HW5Solution_Fall23

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Homework 5 Solutions, Fall 2023 Exercise 1 (a) Frequency tables comparing city fuel efficiency over 30 mpg & the 3 possible categorical predictors follow. Table of cylinders by cityover30mpg cylinders cityover30mpg Frequency Expected 0 1 Total eight 5 3.674 0 1.326 5 four 104 111.69 48 40.309 152 six 24 17.635 0 6.3646 24 Total 133 48 181 Table of fuel by cityover30mpg fuel cityover30mpg Frequency Expected 0 1 Total diesel 7 11.757 9 4.2431 16 gas 126 121.24 39 43.757 165 Total 133 48 181 Table of drive by cityover30mpg drive cityover30mpg Frequency Expected 0 1 Total fwd 67 83.768 47 30.232 114 rwd 66 49.232 1 17.768 67 Total 133 48 181 From the cylinders table, I can see no six or eight cylinder vehicles have city fuel efficiency over 30 mpg, & roughly 1/3 of the four cylinder vehicles do. From the fuel table, I can see more than 1/2 of the diesel-powered vehicles had city fuel efficiency over 30 mpg, & only about 1/4 of gas-powered vehicles did. In terms of drive train, nearly all of the rear wheel drive vehicles had city fuel
efficiency of 30 mpg or less, while just over 40% of the front wheel drive vehicles had city fuel efficiency over 30 mpg. Based on these results, I expect fuel and drive will be useful predictors of city fuel efficiency over 30 mpg. The complete separation for six & eight cylinder vehicles would be problematic for using the cylinders variables, though based on the results I should expect six & eight cylinder vehicles would be less fuel efficient. (b) Here I use backward elimination starting with all 3 categorical predictors in model. With all predictors in the model, the algorithm fails to converge due to the separation in the six & eight cylinder vehicles. This is the reason for the warning. The cylinders variables can be removed without significant loss of information as indicated by the residual chi-square test & the Wald chi-square test used in backward elimination. Neither of the other 2 terms could be removed without losing significantly more information than expected due to chance, so my final model will contain fuel & drive. WARNING: The validity of the model fit is questionable. Residual Chi-Square Test Chi-Square DF Pr > ChiSq 2.1211 2 0.3463 Summary of Backward Elimination Step Effect Removed DF Number In Wald Chi-Square Pr > ChiSq 1 cylinders 2 2 0.0025 0.9987 (c) Results for my final model follow. Model Information Data Set WORK.AUTOS Response Variable cityover30mpg Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring For the final model containing the fuel & cylinder predictors I see both the fuel & drive parameter estimates are statistical significant & positive. The global tests for non-zero betas also concur at least 1 of the betas should be significantly different from 0. The fuel coefficient compares diesel to gas & the drive coefficient compares front wheel drive to rear wheel drive.
I can also see the AIC for this model is much lower than for the intercept only model, indicating a better fit than a constant model. Model Fit Statistics Criterion Intercept Only Intercept and Covariates AIC 211.388 157.469 SC 214.586 167.065 -2 Log L 209.388 151.469 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 57.9184 2 <.0001 Score 43.6686 2 <.0001 Wald 14.4814 2 0.0007 Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Wald Chi-Square Pr > ChiSq Intercept 1 -5.2504 1.2690 17.1172 <.0001 fuel diesel 1 3.0322 1.1005 7.5918 0.0059 drive fwd 1 4.7115 1.2690 13.7849 0.0002 Hosmer & Lemeshow’s test is insignificant at a 0.05 level, so I conclude there are no issues of lack of fit in this model. As with the parameter estimates, the odds ratios are both statistically significant. For fuel, I estimate the odds of a diesel fuel car to have city fuel efficiency over 30 mpg was about 20.74 times that of a gas fuel car in 1985. Front wheel drive cars are estimated to have had odds of over 30 mpg city fuel efficiency about 111 times that of rear wheel drive cars. Neither confidence interval contains 1, but the intervals are pretty wide. While I can determine the odds of city fuel efficiency over 30 mpg was significantly higher for diesel cars than gas cars & significantly higher for front wheel drive cars than for rear wheel drive cars, I don ’ t have a very precise measure for the actual odds ratio. Odds Ratio Estimates Effect Point Estimate 95% Wald Confidence Limits fuel diesel vs gas 20.744 2.400 179.323 drive fwd vs rwd 111.220 9.247 >999.999
