HW6Solution_Fall23

Homework 6 Solutions. Fall 2023 Exercise 1 (a) Note I need to estimate a dispersion parameter to account for greater dispersion than expected for a Poisson distribution. When an overdispersed Poisson log-linear model for rings is fitted using all other predictors, the results are as follows. From the table with parameter estimates, I can see all predictors except length are significant in the model with p-values less than 0.05. Note although indicator variable (sex = F ) has p-value greater than 0.05, it should be kept in the model because indicator variable for sex = I is significant. Based on type 1 & type 3 analysis results, I can also tell which predictors are significant. Specifically type 1 analysis uses the amount of information additionally explained by the model when that term is added sequentially, & type 3 analysis is based on the amount of information lost if I drop that term from the model. 1tly, the type 3 table shows the same result that all predictors except length are significant with p-values less than 0.05. The table from type 1 analysis shows all are significant. Thus, I can conclude sex, diameter, height, whole_weight, meat_weight, gut_weight & shell_weight are likely related to an abalone’s age. Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 4167 1835.0737 0.4404 Scaled Deviance 4167 4167.0000 1.0000 Pearson Chi-Square 4167 1931.8737 0.4636 Scaled Pearson X2 4167 4386.8088 1.0527 Log Likelihood 124719.3088 Full Log Likelihood -9484.8726 AIC (smaller is better) 18989.7452 AICC (smaller is better) 18989.7980 BIC (smaller is better) 19053.1187 Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald Chi- Square Pr > ChiSq Intercept 1 1.4972 0.0306 1.4372 1.5572 2392.10 <.0001 sex F 1 -0.0089 0.0076 -0.0238 0.0061 1.36 0.2442 sex I 1 -0.1086 0.0095 -0.1273 -0.0899 129.69 <.0001 sex M 0 0.0000 0.0000 0.0000 0.0000 . . length 1 0.2795 0.1719 -0.0575 0.6165 2.64 0.1040

Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald Chi- Square Pr > ChiSq diameter 1 1.4416 0.2088 1.0323 1.8509 47.65 <.0001 height 1 1.0201 0.1149 0.7949 1.2454 78.80 <.0001 whole_weight 1 0.7455 0.0635 0.6210 0.8700 137.72 <.0001 meat_weight 1 -1.8116 0.0737 -1.9560 -1.6673 604.81 <.0001 gut_weight 1 -0.8893 0.1150 -1.1146 -0.6639 59.81 <.0001 shell_weight 1 0.5264 0.0964 0.3376 0.7153 29.84 <.0001 Scale 0 0.6636 0.0000 0.6636 0.6636 Note: The scale parameter was estimated by the square root of DEVIANCE/DOF. LR Statistics For Type 1 Analysis Source Deviance Num DF Den DF F Value Pr > F Chi-Square Pr > ChiSq Intercept 4139.3003 sex 3261.7861 2 4167 996.31 <.0001 1992.62 <.0001 length 2613.1519 1 4167 1472.89 <.0001 1472.89 <.0001 diameter 2536.7855 1 4167 173.41 <.0001 173.41 <.0001 height 2480.5560 1 4167 127.68 <.0001 127.68 <.0001 whole_weight 2470.3512 1 4167 23.17 <.0001 23.17 <.0001 meat_weight 1894.7813 1 4167 1306.98 <.0001 1306.98 <.0001 gut_weight 1848.2252 1 4167 105.72 <.0001 105.72 <.0001 shell_weight 1835.0737 1 4167 29.86 <.0001 29.86 <.0001 LR Statistics For Type 3 Analysis Source Num DF Den DF F Value Pr > F Chi-Square Pr > ChiSq sex 2 4167 69.07 <.0001 138.15 <.0001 length 1 4167 2.65 0.1038 2.65 0.1037 diameter 1 4167 47.46 <.0001 47.46 <.0001 height 1 4167 64.27 <.0001 64.27 <.0001 whole_weight 1 4167 135.06 <.0001 135.06 <.0001 meat_weight 1 4167 597.00 <.0001 597.00 <.0001 gut_weight 1 4167 59.76 <.0001 59.76 <.0001 shell_weight 1 4167 29.86 <.0001 29.86 <.0001

Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald Chi- Square Pr > ChiSq meat_weight 1 -1.8009 0.0734 -1.9447 -1.6571 602.31 <.0001 gut_weight 1 -0.8691 0.1143 -1.0932 -0.6450 57.77 <.0001 shell_weight 1 0.5237 0.0964 0.3348 0.7127 29.52 <.0001 Scale 0 0.6637 0.0000 0.6637 0.6637 Note: The scale parameter was estimated by the square root of DEVIANCE/DOF. LR Statistics For Type 1 Analysis Source Deviance Num DF Den DF F Value Pr > F Chi-Square Pr > ChiSq Intercept 4139.3003 sex 3261.7861 2 4168 995.92 <.0001 1991.83 <.0001 diameter 2547.7136 1 4168 1620.84 <.0001 1620.84 <.0001 height 2493.5982 1 4168 122.83 <.0001 122.83 <.0001 whole_weight 2478.1688 1 4168 35.02 <.0001 35.02 <.0001 meat_weight 1894.8180 1 4168 1324.12 <.0001 1324.12 <.0001 gut_weight 1849.2551 1 4168 103.42 <.0001 103.42 <.0001 shell_weight 1836.2395 1 4168 29.54 <.0001 29.54 <.0001 LR Statistics For Type 3 Analysis Source Num DF Den DF F Value Pr > F Chi-Square Pr > ChiSq sex 2 4168 68.22 <.0001 136.43 <.0001 diameter 1 4168 320.49 <.0001 320.49 <.0001 height 1 4168 64.56 <.0001 64.56 <.0001 whole_weight 1 4168 134.54 <.0001 134.54 <.0001 meat_weight 1 4168 594.38 <.0001 594.38 <.0001 gut_weight 1 4168 57.73 <.0001 57.73 <.0001 shell_weight 1 4168 29.54 <.0001 29.54 <.0001 (c) From residual plots against predicted values, I see some slight curvature & possible spread in the middle of the residuals. The model does appear to over-predict for some of the smaller prediction values. All in all, the deviations in the residuals don ’ t look too bad, so issues with the model assumptions are minor. The parameter estimates are 1.74, 1.02, 0.74, -1.8, -0.87 & 0.52 for diameter, height, whole_weight, meat_weight, gut_weight & shell_weight, respectively. Also -0.0089 & -0.1079 for Sex = F & Sex = I. It implies for diameter, expected increase in log rings for a 1-unit increase

in diameter is 1.74 (Multiplicative increase of e^1.74 in count). I can interpret for other predictors having positive estimates (height, whole_weight & shell_weight) in the same way. Now for meat_weight which has negative estimates, -1.8, I can interpret the expected decrease in log rings for a 1-unit increase in meat_weight is -1.8, consequently the expected rings would decrease by a multiplicative factor of e^-1.8. I can interpret the estimate from gut_weight in the same way & conclude 2 have negative relationship with the count of rings. For sex variable, the expected decrease in log (rings) by changing sex = M to sex = F is -0.0089 & to sex = I is - 0.1079. Consequently, the expected count would decrease by a multiplicative factor of e^- 0.0089 & e^-0.1079. The decrease for male to female is insignificant, but the decrease from male to infant is significant. The diagnostic plots don ’ t look too bad, thus the model seems acceptable. Exercise 2 (a) I can see all predictors are significant in a gamma model with p-values less than 0.05. Type 1 & type 3 analyses give the same result. Thus, I can say sex, length, diameter, height, whole_weight, meat_weight, gut_weight & shell_weight are likely related to an abalone’s age. Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 4167 174.9773 0.0420 Scaled Deviance 4167 4205.9571 1.0093 Pearson Chi-Square 4167 188.6534 0.0453 Scaled Pearson X2 4167 4534.6922 1.0882 Log Likelihood -8695.4689 Full Log Likelihood -8695.4689 AIC (smaller is better) 17412.9378 AICC (smaller is better) 17413.0012 BIC (smaller is better) 17482.6487

