ELECTION DATA Competence Vote- Difference 0.2 -0.7 0.23 -0.4 0.4 -0.35 0.35 0.18 0.4 0.38 0.45 -0.1 0.5 0.2 0.55 -0.3 0.6 0.3 0.68 0.18 0.7 0.5 0.76 0.22 RABIT DATA Age Lens 15 15 15 18 28 29 37 37 44 50 50 60 61 64 65 65 72 75 75 82 85 91 91 97 98 125 142 142 147 147 150 159 165 183 192 195 218 218 219 224 225 227 232 232 21.66 22.75 22.3 31.25 44.79 40.55 50.25 46.88 52.03 63.47 61.13 81 73.09 79.09 79.51 65.31 71.9 86.1 94.6 92.5 105 101.7 102.9 110 104.3 134.9 130.68 140.58 155.3 152.2 144.5 142.15 139.81 153.22 145.72 161.1 174.18 173.03 173.54 178.86 177.68 173.73 159.98 161.29 237 246 258 276 285 300 301 305 312 317 338 347 354 357 375 394 513 535 554 591 648 660 705 723 756 768 860 187.07 176.13 183.4 186.26 189.66 186.09 186.7 186.8 195.1 216.41 203.23 188.38 189.7 195.31 202.63 224.82 203.3 209.7 233.9 234.7 244.3 231 242.4 230.77 242.57 232.12 246.7 Homework 8 Math 161B - Spring 2017 1. In the paper \"Inferences of Competence from Faces Predict Election Outcomes\" (Science, 2005, available on the homework web site) the authors found that they could successfully predict the outcome of a U.S. congressional election substantially more than half of the time based on the facial appearance of the candidates (and nothing else). In the study, participants were shown photos of two candidates (A and B) for a U.S. Senate or House of Representatives election. Each participant was asked to judge which candidate looked more competent. (If a participant recognized either candidate, the value was excluded from the study). The proportion of participants who chose candidate A as more competent is recorded as the variable Competence. After the election, the difference in votes (A B) expressed as a proportion of total votes was recorded as the variable Vote Difference. This process was carried out for a large number of congressional races. Only a portion of the results (read from a graph in the paper) is contained in the data set \"Election.txt\" available on the homework web site. (a) Draw a scatterplot of Vote Difference against perceived Competence, include the regression line in your plot. Compute the sample correlation between the two variables. Assume that a simple linear regression is meaningful and that residual assumptions are approximately satsfied (they are). What percentage of variation in Vote Difference is explained by the perceived Competence of a candidate? (b) Produce a 95% confidence interval for the regression slope in this problem. Interpret the confidence interval in the context of the problem. In addition, describe how you can use this confidence interval to decide whether the perceived competence of candidate's face is a signifcant predictor for an election outcome. (c) Suppose two new candidates (C and D) compete for a seat in congress. If 55% of people shown photos of the two candidates find candidate C more competent looking than candidate D, come up with a prediction interval for the difference in votes (C D). (d) Find a 95% confidence interval for the average vote difference for pairs of candidates for which people judge both candidates equally competent based on their photographs. 2. Rabbits are major pests in some countries. Researchers, who are studying these animals, would like a reliable method to estimate the age of a free-living rabbit. Body weight is not a good predictor for age, since the environmental conditions in which the rabbits live have a large influence on the available food supply and therefore the weight. It turns out, that the dry weight of the eye lens is a much more reliable predictor for age. The file \"Rabbit.txt\" available in the homework website contains measurements on the age (in days) and the dry weight of the eye lens in milligrams for 71 rabbits. The rabbits were born and lived free in a large enclosure and depended on natural food supply. (a) Fit quadratic, power and exponential regression functions for age as a function of dry weight of eye lens. Include the scatterplot with the three curves fitted to the data with your work. Among these three regression models select the 1 Homework 8 Math 161B - Spring 2017 one that is the most appropriate fit for this data set. Explain (using output as necessary) why your chosen model is superior to the other two. (b) Write down the estimated regression equation for your selected model. Predict the age of a rabbit whose dry eye lens weights 200mg. (c) Save the residuals and standardized predicted values for the exponential model. Create a qq-plot of the residuals and a residual plot (scatter plot of \u000f against y). Comment on the appearance of the plots. Are the residual assumptions more or less satisfied in this model? Explain how you come to your conclusions. 