March 19, 2018
Examining the affect of another variable on a relationship
Section 7.6 of Modern Data Science with R
Section 1.4.1 of Introductory Statistics with Randomization and Simulation from OpenIntro
… other than the quality of the course …
score is the average student evaluation for the course.bty_avg is the average beauty rating of the professor, based on ratings of physical appear from 6 students in the course.download.file("http://www.openintro.org/stat/data/evals.RData", destfile = "evals.RData")
load("evals.RData")
glimpse(evals)
## Observations: 463 ## Variables: 21 ## $ score <dbl> 4.7, 4.1, 3.9, 4.8, 4.6, 4.3, 2.8, 4.1, 3.4, 4.5... ## $ rank <fctr> tenure track, tenure track, tenure track, tenur... ## $ ethnicity <fctr> minority, minority, minority, minority, not min... ## $ gender <fctr> female, female, female, female, male, male, mal... ## $ language <fctr> english, english, english, english, english, en... ## $ age <int> 36, 36, 36, 36, 59, 59, 59, 51, 51, 40, 40, 40, ... ## $ cls_perc_eval <dbl> 55.81395, 68.80000, 60.80000, 62.60163, 85.00000... ## $ cls_did_eval <int> 24, 86, 76, 77, 17, 35, 39, 55, 111, 40, 24, 24,... ## $ cls_students <int> 43, 125, 125, 123, 20, 40, 44, 55, 195, 46, 27, ... ## $ cls_level <fctr> upper, upper, upper, upper, upper, upper, upper... ## $ cls_profs <fctr> single, single, single, single, multiple, multi... ## $ cls_credits <fctr> multi credit, multi credit, multi credit, multi... ## $ bty_f1lower <int> 5, 5, 5, 5, 4, 4, 4, 5, 5, 2, 2, 2, 2, 2, 2, 2, ... ## $ bty_f1upper <int> 7, 7, 7, 7, 4, 4, 4, 2, 2, 5, 5, 5, 5, 5, 5, 5, ... ## $ bty_f2upper <int> 6, 6, 6, 6, 2, 2, 2, 5, 5, 4, 4, 4, 4, 4, 4, 4, ... ## $ bty_m1lower <int> 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, ... ## $ bty_m1upper <int> 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ... ## $ bty_m2upper <int> 6, 6, 6, 6, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, ... ## $ bty_avg <dbl> 5.000, 5.000, 5.000, 5.000, 3.000, 3.000, 3.000,... ## $ pic_outfit <fctr> not formal, not formal, not formal, not formal,... ## $ pic_color <fctr> color, color, color, color, color, color, color...
score and bty_avg?ggplot(evals, aes(x=bty_avg, y=score)) + geom_point() + theme_bw()
Use some transparency so we can see where there are overlapping points
ggplot(evals, aes(x=bty_avg, y=score)) + geom_point(alpha=0.3) + theme_bw()
Is there a relationship between score and bty_avg?
ggplot(evals, aes(x=bty_avg, y=score)) + geom_point(alpha=0.3) + theme_bw() + geom_smooth(method = "lm", fill=NA)
What would the slope be if there was no relationship?
By default, geom_smooth gives a confidence interval for the fitted line (or curve)
ggplot(evals, aes(x=bty_avg, y=score)) + geom_point(alpha=0.3) + theme_bw() + geom_smooth(method = "lm")
The grey shaded area around the fitted regression line is a 95% confidence interval for the slope.
bty_avg.
Does the confidence interval indicate that 0 is a possible value for \(\beta_1\) (the parameter for the slope)?
Output from the summary command for the estimated regression coefficients:
summary(lm(score ~ bty_avg, data=evals))$coefficients
## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 3.88033795 0.07614297 50.961212 1.561043e-191 ## bty_avg 0.06663704 0.01629115 4.090382 5.082731e-05
R gives results for an hypothesis test with hypotheses:
\[H_0: \beta_1 = 0 \mbox{ versus } H_a: \beta_1 \ne 0\]
## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 3.88033795 0.07614297 50.961212 1.561043e-191 ## bty_avg 0.06663704 0.01629115 4.090382 5.082731e-05
lm function, by default the P-value is calculated based on a probability model that assumes all observations are independent and that the error terms have a symmetric, bell-shaped distribution.Does the hypothesis test for the slope indicate that the slope is different from 0?