LR Statistics For Type 3 Analysis Source DF Chi-Square Pr > ChiSq meat_weight 1 511.84 <.0001 gut_weight 1 59.04 <.0001 shell_weight 1 32.62 <.0001 (b) As I checked in part (a), all terms are significant in type3 analysis. Thus, all predictors should be kept in the model. (c) From residual plots against predicted values, I can find a very clear quadratic pattern. Pearson & deviance residuals should remove any trends in the residuals if the chosen model & link are reasonable, thus there is a strong indication of problems with model assumption. Either I should choose a distribution with smaller variance function (Gamma has a variance proportional to the expected values squared, so perhaps linear relationship would be better) or a different link function should be considered. In terms of parameter estimates, meat_weight, gut_weight & sex dummy variables (sex = F, sex = I) have negative relationships with log (rings). The estimates are -1.71, -0.92, -0.0067 & - 0.1036, respectively. It implies for a 1-unit increase in meat_weight & gut_weight, I expect a multiplicative change in rings of e^(-1.71) & e(^-0.92). Also, by changing sex = M to sex = F or sex = I, I expect a multiplicative change in rings of e^(-0.0067) & e(^-0.1036). Indeed, all are less than 1, thus it means number of rings will decrease. Otherwise for predictors which have positive estimates, a 1-unit increase in those terms gives an increase in the number of rings. Specifically, length, diameter, height, whole_weight & shell_weight are positively related to log of rings with estimates 0.38, 1.39, 1.67, 0.66 & 0.6, respectively. To interpret it, for length, I expect a multiplicative increase in rings of e^0.38 for a 1-unit increase in length. I can interpret diameter, height, whole_weight & shell_weight in a similar way. Despite the fact I have a lower AIC value (17412.93) & a much lower deviance ( 174.9773) than the corresponding degrees of freedom (4167) for the gamma model, I could find some issues on diagnostic plots, thus it ’ s not a good model & I can conclude Poisson model is better. In particular, the apparent curvature & non-constant variation which appeared to be slight in the Poisson model is much more pronounced in the gamma model. Therefore, I should choose the overdispersed Poisson model from exercise 1 as the better model.

Exercise 3 a) The principal component analysis of the data follows. To retain 85% of the variation, 4 components would be needed. This agrees with the average eigenvalue as the 4th principal component is the last with an eigenvalue of at least 1. It also agrees with the scree plot. In the scree plot, the elbow appears to be at 5, so 4 components would be chosen based on that criterion as well. 0 10 20 30 40 Predicted Value -6 -4 -2 0 2 4 6 Standardized Pearson Residual 0 10 20 30 40 Predicted Value -7.5 -5.0 -2.5 0.0 2.5 5.0 Standardized Deviance Residual

Eigenvectors Prin1 Prin2 Prin3 Prin4 Prin5 Prin6 compactness 0.3131 56 - .22530 4 - .08794 6 - .07265 6 0.6825 34 0.5607 86 circularity 0.3526 62 0.0789 70 - .10537 6 - .12117 3 - .44824 7 0.3801 70 radiusRatio 0.3159 70 - .08129 4 0.4297 05 0.0093 60 0.0555 96 - .22150 1 prAxisAspectRatio 0.1205 20 0.1478 44 0.7500 33 - .06055 3 - .05489 9 0.0187 53 scatterRatio 0.3707 32 - .00262 2 - .17141 1 0.0821 71 0.0401 89 - .24158 0 elongatedness - .37043 9 0.0695 23 0.0745 99 - .02793 2 0.0218 59 0.3640 35 prAxisRectangularity 0.3639 85 0.0170 10 - .20830 3 0.0983 59 0.0539 93 - .20878 2 scaledVarianceAlongMajorAxis 0.3653 52 0.1331 40 0.0578 26 0.0957 37 0.1921 55 - .18450 5 scaledRadiusOfGyration 0.3451 73 0.1560 62 - .10381 5 - .15315 4 - .44942 8 0.3243 36 skewnessAboutMajorAxis 0.0070 85 0.6844 76 0.1840 28 0.0582 33 0.2171 46 0.2115 14 kurtosisAboutMajorAxis 0.0101 82 - .01590 9 0.0224 27 0.9606 31 - .11876 9 0.2006 14 hollowsRatio 0.0571 66 - .63131 9 0.3273 21 0.0158 78 - .14812 3 0.1813 37

