BME380J.5 Exam 2 Spring 2017

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University of Texas *

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380J

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Mechanical Engineering

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Dec 6, 2023

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Student# _______ BME 380J.5 Exam 2 Spring 2017 I understand that the exam proctors will watch for violations of the exam rules and that offending students will be reported to the Dean of Students. I understand that Scholastic Dishonesty can result in an automatic F in the course. I understand the exam rules. I know that … • I have 75 minutes to complete this exam • I may not communicate in any way with anyone other than exam proctors during the exam time • I may not consult textbooks, notes, or other reference material during the exam other that that which is provided by the exam proctor • I may not use any electronic devices during the exam including but not limited to laptops, tablets, calculators, cell phones, or smart watches • My answers must be legibly written and clearly demonstrate my thought process in order to earn full credit on the exam. I know that mathematical notation has been consistently used in this course and that I must follow that same convention on the exam. • I should not write my name or any other identifying information on any portion of the exam other than this face page. Writing my name or any other identifying information on any portion of the exam other than this face page will be interpreted as an attempt to gain any unfair advantage and may result in automatic failure of the exam. Print Name ____________________________________ Sign Name ____________________________________ Date _________________________________________

Student# _______ Problem #1 _______ / 10 points Problem #2 _______ / 10 points Problem #3 _______ / 10 points Problem #4 _______ / 10 points Problem #5 _______ / 10 points Problem #6 _______ / 10 points Problem #7 _______ / 10 points Problem #8 _______ / 10 points Problem #9 _______ / 10 points Problem #10 _______ / 10 points TOTAL _______ / 100 points

Student# _______ Problem # Problem Total = ___ / 10 1 Problem #1. [10 points] Recall from the textbook by Rosner an example in which the Wilcoxon Rank Sum test was used to compare the visual acuity of people with different forms (dominant, sex-linked) of an eye disease (retinitis pigmentosa). (a) The key idea of non-parametric methods such as the Wilcoxon Rank Sum test is to base the analysis on the ranks of the observations rather than their actual values. Complete the table below to demonstrate your understanding of the ranking procedure. Visual acuity Dominant Sex-linked Combined sample Range of ranks Average rank 20-20 5 1 A1= 20-25 9 5 A2= 20-30 6 4 10 21-30 A3=(21+30)/2 20-40 3 4 7 31-37 A4=(31+37)/2 20-50 2 8 10 38-47 A5=(38+47)/2 20-60 0 5 5 48-52 A6=(48+52)/2 20-70 0 2 2 53-54 A7=(53+54)/2 20-80 0 1 1 55 A8=55 (b) Set up the computation of the rank sum for the dominant form of the disease. (c) Explain what is meant by a “tie” in the context of the Wilcoxon Rank Sum test.

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Student# _______ Problem # Problem Total = ___ / 10 2 Problem #2. [10 points] Suppose that we wish to assess if the probability of having an invasive breast cancer is significantly different for women who have used hormone replacement therapy (HRT) compared to that of women who have not used hormone replacement therapy. A key step in such a contingency table analysis is computation of the expected values. (a) Set up the computation of the expected table corresponding to the observed data table provided below. Observed Pathology HRT In situ Invasive Total No 27 16 43 Yes 31 11 42 Total 58 27 85 Expected Pathology HRT In situ Invasive Total No 43 Yes 42 Total 58 27 85 (b) Recall that the test statistic for a chi-square test for this type of analysis is Χ ! = ! !" ! ! !" ! ! . ! ! ! !" ! ! ! ! ! ! ! ! Explain the purpose of the “-0.5” term in the test statistic.

Student# _______ Problem # Problem Total = ___ / 10 3 Problem #3. [10 points] Suppose that two medications, A and B, are compared for treatment of a disease. Each medication is assessed as effective (1) or ineffective (0). Participants are recruited as pairs matched carefully for relevant factors such as disease stage. Recall that the p-value for McNemar’s exact test is given by: Set up the calculation of the p-value for McNemar’s exact test applied to the data table provided. Define the “event” of interest as ineffective treatment. Pair# Treatment A Treatment B 1 0 1 2 0 0 3 0 0 4 1 0 5 0 1 6 1 0 7 1 0 8 0 1 9 1 1 10 1 1 11 1 0 12 0 1 13 1 1 14 0 1 15 0 1 16 0 0 17 1 1 18 0 1 19 0 1 20 1 1 p = 2* n D k " # $ % & ' 1 2 " # $ % & ' n D k = 0 n A ∑ if n A < n D 2 p = 2* n D k " # $ % & ' 1 2 " # $ % & ' n D k = n A n D ∑ if n A > n D 2 p = 1 if n A = n D 2

Student# _______ Problem # Problem Total = ___ / 10 4 Problem #4. [10 points] Recall from the textbook by Rosner an example on the association between conduct of a bacterial culture and administration of antibiotics. The smallest expected value is less than 5 such that Fisher’s exact test must be used to analyze the data. Enumerate all possible tables with the same row and column margins as observed in the data set provided. Participant # Antibiotics? (1 = Yes) Culture? (1 = Yes) 1 0 0 2 0 1 3 0 0 4 0 0 5 0 0 6 1 0 7 1 1 8 0 0 9 0 0 10 0 1 11 0 1 12 0 0 13 1 0 14 1 1 15 0 1 16 0 0 17 1 0 18 0 0 19 1 0 20 0 0 21 0 0 22 1 0 23 0 0 24 0 0 25 0 0