score and gender?ggplot(evals, aes(x=gender, y=score)) + geom_point(alpha=0.3) + theme_bw()
gender as the independent variablelm(score ~ gender, data=evals)$coefficients
## (Intercept) gendermale ## 4.0928205 0.1415078
\[ \widehat{score} = 4.09 + 0.14 \, gender\_is\_male \]
How to interpret the slope:
On average, course evaluation scores for male professors are \(0.14\) higher than for female professors.
\[ \widehat{score} = 4.09 + 0.14 \, gender\_is\_male \]
R encodes categorical independent variables as indicator variables (also called dummy variables).R picks a baseline value of the categorical variable. Here the baseline level is female.gender_is_male is 1 for observations for which gender is male and 0 otherwise.Could the difference between the mean score for males and females just be due to chance?
The regression model is \[score = \beta_0 + \beta_1 \, gender\_is\_male + \epsilon\] where
\[ gender\_is\_male = \begin{cases} 1 & \mbox{if } gender\mbox{ is } male\\ 0 & \mbox{if } gender\mbox{ is } female \end{cases}\]
We can answer the question with an hypothesis test with hypotheses \[H_0: \beta_1 = 0 \mbox{ versus } H_a: \beta_1 \ne 0\]
summary(lm(score ~ gender, data=evals))$coefficients
## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 4.0928205 0.03866539 105.852305 0.000000000 ## gendermale 0.1415078 0.05082127 2.784422 0.005582967
What conclusion do we make?
score and bty_avg the same for male and female professors?ggplot(evals, aes(x=bty_avg, y=score, colour=gender)) + geom_point(alpha=0.5) + theme_bw()
Model 1: \[score = \beta_0 + \beta_1 \, gender\_is\_male + \beta_2 \, bty\_avg + \epsilon\]
Model 1 for male professors: \[score = \beta_0 + \beta_1 + \beta_2 \, bty\_avg + \epsilon\]
Model 1 for female professors: \[score = \beta_0 + \beta_2 \, bty\_avg + \epsilon\]
How would you describe these two lines?
parallel_lines <- lm(score ~ gender + bty_avg, data=evals) parallel_lines$coefficients
## (Intercept) gendermale bty_avg ## 3.74733824 0.17238955 0.07415537
The augment function (in the library broom) creates a data frame with predicted values (.fitted), residuals, etc. for linear model output.
library(broom) augment(parallel_lines)
## score gender bty_avg .fitted .se.fit .resid .hat ## 1 4.7 female 5.000 4.118115 0.03826383 0.581884899 0.005238155 ## 2 4.1 female 5.000 4.118115 0.03826383 -0.018115101 0.005238155 ## 3 3.9 female 5.000 4.118115 0.03826383 -0.218115101 0.005238155 ## 4 4.8 female 5.000 4.118115 0.03826383 0.681884899 0.005238155 ## 5 4.6 male 3.000 4.142194 0.03808791 0.457806096 0.005190100 ## 6 4.3 male 3.000 4.142194 0.03808791 0.157806096 0.005190100 ## 7 2.8 male 3.000 4.142194 0.03808791 -1.342193904 0.005190100 ## 8 4.1 male 3.333 4.166888 0.03551641 -0.066887644 0.004512941 ## 9 3.4 male 3.333 4.166888 0.03551641 -0.766887644 0.004512941 ## 10 4.5 female 3.167 3.982188 0.04495870 0.517811698 0.007231509 ## 11 3.8 female 3.167 3.982188 0.04495870 -0.182188302 0.007231509 ## 12 4.5 female 3.167 3.982188 0.04495870 0.517811698 0.007231509 ## 13 4.6 female 3.167 3.982188 0.04495870 0.617811698 0.007231509 ## 14 3.9 female 3.167 3.982188 0.04495870 -0.082188302 0.007231509 ## 15 3.9 female 3.167 3.982188 0.04495870 -0.082188302 0.007231509 ## 16 4.3 female 3.167 3.982188 0.04495870 0.317811698 0.007231509 ## 17 4.5 female 3.167 3.982188 0.04495870 0.517811698 0.007231509 ## 18 4.8 female 7.333 4.291120 0.05763815 0.508880414 0.011885613 ## 19 4.6 female 7.333 4.291120 0.05763815 0.308880414 0.011885613 ## 20 4.6 female 7.333 4.291120 0.05763815 0.308880414 0.011885613 ## 21 4.9 female 7.333 4.291120 0.05763815 0.608880414 0.011885613 ## 22 4.6 female 7.333 4.291120 0.05763815 0.308880414 0.011885613 ## 23 4.5 female 7.333 4.291120 0.05763815 0.208880414 0.011885613 ## 24 4.4 male 5.500 4.327582 0.03821855 0.072417663 0.005225766 ## 25 4.6 male 5.500 4.327582 0.03821855 0.272417663 0.005225766 ## 26 4.7 male 5.500 4.327582 0.03821855 0.372417663 0.005225766 ## 27 4.5 male 5.500 4.327582 0.03821855 0.172417663 0.005225766 ## 28 4.8 male 5.500 4.327582 0.03821855 0.472417663 0.005225766 ## 29 4.9 male 5.500 4.327582 0.03821855 0.572417663 0.005225766 ## 30 4.5 male 5.500 4.327582 0.03821855 0.172417663 0.005225766 ## 31 4.4 female 4.167 4.056344 0.03869495 0.343656325 0.005356856 ## 32 4.3 female 4.167 4.056344 0.03869495 0.243656325 0.005356856 ## 33 4.1 female 4.167 4.056344 0.03869495 0.043656325 0.005356856 ## 34 4.2 female 4.167 4.056344 0.03869495 0.143656325 0.005356856 ## 35 3.5 female 4.167 4.056344 0.03869495 -0.556343675 0.005356856 ## 36 3.4 female 4.000 4.043960 0.03934540 -0.643959728 0.005538466 ## 37 4.5 female 4.000 4.043960 0.03934540 0.456040272 0.005538466 ## 38 4.4 female 4.000 4.043960 0.03934540 0.356040272 0.005538466 ## 39 4.4 female 4.000 4.043960 0.03934540 0.356040272 0.005538466 ## 40 2.5 female 4.000 4.043960 0.03934540 -1.543959728 0.005538466 ## 41 4.3 female 4.000 4.043960 0.03934540 0.256040272 0.005538466 ## 42 4.5 female 4.000 4.043960 0.03934540 0.456040272 0.005538466 ## 43 4.8 female 4.667 4.093421 0.03786035 0.706578639 0.005128267 ## 44 4.8 female 4.667 4.093421 0.03786035 0.706578639 0.005128267 ## 45 4.4 female 4.667 4.093421 0.03786035 0.306578639 0.005128267 ## 46 4.7 female 4.667 4.093421 0.03786035 0.606578639 0.005128267 ## 47 4.4 female 4.667 4.093421 0.03786035 0.306578639 0.005128267 ## 48 4.7 female 4.667 4.093421 0.03786035 0.606578639 0.005128267 ## 49 4.5 female 4.667 4.093421 0.03786035 0.406578639 0.005128267 ## 50 4.0 male 5.500 4.327582 0.03821855 -0.327582337 0.005225766 ## 51 4.3 male 5.500 4.327582 0.03821855 -0.027582337 0.005225766 ## 52 4.4 male 5.500 4.327582 0.03821855 0.072417663 0.005225766 ## 53 4.5 male 5.500 4.327582 0.03821855 0.172417663 0.005225766 ## 54 5.0 male 5.500 4.327582 0.03821855 0.672417663 0.005225766 ## 55 4.9 male 5.500 4.327582 0.03821855 0.572417663 0.005225766 ## 56 4.6 male 5.500 4.327582 0.03821855 0.272417663 0.005225766 ## 57 5.0 male 5.500 4.327582 0.03821855 0.672417663 0.005225766 ## 58 4.7 male 5.500 4.327582 0.03821855 