Eigenvectors Prin7 Prin8 Prin9 Prin10 Prin11 Prin12 compactness - .18642 0 - .11024 2 - .09116 5 0.0089 93 - .04921 5 - .01023 3 circularity - .13021 4 0.5136 61 - .20429 2 - .41707 8 0.0175 57 - .03673 7 radiusRatio - .35682 8 - .03880 4 0.3480 87 - .20266 2 0.6046 47 0.0008 08 prAxisAspectRatio - .22342 3 0.0428 15 - .24113 3 0.2534 81 - .46862 2 0.0167 31 scatterRatio 0.0291 57 0.1372 04 - .09441 8 0.2007 84 - .01918 6 0.8349 66 elongatedness - .10972 0 0.4209 80 0.6503 80 0.2076 28 - .06119 2 0.2438 99 prAxisRectangularity - .01601 2 0.4081 82 0.0740 09 0.6026 05 0.0539 01 - .47882 3 scaledVarianceAlongMajorAxis 0.4017 02 0.0304 58 0.4607 02 - .40709 7 - .46337 2 - .10167 3 scaledRadiusOfGyration 0.0416 10 - .57894 1 0.3003 93 0.2961 39 - .00939 6 0.0126 82 skewnessAboutMajorAxis 0.4491 54 0.0608 01 - .17108 1 0.0643 27 0.4031 34 0.0305 71 kurtosisAboutMajorAxis - .11598 8 - .08254 4 - .01893 7 - .02501 4 - .02507 2 - .00983 3 hollowsRatio 0.6161 29 0.1067 16 - .03389 7 0.1260 70 0.1646 16 0.0284 46

b) For the 1st principal component, elongatedness has the only negative value. The largest positive values include circular & rectangular shape related features (Compactness, radius ratio, scatter ratio, principal axis rectangularity, scaled variance along major axis, & scaled radius of gyration). This feature seems to contrast elongation with the other listed shape measures. A silhouette with a smaller or more negative principal component 1 will tend to be more elongated relative to the other shape measures than the typical vehicle silhouette. A larger positive value would tend to indicate a larger amount of some subset of those shape features relative to elongation than the typical silhouette. The 2nd component has many small but potentially noticeable contributors with coefficient magnitudes around 0.15 or 0.2. These are big enough to have noticeable impacts in some cases, but mostly the principal component is picking up on a contrast between skewness about the major axis & hollows ratio. Silhouettes with a more positive principal component 2 will tend to have greater skewness about the major axis relative to the ratio of hollows than the typical silhouette. Those with a more negative value will tend to have a higher hollows ratio relative to skewness as compared to the typical silhouette. c) The score plot shows a couple of buses which are slightly higher in principal component 1 than the rest. These bus silhouettes might be more circular, rectangular, or compact & less elongated than other silhouettes. There ’ s also a bus & a van with much higher principal component 2. These 2 vehicle silhouettes most likely have a higher contrast of major axis skewness to hollows ratio than a typical vehicle silhouette, with greater skewness &/or smaller hollows ratios than is typical. Looking at the plot overall, the vehicles seem to be pretty mixed up. These 2 components wouldn ’ t be very useful for separating the 4 vehicle silhouette types. van van saab van bus bus bus van van saab van saab bus van bus opel van bus saab opel bus van bus bus saab van saab saab bus saab van saab opel opel opel van bus van saab bus opel van van saab saab van van bus van saab saab saab opel bus bus van saab van opel van opel opel van bus bus opel bus opel van bus opel opel opel opel van opel saab saab bus bus bus bus van opel bus bus van van bus opel saab opel saab van -8 -6 -4 -2 0 2 4 6 8 Component 1 (56.37%) -2 0 2 4 6 Component 2 (15.94%) Component Scores

