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theory of statistics

Questions and Answers of Theory of Statistics

The following printout provides information for the fitting of two logistic models based on data obtained from a matched case-control study of cervical cancer in 313 women from Sydney, Australia
12. Using the output for model 11, give a formula for the point estimate of the odds ratio far the effect of CAT on CHD, which adjusts for the confounding effects of AGE, CHL, ECG, SMK, and HPT and
11. Carry out the Wald test for the effect of CC on outcome, controlling for the other variables in model 11. In carrying out this test, provide the same information as requested in Exercise 10. Is
10. Now consider model 11: Carry out the likelihood ratio test for the effect of the product term CC on the outcome, controlling for the other variables in the model. Make sure to state the null
9. Using the printout for model I, compute the point estimate and a 95%confidence interval far the odds ratio for the effect of CAT on CHD controlling for the other variables in the model.
8. For model I, test the hypothesis for the effect of CAT on the development of CHD. State the null hypothesis in terms of an odds ratio parameter, give the formula for the test statistic, state the
7. Give a formula for a 95% confidence interval for the odds ratio describing the effect of HT controlling for the other variables in the no interaction model.(Note: In answering all of the above
6. Based on the study description preceding Exercise 1, do you think that the likelihood ratio and Wald test resuIts will be approximately the same? Explain.A prevalence study of predictors of
5. For the same question as described in Exercise 4, that is, concerning the effect of HT controlling for the other variables in the model, describe the Wald test for this effect by providing the
4. Consider a test for the effect of hospital type (HT) adjusted for the other variables in the no interaction model. Describe the likelihood ratio test for this effect by stating the following: the
3. Suppose you want to carry out a (global) test for whether any of the two-way product terms (considered collectively) in your interaction model of Exercise 2 are significant. State the null
2. State the logit form of a model that extends the model of Exercise 1 byadding all possible pairwise products of different variables.A prevalence study of predictors of surgical wo und infection in
1. State the logit form of a no interaction model that includes all of the above predictor variables.A prevalence study of predictors of surgical wo und infection in 265 hospitals throughout
A prevalence study of predictors of surgical wo und infection in 265 hospitals throughout Australia collected data on 12,742 surgical patients (McLaws et al., Med. J. of Australia, 1988). For each
19. The P-values given in the table correspond to Wald test statistics for each variable adjusted for the others in the model. The apprapriate Z statistic is computed by dividing the estimated
17. State two alternative ways to describe the null hypo thesis apprapriate for testing whether the odds ratio described in Question 16 is significant.
The printout given below comes from a matched case-control study of 313 women in Sydney, Australia (Brack et al. , J. Nat. Cancer Inst., 1988), to assess the etiologic role of sexual behaviors and
T F 12. Confidence intervals for odds ratio estimates obtained from the fit of a logistic model use large sampIe formulae that involve variances and possibly covariances from the variance-covariance
T F 11. The variance-covariance matrix printed out for a fitted logistic model gives the variances of each variable in the model and the covariances of each pair of variables in the model.
T F 10. Thc Wald test and thc likelihood ratio test of the same hypothesis give approximately the same results in large sampIes.
T F 9. The likelihood ratio test is a chi-square test that uses the maximized likelihood value i in its computation.
T F 8. The nuisance parameter a is not estimated using an unconditional ML program.
T F 7. The conditional likelihood function reflects the probability of the observed data configuration relative to the probability of all possible configurations of the data.
T F 6. If a likelihood function for a logistic model contains 10 parameters, then the ML solution solves a system of 10 equations in 10 unknowns by using an iterative procedure.
T F 5. In a matched case-control study involving 50 cases and 2-to-l matching, a logistic model used to analyze the data will contain a small number of parameters relative to the total number of
T F 4. Until recently, the most widely available computer packages for fitting the logistic model have used unconditional procedures.
T F 3. In a case-control study involving 1200 subjects, a logistic model involving 1 exposure variable, 3 potential confounders, and 3 potential effect modifiers is to be estimated. Assuming no
T F 2. If discriminant function analysis is used to estimate logistic model parameters, biased estimates can be obtained that result in estimated odds ratios that are too high.
T F 1. Maximum likelihood estimation is preferred to least squares estimation for estimating the parameters of the logistic and other nonlinear models.
T F 4. Conditional ML estimation should be used to estimate logistic model parameters if matching has been carried out in one's study.Unconditional ML estimation gives unbiased results always.The
T F 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.
T F 2. Two alternative maximum likelihood approaches are called unconditional and conditional methods of estimation.
T F 1. When estlmating the parameters of the logistic model, least squares estimation is the preferred method of estimation.