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Hosmer & Lemeshow Goodness-of-Fit Test Chi-Square DF Pr > ChiSq 0.4729 1 0.4916 Exercise 2 (a) The model with all 4 continuous predictors definitely fits the data better than a constant model. The AIC for this model is 52.745, which is quite a bit smaller than the AIC of 204.216 for the constant model. Model Information Data Set WORK.AUTOS Response Variable cityover30mpg Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring Model Fit Statistics Criterion Intercept Only Intercept and Covariates AIC 204.216 52.745 SC 207.387 68.598 -2 Log L 202.216 42.745 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 159.4713 4 <.0001 Score 62.2535 4 <.0001 Wald 11.9662 4 0.0176 Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Wald Chi-Square Pr > ChiSq Intercept 1 16.0666 10.8570 2.1899 0.1389 price1k 1 0.0380 0.2551 0.0222 0.8815 rpm 1 0.00340 0.00165 4.2583 0.0391 enginesize 1 0.1425 0.1161 1.5068 0.2196 hp 1 -0.6630 0.2083 10.1288 0.0015
The global tests are all significant, indicating the present of 1 or more significant parameter estimates. From the parameter estimates & odds ratios, I can see rpm & hp are both significant & so I ’ d likely want to retain them. price1k & enginesize are both insignificant both in their parameter estimates & odds ratios, so I may be able to remove 1 or both of those terms. Odds Ratio Estimates Effect Point Estimate 95% Wald Confidence Limits price1k 1.039 0.630 1.712 rpm 1.003 1.000 1.007 enginesize 1.153 0.918 1.448 hp 0.515 0.343 0.775 (b) Using backward selection, price1k, enginesize, & rpm are all removed at the 0.05 level leaving only the hp term in the model. The residual chi-square tests follow showing an insignificant amount of information is removed as price1k, enginesize, & rpm are removed. Residual Chi-Square Test Chi-Square DF Pr > ChiSq 0.0223 1 0.8814 Residual Chi-Square Test Chi-Square DF Pr > ChiSq 3.1063 2 0.2116 Residual Chi-Square Test Chi-Square DF Pr > ChiSq 5.2055 3 0.1574 Summary of Backward Elimination Step Effect Removed DF Number In Wald Chi-Square Pr > ChiSq 1 price1k 1 3 0.0222 0.8815 2 enginesize 1 2 2.0385 0.1534 3 rpm 1 1 2.2338 0.1350 (c) In the final model, I can see hp term is significant & negative with an estimate of -0.396, so the expected log odds for greater than 30 mpg city fuel efficiency decreased as horsepower increased. The global tests also agree there are significant non-constant terms in the model.
The AIC for this model is 53.54, which is a lot smaller than the AIC of 210.15 for the constant model, so this model fits the data far better than a constant model. Model Information Data Set WORK.AUTOS Response Variable cityover30mpg Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring Model Fit Statistics Criterion Intercept Only Intercept and Covariates AIC 210.147 53.538 SC 213.335 59.913 -2 Log L 208.147 49.538 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 158.6087 1 <.0001 Score 56.9695 1 <.0001 Wald 15.4881 1 <.0001 Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Wald Chi-Square Pr > ChiSq Intercept 1 28.8604 7.1140 16.4578 <.0001 hp 1 -0.3961 0.1007 15.4881 <.0001 From Hosmer & Lemeshow’s test, I see no evidence of a lack of fit. The p-value of 0.529 is highly insignificant. From the odds ratio estimate I ’ d expect the odds of having city fuel efficiency over 30 mpg in 1985 to be multiplied by 0.673 for a 1 unit increase in horsepower, & the estimate is clearly significant since the interval is entirely less than 1. Put another way, for a 1 unit increase in horsepower, I ’ d expect a 32.7% decrease in the odds of having city fuel efficiency over 30 mpg.