Exercise 4 a) Repeating the principal component analysis using a covariance matrix, the total variation is 5447 & the average eigenvalue would be 5447 / 12 which is about 454. 2 components would be needed to retain at least 85% of the variation, & 1st 2 components are also the only 2 with an eigenvalue greater than the average. The scree plot also suggests 2 as a good choice for the number of principal components. Cases could also be made for 1 component or 4 components given the large drop from the 1st to the 2nd eigenvalue & the drop off from 4 to 5, but the variation described by components 3 & 4 are pretty small so more than 2 components is more than needed. 2 will be a good choice based on all criteria here. Total Variance 5447.02070 46 Eigenvalues of the Covariance Matrix Eigenvalue Difference Proportion Cumulative 1 4300.0703 4 3680.224 78 0.7894 0.7894 2 619.84556 377.6803 4 0.1138 0.9032 3 242.16522 64.63518 0.0445 0.9477 4 177.53005 129.5533 1 0.0326 0.9803 5 47.97674 19.62957 0.0088 0.9891 6 28.34716 11.50360 0.0052 0.9943 7 16.84356 9.55712 0.0031 0.9974 8 7.28644 4.37464 0.0013 0.9987 9 2.91180 0.50232 0.0005 0.9993 10 2.40948 0.85688 0.0004 0.9997 11 1.55260 1.47084 0.0003 1.0000 12 0.08176 0.0000 1.0000

Eigenvectors Prin7 Prin8 Prin9 Prin10 Prin11 Prin12 compactness 0.8170 84 0.0373 76 - .08780 5 - .13723 1 - .04884 3 - .00038 1 circularity 0.0607 73 0.1822 57 0.8101 91 0.1125 55 - .51903 7 - .01017 1 radiusRatio 0.1000 33 - .24341 2 0.0059 73 0.1788 90 - .00323 5 - .00206 9 prAxisAspectRatio - .01828 5 0.6516 03 0.0014 66 - .59415 6 0.0876 73 0.0071 53 scatterRatio 0.0772 77 0.3636 48 - .00623 1 0.0536 96 0.2069 13 - .09686 4 elongatedness 0.1365 90 - .07081 7 0.5169 68 0.1119 68 0.8067 30 - .08866 2 prAxisRectangularity 0.0237 60 0.0257 72 0.0535 11 0.0216 97 0.0858 26 0.9912 48 scaledVarianceAlongMajorAxis - .28011 9 - .18629 5 0.1011 07 - .13139 9 - .00329 3 0.0024 06 scaledRadiusOfGyration - .02455 8 - .08897 5 - .10722 2 - .01748 8 0.0764 09 0.0009 46 skewnessAboutMajorAxis 0.2989 25 0.2733 75 - .17985 9 0.5838 02 - .06762 2 - .00296 3 kurtosisAboutMajorAxis 0.1572 60 - .07642 9 - .02162 9 - .08199 6 - .04547 9 - .00125 4 hollowsRatio - .31621 1 0.4695 02 - .10558 8 0.4499 62 0.0838 73 0.0028 74

Variance Explained 0.0 0.2 0.4 0.6 0.8 1.0 Proportion 2 4 6 8 10 12 Principal Component Proportion Cumulative Scree Plot 0 1000 2000 3000 4000 Eigenvalue 2 4 6 8 10 12 Principal Component