¦SPECIAL MS WRITTEN EXAMINATION IN BIOSTATISTICSΘ ¦PART IIΘ ¦May 13, 1994: 9:00Ϋ,21:00 pm Θ ‚INSTRUCTIONSƒ: ™a)™This is an open bookƒ examination. ™b)™Answer any threeƒ questions during the four-hour time period. ™c)™Put the answers to different questions on separate sheets of paper. ™d)™Put your CODE LETTER, (notƒ your name) on each page. ώμ™e)™Return the examination with a signed statement of the honor pledge on a page separate from ώμξyour answers. ώμξ™f)™You are required to answer only what is askedƒ in the questions, not to tell all you knowƒ about ώμξthe topics. ώθξ1.™Consider the relationships among the variables sex (S), height (H), and weight (W) for American ώξadults, where S is coded so that Female = 0 and Male = 1. Suppose that on the basis of common sense" we conclude that ™™ (1)™S might be a cause of H and/or W, but not an effect of either; ™™ (2)™W might be an effect of S and/or H, but not a cause of either. In addition, suppose it is known as an empirical fact that ™™ (3) ™S, H, and W all have strongly positive (total) correlations with each other. ώξ [9]™(a)™ What causal models that do not introduce any fourth variable are consistent with (1), ώξ (2), and (3) above? (Distinguish between common effect models with related or unrelated causes.) Name the models, and diagram them. Include +", Ϋ,2", and 0", as appropriate, for all total and partial correlations. ώξ [4]™(b)™ On the basis of common knowledge about the distribution of S, H, and W among ώξ American adults, guesstimate" the partial correlation between H and W given S. If your guess is correct, which model(s) of (a) would be eliminated? ώξ [8]™(c)™ Tables of ideal" weights have traditionally shown the two sexes separately, with greater ώξ weights for males. However, recently published tables show ideal weights for each height without regard to sex. In a population whose weights agreed with the traditional tables, which model(s) of (a) would be eliminated? If weights agreed with the newer tables? ώξ [4]™(d)™ If S is recoded so that Male = 1 and Female = 2, what changes must be made in the ώξ statements above, and in the answers? ώθξ 2.™To study the effect of career counseling method on career development score, 40 subjects were ώξrandomized, 10 to each of four career counseling methods (A, B, C, D, where A is the control). Assume no baseline differences in career development scores. ™™ The following ANOVA table was derived from the analyses: ™™ Source™™df™™(MS™™SS ™™ ™™™ ?™(591.26™™ Regression™™ 3™ ™™ Error™™ ?™4895.30™( ?™2™ ™™ Total™™ ?™6759.08™( ώξ[7]™(a)™ Using reference cell coding and assuming effects are fixed, give a detailed description (using ώξ appropriate mathematical and statistical notation) of the regression model to be used. List all appropriate model assumptions. Assume that career counseling method A is the referenced cell. ώξ [3]™(b)™ What are the values of the cells labeled ?" in the above ANOVA table? [7]™(c)™ Using the notation of (a), carefully state the null hypothesis of no treatment effects (i.e., ώξ no effects due to career counseling methods). Perform a test of this null hypothesis. Be careful to state and/or compute the significance level, the test statistic, distributional assumptions, and critical value. ώξ [5]™(d)™ After further investigation, one of the researchers requested a simultaneous comparison of ώξ each pair of career counseling methods using the model stated in (a). The level of significance for testing this group of comparisons was 0.05. The results were as follows: ‚Comparison™™ ™™™P-value ƒControl (A) vs B™™ ™™™0.509 Control vs C™™ ™™™0.309 Control vs D™™ ™™™0.003 B vs C™™ ™™™0.683 B vs D™™ ™™™0.014 C vs D™™ ™™™0.049 The researcher concluded that Control vs D, B vs D, and C vs D were all statistically significant. Carefully explain to the researcher why B vs D and C vs D may not be statistically significant. ώξ [3]™(e)™ The mean and standard deviation of career development score for each career counseling ώξ method follows: ‚Method™™ ™™™Mean™(™2Standard Deviation ƒ A™™ ™™™114.67™(™2™711.61 B™™ ™™™110.90™(™2™712.60 C™™ ™™™108.50™(™2™711.39 D™™ ™™™ 96.60™(™2™713.21 Using these statistics, provide estimates of the regression coefficients of the model given in (a). ώθξ 3.