BIOSTATS Intermediate Biostatistics Spring 2017 Exam 2 (Units 3, 4 & 5) Practice Problems SOLUTIONS

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1 BIOSTATS Intermediate Biostatistics Spring 2017 Exam 2 (Units 3, 4 & 5) Practice Problems SOLUTIONS Practice Question 1 Both the Binomial and Poisson distributions have been used to model the quantal nature of synaptic transmission. Crudely, the quantal hypothesis says that a nerve terminal contains a very large number of quanta, each with a small probability of releasing acetylcholine (ACh) in response to a nerve stimulus. Suppose it is known that, for a given stimulus, the probability of Ach release is 0.01 for each quantum and is the same for all quanta. You may assume the quantal responses are independent. (a) Using the Binomial distribution, what is the probability that in a nerve terminal containing 200 quanta zero Ach is released in response to stimulus? (b) Using the Poisson distribution, what is the probability that in a nerve terminal containing 200 quanta zero Ach is released in response to stimulus? SOLUTION (a) Binomial answer: Solution: # trials = 200 π = Pr[ X = 0 ] = 0 R Solution ( )(0.01) ( ) 200 = Stata Solution (note Must have installed probcalc) 2017\Exam II\BIOSTATS 640 spring 2017 Exam 2 Study Guide SOLUTIONS.docx Page 1 of 6

2 (b) Poisson answer: Solution: Poisson parameter µ = (T) (λ) = (200) (.01) = 2 0 (-µ) 0 (-2) (µ) e 2 e Pr( X=0 ) = = = ! 0! R Solution Stata Solution (note Must have installed probcalc) 2017\Exam II\BIOSTATS 640 spring 2017 Exam 2 Study Guide SOLUTIONS.docx Page 2 of 6

3 Practice Question 2 A logistic regression analysis was used to explore the relationship between the diabetes (presence or absence) and body mass index (BMI). The Y-variable for this analysis was Y=Diabetes and was coded Y=1 for persons with diabetes and Y=0 for persons without diabetes. The X-variable for this analysis was X=BMI where BMI is measured as kg/m 2. The following fitted model was obtained: With the following values of ln-likelihood: ˆπ ln = X 1 - πˆ Ln-Likihood (intercept only) = Ln-Likelihood (intercept + BMI) = (a) Using the information given in the fitted model, together with your understanding of logistic regression, complete the following table by filling in the five blanks in the 2 nd row. Coefficient Standard Error Wald Statistic p-value OR 95% CI for OR Intercept Not asked Not asked Not asked - - = BMI (1.01, 1.15) (b) Using the information given in the fitted model, calculate the value of the estimated odds ratio for the outcome of diabetes in relationship to a 5 kg/m 2 increase in BMI. OR = exp [ (5) beta for BMI ] = exp [ (5) (.075) ] = \Exam II\BIOSTATS 640 spring 2017 Exam 2 Study Guide SOLUTIONS.docx Page 3 of 6

4 Practice Questions 3 and 4 Consider again the same logistic regression analysis setting of practice question 2. Further analysis of diabetes explored two additional predictors: treatment with digoxin (X 2 ) and non-white race (X 3 ). The following is a full coding manual. Variable Label Codings Outcome Y Diabetes 1 = yes, 0 = no Predictors X 1 BMI Continuous kg/m 2 X 2 Digoxin 1 = yes, 0 = no X 3 Race 1 = non-white, 0=other The fitted logit model is now the following. ˆπ ln = X X X 1 - πˆ The following ln-likelhood values are provided for you: Ln-Likihood (intercept only model) = Ln-Likelihood (intercept + X 1 model) = Ln-Likelihood (intercept + X 1 + X 2 + X 3 model) = Practice Question 3 Using the fitted logit model, calculate the estimated probability of diabetes for a person with BMI of 24 kg/m 2, on digoxin treatment, and being of non-white race. Answer: 0.29 ˆp = = = exp ˆb 0 +ˆb 1 X 1 +ˆb 2 X 2 +ˆb 3 X 3 1+exp ˆb 0 +ˆb 1 X 1 +ˆb 2 X 2 +ˆb 3 X 3 exp[ (.081)(24)-(.796)(1)+(.904)(1)] 1+exp (.081)(24)-(.796)(1)+(.904)(1) exp exp = \Exam II\BIOSTATS 640 spring 2017 Exam 2 Study Guide SOLUTIONS.docx Page 4 of 6

5 Practice Question 4 Carry out the appropriate likelihood ratio test to compare the reduced model containing X 1 = BMI with a full model containing all three predictors X 1 = BMI, X 2 = Digoxin and X 3 = Race Likelihood Ratio Chi Square Test Statistic Value: 9.87 Degrees of freedom: 2 P-value: = Pr [ Chi Square DF=2 > 9.87] = Interpretation: Reject the null. There is statistically significant evidence of an association of events of diabetes with increasing BMI. LR Test = ΔDeviance R Solution - [(-2)lnL FULL ] = (-2)lnL REDUCED = [(-2)*( )]- (-2)*( ) = = 9.87 Stata Solution 2017\Exam II\BIOSTATS 640 spring 2017 Exam 2 Study Guide SOLUTIONS.docx Page 5 of 6

6 Practice Question 5 A logistic regression analysis of likelihood (π) of mortality considered several variables: shock (SHOCK: coded 1=shock, 0=NO shock), malnutrition (MALNUT; coded 1=malnourished, 0 = NOT malnourished), alcoholism (ALC: coded 1=alcoholic 0=NOT alcoholic), age (AGE: continuous), and bowel infarction (INFARCT: coded 1=infarction, 0=NO infarction). The following fitted model was obtained: ˆ ˆ logit[π] = [SHOCK] [MALNUT] [ALC] [AGE] [INFARCT] What is the estimated probability of death for a 60-year old malnourished patient with no evidence of shock, but with symptoms of alcoholism and prior bowel infarction? In developing your answer write out the formula you use and provide the numeric estimate. Answer:.9587 or 96%, approx Solution: logit[π] ˆ ˆ = [SHOCK] [MALNUT] [ALC] [AGE] [INFARCT] = [1] [1] + = [60] [1] e ˆπ= = e 2017\Exam II\BIOSTATS 640 spring 2017 Exam 2 Study Guide SOLUTIONS.docx Page 6 of 6