6. Suppose the model in Ouestion 5 is revised to contain interaction terms:logit P(X) = a + ß1NS + ß20C + ß3AFS + YIAGE + Y2RACE+Ölt(NS x AGE)+ Ö12(NS x RACE) +Ö21 (OC x AGE)+822 (OC x RACE) +
5. Given the following model logit P(X) = a + ß1NS + ß20C + ß3AFS + 'YtAGE + 'Y2RACE, where NS denotes nu mb er of sex partners in one's lifetime, OC denotes oral contraceptive use (yes/no), and
4. Suppose the variable SSU in Ouestion 3 is partitioned into three categories denoted as low, medium, and high.a. Revise the model of Ouestion 3 to give the logit form of a logistic model that
3. Given the model logit P(X) = a + ßSSU + 'YIAGE + 'Y2SEX + SSU(o\AGE + 02SEX), where SSU denotes "social support score" and is an ordinal variable ranging from ° to 50, answer the following
2. Suppose the model in Question 1 is revised as follows:logit P(X) = a + ßCAT + 'YIAGE + 'Y2CHL + CAT(olAGE + 02CHL).For this revised model, answer the same questions as given in parts a-e of
1. Given the following logistic model logit P(X) = a + ßCAT + 'YIAGE + 'Y2CHL, where CAT is a dichotomous exposure variable and AGE and CHL are continuous, answer the following questions concerning
11. Given the model logit P(X) = u + ßI(SMK) + ß2(ASB) + 'Yl(AGE)+ 81 (SMK X AGE) + 82(ASB X AGE), where SMK is a (0, 1) variable for smoking status, ASB is a (0, 1) variable for asbestos exposure
10. For the model of Exercise 9, give an expression for the odds ratio far the E, D relationship that compares urban with rural persons, controlling for the confounding and effect modifying cffects
9. Revise your model of Excrcise 7 to allow effect modification of each covariate with the exposure variable. State the logit form of this revised model.
8. Far 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. (Assurne no interaction involving
T F 6. If we assurne no interaction in the above model, the expression exp(ß) gives the odds ratio for describing the effect of one unit change in CHL value, controlling for AGE.Suppose a study is
T F 5. The odds ratio that compares a person with CHL=200 to a person with CHL=140 controlling for AGE is given by exp(60ß).
T F 4. If the correct odds ratio formula for a given coding scheme for E is used, then the estimated odds ratio will be the same regardless of the coding scheme used.Given the model logit P(X) = a+
T F 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(ß).
T F 2. If Eis coded as (-1, 1), then the odds ratio for the E, D relationship that controls for SMK and HPT is given by exp[2ß + 2°1 (SMK) + 202(HPT)].
T F 1. If Eis coded as (O=unexposed, 1=exposed), then the odds ratio for the E, D relationship that controls for SMK and HPT is given by exp[ß + °1 (E X SMK) + 02(E X HPT)].
15. Using the model in Question 13, give an expression for the same odds ratio for the effect of CON, controlling for the confounding effects of NP, ASCM, and PAR and for the interaction effect of
14. Using the model in Question 13, give an expression for the risk of an exposed person (CON = 1) who is in the first matched pair and whose value for PAR is 1.
13. Within the matched pairs case-control framework, state the logit form of a logistic model for assessing the effect of CON on HIV acquisition, controlling for PAR, NP, and ASCM as potential
12. Using the model in Question 11, give an expression for the odds ratio that compares an exposed person (CON = 1) with an unexposed person(CON = 0) who has the same values for PAR, NP, and
11. 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
T F 10. Given the model in Ouestion 9, the odds ratio for the exposure-disease relations hip that controls for matching and for the confounding and interactive effects of OBS and PAR is given by
T F 9. For the matched pairs study above, a logistic model assessing the effect of a (0, 1) exposure E, controlling for the confounding effects of the matched variables and the unmatched variables
T F 8. For the model in Ouestion 7, if Cl = 5 and C2 = 20, then the odds ratio for the E, D relationship has the form exp(ß + 200).Given a 4-to-1 case-control study with 100 subjects (i.e., 20
T F 7. Given E, Cl' and C2, and letting VI = Cl = Wl , V2 = (C1)2, and V3 = C2, then the corresponding logistic model is given by logit P(X) = a + ß E + 'YlCI + 'Y2C12 + 'Y3C2 + 8ECl .
T F 6. Given the modellogit P(X) = a + ßE + HPT +t/J ECG, where E, HPT, and ECG are (0, 1) variables, then the odds ratio for es timating the effect of ECG on the disease, controlling for E and HPT,
T F 5. For the model in Ouestion 4, the odds ratio that describes the effect of HPT on disease status, controlling for ECG, is given by exp( t/J +rr ECG).
T F 4. Given the modellogit P(X) = + tjJHPT + pECG + 1THPTXECG, where HPT is a (0, 1) exposure variable denoting hypertension status and ECG is a (0, 1) variable for electrocardiogram status, the
T F 3. Given the model in Ouestion 2, the odds of getting the disease for unexposed persons (E = 0) is given by exp(
T F 2. Given the modellogit P(X) = IX + ßE, where E denotes a (0, 1)exposure variable, the risk for unexposed persons (E = 0) is expressible as lIexp( - IX).