™The accompanying text and tables are from a draft of a paper examining the relationship of ώξeating disorders (such as anorexia, bulimia, and obesity) to history of abuse in adolescents. The author asked a faculty member of this Department to review the paper before submission for publication. ώξ[10]™(a)™ Comment on the presentation of Table 5 and its accompanying text. Suggest ώξ improvements, but also give praise good points. Assume that the variables have been explained adequately earlier in the paper, that the choice of analysis is reasonable, and that the calculations are accurate. ώξ [15]™(b)™ Comment similarly on both the analysis and the presentation of Table 6 and its ώξ accompanying text, and especially its title. The X2" in the first sentence of the text is to be read chi-squared"; the 2754" is the total number of males in the study for whom information on incest and weight was not missing (and others similarly: the n's vary because of missing data). Again assume the calculations are accurate. ώξ TEXT TO ACCOMPANY TABLE 5 Adolescents with eating disorders, both male and female, reported lower self-esteem, more stress, more anxiety, more hopelessness, and more suicidal thoughts (see Table 5). For males and for females, presence of an eating disorder was associated with more cigarette use, alcohol consumption, and hard drug use. Prevalence of an eating disorder was also correlated with another addictive behavior, exercising for weight control, and with more sexual activity. In addition, males and females with eating disorders were more likely to have family histories of alcoholism and drug addiction. All the above variables are also significantly related to histories of physical abuse, extrafamilial sexual abuse, and incest. TEXT TO ACCOMPANY TABLE 6 More males experiencing incest were above normal weight than their counterparts (X2(2754) =10.06, p < .0001) (see Table 6); the same was true for females (X2(2359) =5.02, p < .03). Also, more males experiencing extrafamilial abuse were above normal weight than their counterparts (X2(2754 =5.10, p < .02); the same was true for females (X2(2356) =3.80, p < 5). Obesity alone was related to incest among females (X2(2154) =11.87, p < .001) but not among males. Males who were extremely underweight were more likely to have reported extrafamilial sexual abuse (X2(2492) =8.82, p < .003) or physical abuse (X2(2495) =8.31, p < .004). Females who were extremely underweight were more likely to be physically abused (X2(2510) =4.55, p < .03). Table 5 -- Student's t-test means and standard deviations on psychological, behavioral and family variables for adolescents who did and did not experience eating disorders, physical abuse, incest and extrafamilial sex abuse (EFA) ™™ ™™ Eating disorders™™™#™(Physical abuse™-™2™7 Incest™<™A EFA ™™ ™™n =™ 6224™™#™( 6141™-™2™7 6116™<™A 6117 ώοLP™™ ™™ no™™ yes™#™( no™- yes™2™7 no™< yes™A no™F yes ώοPP™™ ™™n =  5822™™ 402™#  5335™- 806™2 5857™7™< 259™A 5614™F 503 ™™ ™™ ώοPRstressΫ*8Ϋ*8™™ ™™ 14.1™™ 11.2™#™(14.2™- 11.6™2 13.9™7™< 11.7™A 14.0™F 11.6™™7™< ώοRR™™ ™™sd (3.3) (3.5)™™™#™((3.2)™- (3.6) (3.3) (3.6) (3.3) (3.5) anxiety**™™ ™™ 3.4 2.8 3.4 2.9 3.4 2.9 3.4 2.8 ™™ ™™sd ( .9)™™ (1.0) ( .9) (1.0) ( .9) (1.0) ( .9) (1.0) hopelessness**™™ ™™ 3.9 2.9 3.9 3.1 3.9 3.2 3.9 3.2 ™™ ™™sd (1.1) (1.3) (1.1) (1.3) (1.2) (1.2) (1.1) (1.3) self esteem™™ ™™ 12.3 10.3 12.4 10.4 12.2 10.7 13.3 10.6 ™™ ™™sd (3.6) (3.5) (3.5) (3.6) (3.6) (3.5) (3.6) (3.6) suicide thoughts ™™ ™™™1.3 1.7 1.3 1.6 1.3 1.6 1.3 1.6 ™™ ™™sd ( .5) ( .7) ( .5) ( .7) ( .5) ( .7) ( .5) ( .7) cigarette use™™ ™™™2.2 3.5 2.2 3.0 2.3 3.0 2.2 3.0 ™™ ™™sd (1.8) (2.1) (1.7) (2.1) (1.8) (2.0) (1.8) (2.1) alcohol use™™ ™™™2.9 3.8 2.9 3.3 3.0 3.3 2.9 3.4 ™™ ™™sd (1.5) (1.4) (1.5) (1.5) (1.6) (1.6) (1.6) (1.5) hard drug use™™ ™™™6.5 8.1 6.5 7.3 6.6 7.9 6.5 7.7 ™™ ™™sd (1.8) (4.3) (1.8) (3.3) (1.9) (4.2) (1.9) (3.8) sexual