.05.
An analyst plans to construct family-wise confidence intervals for normal linear model parameters {????(1),…, ????(g)} in estimating an effect as part of a metaanalysis with g independent studies.
For the normal linear model for the r × c two-way layout with n observations per cell, explain how to use the Tukey method for family-wise comparisons of all pairs of the r row means with confidence
Based on the expression for a squared partial correlation in Section 3.4.4, show how it relates to a partial SS for the full model and SSE for the model without that predictor.
Consider the null model, for simplicity with known ????2. After estimating ???? =E(y) by ȳ, you plan to predict a future y from the N(????, ????2) distribution. State the formula for a 95%
When there are no explanatory variables, show how the confidence interval in Section 3.3.2 simplifies to a confidence interval for the marginal E( y).
Mimicking the derivation in Section 3.3.2, derive a confidence interval for the linear combination ????????. Explain how it simplifies for the case ????j − ????k.
Suppose a one-way layout has ordered levels for the c groups, such as dose levels in a dose–response assessment. The model E( yij) = ????0 + ????i treats the groups as a qualitative factor. The
Explain how to use the F test for the general linear hypothesis H0: ???????? = c to invert a test of H0: ???? = ????0 to form a confidence ellipsoid for ????. For p = 2, describe how this could give
For a normal linear model with p parameters and n observations, explain how to test H0: ????j = ????k in the context of the (a) general linear hypothesis and (b)F test comparing two nested linear
For the linear model E( yij) = ????0 + ????i for the one-way layout, explain how H0:????1 = ⋯ = ????c is a special case of the general linear hypothesis.
Using the F formula for comparing models in the previous exercise, show that adjusted R2 being larger for the more complex model is equivalent to F > 1.
a. Show that the F statistic in Section 3.2.4 for testing that all effects equal 0 has expression in terms of the R2 value as F = R2∕(p − 1)(1 − R2)∕(n − p)b. Show that the F statistic
Refer to the previous exercise. Now consider the model permitting interaction.Table 3.4 shows the resulting ANOVA table.a. Argue intuitively and in analogy with results for one-way ANOVA that the SS
For the balanced two-way r × c layout with n observations {yijk} in each cell, denote the sample means by {ȳij.} in the cells, ȳi.. in level i of A, ȳ.j. in level j of B, and ȳ overall for
Section 2.3.4 considered the projection matrices and ANOVA table for the two-way layout with one observation per cell. For testing each main effect in that model, show how to construct test
Refer to the previous exercise. Based on inverting significance tests with nonzero null values, show how to construct a confidence interval for ????1 − ????2.
Using principles from this chapter, inferentially compare ????1 and ????2 from N(????1, ????2) and N(????2, ????2) populations, based on independent random samples of sizes n1 and n2.a. Put the
Based on the formula s2(XTX)−1 for the estimated var(????̂), explain why the standard errors of {????̂j} tend to decrease as n increases.
A one-way ANOVA uses ni observations from group i, i = 1,…, c.a. Verify the noncentrality parameter for the scaled between-groups sum of squares.b. Suppose c = 3, with ????1 − ????2 = ????2 −
Consider the normal linear model for the one-way layout (Section 3.2.1).a. Explain why the F statistic used to test H0: ????1 = ⋯ = ????c has, under H0, an F distribution.b. Why is the test is
For y1,…, yn independent from N(????, ????2), apply Cochran’s theorem to construct a F test of H0: ???? = ????0 against H1: ???? ≠ ????0 by applying the SS decomposition with the projection

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