0.372417663 0.005225766 ## 59 5.0 male 5.500 4.327582 0.03821855 0.672417663 0.005225766 ## 60 3.6 male 4.833 4.278121 0.03369075 -0.678120703 0.004060904 ## 61 3.7 male 4.833 4.278121 0.03369075 -0.578120703 0.004060904 ## 62 4.3 male 4.833 4.278121 0.03369075 0.021879297 0.004060904 ## 63 4.1 male 4.333 4.241043 0.03232826 -0.141043017 0.003739091 ## 64 4.2 male 4.333 4.241043 0.03232826 -0.041043017 0.003739091 ## 65 4.7 male 4.333 4.241043 0.03232826 0.458956983 0.003739091 ## 66 4.7 male 4.333 4.241043 0.03232826 0.458956983 0.003739091 ## 67 3.5 male 4.333 4.241043 0.03232826 -0.741043017 0.003739091 ## 68 4.1 male 4.833 4.278121 0.03369075 -0.178120703 0.004060904 ## 69 4.2 male 4.833 4.278121 0.03369075 -0.078120703 0.004060904 ## 70 4.0 male 4.833 4.278121 0.03369075 -0.278120703 0.004060904 ## 71 4.0 male 4.833 4.278121 0.03369075 -0.278120703 0.004060904 ## 72 3.9 male 4.833 4.278121 0.03369075 -0.378120703 0.004060904 ## 73 4.4 male 4.833 4.278121 0.03369075 0.121879297 0.004060904 ## 74 3.8 male 4.833 4.278121 0.03369075 -0.478120703 0.004060904 ## 75 3.5 male 4.000 4.216349 0.03253424 -0.716349277 0.003786892 ## 76 4.2 male 4.000 4.216349 0.03253424 -0.016349277 0.003786892 ## 77 3.5 male 4.000 4.216349 0.03253424 -0.716349277 0.003786892 ## 78 3.6 male 4.000 4.216349 0.03253424 -0.616349277 0.003786892 ## 79 2.9 female 5.500 4.155193 0.04025233 -1.255192787 0.005796737 ## 80 3.3 female 5.500 4.155193 0.04025233 -0.855192787 0.005796737 ## 81 3.3 female 5.500 4.155193 0.04025233 -0.855192787 0.005796737 ## 82 3.2 female 5.500 4.155193 0.04025233 -0.955192787 0.005796737 ## 83 4.6 male 4.167 4.228733 0.03231802 0.371266775 0.003736723 ## 84 4.2 male 4.167 4.228733 0.03231802 -0.028733225 0.003736723 ## 85 4.3 male 4.167 4.228733 0.03231802 0.071266775 0.003736723 ## 86 4.4 male 4.167 4.228733 0.03231802 0.171266775 0.003736723 ## 87 4.1 male 4.167 4.228733 0.03231802 -0.128733225 0.003736723 ## 88 4.6 male 4.167 4.228733 0.03231802 0.371266775 0.003736723 ## 89 4.4 female 2.500 3.932727 0.05161850 0.467273332 0.009532618 ## 90 4.8 female 2.500 3.932727 0.05161850 0.867273332 0.009532618 ## 91 4.3 female 2.500 3.932727 0.05161850 0.367273332 0.009532618 ## 92 3.6 female 2.500 3.932727 0.05161850 -0.332726668 0.009532618 ## 93 4.3 female 2.500 3.932727 0.05161850 0.367273332 0.009532618 ## 94 4.0 male 4.333 4.241043 0.03232826 -0.241043017 0.003739091 ## 95 4.2 male 4.333 4.241043 0.03232826 -0.041043017 0.003739091 ## 96 4.1 male 4.333 4.241043 0.03232826 -0.141043017 0.003739091 ## 97 4.1 male 4.333 4.241043 0.03232826 -0.141043017 0.003739091 ## 98 4.4 male 4.333 4.241043 0.03232826 0.158956983 0.003739091 ## 99 4.3 male 4.333 4.241043 0.03232826 0.058956983 0.003739091 ## 100 4.4 male 4.333 4.241043 0.03232826 0.158956983 0.003739091 ## 101 4.4 male 4.333 4.241043 0.03232826 0.158956983 0.003739091 ## 102 4.9 female 4.333 4.068653 0.03822881 0.831346533 0.005228571 ## 103 5.0 female 4.333 4.068653 0.03822881 0.931346533 0.005228571 ## 104 4.4 female 4.333 4.068653 