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Odds Ratio Estimates Effect Point Estimate 95% Wald Confidence Limits hp 0.673 0.552 0.820 Hosmer and Lemeshow Goodness-of-Fit Test Chi-Square DF Pr > ChiSq 5.1159 6 0.5290 Exercise 3 (a) I start by considering the full model — the model with all of the potential predictors included. Based on these results, I conclude I need to account for underdispersion because the scaled deviance of 0.2826 is much less than 1. I can estimate the overdispersion parameter from the data using the deviance. Model Information Data Set WORK.AUTOS Distribution Poisson Link Function Log Dependent Variable hwaympg Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 167 47.2017 0.2826 Scaled Deviance 167 47.2017 0.2826 Pearson Chi-Square 167 48.0247 0.2876 Scaled Pearson X2 167 48.0247 0.2876 Log Likelihood 13538.0357 Full Log Likelihood -486.7354 AIC (smaller is better) 991.4708 AICC (smaller is better) 992.5551 BIC (smaller is better) 1020.0051 After accounting for the additional dispersion parameter, I see numerous significant terms in the type 3 analysis & insignificant price1k & rpm terms. I should consider removing each of them separately. The type 1 analysis would lead me to a similar conclusion about terms I might consider removing. Here, the scale has been estimated from the data, so I need to focus on the F tests in the type 1 & type 3 analyses.
Model Information Data Set WORK.AUTOS Distribution Poisson Link Function Log Dependent Variable hwaympg Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 167 47.2017 0.2826 Scaled Deviance 167 167.0000 1.0000 Pearson Chi-Square 167 48.0247 0.2876 Scaled Pearson X2 167 169.9118 1.0174 Log Likelihood 47897.6940 Full Log Likelihood -486.7354 AIC (smaller is better) 991.4708 AICC (smaller is better) 992.5551 BIC (smaller is better) 1020.0051 LR Statistics For Type 1 Analysis Source Deviance Num DF Den DF F Value Pr > F Chi-Square Pr > ChiSq Intercept 247.2911 fuel 228.5468 1 167 66.32 <.0001 66.32 <.0001 drive 128.2710 1 167 354.78 <.0001 354.78 <.0001 hp 55.9285 1 167 255.95 <.0001 255.95 <.0001 enginesize 51.0173 1 167 17.38 <.0001 17.38 <.0001 cylinders 47.3348 2 167 6.51 0.0019 13.03 0.0015 price1k 47.2817 1 167 0.19 0.6651 0.19 0.6645 rpm 47.2017 1 167 0.28 0.5955 0.28 0.5948
LR Statistics For Type 3 Analysis Source Num DF Den DF F Value Pr > F Chi-Square Pr > ChiSq fuel 1 167 14.92 0.0002 14.92 0.0001 drive 1 167 8.98 0.0032 8.98 0.0027 hp 1 167 40.17 <.0001 40.17 <.0001 enginesize 1 167 5.54 0.0197 5.54 0.0186 cylinders 2 167 6.34 0.0022 12.69 0.0018 price1k 1 167 0.25 0.6158 0.25 0.6151 rpm 1 167 0.28 0.5955 0.28 0.5948 When I remove price1k from the model, I see rpm is still highly insignificant. My final model will contain fuel, drive, hp, enginesize, & cylinders. Model Information Data Set WORK.AUTOS Distribution Poisson Link Function Log Dependent Variable hwaympg Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 171 56.8097 0.3322 Scaled Deviance 171 171.0000 1.0000 Pearson Chi-Square 171 58.0791 0.3396 Scaled Pearson X2 171 174.8209 1.0223 Log Likelihood 41647.0350 Full Log Likelihood -499.7304 AIC (smaller is better) 1015.4607 AICC (smaller is better) 1016.3078 BIC (smaller is better) 1040.9598 LR Statistics For Type 1 Analysis Source Deviance Num DF Den DF F Value Pr > F Chi-Square Pr > ChiSq Intercept 255.4529 fuel 237.4928 1 171 54.06 <.0001 54.06 <.0001 drive 133.0720 1 171 314.31 <.0001 314.31 <.0001 hp 70.3837 1 171 188.69 <.0001 188.69 <.0001 enginesize 61.2341 1 171 27.54 <.0001 27.54 <.0001
