Logistic Regression A Self Learning Text 3rd Edition David G. Kleinbaum, Mitchel Klein - Solutions

4. Until recently, the most widely available com- puter packages for fitting the logistic model have used unconditional procedures.
3. In a case-control study involving 1,200 subjects, a logistic model involving 1 exposure variable, 3 potential confounders, and 3 potential effect modifiers is to be estimated. Assuming no matching has been done, the preferred method of estimation for this model is conditional ML estimation.
2. If discriminant function analysis is used to esti- mate logistic model parameters, biased esti- mates can be obtained that result in estimated odds ratios that are too high.
1. Maximum likelihood estimation is preferred to least squares estimation for estimating the parameters of the logistic and other nonlinear models.
10. The likelihood ratio test is the preferred method for testing hypotheses about parameters in the logistic model.
9. The maximized likelihood value L() is used for confidence interval estimation of parameters in the logistic model.
8. The likelihood function formulae for both the unconditional and conditional approaches are the same.
7. The maximum likelihood method maximizes the function In L(0).
6. The likelihood function L(0) represents the joint probability of observing the data that has been collected for analysis.
5. Unconditional ML estimation gives unbiased results always.
4. Conditional ML estimation should be used to estimate logistic model parameters if matching has been carried out in one's study.
3. The conditional approach is preferred if the number of parameters in one's model is small relative to the number of subjects in one's data set.
2. Two alternative maximum likelihood approaches are called unconditional and conditional meth- ods of estimation.
1. When estimating the parameters of the logistic model, least squares estimation is the preferred method of estimation.
6. Suppose the model in Question 5 is revised to contain interaction terms: logit P(X)=x+BNS+BOC+B3AFS+7,AGE+72RACE +611(NSX AGE) +812(NS RACE) +821 (OC AGE) +822 (OCX RACE) +31 (AFS AGE) +832 (AFSX RACE). For this revised model, answer the same questions as given in parts a and b of Question 5.a.
5. Given the following model logit P(X) + BNS + BOC + 3AFS+AGE + 72 RACE, where NS denotes number of sex partners in one's lifetime, OC denotes oral contraceptive use (yes/no), and AFS denotes age at first sexual intercourse experience, answer the following questions about the above model:a. Give
4. Suppose the variable SSU in Question 3 is partitioned into three categories denoted as low, medium, and high.a. Revise the model of Question 3 to give the logit form of a logistic model that treats SSU as a nomi- nal variable with three categories (assume no interaction).b. Using your model of
3. Given the model logit P(X) + BSSU+1AGE + 2 SEX + SSU(6AGE + 8SEX), where SSU denotes "social support score" and is an ordinal variable ranging from 0 to 5, answer the following questions about the above model:a. Give an expression for the odds ratio that compares a person who has SSU = 5 to a
2. Suppose the model in Question 1 is revised as follows: logit P(X)=x+BCAT+AGE +2CHL + CAT(1AGE +8CHL).
1. Given the following logistic model logit P(X)=x+BCAT +AGE + CHL, where CAT is a dichotomous exposure variable and AGE and CHL are continuous, answer the following questions concerning the odds ratio that compares exposed to unexposed persons controlling for the effects of AGE and CHL:a. Give an
11. Given the model logit P(X) + B(SMK) + B (ASB) +71 (AGE) +8 (SMK AGE) + 2(ASB x AGE),
10. For the model of Exercise 9, give an expression for the odds ratio for the E, D relationship that compares urban with rural persons, controlling for the con- founding and effect-modifying effects of the four covariates.
9. Revise your model of Exercise 7 to allow effect modi- fication of each covariate with the exposure variable. State the logit form of this revised model.
8. For the model of Exercise 7, give an expression for the odds ratio for the E, D relationship that compares urban with rural persons, controlling for the four covariates.
7. State the logit form of a logistic model that treats region as a polytomous exposure variable and controls for the confounding effects of AGE, SMK, RACE, and SEX. (Assume no interaction involving any covariates with exposure.)
6. If we assume no interaction in the above model, the expression exp() gives the odds ratio for describing the effect of one unit change in CHL value, controlling for AGE.