0.03822881 0.331346533 0.005228571 ## 105 4.8 female 4.333 4.068653 0.03822881 0.731346533 0.005228571 ## 106 4.9 female 4.333 4.068653 0.03822881 0.831346533 0.005228571 ## 107 4.3 female 4.333 4.068653 0.03822881 0.231346533 0.005228571 ## 108 5.0 female 4.333 4.068653 0.03822881 0.931346533 0.005228571 ## 109 4.7 female 4.333 4.068653 0.03822881 0.631346533 0.005228571 ## 110 4.5 female 4.333 4.068653 0.03822881 0.431346533 0.005228571 ## 111 3.5 female 4.333 4.068653 0.03822881 -0.568653467 0.005228571 ## 112 3.9 female 4.333 4.068653 0.03822881 -0.168653467 0.005228571 ## 113 4.0 female 4.333 4.068653 0.03822881 -0.068653467 0.005228571 ## 114 4.0 female 4.333 4.068653 0.03822881 -0.068653467 0.005228571 ## 115 3.7 female 4.333 4.068653 0.03822881 -0.368653467 0.005228571 ## 116 3.4 female 4.333 4.068653 0.03822881 -0.668653467 0.005228571 ## 117 3.3 female 4.333 4.068653 0.03822881 -0.768653467 0.005228571 ## 118 3.8 female 4.333 4.068653 0.03822881 -0.268653467 0.005228571 ## 119 3.9 female 4.333 4.068653 0.03822881 -0.168653467 0.005228571 ## 120 3.4 female 4.333 4.068653 0.03822881 -0.668653467 0.005228571 ## 121 3.7 female 4.833 4.105731 0.03796571 -0.405731153 0.005156849 ## 122 4.1 female 4.833 4.105731 0.03796571 -0.005731153 0.005156849 ## 123 3.7 female 4.833 4.105731 0.03796571 -0.405731153 0.005156849 ## 124 3.5 female 4.833 4.105731 0.03796571 -0.605731153 0.005156849 ## 125 3.5 female 4.833 4.105731 0.03796571 -0.605731153 0.005156849 ## 126 4.4 female 4.833 4.105731 0.03796571 0.294268847 0.005156849 ## 127 3.4 female 2.833 3.957420 0.04810386 -0.557420407 0.008278685 ## 128 4.3 male 3.000 4.142194 0.03808791 0.157806096 0.005190100 ## 129 3.7 male 3.000 4.142194 0.03808791 -0.442193904 0.005190100 ## 130 4.7 male 3.000 4.142194 0.03808791 0.557806096 0.005190100 ## 131 3.9 male 3.000 4.142194 0.03808791 -0.242193904 0.005190100 ## 132 3.6 male 3.000 4.142194 0.03808791 -0.542193904 0.005190100 ## 133 4.5 male 4.167 4.228733 0.03231802 0.271266775 0.003736723 ## 134 4.5 male 4.167 4.228733 0.03231802 0.271266775 0.003736723 ## 135 4.8 male 4.167 4.228733 0.03231802 0.571266775 0.003736723 ## 136 4.8 male 4.167 4.228733 0.03231802 0.571266775 0.003736723 ## 137 4.7 male 4.167 4.228733 0.03231802 0.471266775 0.003736723 ## 138 4.5 male 4.167 4.228733 0.03231802 0.271266775 0.003736723 ## 139 4.3 male 4.167 4.228733 0.03231802 0.071266775 0.003736723 ## 140 4.8 female 7.833 4.328197 0.06398835 0.471802728 0.014648840 ## 141 4.1 female 7.833 4.328197 0.06398835 -0.228197272 0.014648840 ## 142 4.4 male 3.833 4.203965 0.03297321 0.196034670 0.003889770 ## 143 4.3 male 3.833 4.203965 0.03297321 0.096034670 0.003889770 ## 144 3.6 male 3.833 4.203965 0.03297321 -0.603965330 0.003889770 ## 145 4.5 male 3.833 4.203965 0.03297321 0.296034670 0.003889770 ## 146 4.3 male 3.833 4.203965 0.03297321 0.096034670 0.003889770 ## 147 4.4 male 4.833 4.278121 0.03369075 0.121879297 0.004060904 ## 148 4.7 male 4.833 4.278121 0.03369075 0.421879297 0.004060904 ## 149 4.8 male 