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LR Statistics For Type 1 Analysis Source Deviance Num DF Den DF F Value Pr > F Chi-Square Pr > ChiSq cylinders 56.8497 2 171 6.60 0.0017 13.20 0.0014 rpm 56.8097 1 171 0.12 0.7291 0.12 0.7286 LR Statistics For Type 3 Analysis Source Num DF Den DF F Value Pr > F Chi-Square Pr > ChiSq fuel 1 171 20.98 <.0001 20.98 <.0001 drive 1 171 9.38 0.0026 9.38 0.0022 hp 1 171 19.22 <.0001 19.22 <.0001 enginesize 1 171 31.24 <.0001 31.24 <.0001 cylinders 2 171 6.61 0.0017 13.22 0.0013 rpm 1 171 0.12 0.7291 0.12 0.7286 (b) The type 1 & type 3 analysis F statistics clearly show the terms in the model are statistically significant. There are some noticeable changes in the scale estimates as I remove terms in this case, so I can ’ t compare the AIC for this model with the previous ones. If I had obtained a scale estimate & held that constant across models, I could compare directly. Model Information Data Set WORK.AUTOS Distribution Poisson Link Function Log Dependent Variable hwaympg Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 172 56.8497 0.3305 Scaled Deviance 172 172.0000 1.0000 Pearson Chi-Square 172 58.1627 0.3382 Scaled Pearson X2 172 175.9726 1.0231 Log Likelihood 41861.0578 Full Log Likelihood -499.7504 AIC (smaller is better) 1013.5007 AICC (smaller is better) 1014.1557 BIC (smaller is better) 1035.8124
Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald Chi- Square Pr > ChiSq Intercept 1 4.1363 0.0858 3.9681 4.3045 2322.81 <.0001 fuel diesel 1 0.1389 0.0279 0.0842 0.1936 24.80 <.0001 fuel gas 0 0.0000 0.0000 0.0000 0.0000 . . drive fwd 1 0.0678 0.0222 0.0244 0.1113 9.37 0.0022 drive rwd 0 0.0000 0.0000 0.0000 0.0000 . . hp 1 -0.0022 0.0004 -0.0031 -0.0013 25.41 <.0001 enginesize 1 -0.0034 0.0005 -0.0045 -0.0023 39.02 <.0001 cylinders eight 1 0.1135 0.0709 -0.0256 0.2525 2.56 0.1097 cylinders four 1 -0.1260 0.0375 -0.1995 -0.0526 11.32 0.0008 cylinders six 0 0.0000 0.0000 0.0000 0.0000 . . Scale 0 0.5749 0.0000 0.5749 0.5749 Note: The scale parameter was estimated by the square root of DEVIANCE/DOF. In my final model, I can see all parameter estimates are significantly different from 0 with the exception of the estimate comparing eight & six cylinder cars. The positive estimate for diesel fuel indicates higher highway fuel efficiency for diesels than gas powered cars. The positive estimate for front wheel drive indicates higher highway fuel efficiency for front wheel drive cars than rear wheel drive cars. The negative estimates for hp & engine size indicate reduced fuel efficiency as horsepower & engine size increase. The negative estimate for four cylinders indicates a reduction in expected highway fuel efficiency compared to six cylinder vehicles. Quantitatively, I need to exponentiate the parameter estimates to get the expected multiplicative change in highway fuel efficiency as these predictors change. After taking the exponential of the significant parameter estimates, I see I expect multiplicative factors of 1.15 for diesel as compared to gas, 1.07 for front wheel drive as compared to rear wheel drive, 0.9978 for a 1 unit increase in horsepower, 0.997 for a 1 unit increase in engine size, & 0.882 for four cylinders as compared to six cylinders. LR Statistics For Type 1 Analysis Source Deviance Num DF Den DF F Value Pr > F Chi-Square Pr > ChiSq Intercept 255.4529 fuel 237.4928 1 172 54.34 <.0001 54.34 <.0001 drive 133.0720 1 172 315.93 <.0001 315.93 <.0001 hp 70.3837 1 172 189.66 <.0001 189.66 <.0001 enginesize 61.2341 1 172 27.68 <.0001 27.68 <.0001 cylinders 56.8497 2 172 6.63 0.0017 13.27 0.0013
LR Statistics For Type 3 Analysis Source Num DF Den DF F Value Pr > F Chi-Square Pr > ChiSq fuel 1 172 24.18 <.0001 24.18 <.0001 drive 1 172 9.41 0.0025 9.41 0.0022 hp 1 172 26.19 <.0001 26.19 <.0001 enginesize 1 172 38.93 <.0001 38.93 <.0001 cylinders 2 172 6.63 0.0017 13.27 0.0013
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