activity™™ ™™™1.6 1.8 1.6 1.7 1.6 1.7 1.6 1.8 ™™ ™™sd ( .5) ( .4) ( .5) ( .4) ( .5) ( .4) ( .5) ( .4) exercises™™ ™™™1.4 1.6 1.4 1.5 1.4 1.5 1.4 1.5 ™™ ™™sd ( .5) ( .5) ( .5) ( .5) ( .5) ( .5) ( .5) ( .5) family alcoholism™™ ™™™1.2 1.4 1.2 1.4 1.2 1.5 1.2 1.4 ™™ ™™sd ( .4) ( .5) ( .4) ( .5) ( .4) ( .5) ( .4) ( .5) family drug use™™ ™™™1.1 1.2 1.1 1.2 1.1 1.4 1.1 1.3 ™™ ™™sd ( .3) ( .4) ( .3) ( .4) ( .3) ( .5) ( .3) ( .4) * -- all differences are significant at p < .01 ** -- reverse scoring Table 6 -- Male and female adolescents with weight problems who have experienced physical abuse, incest, extrafamilial abuse, and eating disorders ™™ ™™™Physical abuse™™#™(™- Incest™2™7™<™A EFA ™™ ™™ no™ yes™#™( no™-™2 yes™7™< no™A™F yes ™ M F M F M F M F M F M F™#™# 1. Underweight™™ ™™ 10% 14% 15%«Ϋ*8Θ 17%«Ϋ*8Θ 10% 14% 13% 16% 10% 14% 17%«Ϋ*8Θ 16% 2. Normal™™ ™™ 72% 77% 66% 72%«Ϋ*8Θ 72% 77% 48%«Ϋ*8Θ 70% 72% 77% 58%«Ϋ*8Θ 72% 3. Overweight™™ ™™ 15% 7% 15% 8% 15% 7% 34% 9% 15% 7% 19% 9% 4. Obese™™ ™™ 3% 2% 4% 3% 3% 2% 5% 5%«Ϋ*8Θ 3% 2% 6% 3% ±Ϋ*8Á-- significant at p < .05 Θ ώθ4.™The contingency table shown below provides frequencies for relief of nasal congestion from allergic ώξrhinitis; (yes or no) for patients in a clinical trial to compare four treatments (i.e., placebo, Drug A, Drug B, and Drug AB). Here Drug AB is the combination of both Drug A and Drug B in a single dose. ™™ ™™™™Relief™#™(™-™2Sample Treatment™™ ™™™ No™™#Yes™(™-™2Size AB™™ ™™™ 1™™# 58™(™-™2 60 B™™ ™™™ 6™™# 54™(™-™2 60 ™ A™ ™™™ 7 ™™# 53™(™-™2 60 Placebo™™ ™™™ 15™™# 45™(™-™2 60 ώξ[2]™(a)™ Provide estimates for the probabilities of relief for Drug A and for placebo and their standard ώξ errors. ώξ [2]™(b)™™ Provide a two-sided 0.90 confidence interval for the difference between the probabilities of relief for ώξ Drug A and for placebo. Interpret this result. ώξ [3]™(c)™™ Apply a statistical test to compare the probabilities of relief for Drug B and Drug AB. Interpret ώξ this result at the two-sided Ϋa; = 0.10 significance level. (Note that B is present in B and AB and ™™ that B is absent from placebo and A). ώξ [3]™(d)™™ For all patients, apply a statistical test under minimal assumptions to evaluate the association of ώξ relief with the presence or absence of component B in a treatment. Interpret this result at the two-ώξ sided Ϋa; = 0.10 significance level. (Note that B is present in B and AB and that B is absent from placebo and A). ώξ [3]™(e)™™ Apply a statistical test to evaluate whether the association between yes" or no" for relief with ώξ the presence or absence of component B is or is not affected by the presence or absence of ώξ component A. Interpret this result at the two-sided Ϋa; = 0.10 significance level. ώθξ [3]™(f)™™ A logistic regression model was fit to describe the relationship of probability of relief for to presence ώξ or absence or absence of component B (1 if present, 0 if absent) and age in years. The ώξ estimated parameters for that model and their standard errors were as follows: ™™ ™™™™™#™(Ϋb;΅Ϋ^8Θ™-™2s.e. (Ϋb;΅Ϋ^8Θ) ™™ ™™Intercept™™™# 1.50™(™-™21.20 ™™ ™™Component B present™™™# 2.20™(™-™2™(™-™20.80 ™™ ™™Component A present™™™# 1.80™(™-™20.70 State the relevant assumptions for this model. Specify the mathematical structure of the model. Indicate the interpretation of the model parameters Ϋb;. ώξ [3]™(g)™™ On the basis of the estimated parameters in (f), provide a two-sided 0.95 confidence interval for the ώξ odds ratio corresponding to the association of yes" for relief with the presence of component B. ώξ [3]™(h)™™ Describe how the goodness of fit of the model might be evaluated. What additional information ώξ would be needed for this purpose. ώξ [3]™(i)™™ How large a sample size is necessary for a two-sided test at the 0.05 significance level to have at ώξ least 0.80 power to detect the difference between treatments for which the rates of relief are 75% and 90%? Comment on how this consideration for sample size influences the interpretation of the analyses of relief for the clinical trial in this exercise.