T F 1. Given the simple analysis model, logit P(X) = +tjJQ, where and tjJ are unknown parameters and 0 is a (0, 1) exposure variable, the odds ratio for describing the exposure-disease relationship
T F 20. Again, given the model in Exercise 18, the null hypothesis for a test of no interaction on a multiplicative scale can be stated as Ho: ß = O.True or False (Circle T or F)
T F 19. Given the model in Exercise 18, the odds ratio for the exposure-disease relationship that controls for matching and for the confounding and interactive effect of PAL is given by exp(ß +
T F 18. For the matched pairs study above, a logistic model assessing the effect of a (0, 1) exposure E and controlling for the confounding effects of the matched variables and the unmatched variable
T F 17. For the matched pairs study described above, assuming no pooling of matched pairs into larger strata, a logistic model for a matched analysis that contains an intercept term requires 49 dummy
Given a matched pairs case-control study with 100 subjects (50 pairs), suppose that in addition to the variables involved in the matching, the variable physical activity level (PAL) was measured but
T F 16. For the model in Exercise 15, if Cl = 20 and C2 = 5, then the odds ratio for the E, D relationship has the form exp(ß + 20ö\ + 5ö2).
T F 15. Given E, Cl' and C2, and letting VI = Cl = WI and V2 = C2 = W2, then the corresponding logistic model is given by logit P(X) = a + ßE + ')'1 Cl + ')'2 C2 + E(ölCl + ö2C2)·
T F 14. Given the modellogit P(X) = a + ßE + ')'\ SMK + ')'2 SBP, 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,
T F 13. If a logistic model contains interaction terms expressible as products of the form EWj where Wj are potential effect modifiers, then the value of the odds ratio for the E, D relationship will
T F 12. Given a logistic model ofthe form logit P(X) = a + ßE + ')'\ AGE + ')'2 SBP + ')'3 CHL, where Eis a (0, 1) exposure variable, the odds ratio for the effect of E adjusted for the confounding
T F 11. Given an exposure variable E and contral variables AGE, SBP, and CHL, suppose it is of interest to fit a model that adjusts for thc potential confounding effects of all three control
T F 10. For the model in Exercise 14, the odds ratio that describes the exposure disease effect controlling for smoking is given by exp(ß + 8).
T F 9. Given the model logit P(X) =:: a + ßE + ')'SMK + öE X SMK, where E is a (0, 1) exposure variable and SMK is a (0, 1) variable for smoking status, the null hypothesis for a test of no
T F 8. An equation that describes "no interaction on a multiplicative scale" is given by ORll = ORIO I OROl .
T F 7. A logistic model that incorporates a multiplicative interaction effect involving two (0, 1) independent variables X\ and X2 is given by logit P(X) = a + ß1X1 + ß2X2 + ß3X\X2.
T F 6. Given the modellogit P(X) == a + ßE, as in Exercise 5, the odds of getting the disease for exposed persons CE == 1) is given by ea+ß.
T F 5. Given the model logit P(X) :::: a + ßE, where E denotes a (0, 1)exposure variable, the risk for exposed persons (E == 1) is expressible as eß.
T F 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, where a,b,c, and d are the cell frequencies in the corresponding
T F 3. The null hypothesis of no exposure-disease effect in a logistic model for a simple analysis is given by Ho: ß == 1, where ß is the coefficient of the exposure variable.
T F 2. The odds ratio for the exposure-disease relations hip in a logistic model for a simple analysis involving a (0, 1) exposure variable is given by ß, where ß is the coefficient of the exposure
T F 1. A logistic model for a simple analysis involving a (0, 1) exposure variable is given by logit P(X) :::: a + ßE, where E denotes the (0, 1) exposure variable.
34. Why can you not use the fonnula exp(ßi ) 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(ßi) formula for estimating the effect of OCC controlling for AGE, SMK, SEX, and CHOL.
31. lf you could not conclude that the odds ratio computed in Ouestion 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 Ouestion 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 OCe. (lf you do not have a calculator, just state the computational formula required.)
28. Would the risk ratio computation of Ouestion 27 have been appropriate if the study design had been either cross-sectional or case-contro!?Explain.
25. Assuming the study design used was a follow-up design, compute the estimated risk for a 40-year-old male (SEX=l) smoker (SMK=l) with CHOL=200 and OCC=I. (You need a calculator to answer this
22. State the fonn of the logistic model that was fit to these data (Le., state the model in tenns of the unknown population parameters and the independent variables being considered).
Suppose a logistic model involving the variables D=HPT[hypertension status(0, 1)], X}=AGE(continuous), X2=SMK(0, 1), X3=SEX(0, 1), X4=CHOL (cholesterollevel, continuous), and Xs=OCC[occupation (0,
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)=lI(l +exp[ -(a+kßiXi)]), where "exp(.}" denotes the quantity e raised
T F 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
T F 19. Given independent variables AGE, SMK, and RACE, as before, plus the product terms SMK X RACE and SMK x AGE, an adjusted odds ratio for the effect of SMK is obtained by exponentiating the
T F 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
T F 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

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