STA3701_2023_TL_014_0_E

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STA3701/014/0/2023 Tutorial Letter 014/0/2023 Applied Statistics III STA3701 Year module Department of Statistics ASSIGNMENT 4 QUESTIONS

STA3701/014/0 1 Daniel, W. W., & Cross, C. L. (2013). Biostatistics: a foundation for analysis in the health sciences. Tenth edition. Hoboken, NJ, Wiley. 2 ASSIGNMENT 04 Unique Nr.: 859766 Due date: 18 September 2023 Instructions 1. Do not PLAGIARISE. Students suspected of plagiarism will be subjected to disciplinary processes. 2. Use R to answer all the questions. Present or attach R outputs. Label all the figures and tables. Question 1 [10] 1.1 Define analysis of variance (ANOVA). (1) 1.2 How is ANOVA different from regression analysis? (2) 1.3 Explain the difference between one-factor (one-way) ANOVA and two-factor (two-way) ANOVA. (2) 1.4 The objective of a study was to determine whether there are a different responses to different calcium channel blockers 1 . Two hundred and fifty patients with mild-to-moderate hypertension were randomly assigned to 4 weeks of treatment with once-daily doses of: (i) lercanidipine; (ii) felodipine; or (iii) nifedipine. Prior to treatment and at the end of 4 weeks, each of the subjects had his or her systolic blood pressure measured. Researchers then calculated the change in systolic blood pressure. 1.4.1 What is the treatment variable in this study? (1) 1.4.2 What is the response variable in this study? (1) 1.4.3 Mention or state one extraneous variable that you think its effects would be included in the error term. (1) 1.4.4 What are the degrees of freedom for the F critical value in this study? (2)

STA3701/014/0 1 Daniel, W. W., & Cross, C. L. (2013). Biostatistics: a foundation for analysis in the health sciences. Tenth edition. Hoboken, NJ, Wiley. 3 Question 2 [40] Fifty-five pigs were randomly assigned to five different treatments and the weight gain was recorded 1 . The weight gains of the pigs are given in the table below. Treatment 1 Treatment 2 Treatment 3 Treatment 4 Treatment 5 165 168 164 185 201 156 180 156 195 189 159 180 156 195 189 159 180 189 184 173 167 166 138 201 193 170 170 153 165 164 146 161 190 175 160 130 171 160 187 200 151 169 172 177 142 164 179 142 166 184 158 191 155 165 149 Table 1 :Table 1: Weight gain in pigs 2.1 Create a boxplot of the data. (4) 2.2 Does the boxplot suggest any differences among the means? Comment on the homogeneity of variances. (2) 2.3 The model for this data is 𝑦𝑦 𝑖𝑖𝑖𝑖 = 𝜇𝜇 + 𝛼𝛼 𝑖𝑖 + 𝜖𝜖 𝑖𝑖𝑖𝑖 . Define all the terms/notations in the model. (2) 2.4 What are the assumptions underlying the model in part 2.3? (2) 2.5 Use the Anova to test the hypothesis of equal treatment means ( 𝐻𝐻 0 ∶ 𝜇𝜇 1 = 𝜇𝜇 2 = 𝜇𝜇 3 = 𝜇𝜇 4 = 𝜇𝜇 5 ) at five percent level of significance. (5) 2.6 Use Tukey’s honestly significant differences (HSD) test to test all possible pairwise mean differences at 5% level of significance. (15) 2.7 Perform residual analyses to investigate whether any model assumptions were violated. (10)

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STA3701/014/0 1 Daniel, W. W., & Cross, C. L. (2013). Biostatistics: a foundation for analysis in the health sciences. Tenth edition. Hoboken, NJ, Wiley. 4 Question 3 [50] Researchers at a trauma center wished to develop a program to help brain-damaged trauma victims regain an acceptable level of independence 1 . An experiment involving 72 subjects with the same degree of brain damage was conducted. The objective was to compare different combinations of psychiatric treatment and physical therapy. Each subject was randomly assigned to one of 24 different combinations of four types of psychiatric treatment and six physical therapy programs. There were three subjects in each combination. The response variable is the number of months elapsing between initiation of therapy and time at which the patient was able to function independently. Use in 𝑅𝑅 to analyze the data below and answer the questions that follow. Physical Therapy Program Psychiatric treatment A B C D 1 11.0 9.6 10.8 9.4 9.6 9.6 12.5 11.5 10.5 13.2 13.2 13.5 2 10.5 11.5 12.0 10.8 10.5 10.5 10.5 11.8 11.5 15.0 14.6 14.0 3 12.0 11.5 11.8 11.5 11.5 12.3 11.8 11.8 12.3 12.8 13.7 13.1 4 11.5 11.8 10.5 9.4 9.1 10.8 13.7 13.5 12.5 14.0 15.0 14.0 5 11.0 11.2 10.0 11.2 11.8 10.2 14.4 14.2 13.5 13.0 14.2 13.7 6 11.2 10.8 11.8 10.8 11.5 10.2 11.5 10.2 11.5 11.8 12.8 12.0 Table 2:the number of months elapsing between initiation of therapy and time at which the patient was able to function independently. 3.1 Use interaction plots to investigate whether there are significant psychiatric treatment type and physical therapy program. (6)

STA3701/014/0 1 Daniel, W. W., & Cross, C. L. (2013). Biostatistics: a foundation for analysis in the health sciences. Tenth edition. Hoboken, NJ, Wiley. 5 3.2 Give the appropriate model and interaction effects assumptions. (Define the terms/notations used in the model using the given information). (8) 3.3 Conduct a two-way analysis of variance (ANOVA) and present the results in an ANOVA table. (8) 3.4 Using a 5% level of significance, test the hypotheses, which follow. 3.4.1 Can you conclude based on these data that the different psychiatric treatment programs have different effects? In other words, does the mean number of months elapsing between initiation of therapy and time at which the patient was able to function independently differ among different types of psychiatric treatments? ( 𝐻𝐻 0 : 𝛼𝛼 𝐴𝐴 = 𝛼𝛼 𝐵𝐵 = 𝛼𝛼 𝐶𝐶 = 𝛼𝛼 𝐷𝐷 )? (6) 3.4.2 Can you conclude that the physical therapy programs differ in effectiveness? In other words, does the type of physical therapy program affect the mean number of months elapsing between initiation of therapy and time at which the patient was able to function independently? ( 𝐻𝐻 0 : 𝛽𝛽 1 = 𝛽𝛽 2 = 𝛽𝛽 3 = 𝛽𝛽 4 = 𝛽𝛽 5 = 𝛽𝛽 6 )? (6) 3.4.3 Can you conclude that there are significant interaction effects between psychiatric treatment type and physical therapy program ( 𝐻𝐻 0 : ( 𝛼𝛼𝛽𝛽 ) 𝑖𝑖𝑖𝑖 = 0 )? Does your conclusion corroborate your finding in part 3.1? (6) NB: Provide the null and alternative hypotheses, critical regions (or rejection regions), test statistics and conclusions. 3.5 Perform a residual analysis of your model and report your observations. (10) Grand total = [100]

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