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Student# _______ Problem # Problem Total = ___ / 10 5 Problem #5. [10 points] Annotate the drawing to explain the quantities (a) ResSS = y i − ˆ y i ( ) 2 i = 1 n ∑ (b) RegSS = ˆ y i − y ( ) 2 i = 1 n ∑ and (c) R 2 = RegSS TotalSS . (a) (b) (c)

Student# _______ Problem # Problem Total = ___ / 10 6 Problem #6. [10 points] As part of an ultrasound assessment of a breast lesion, a radiologist categorizes the breast composition as homogeneous –fat (A); homogeneous – fibroglandular (B); or heterogeneous (C). Suppose that the same radiologist repeats their assessment of the recorded images after a month delay in order to evaluate the reproducibility of the assessment. Recall that ࠵? = ! ! ! ! ! ! ! ! ! where ࠵? ! is computed from the marginal probabilities as ࠵? ! = ࠵? ! ࠵? ! . In answering the following questions, simply set up the calculation, e.g., your answers can be fractions such as ! ! ! !! ! !" ! !" . (a) What is the marginal probability of the radiologist categorizing a breast ultrasound image as homogeneous –fat (A) in the first assessment? (b) What is the contribution to ࠵? ! for the category homogeneous – fibroglandular (B)? (c) What is the value of ࠵? ! ? Image# Assessment #1 Assessment #2 1 B B 2 C C 3 A B 4 A A 5 B B 6 B C 7 C B 8 B B 9 B A 10 A A 11 C C 12 C B 13 B B 14 A C 15 C C

Student# _______ Problem # Problem Total = ___ / 10 7 Problem #7. [10 points] In each blank, write the number from the flowchart that indicates the best location for the question/statement. ____ Ordinal data? ____ Both variables normal? ____ Interested in the relationship between two variables? ____ Association ____ Interested in studying the correlation between two variables ____ Both variables categorical ____ Both variables continuous? ____ Interested in predicting one variable from another? ____ Reproducibility ____ One variable continuous and one variable categorical?

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Student# _______ Problem # Problem Total = ___ / 10 8 Problem #8. [10 points] Recall from the textbook by Rosner an example on the relationship between five-year lung cancer mortality and annual cigarette consumption (each expressed on log10 scale). (a) Recall that ࠵? = ! !" ! !! ! !! . Use the data provided to explain the calculation of L xy . (b) Suppose that you conduct a one-sample t-test for the correlation coefficient. Recall that the test statistic is t = r n − 2 1 − r 2 , which follows t n − 2 when the null hypothesis is true. For this data set, what is the numerical value of n ? (c) Use a clearly labeled diagram of the distribution of the test statistic under the null hypothesis to explain the p value calculation for the one-sample t-test for the correlation coefficient. For the data provided, ࠵? = 6 . 20 . (d) Suppose that after conducting the one-sample t-test for the correlation coefficient, you decide to fit a regression line to predict mortality to cigarette consumption using these data. Are additional tests needed to assess the significance of the regression line? Explain. Time Period Mortality Cigarette Consumption 1 -2.35 -0.26 2 -2.20 -0.03 3 -2.12 0.30 4 -1.95 0.37 5 -1.85 0.40 6 -1.80 0.50 7 -1.70 0.55 8 -1.58 0.55

Student# _______ Problem # Problem Total = ___ / 10 9 Problem #9. [10 points] In each blank write the letter corresponding to the best answer. Each letter may be used 0, 1, or more than 1 time. ____ Non-parametric analog of the t test on the means for two independent samples A. Chi-square test B. Fisher’s exact test C. Kappa statistic D. McNemar’s test E. Paired t test F. Pearson correlation coefficient G. Simple linear regression H. Spearman rank correlation coefficient I. Wilcoxon Signed Rank test J. Wilcoxon Rank Sum test ____ The calculation of this test is similar to that of the Sign test because both are special cases of the general one-sample binomial test. ____ Whereas a chi-square test for binomial proportions is appropriate for independent samples, this test is used when the samples are dependent. ____ If a chi-square test for binomial proportions is planned, but one or more of the expected values is less than 5, then this test is performed instead. ____ Whereas a chi-square test is used to assess the association between two categorical variables, this method is used to assess agreement or reproducibility of categorical variables. ___ If the scores of the groups are defined as the average ranks, then the chi-square test for trend is equivalent to this test. ____ Non-parametric analog of the paired t-test on the mean difference ____ Whereas the Pearson correlation coefficient assesses the association between two normally distributed random variables, this method is used to predict the value of a continuous variable from another random variable. ____ Whereas the Pearson correlation coefficient assesses the association between two normally distributed random variables, this test is used to assess the association between two categorical variables. ____ Whereas the Pearson correlation coefficient assesses the association between two normally distributed random variables, this test is used to assess the association between two ordinal variables.

Student# _______ Problem # Problem Total = ___ / 10 10 Problem #10. [10 points] Recall that in class that we used a chi-square test to address the question: is the probability that a breast lesion is malignant significantly different for women who have a prior history of cancer compared to those who don’t have a prior history of cancer? Suppose that you are going to consult with a statistician to plan the sample size needed for such an analysis. What information will you need to discuss with the statistician? In other words, explain what ࠵? ! , ࠵? ! , ࠵? , ࠵? , and ࠵? mean in this specific context.

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