In a study of game system preferences, a number of users were asked to rank three...

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In a study of game system preferences, a number of users were asked to rank three choices: PC (1), PlayStation (2), and Xbox (3). Each user was also asked to report their age, number of hours they play per week, and which systems they own. Suppose now that the probability of a user selecting the j-th system is j, j = 1, ..., K = 3. The probability of the ranking r = (r1,,K) is then modeled as a sequence of successive choices, K P(r) = II rj ; For example, the ranking 3 > 1> 2, that is, r = 3, r2 = 1, and r3 = 2, is P(3 > 1 > 2) = 2 =31 1 +2 2 1 +2 The first-choice probabilities are in turn represented as in a multinomial model with the predictors described above, with an intercept for each choice (having PC, the first level, as reference) and interactions with hours and age, but not for own. (a) Show that with the canonical link for the multinomial distribution the probabil- ity of a ranking can be expressed as P(r) = K II; exp{x,B} exp{x} j=1 = where x; has the regressors for choice j. (b) Here is the summary from fitting the model described above: Estimate Std. Error z-value Pr(>|z|) (Intercept): PlayStation 1.962800 (Intercept): Xbox own 1.765005 1.1121 0.266110 2.547272 1.671729 1.5237 0.127575 0.929688 0.292422 3.1793 0.001476 hours: PlayStation hours: Xbox age: PlayStation age: Xbox Deviance: 304.57 -0.094577 0.046603 -2.0294 0.042416 -0.142441 0.048780 -2.9201 0.003500 -0.059558 0.086579 -0.6879 0.491510 -0.064226 0.080645 -0.7964 0.425796 Interpret the coefficient for own. In particular, explain why the probability of selecting any choice as top rank is the same if the user owns all systems or none of them. (c) Here is a simpler model fit after removing age as a covariate. Estimate Std. Error z-value Pr(>z) 0.324045 2.3796 0.0173337 0.356881 3.5033 0.0004595 (Intercept): PlayStation 0.771083 (Intercept): Xbox own hours: PlayStation hours: Xbox 1.250263 0.928585 -0.096381 0.291919 3.1810 0.0014678 -0.143285 0.045745 -2.1069 0.0351234 0.048152 -2.9757 0.0029236 Deviance: 305.30 Conduct a test to check if this simpler model is adequate. State the test statistic and its distribution under the null. (d) A new user is considering buying their first gaming system. What is their most likely ranking of the systems? What is the estimated probability of this ranking? (e) Fitting these rank-ordered models can be done as in a regular multinomial re- gression, but requires expanding the data to a new format. How would you design a specialized Newton method (or iteratively reweighted least squares) to avoid this extra amount of work and memory? In particular, how would the score calculation change? In a study of game system preferences, a number of users were asked to rank three choices: PC (1), PlayStation (2), and Xbox (3). Each user was also asked to report their age, number of hours they play per week, and which systems they own. Suppose now that the probability of a user selecting the j-th system is j, j = 1, ..., K = 3. The probability of the ranking r = (r1,,K) is then modeled as a sequence of successive choices, K P(r) = II rj ; For example, the ranking 3 > 1> 2, that is, r = 3, r2 = 1, and r3 = 2, is P(3 > 1 > 2) = 2 =31 1 +2 2 1 +2 The first-choice probabilities are in turn represented as in a multinomial model with the predictors described above, with an intercept for each choice (having PC, the first level, as reference) and interactions with hours and age, but not for own. (a) Show that with the canonical link for the multinomial distribution the probabil- ity of a ranking can be expressed as P(r) = K II; exp{x,B} exp{x} j=1 = where x; has the regressors for choice j. (b) Here is the summary from fitting the model described above: Estimate Std. Error z-value Pr(>|z|) (Intercept): PlayStation 1.962800 (Intercept): Xbox own 1.765005 1.1121 0.266110 2.547272 1.671729 1.5237 0.127575 0.929688 0.292422 3.1793 0.001476 hours: PlayStation hours: Xbox age: PlayStation age: Xbox Deviance: 304.57 -0.094577 0.046603 -2.0294 0.042416 -0.142441 0.048780 -2.9201 0.003500 -0.059558 0.086579 -0.6879 0.491510 -0.064226 0.080645 -0.7964 0.425796 Interpret the coefficient for own. In particular, explain why the probability of selecting any choice as top rank is the same if the user owns all systems or none of them. (c) Here is a simpler model fit after removing age as a covariate. Estimate Std. Error z-value Pr(>z) 0.324045 2.3796 0.0173337 0.356881 3.5033 0.0004595 (Intercept): PlayStation 0.771083 (Intercept): Xbox own hours: PlayStation hours: Xbox 1.250263 0.928585 -0.096381 0.291919 3.1810 0.0014678 -0.143285 0.045745 -2.1069 0.0351234 0.048152 -2.9757 0.0029236 Deviance: 305.30 Conduct a test to check if this simpler model is adequate. State the test statistic and its distribution under the null. (d) A new user is considering buying their first gaming system. What is their most likely ranking of the systems? What is the estimated probability of this ranking? (e) Fitting these rank-ordered models can be done as in a regular multinomial re- gression, but requires expanding the data to a new format. How would you design a specialized Newton method (or iteratively reweighted least squares) to avoid this extra amount of work and memory? In particular, how would the score calculation change?