[28] Question 5 Consider the following data which report the method of public transport most frequently...

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[28] Question 5 Consider the following data which report the method of public transport most frequently used (either train or tram) classified according to age group. Age groups are indexed by i. The number of people surveyed in age group i is denoted by n. The number of people in age group i, out of the n, people surveyed, who most often use the tram is denoted by y₁. Age Group i <25 (i=1) 25-29 (i=2) 30-39 (i=3) 40-49 (i=4) Totals [5] (a) [4] (b) [3] (c) Use tram most often yi 72 105 237 93 507 Use train most often n₁-yi 325 299 375 101 1100 We define to be the probability that a person in age group i uses the tram most often. We also define the four age groups as i=1,2,3,4 for <25, 25-29, 30-39 and 40-49 respectively (as in the table above). Finally we define n to be the number of age groups. For these data n = 4. Suppose you fit a binomial model to these data Iti - π, of the form logit (7₁) = In =B₂. Σ (1")+ y.,P. i=1 By first writing the likelihood function as a product of n=4 binomial probabilities, show that the log-likelihood function is In +y₁B₁-n₂ (B₁+ +In (1+ 1+e™ + e²^))} Number of people surveyed n, 397 404 612 194 1607 Using the log-likelihood function from part (a), derive the maximum likelihood estimate of Bo. Use the data from the table to give a numerical answer. Make sure you also use the second-order derivative test to confirm that you have indeed located a local maximum of the log-likelihood function. Suppose instead you decide to fit a model of the form logit (7)= In T₁ 1-T₁ | = B₁ + B₁X₁ + B₂X₂ + B3X39 where X₁ = 1 for age group 25 - 29 and 0 otherwise X₂ = 1 for age group 30-39 and 0 otherwise X3 = 1 for age group 40 - 49 and 0 otherwise. (i) Explain why this model is the saturated model for these data. [2] (d) (ii) Find the estimates of the parameters B, B₁, B₂ and 3. Do NOT attempt to derive any likelihood functions or score equations for this part. Use the fact that this is the saturated model to work out your answer. Based on the results from the modelling in part (c), you decide to estimate a model =B₁ + Bx, where x, denotes a representative of the form logit (7₁) = In = 6 age for people in age group i. We will use x₁ = 20, x₂ = 27.5, x3 = 35 and x₁4 = 45. Explain why the result from part (c) suggests trying this model. TT₁ 1-π The model from part (d) is estimated using R. Some relevant R code and output is given below. > transport=read.csv('transport.csv',header=T) > transport prop age tram total 1 20.0 72 2 27.5 105 3 35.0 237 612 0.3872549 4 45.0 93 194 0.4793814 397 0.1813602 404 0.2599010 > attach (transport) > mod1=g1m (prop-age, family-binomial (link=logit), weights=total) > mod1 Coefficients: (Intercept) -2.67267 Call: glm (formula = prop~ age, family = binomial (link = logi t), weights = total) age 0.06067 Degrees of Freedom: 3 Total (i.e. Null); 2 Residual 79.19 Nu11 Deviance: Residual Deviance: 2.403 AIC: 31.05 [28] Question 5 Consider the following data which report the method of public transport most frequently used (either train or tram) classified according to age group. Age groups are indexed by i. The number of people surveyed in age group i is denoted by n. The number of people in age group i, out of the n, people surveyed, who most often use the tram is denoted by y₁. Age Group i <25 (i=1) 25-29 (i=2) 30-39 (i=3) 40-49 (i=4) Totals [5] (a) [4] (b) [3] (c) Use tram most often yi 72 105 237 93 507 Use train most often n₁-yi 325 299 375 101 1100 We define to be the probability that a person in age group i uses the tram most often. We also define the four age groups as i=1,2,3,4 for <25, 25-29, 30-39 and 40-49 respectively (as in the table above). Finally we define n to be the number of age groups. For these data n = 4. Suppose you fit a binomial model to these data Iti - π, of the form logit (7₁) = In =B₂. Σ (1")+ y.,P. i=1 By first writing the likelihood function as a product of n=4 binomial probabilities, show that the log-likelihood function is In +y₁B₁-n₂ (B₁+ +In (1+ 1+e™ + e²^))} Number of people surveyed n, 397 404 612 194 1607 Using the log-likelihood function from part (a), derive the maximum likelihood estimate of Bo. Use the data from the table to give a numerical answer. Make sure you also use the second-order derivative test to confirm that you have indeed located a local maximum of the log-likelihood function. Suppose instead you decide to fit a model of the form logit (7)= In T₁ 1-T₁ | = B₁ + B₁X₁ + B₂X₂ + B3X39 where X₁ = 1 for age group 25 - 29 and 0 otherwise X₂ = 1 for age group 30-39 and 0 otherwise X3 = 1 for age group 40 - 49 and 0 otherwise. (i) Explain why this model is the saturated model for these data. [2] (d) (ii) Find the estimates of the parameters B, B₁, B₂ and 3. Do NOT attempt to derive any likelihood functions or score equations for this part. Use the fact that this is the saturated model to work out your answer. Based on the results from the modelling in part (c), you decide to estimate a model =B₁ + Bx, where x, denotes a representative of the form logit (7₁) = In = 6 age for people in age group i. We will use x₁ = 20, x₂ = 27.5, x3 = 35 and x₁4 = 45. Explain why the result from part (c) suggests trying this model. TT₁ 1-π The model from part (d) is estimated using R. Some relevant R code and output is given below. > transport=read.csv('transport.csv',header=T) > transport prop age tram total 1 20.0 72 2 27.5 105 3 35.0 237 612 0.3872549 4 45.0 93 194 0.4793814 397 0.1813602 404 0.2599010 > attach (transport) > mod1=g1m (prop-age, family-binomial (link=logit), weights=total) > mod1 Coefficients: (Intercept) -2.67267 Call: glm (formula = prop~ age, family = binomial (link = logi t), weights = total) age 0.06067 Degrees of Freedom: 3 Total (i.e. Null); 2 Residual 79.19 Nu11 Deviance: Residual Deviance: 2.403 AIC: 31.05

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