2 1) a) Correlations competence vote_differen ce Pearson 1 Correlation competence Sig. (2-tailed) N 12 Pearson .680* Correlation vote_differenc e Sig. (2-tailed) .015 N 12 *. Correlation is significant at the 0.05 level (2-tailed). .680* .015 12 1 12 The correlation coefficient between competence and vote difference is 0.680. Coefficient of determination, R^2 = Correlation coefficient^2, r^2 = 0.68^2 = 0.4624. This implies 46.24% variation in the dependent variable Vote Difference is explained by the perceived Competence of a candidate b) 95% confidence interval for the regression slope, competence is (.334, 2.458). This implies that I am 95% confident that estimated values of coefficient of competence lies in this interval. Since this confidence interval doesn't contain zero, I can say that perceived competence of candidate's face is a significant predictor for an election outcome. c) If 55% of people shown photos of the two candidates find candidate C more competent looking than candidate D, the prediction interval for the difference in votes (C D) is given by: For Individual Response Y Interval Half Width 0.6646 Prediction Interval Lower Limit -0.5647 0.76444 Prediction Interval Upper Limit 2 d) 95% confidence interval for average vote difference for pairs of candidates for which people judge both candidates equally competent based on their photographs is given by: For Average Y Interval Half Width Confidence Interval Lower Limit Confidence Interval Upper Limit 0.1840 -0.1539 0.21411 Appendix Variables Entered/Removeda Model Variables Variables Method Entered Removed b 1 competence . Enter a. Dependent Variable: vote_difference b. All requested variables entered. Model Summary Model R R Square Adjusted R Std. Error of Square the Estimate 1 .680a .462 .408 .28500 a. Predictors: (Constant), competence ANOVAa df Mean Square Model Sum of Squares Regression .697 1 Residual .812 Total 1.509 a. Dependent Variable: vote_difference b. Predictors: (Constant), competence Model 1 (Constant) Unstandardized Coefficients B Std. Error -.668 .245 competence 1.396 .477 a. Dependent Variable: vote_difference 1 10 11 F .697 .081 Coefficientsa Standardized Coefficients Beta 8.579 t Sig. .021 2.722 .680 2.929 .015 Sig. .015b 95.0% Confidence Interval for B Lower Upper Bound Bound -1.214 -.121 .334 2.458 2) a) I prefer quadratic model as compared to exponential and power model. This is so because quadratic model has highest value of R^2 (95%) as compared to that with models exponential (55%) and power model (93.8%) b) y = -0.0006x2 + 0.6978x + 35.584 When, x = 200 mg, predicted value of y: =-0.0006*200^2+0.6978*200+35.584 = 151.144 c) The graph of residuals is approximately normally distributed since QQ plot is S shaped. Points are not randomly distributed in residual plot. Hence the variance is not constant and I can say that assumption of homogeneity of error variance is not satisfied. APPENDIX Power: Model Description Model Name MOD_3 Dependent Variable 1 lens Equation 1 Powera Independent Variable age Constant Included Variable Whose Values Label Observations in Plots Unspecified a. The model requires all non-missing values to be positive. Case Processing Summary N Total Cases 71 Excluded Cases a 0 Forecasted Cases 0 Newly Created Cases 0 a. Cases with a missing value in any variable are excluded from the analysis. Variable Processing Summary Variables Dependent Independent lens age Number of Positive Values 71 71 Number of Zeros 0 0 Number of Negative Values 0 0 User-Missing 0 0 System-Missing 0 0 Number of Missing Values Model Summary and Parameter Estimates Dependent Variable: lens Equation Model Summary R Square F df1 Parameter Estimates df2 Sig. Constant b1 Power .938 1039.435 1 69 .000 6.531 The independent variable is age. Exponential model: Model Description Model Name MOD_2 Dependent Variable 1 lens Equation 1 Exponentiala Independent Variable age Constant Included Variable Whose Values Label Observations in Plots Unspecified a. The model requires all non-missing values to be positive. Case Processing Summary N Total Cases Excluded Cases 71 a 0 Forecasted Cases 0 Newly Created Cases 0 a. Cases with a missing value in any variable are excluded from the analysis. Variable Processing Summary Variables Number of Positive Values Dependent Independent lens age 71 71 Number of Zeros 0 0 Number of Negative Values 0 0 User-Missing 0 0 System-Missing 0 0 Number of Missing Values Model Summary and Parameter Estimates .584 Dependent Variable: lens Equation Model Summary R Square Exponential F .550 Parameter Estimates df1 84.322 df2 Sig. 1 69 Constant .000 73.720 The independent variable is age. QUADRATI MODEL: Model Description Model Name MOD_4 Dependent Variable 1 lens Equation 1 Quadratic Independent Variable age Constant Included Variable Whose Values Label Observations in Plots Unspecified Tolerance for Entering Terms in Equations .0001 Case Processing Summary N Total Cases Excluded Cases 71 a 0 Forecasted Cases 0 Newly Created Cases 0 a. Cases with a missing value in any variable are excluded from the analysis. Variable Processing Summary Variables Number of Positive Values Dependent Independent lens age 71 71 Number of Zeros 0 0 Number of Negative Values 0 0 b1 .002 Number of Missing Values User-Missing 0 0 System-Missing 0 0 Model Summary and Parameter Estimates Dependent Variable: lens Equation Model Summary R Square Quadratic .950 F 652.858 The independent variable is age. df1 Parameter Estimates df2 2 Sig. 68 .000 Constant 35.584 b1 .698 b2 -.001 1) a) Correlations competence vote_differen ce Pearson 1 Correlation competence Sig. (2-tailed) N 12 Pearson .680* Correlation vote_differenc e Sig. (2-tailed) .015 N 12 *. Correlation is significant at the 0.05 level (2-tailed). .680* .015 12 1 12 The correlation