5. The odds ratio that compares a person with CHL = 200 to a person with CHL = 140 controlling for AGE is given by exp(60).
4. If the correct odds ratio formula for a given cod- ing scheme for E is used, then the estimated odds ratio will be the same regardless of the coding scheme used.
3. If there is no interaction in the above model and E is coded as (-1, 1), then the odds ratio for the E, D relationship that controls for SMK and HPT is given by exp().
2. If E is coded as (-1, 1), then the odds ratio for the E, D relationship that controls for SMK and HPT is given by exp[2 + 281(SMK) + 282(HPT)].
1. If E is coded as (0 = unexposed, 1 = exposed), then the odds ratio for the E, D relationship that controls for SMK and HPT is given by exp[+8,(Ex SMK) + 2(Ex HPT)].
10. Using the model in Question 9, give an expression for the odds ratio that compares an exposed person (CON ¼ 1) with an unexposed person (CON ¼ 0) who has the same values for PAR, NP, and ASCM.
9. Within the above study framework, state the logit form of a logistic model for assessing the effect of CON on HIV acquisition, controlling for each of the other three risk factors as both potential confounders and potential effect modifiers. (Note: In defining your model, only use interaction
8. For the model in Question 7, if C = 5 and C = 20, then the odds ratio for the E, D relation- ship has the form exp( + 208).
7. Given E, C, and C2, and letting V = C = W, V2 (C1), and V3 C2, then the corresponding logistic model is given by logit P(X) = x + BE +71C1+2C12 + 3C2 + SEC1.
6. Given the model logit P(X) = a + BE + HPT + ECG, where E, HPT, and ECG are (0, 1) vari- ables, then the odds ratio for estimating the effect of ECG on the disease, controlling for E and HPT, is given by exp().
5. For the model in Question 4, the odds ratio that describes the effect of HPT on disease status, controlling for ECG, is given by exp( + ECG).
4. Given the model logit P(X) = + HPT + PECG + HPT x ECG, where HPT is a (0, 1) exposure variable denoting hypertension status and ECG is a (0, 1) variable for electrocardio- gram status, the null hypothesis for a test of no interaction on a multiplicative scale is given by Ho: exp(x) = 1.
3. Given the model in Question 2, the odds of get- ting the disease for unexposed persons (E = 0) is given by exp(x).
2. Given the model logit P(X) = x + BE, where E denotes a (0, 1) exposure variable, the risk for unexposed persons (E = 0) is expressible as 1/exp(-2).
1. Given the simple analysis model, logit P(X) = + Q, where and are unknown parameters and Q is a (0, 1) exposure variable, the odds ratio for describing the exposure-disease relationship is given by exp().
8. For the model in Question 7, if C = 5 and C = 20, then the odds ratio for the E, D relation- ship has the form exp( + 208).
7. Given E, C, and C2, and letting V = C = W, V2 (C1), and V3 C2, then the corresponding logistic model is given by logit P(X) = x + BE +71C1+2C12 + 3C2 + SEC1.
6. Given the model logit P(X) = a + BE + HPT + ECG, where E, HPT, and ECG are (0, 1) vari- ables, then the odds ratio for estimating the effect of ECG on the disease, controlling for E and HPT, is given by exp().
5. For the model in Question 4, the odds ratio that describes the effect of HPT on disease status, controlling for ECG, is given by exp( + ECG).
4. Given the model logit P(X) = + HPT + PECG + HPT x ECG, where HPT is a (0, 1) exposure variable denoting hypertension status and ECG is a (0, 1) variable for electrocardio- gram status, the null hypothesis for a test of no interaction on a multiplicative scale is given by Ho: exp(x) = 1.
3. Given the model in Question 2, the odds of get- ting the disease for unexposed persons (E = 0) is given by exp(x).
2. Given the model logit P(X) = x + BE, where E denotes a (0, 1) exposure variable, the risk for unexposed persons (E = 0) is expressible as 1/exp(-2).
1. Given the simple analysis model, logit P(X) = + Q, where and are unknown parameters and Q is a (0, 1) exposure variable, the odds ratio for describing the exposure-disease relationship is given by exp().
