ώδ<> ¦MS WRITTEN EXAMINATION IN BIOSTATISTICSΘ ¦PART IIΘ ¦March 4, 1995: 8:30 AM to 12:30 PMΘ ‚INSTRUCTIONSƒ: ώμ™a)™ This is an open bookƒ examination. ™b)™ Answer any threeƒ questions. ™c)™ Put the answers to different questions on separate sets of papers. ™d)™ Put your CODE LETTER (not your name) on each page. ™e)™ Return the examination with a signed statement of the honor pledge on a page ™™ separate from your answers. ™f)™ You are required to answer only what is askedƒ in the question, not all you know ƒ ™™ about the topic. ώθμQuestion 1. The data shown below are the numbers of rats dead and alive after an experiment to assess the survival at 96 hours of animals given a bacterial challenge subsequent to removal of a specific fraction of the spleen. ™™ ™™™™™#™(™-Bacterial Dose ώμ›K Fraction of spleen™™ ™™™1.2Ϋ-210«Ϋ47Θ™™#™( 1.2Ϋ-210«Ϋ57Θ™-™2™7™<1.2Ϋ-210«Ϋ67Θ removed™™ ™™ Dead Alive™™™# Dead™- Alive™-™2™7 Dead Alive ›K Sham (none)™™ ™™™1™ 4™#™( 0™- 5™2™7™<0™A 5 One-fourth™™ ™™™0™ 5™#™( 2™- 3™2™7™<4™A 2 One-half™™ ™™™1™ 4™#™( 5™- 1™2™7™<6™A 0 Three-fourths™™ ™™™2™ 4™#™( 5™- 0™2™7™<5™A 0 Entire Spleen™™ ™™™5™ 1™#™( 4™- 1™2™7™<5™A 0 ώμ›K a)™For each bacterial dose, display the estimated proportion of dead rats for the group with one-half ώξthe spleen removed. ώξ b)™Apply a statistical test to compare the 1.2Ϋ-210«Ϋ47Θ dose and the 1.2Ϋ-2Ϋ18Ϋ08«Ϋ57Θ dose for rats with one-half ώξthe spleen removed. Justify the method and interpret results. ώξc)™For rats with one-half of the spleen removed, provide a 0.90 two-sided confidence interval for the ώξodds ratio which compares the dead versus alive distributions for the 1.2Ϋ-2Ϋ18Ϋ08«Ϋ47Θ dose and the 1.2Ϋ-2Ϋ18Ϋ08«Ϋ57Θ dose. ώξd)™Apply a two-sided statistical test which powerfully compares the three bacterial doses for rats ώξwith one-fourth of the spleen removed. Justify the method and interpret results. ώξe)™Under minimal assumptions, apply a statistical test to compare the 1.2Ϋ-2Ϋ18Ϋ08«Ϋ47Θ dose and the ώξ1.2Ϋ-2Ϋ18Ϋ08«Ϋ57Θ dose for all rats from the combined groups according to fraction of spleen removed. Justify the method and interpret the results. ώξf)™The study additionally included rats at bacterial doses of 1.2Ϋ-2Ϋ18Ϋ08«Ϋ37Θ, 1.2Ϋ-2Ϋ18Ϋ08«Ϋ77Θ, and 1.2Ϋ-2Ϋ18Ϋ08«Ϋ87Θ for ώξeach of the five groups according to fraction of spleen removed. Thus it had 30 experimental groups in all. A logistic regression model, M, was used to describe the variation in the probability of death across the groups. In addition to an intercept, Model M included four parameters for the comparison of the group with no spleen removed to the groups with one-fourth, one-half, three-fourths, and the entire spleen removed, respectively; also a slope parameter for log¬Ϋe7Θ{bacterial dose}. State the relevant assumptions for Model M, specify its mathematical structure, and interpret its parameters. ώξg)™The estimated parameters for Model M are as follows: ™‚Parameter™ ™™™™™#™(Estimate™-™2™7™<™AStd. errorƒ ώμ™Intercept™ ™™™™™#™( Ϋ,210.9™-™2™7™<™A 1.9 ™One-fourth vs. sham™ ™™™™™#™( 1.9™-™2™7™<™A 0.8 ™One-half vs. sham™ ™™™™™#™( 3.5™-™2™7™<™A 0.9 ™Three-fourths vs. sham™ ™™™™™#™( 4.2™-™2™7™<™A 1.0 ™Entire spleen vs. sham™ ™™™™™#™( 6.1™-™2™7™<™A 1.2 ™Log¬Ϋe7Θ{bacterial dose} slope™ ™™™™™#™( 0.68™-™2™7™<™A 0.11 ώμξProvide a two-sided 0.95 confidence interval for the extent to which the odds of dead versus alive are greater for rats with one-half of the spleen removed relative to those with none removed. Also, provide a two sided 0.95 confidence interval for a quantity which expresses the extent