1 The Hang Seng University of Hong Kong Department of Mathematics, Statistics and Insurance AMS3328 Categorical Data Analysis Midterm Test Question 1. Suppose that all customers of an Internet service provider either enrolled in Plan A service or Plan B service. They are allowed to switch the plan. The sampled customers are asked the plans they have used in Year 2020 and Year 2021. The data is as follows. Year 2021 Branch Year 2020 Plan A Plan B Shatin Plan A Plan B 35 30 75 80 Taipo Plan A Plan B 25 20 30 35 Tsuen Wan Plan A Plan B 35 18 6 3 (a) Find the marginal odds-ratio (involving the plans used in Year 2020 and Year 2021, but not controlled for branch). [5marks] [Ans] The marginal odds-ratio is 95×118 68×111 = 1.49 (or the reciprocal 1/1.49=0.67). (b) For the partial table of Shatin, test the independence between the plans used in Year 2020 and Year 2021. State clearly the test method, null hypothesis, and alternative hypothesis that you are using. [5 marks] [Ans] Chi-squared test can be used. Set the null hypothesis H0: Plan used in 2020 and Plan used in 2021 are independent and alternative H1: not H0. The test statistic is (35 65×110 220 ) 2 65×110 220 + (75 155×110 220 ) 2 115×110 220 + (30 65×110 220 ) 2 65×110 220 + (80 155×110 220 ) 2 155×110 220 = 0.55. It is less than 1,5% 2 = 3.841. Therefore, the null hypothesis is not rejected. (c) Explain the meanings
of “conditional independence” and “homogenous association”. [5 marks] [Ans] Conditional independence means that the odds-ratios of all three towns are one. Homogenous association means that the odds-ratios of all three towns are the same. 2 (d) For the partial table of Tsuen Wan, construct the 95% confidence interval for the odds-ratio. [5 marks] [Ans] The odds-ratio is = 35×3 18×6 = 0.97 . The confidence interval is therefore exp {log ± 1.96√ 1 35 + 1 18 + 1 6 + 1 3 } = (0.22,4.34) Question 2. [10 marks] Suppose that there are three schools in a university, namely science, humanity, and engineering. The university would like to understand the career status of students graduated for one year. The record shows (i) the school that the student belongs to, (ii) whether the student participated in summer internship, and (iii) whether the student get a job within one year. Suggest a model for the prediction of the probability that a student get a job within one year. State clearly the meaning of each symbol that you have introduced. [Ans] Let 1 = 1 if the student belongs to science school and 0 otherwise, 2 = 1 if the student belongs to humanity school and 0 otherwise, 3 = 1 if the student participated in summer internship and 0 otherwise, = 1 if the student got a job within one year and 0 otherwise. Model: ( = 1) = 1 + , = 0 + 11 + 22 + 33. Here, 0, 1, 2, 3 are some unknown parameters.