coefficient between competence and vote difference is 0.680. Coefficient of determination, R^2 = Correlation coefficient^2, r^2 = 0.68^2 = 0.4624. This implies 46.24% variation in the dependent variable Vote Difference is explained by the perceived Competence of a candidate b) 95% confidence interval for the regression slope, competence is (.334, 2.458). This implies that I am 95% confident that estimated values of coefficient of competence lies in this interval. Since this confidence interval doesn't contain zero, I can say that perceived competence of candidate's face is a significant predictor for an election outcome. c) If 55% of people shown photos of the two candidates find candidate C more competent looking than candidate D, the prediction interval for the difference in votes (C D) is given by: For Individual Response Y Interval Half Width 0.6646 Prediction Interval Lower Limit -0.5647 0.76444 Prediction Interval Upper Limit 2 d) 95% confidence interval for average vote difference for pairs of candidates for which people judge both candidates equally competent based on their photographs is given by: For Average Y Interval Half Width Confidence Interval Lower Limit Confidence Interval Upper Limit 0.1840 -0.1539 0.21411 Appendix Variables Entered/Removeda Model Variables Variables Method Entered Removed b 1 competence . Enter a. Dependent Variable: vote_difference b. All requested variables entered. Model Summary Model R R Square Adjusted R Std. Error of Square the Estimate 1 .680a .462 .408 .28500 a. Predictors: (Constant), competence ANOVAa df Mean Square Model Sum of Squares Regression .697 1 Residual .812 Total 1.509 a. Dependent Variable: vote_difference b. Predictors: (Constant), competence Model 1 (Constant) Unstandardized Coefficients B Std. Error -.668 .245 competence 1.396 .477 a. Dependent Variable: vote_difference 1 10 11 F .697 .081 Coefficientsa Standardized Coefficients Beta 8.579 t Sig. .021 2.722 .680 2.929 .015 Sig. .015b 95.0% Confidence Interval for B Lower Upper Bound Bound -1.214 -.121 .334 2.458 2) a) I prefer quadratic model as compared to exponential and power model. This is so because quadratic model has highest value of R^2 (95%) as compared to that with models exponential (55%) and power model (93.8%) b) y = -0.0006x2 + 0.6978x + 35.584 When, x = 200 mg, predicted value of y: =-0.0006*200^2+0.6978*200+35.584 = 151.144 c) The graph of residuals is approximately normally distributed since QQ plot is S shaped. Points are not randomly distributed in residual plot. Hence the variance is not constant and I can say that assumption of homogeneity of error variance is not satisfied. APPENDIX Power: Model Description Model Name MOD_3 Dependent Variable 1 lens Equation 1 Powera Independent Variable age Constant Included Variable Whose Values Label Observations in Plots Unspecified a. The model requires all non-missing values to be positive. Case Processing Summary N Total Cases 71 Excluded Cases a 0 Forecasted Cases 0 Newly Created Cases 0 a. Cases with a missing value in any variable are excluded from the analysis. Variable Processing Summary Variables Dependent Independent lens age Number of Positive Values 71 71 Number of Zeros 0 0 Number of Negative Values 0 0 User-Missing 0 0 System-Missing 0 0 Number of Missing Values Model Summary and Parameter Estimates Dependent Variable: lens Equation Model Summary R Square F df1 Parameter Estimates df2 Sig. Constant b1 Power .938 1039.435 1 69 .000 6.531 The independent variable is age. Exponential model: Model Description Model Name MOD_2 Dependent Variable 1 lens Equation 1 Exponentiala Independent Variable age Constant Included Variable Whose Values Label Observations in Plots Unspecified a. The model requires all non-missing values to be positive. Case Processing Summary N Total Cases Excluded Cases 71 a 0 Forecasted Cases 0 Newly Created Cases 0 a. Cases with a missing value in any variable are excluded from the analysis. Variable Processing Summary Variables Number of Positive Values Dependent Independent lens age 71 71 Number of Zeros 0 0 Number of Negative Values 0 0 User-Missing 0 0 System-Missing 0 0 Number of Missing Values Model Summary and Parameter Estimates .584 Dependent Variable: lens Equation Model Summary R Square Exponential F .550 Parameter Estimates df1 84.322 df2 Sig. 1 69 Constant .000 73.720 The independent variable is age. QUADRATI MODEL: Model Description Model Name MOD_4 Dependent Variable 1 lens Equation 1 Quadratic Independent Variable age Constant Included Variable Whose Values Label Observations in Plots Unspecified Tolerance for Entering Terms in Equations .0001 Case Processing Summary N Total Cases Excluded Cases 71 a 0 Forecasted Cases 0 Newly Created Cases 0 a. Cases with a missing value in any variable are excluded from the analysis. Variable Processing Summary Variables Number of Positive Values Dependent Independent lens age 71 71 Number of Zeros 0 0 Number of Negative Values 0 0 b1 .002 Number of Missing Values User-Missing 0 0 System-Missing 0 0 Model Summary and Parameter Estimates Dependent Variable: lens Equation Model Summary R Square Quadratic .950 F 652.858 The independent variable is age. df1 Parameter Estimates df2 2 Sig. 68 .000 Constant 35.584 b1 .698 b2 -.001