to which multiplication of the bacterial dose by 10 increases the odds of dead versus alive for the rats in this study. ώξh)™From (g), one can see that the estimated parameters for one-fourth versus sham, one-half versus ώξsham, three-fourths versus sham and entire spleen versus sham tend to increase in a nearly linear way. Describe how to test the hypothesis of such a linear relationship for fraction of spleen removed through the fitting of a simplified model which incorporates it. Specify the structure of this simplified model and the degrees of freedom for the chi-squared statistic to test it. Given that the result for this chi-squared statistic is 1.44, what is the interpretation of the statistical test? ώξi)™Describe how to evaluate the goodness of fit of Model M by fitting an expanded model in which ώξthere are different slopes for log¬Ϋe7Θ{bacterial dose} for each group according to fraction of spleen removed. Specify the structure of this expanded model and the degrees of freedom for the chi-squared statistic for testing its simplification to Model M. Given that the result for this chi-squared statistic is 6.83, what is the interpretation concerning goodness of fit of Model M? ώξScoring:™™ a) 2; b) 3; c) 2; d) through i) 3 each. Question 2. H. L. Self reported the back-fat thickness, Y, and slaughter weight, X, for Poland China pigs fed different rations. The data for two treatments, A and B, are: ώμ›(K™™ ™Treatment A™™™™™#™(™-™2Treatment B ›K™ Y (mm) ›$™ ™™™ ›$ ™#™ ™™ ™™ X (lb)™™™#™( Y (mm)™2 ›$™7™< X (lb) ›K™ 42™ ™™ 206™™™#™( 33™2™7™< 167 ›K™ 38™ ™™ 261™™™#™( 34™2™7™< 192 ›K™ 53™ ™™ 279™™™#™( 38™2™7™< 204 ›K™ 34™ ™™ 221™™™#™( 33™2™7™< 197 ›K™ 35™ ™™ 216™™™#™( 26™2™7™< 181 ›K™ 31™ ™™ 198™™™#™( 28™2™7™< 178 ›K™ 45™ ™™ 277™™™#™( 37™2™7™< 236 ›K™ 43™ ™™ 250™™™#™( 31™2™7™< 204™™ ώμa)™Fit a simple linear regression to the Treatment A data and another to the Treatment B data, ώξusing Y as the dependent variable and X as the independent variable in each case. Present appropriate statistics for these regressions. ώξb)™Test the null hypothesis that the slope from Treatment A is equal to the slope from Treatment B ώξversus the alternative that the slopes are not equal. ώξ Scoring:™™ a) 16, b) 9 . ώθQuestion 3. Below is shown part of the output produced by SAS PROC UNIVARIATE given data on a certain continuous variable whose values have been rounded DOWN to integers. Using the output provided wherever convenient, find the P-value for testing the hypothesis that the median is 117. Use (a) a sign test; (b) a signed-ranks test; (c) a t-test. (d) Which seems the most suitable test for these data? ώμ™™ ™™™MOMENTS™™#™(™-™2™7QUANTILES (DEF=4) N™™ ™™100™SUM WGTS™™#™( ™-100™- ™2100% MAX™< 131™A99%™F 130.99 MEAN™™ ™ 117.21™™SUM™™#™( 11721™-™2 75% Q3™7 ™< 120™A95%™F 123 STD DEV™™ ™ 4.38154™™VARIANCE™™#™( 19.1979™-™2 50% MED™7™< 117™A90%™F 122.9 SKEWNESS™™ ™ 0.493901™™KURTOSIS™™#™( 0.35883™-™2 25% Q1™7™< 114™A10%™F 112 USS™™ ™ 1375719™™CSS™™#™( 1900.59™-™2 0% MIN™7™< 108™A 5%™F 111 CV™™ ™ 3.7382™™STD MEAN™™#™( 0.438154™-™2 ™7™<™A 1%™F 108.02 T: MEAN=0™™ ™ 267.509™™PROB>|T|™™#™( 0.0001™-™2 RANGE™7™< 23™A SGN RANK™™ ™ 2525™™PROB>|S|™™#™( 0.0001™-™2 Q3-Q1™7™< 6 NUM« ^Θ= 0™™ ™ 100™™™™#™(™-™2 MODE™7™< 114 D: NORMAL™™ ™ 0.0830058™™PROB>D™™#™( 0.089™™-™2 FREQUENCY TABLE ώοLP™™ ™PERCENTS™™™™#™( PERCENTS™-™2™7™<™A™FPERCENTS VALUE COUNT CELL CUM VALUE COUNT CELL CUM VALUE COUNT CELL CUM 108 1 1.0 1.0 115 8 8.0 39.0 121 7 7.0 83.0 110™ 1 1.0 2.0 116 7 7.0 46.0 122 7 7.0 90.0 111 7 7.0 9.0 117 8 8.0 54.0 123 7 7.0 97.0 112 7 7.0 