6. Given the model logit P(X) = a+ BE + HPT + ECG, where E, HPT, and ECG are (0, 1) vari- ables, then the odds ratio for estimating the effect of ECG on the disease, controlling for E and HPT, is given by exp().
5. For the model in Question 4, the odds ratio that describes the effect of HPT on disease status, controlling for ECG, is given by exp( + ECG).
4. Given the model logit P(X) = + HPT + PECG + HPT x ECG, where HPT is a (0, 1) exposure variable denoting hypertension status and ECG is a (0, 1) variable for electrocardio- gram status, the null hypothesis for a test of no interaction on a multiplicative scale is given by Ho: exp(x) = 1.
3. Given the model in Question 2, the odds of get- ting the disease for unexposed persons (E = 0) is given by exp(a).
2. Given the model logit P(X) = a + BE, where E denotes a (0, 1) exposure variable, the risk for unexposed persons (E = 0) is expressible as 1/exp(-x).
1. Given the simple analysis model, logit P(X) = + Q, where and are unknown parameters and Q is a (0, 1) exposure variable, the odds ratio for describing the exposure-disease relationship is given by exp().
15. Given E, C1, and C2, and letting V = C = W and V2 C2 W2, then the corresponding logistic model is given by logit P(X) = x + BE +71C1+%2C2 + E(8C + 82C2). 16. For the model in Exercise 15, if C = 20 and C2=5, then the odds ratio for the E, D relation- ship has the form exp(+ 2081 +582).
14. Given the model logit P(X) = x + BE + SMK +72SBP, where E and SMK are (0, 1) variables, and SBP is continuous, then the odds ratio for estimating the effect of SMK on the disease, controlling for E and SBP is given by exp(71).
13. If a logistic model contains interaction terms expressible as products of the form EW; where W; are potential effect modifiers, then the value of the odds ratio for the E, D relationship will be different, depending on the values specified for the W, variables.
12. Given a logistic model of the form logit P(X) = + BE +71AGE +2SBP + 3CHL, where E is a (0, 1) exposure variable, the odds ratio for the effect of E adjusted for the confounding of AGE, CHL, and SBP is given by exp(B).
11. Given an exposure variable E and control vari- ables AGE, SBP, and CHL, suppose it is of inter- est to fit a model that adjusts for the potential confounding effects of all three control vari- ables considered as main effect terms and for the potential interaction effects with E of all three
10. For the model in Exercise 9, the odds ratio that describes the exposure disease effect controlling for smoking is given by exp( + 8).
9. Given the model logit P(X) = x + BE + SMK + SE x SMK, where E is a (0, 1) exposure vari- able and SMK is a (0, 1) variable for smoking status, the null hypothesis for a test of no inter- action on a multiplicative scale is given by Ho: 8 = 0.
8. An equation that describes "no interaction on a multiplicative scale" is given by OR 11 OR 10/OR01-
7. A logistic model that incorporates a multiplica- tive interaction effect involving two (0, 1) inde- pendent variables X and X2 is given by logit P(X)=x+X1 + B2X2 + B3X1X2.
6. Given the model logit P(X) = x + BE, as in Exercise 5, the odds of getting the disease for exposed persons (E = 1) is given by e+
5. Given the model logit P(X) = x + BE, where E denotes a (0, 1) exposure variable, the risk for exposed persons (E = 1) is expressible as e".
4. The log of the estimated coefficient of a (0, 1) exposure variable in a logistic model for simple analysis is equal to ad/bc, wherea, b,c, and d are the cell frequencies in the corresponding fourfold table for simple analysis.
3. The null hypothesis of no exposure-disease effect in a logistic model for a simple analysis is given by Ho: = 1, where is the coefficient of the exposure variable.
2. The odds ratio for the exposure-disease rela- tionship in a logistic model for a simple analysis involving a (0, 1) exposure variable is given by , where is the coefficient of the exposure variable.
1. A logistic model for a simple analysis involving a (0, 1) exposure variable is given by logit P(X)=x+BE, where E denotes the (0, 1) expo- sure variable.
34. Why can you not use the formula exp(bi) formula to obtain an adjusted odds ratio for the effect of AGE, controlling for the other four variables?
33. State two characteristics of the variables being considered in this example that allow you to use the exp(bi) formula for estimating the effect of OCC controlling for AGE, SMK, SEX, and CHOL.
