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INTRODUCTION TO BIOSTATISTICS Page 1 of 11 Introduction to Biostatistics 2020 Assignment 3 [5 questions in total] This assignment is due to be submitted by 4pm Monday 1st June 2020. All assignments must be submitted via MyUni in one single pdf or Microsoft Word document. Please include your student ID in the Header or Footer on each page and number pages in your assignment. This assignment is worth 30% of the total credit for this course. [There is a total of 40 marks in this assignment, which will be rescaled in the final course mark calculations.] Answer all questions. You may use a computer or calculator to assist with summarising data and doing intermediate calculations, but you may lose points if your answer is incorrect and you have not provided evidence of your working. Question 1 [5 marks; 1 mark for each] We will start with some more practise in looking up tables. State the following probability values – either exact values or the range of possible values indicated on the Statistical Tables. For each part, include a diagram to indicate the area of interest in the relevant distribution. [Blank curves are appended at the back of this assignment for your reference.] (a) P(t > 2.31) (for the t distribution on 18 degrees of freedom) (b) P(t > -1.8727) (for t on 5 df) (c) P(t < -1.7247)="" (for="" t="" on="" 20="" df)="" (d)="" p(2=""> 6.68) (arising from a test of association on a 2x2 contingency table) (e) P(2 > 9.50) (arising from a test of association on a 4x3 contingency table) Question 2 [7 marks] This question concerns a study that was introduced in Assignment 2. In a study concerning the effectiveness of Ginkgo biloba in treating tinnitus, 24 participants were recruited through advertisements in the national press in the United Kingdom. Once enrolled in the study, participants were asked to complete a number of questionnaires that allowed for the calculation of a severity of tinnitus score. Participants were then instructed to take three tablets a day (each containing 50mg of Ginkgo biloba) over a 12 week period. After this time participants completed the same questionnaires so that their severity of tinnitus could again be calculated. All participants are assumed to have complied with the treatment regimen. (a) If the mean severity of tinnitus score for patients at entry to the study was found to be 30.07 units (with a standard deviation of 6.10 units) and after the 12 week period was found to be 27.26 units (with a standard deviation of 5.20 units), using a Type 1 error level of 0.05 (i.e. = 0.05), determine if there is a “statistically significant” difference in mean score before and after treatment. State the appropriate null and alternative hypotheses, and show all working. [5 marks] [Note: if needed, the standard deviation of the differences (score at entry to study – score after treatment) was 7.10 units.] (b) State and justify your conclusion clearly so that an individual without your statistical knowledge could understand the results. [2 marks] Remember to show your working and justify your conclusions. Page 2 of 11 Question 3 [8 marks] This question also concerns a study that was introduced in Assignment 2. An article by Holland et al “Does home based medication review keep older people out of hospital?” (British Medical Journal 2005; doi:10.1136) concerned the reporting of evidence obtained through a randomised controlled trial intended to investigate this question. Participants were all patients aged over 80 who had experienced an emergency admission to hospital (for any cause), were prescribed two or more drugs on discharge and were returning to their own home or warden controlled accommodation. Participants were randomised to receive either: Intervention: two home visits by a pharmacist within two weeks and eight weeks of discharge to educate and aid patients with their medications, or Control: standard care. Analysis focused on 415 individuals randomised to the intervention and 414 individuals randomised to control. The primary outcome measure was the number of emergency re-admissions to hospital at 6 months. The average number of re-admissions was 0.56 (SD = 0.87) for participants randomised to the intervention and 0.43 (SD = 0.73) for participants randomised to control. (a) Assuming the number of re-admissions is a continuous variable, test for a difference in (population) average number of re-admissions for participants randomised to the intervention and (population) average number of re-admissions for participants randomised to the control. Use a two-tailed test and assume a Type 1 error level of 0.05 (i.e. = 0.05) was pre-set as acceptable. Show your working. [5 marks] (b) State your conclusion clearly so that an individual without your statistical knowledge could understand the results. [2 marks] (c) Without any calculations, briefly explain how the results of a test of a one-tailed alternative hypothesis might differ from those found in (a). [1 marks] Page 3 of 11 Questions 4 [12 marks] In a study by Radelet and Pierce (1991), the relationship between defendant’s ethnicity and sentencing to the death penalty over a 12 year period was investigated among 674 defendants convicted of murder in Florida. Only defendants and victims who were Caucasian or African-American were considered in the study results described in this question. (a) Of the 483 Caucasian defendants, 53 were sentenced to the death penalty. Of the 191 African- American defendants, 15 were sentenced to the death penalty. State an appropriate null and alternative hypothesis for this study. Draw a suitable 2x2 contingency table to display these data and test at the 0.05 level (i.e. = 0.05) for an association between defendant’s ethnicity and sentencing to the death penalty. Show your working, and present your conclusion with respect to your study hypotheses. [3 marks] Radelet and Pierce also presented information concerning the ethnicity of the murder victim. (b) There was a total of 515 Caucasian murder victims. Among the Caucasian victims, 467 defendants were also Caucasian. In these cases, there was a death penalty sentence for 53 defendants. The court determined that a total of 48 Caucasian victims were murdered by African-American defendants, and 11 of these defendants were sentenced to the death penalty. State an appropriate null and alternative hypothesis for this component of the study concerned with Caucasian victims, clearly stating the population of interest. Create a suitable 2x2 table to display these data and test at the 0.05 level (i.e. = 0.05) for an association between defendant’s ethnicity and sentencing to the death penalty. Show your working, and present your conclusion with respect to your study hypotheses. [3 marks] (c) There was a total of 159 African-American victims. Among the African-American victims, 16 of the convictions were for Caucasian defendants, and no death penalties were sentenced for these defendants. Both defendant and victim were African-American for 143 convictions, and the death penalty was sentenced in 4 of these murders. State an appropriate null and alternative hypothesis for this component of the study concerned with African- American victims, clearly stating the population of interest. Create a suitable 2x2 table to display these data and test at the 0.05 level (i.e. = 0.05) for an association between defendant’s ethnicity and sentencing to the death penalty. Show your working, and present your conclusion with respect to your study hypotheses. [3 marks] (d) Comment on your conclusions across parts (a) – (c). [3 marks] Page 4 of 11 Question 5 [8 marks] Metatarsus adductus (MA) is a foot condition wherein the front part of the foot turns in. It is a common condition in adolescents, and usually corrects itself. Hallux abducto valgus (HAV) is a deformation of the big toe that is not usually severe in adolescents, but if severe it usually requires surgery. The severity of each of these foot conditions is measured as the angle of deformity, where higher angles indicate greater deformity. As part of a study concerned with foot health in adolescents, data were collected from 38 patients who had surgery for HAV. The research question of interest in this study was if severity of MA can help to predict the severity of HAV. Output from an analysis of these data using Microsoft Excel is shown below. Included is a scatterplot, summary output from a simple linear regression, a plot showing the line of best fit, and a plot of the residuals versus MA angles. Page 5 of 11 Page 6 of 11 Using this output (above): (a) State the line of best fit obtained from the regression using the output provided. Briefly comment if the assumptions for a simple linear regression model seem reasonable in this analysis from the information you are presented. [2 marks] (b) Using the computer output for some of the values you need in the calculations, derive a 90% confidence interval for the population slope. Show all of your working. [3 marks] (c) Use the line of best fit obtained in part (a) to predict (i) the HAV for an MA angle of 30 degrees and (ii) the HAV an MA angle of 5 degrees. Comment briefly if you have any concerns about the accuracy of these predictions. [3 marks] [Total 40 Marks] END OF ASSIGNMENT 3 Page