16.0 118 7 7.0 61.0 128 1 1.0 98.0 113 6 6.0 22.0 119 7 7.0 68.0 130 1 1.0 99.0 114 9 9.0 31.0 120 8 8.0 76.0 131 1 1.0 100.0  Scoring: a) 7; b) 9; c) 5; d) 4. ώθQuestion 4. ™Two physicians ask for help in designing a clinical trial. Some patients with cystic fibrosis (CF) receive lung transplants. Although only 15-40 years old, CF patients have very low bone mineral density (BMD), similar to that of people 60-70 years old. Such low BMD creates substantial risks of broken bones and other significant complications, including death. Successful transplants currently require massive doses of steroids. Unfortunately, steroids have the side effect of reducing BMD, often substantially. Hence the physicians propose to study the efficacy of three regimens for limiting the reduction in BMD associated with transplantation: 1) standard aggressive care, 2) standard care plus drug A, and 3) standard care plus drug B. The response variable will be BMD measured at a particular location in the spine. Treatment will begin immediately following the transplant, with the response measured six months later. Experience with BMD in other studies indicates that its residual variability among patients is likely to follow a Gaussian distribution. Assume 15 men and 15 women will be recruited for the entire study. (In order to simplify the question, the role of estrogen use has been ignored here, although the actual researchers will not have that luxury.) In the following, General Linear Univariate Model (GLUM) refers to a form like ¦yΆNΫ-2Ϋ18ΘΫV2 XΫb;Ά(NΫ-2q)(qΫ-2Ϋ18Ϋ)8ΘΫ+2 eΆNΫ-2Ϋ18Θ .Θ The first study design suggested by the physicians corresponds to a two-way factorial ANOVA, with Treatment as the first factor and Gender as the second. a) ™Using a reference-cell coding scheme for the first study design, ™with females in treatment 1 as the reference cell, ™1) ™ Clearly specify the dimensions and elements of X. ™2)™™ Clearly specify the dimensions and describe the elements of Ϋb;. ™3) ™™ Specify the contrast matrix, C , needed to test the General Linear Hypothesis (GLH) for the ™ ™™ interaction of Gender and Treatment. b) ™Using a cell-mean coding for the first study design, ™1) ™™ Clearly specify the dimensions and elements of X. ™2) ™™ Clearly specify the dimensions and describe the elements of Ϋb; . ™3) ™™ Specify the contrast matrix, C, needed to test the GLH for the main effect of Treatment. Further discussion reveals that BMD will be measured just before the transplant, providing a baseline measure. In addition, the reliability of the measurement is likely to be extremely high, due to the nature of the measurement and the quality of a recently acquired testing machine. So a second design considered involves using the baseline measure as a covariate. c) ™Consider a design traditionally known as an Analysis of Covariance (ANCOVA). Using a reference cell ™ ™type coding for an ANCOVA, ™1) ™ Clearly specify the dimensions and elements of X. ™2) ™™ Clearly specify the dimensions and describe the elements Ϋb;. ™3) ™™ Specify the contrast matrix, C, needed to test the GLH for the interaction of Gender and ™™ Treatment. d) ™A more general model allows for different slopes for each Treatment by Gender combination. Extend ™the cell-mean model considered in b) to allow unequal slopes for each Treatment by Gender ™ ™combination. ™1) ™™ Clearly specify the dimensions and elements of X. ™2) ™™ Clearly specify the dimensions and describe the elements of Ϋb;. Scoring: a.1) 3, 2) 3, 3) 1; b.1) 3, 2) 3, 3) 1; c.1) 3, 2) 3, 3) 1; d.1) 2, 2) 2 .