32. Compute and interpret the estimated odds ratio for the effect of OCC controlling for AGE, SMK, SEX, and CHOL. (If you do not have a calculator, just state the computational formula required.)
31. If you could not conclude that the odds ratio computed in Question 29 is approximately a risk ratio, what measure of association is appropriate? Explain briefly.
30. What assumption will allow you to conclude that the estimate obtained in Question 29 is approximately a risk ratio estimate?
29. Compute and interpret the estimated odds ratio for the effect of SMK controlling for AGE, SEX, CHOL, and OCC. (If you do not have a calculator, just state the computational formula required.)
28. Would the risk ratio computation of Question 27 have been appropriate if the study design had been either cross-sectional or case-control? Explain.
27. Compute and interpret the estimated risk ratio that compares the risk of a 40-year-old male smoker to a 40-year-old male nonsmoker, both of whom have CHOL ¼ 200 and OCC ¼ 1.
26. Again assuming a follow-up study, compute the estimated risk for a 40-year-old male nonsmoker with CHOL ¼ 200 and OCC ¼ 1. (You need a calculator to answer this question.)
25. Assuming the study design used was a follow-up design, compute the estimated risk for a 40-year-old male (SEX ¼ 1) smoker (SMK ¼ 1) with CHOL ¼ 200 and OCC ¼ 1. (You need a calculator to answer this question.)
24. State the estimated logistic model in logit form.
23. State the form of the estimated logistic model obtained from fitting the model to the data set.
22. State the form of the logistic model that was fit to these data (i.e., state the model in terms of the unknown population parameters and the independent variables being considered).
21. Which of the following is not a property of the logistic model? (Circle one choice.)a. The model form can be written as P(X)=1/{1 þ exp[(a þ ~biXi)]}, where “exp{}” denotes the quantity e raised to the power of the expression inside the brackets.b. logit P(X) ¼ a þ ~biXi is an
20. Given the independent variables AGE, SMK, and RACE as in Question 18, but with SMK coded as (1, -1) instead of (0, 1), then e to the coefficient of the SMK variable gives the adjusted odds ratio for the effect of SMK.
19. Given independent variables AGE, SMK, and RACE, as before, plus the product terms SMK RACE and SMK x AGE, an adjusted odds ratio for the effect of SMK is obtained by exponentiating the coefficient of the SMK variable.
18. Given independent variables AGE, SMK [smoking status (0, 1)], and RACE (0, 1), in a logistic model, an adjusted odds ratio for the effect of SMK is given by the natural log of the coefficient for the SMK variable.
17. Given a (0, 1) independent variable and a model containing only main effect terms, the odds ratio that describes the effect of that variable controlling for the others in the model is given by e to thea, where x is the constant parameter in the model.
16. The product formula for the odds ratio tells us that the joint contribution of different independent variables to the odds ratio is additive.
15. We can compute an odds ratio for a fitted logistic model by identifying two groups to be compared in terms of the independent variables in the fitted model.
14. The coefficient , in the logistic model can be interpreted as the change in log odds cor- responding to a one unit change in the variable X; that ignores the contribution of other variables.
13. The constant term, x, in the logistic model can be interpreted as a baseline log odds for getting the disease.
12. The logit transformation for the logistic model gives the log odds ratio for the comparison of two groups.
11. Given a fitted logistic model from a case- control study, we can estimate a risk ratio if the rare disease assumption is appropriate.
10. Given a fitted logistic model from a case- control study, an odds ratio can be estimated.
9. Given a fitted logistic model from a follow-up study, it is not possible to estimate individual risk as the constant term cannot be estimated.
8. In follow-up studies, we can use a fitted logistic model to estimate a risk ratio comparing two groups whenever all the independent variables in the model are specified for both groups.
7. Given a fitted logistic model from case-control data, we can estimate the disease risk for a specific individual.
6. The study design framework within which the logistic model is defined is a follow-up study.
5. The logistic model describes the probability of disease development, i.e., risk for the disease, for a given set of independent variables.

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Logistic Regression A Self Learning Text 3rd Edition David G. Kleinbaum, Mitchel Klein - Solutions

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