4.833 4.278121 0.03369075 0.521879297 0.004060904 ## 150 3.5 male 4.833 4.278121 0.03369075 -0.778120703 0.004060904 ## 151 3.8 male 4.833 4.278121 0.03369075 -0.478120703 0.004060904 ## 152 3.6 male 4.833 4.278121 0.03369075 -0.678120703 0.004060904 ## 153 4.2 male 4.833 4.278121 0.03369075 -0.078120703 0.004060904 ## 154 3.6 male 3.000 4.142194 0.03808791 -0.542193904 0.005190100 ## 155 4.4 male 3.000 4.142194 0.03808791 0.257806096 0.005190100 ## 156 3.7 male 3.000 4.142194 0.03808791 -0.442193904 0.005190100 ## 157 4.3 male 3.000 4.142194 0.03808791 0.157806096 0.005190100 ## 158 4.6 male 3.000 4.142194 0.03808791 0.457806096 0.005190100 ## 159 4.6 male 3.000 4.142194 0.03808791 0.457806096 0.005190100 ## 160 4.1 male 3.000 4.142194 0.03808791 -0.042193904 0.005190100 ## 161 3.6 male 3.000 4.142194 0.03808791 -0.542193904 0.005190100 ## 162 2.3 female 5.167 4.130499 0.03875022 -1.830499048 0.005372170 ## 163 4.3 male 4.333 4.241043 0.03232826 0.058956983 0.003739091 ## 164 4.4 male 4.333 4.241043 0.03232826 0.158956983 0.003739091 ## 165 3.6 male 4.333 4.241043 0.03232826 -0.641043017 0.003739091 ## 166 4.4 male 4.333 4.241043 0.03232826 0.158956983 0.003739091 ## 167 3.9 male 4.333 4.241043 0.03232826 -0.341043017 0.003739091 ## 168 3.8 male 4.333 4.241043 0.03232826 -0.441043017 0.003739091 ## 169 3.4 male 4.333 4.241043 0.03232826 -0.841043017 0.003739091 ## 170 4.9 male 2.667 4.117500 0.04121336 0.782499835 0.006076836 ## 171 4.1 male 2.667 4.117500 0.04121336 -0.017500165 0.006076836 ## 172 3.2 male 2.667 4.117500 0.04121336 -0.917500165 0.006076836 ## 173 4.2 male 5.500 4.327582 0.03821855 -0.127582337 0.005225766 ## 174 3.9 male 5.500 4.327582 0.03821855 -0.427582337 0.005225766 ## 175 4.9 male 5.500 4.327582 0.03821855 0.572417663 0.005225766 ## 176 4.7 male 5.500 4.327582 0.03821855 0.372417663 0.005225766 ## 177 4.4 male 5.500 4.327582 0.03821855 0.072417663 0.005225766 ## 178 4.2 female 4.333 4.068653 0.03822881 0.131346533 0.005228571 ## 179 4.0 female 4.333 4.068653 0.03822881 -0.068653467 0.005228571 ## 180 4.4 female 4.333 4.068653 0.03822881 0.331346533 0.005228571 ## 181 3.9 female 4.333 4.068653 0.03822881 -0.168653467 0.005228571 ## 182 4.4 female 4.333 4.068653 0.03822881 0.331346533 0.005228571 ## 183 3.0 female 4.333 4.068653 0.03822881 -1.068653467 0.005228571 ## 184 3.5 female 4.333 4.068653 0.03822881 -0.568653467 0.005228571 ## 185 2.8 female 4.333 4.068653 0.03822881 -1.268653467 0.005228571 ## 186 4.6 female 4.333 4.068653 0.03822881 0.531346533 0.005228571 ## 187 4.3 female 4.333 4.068653 0.03822881 0.231346533 0.005228571 ## 188 3.4 female 4.333 4.068653 0.03822881 -0.668653467 0.005228571 ## 189 3.0 female 4.333 4.068653 0.03822881 -1.068653467 0.005228571 ## 190 4.2 female 4.333 4.068653 0.03822881 0.131346533 0.005228571 ## 191 4.3 male 2.333 4.092732 0.04478818 0.207267729 0.007176756 ## 192 4.1 male 2.333 4.092732 0.04478818 0.007267729 0.007176756 ## 193 4.6 male 2.333 4.092732 0.04478818 0.507267729 0.007176756 ## 194 3.9 female 6.500 4.229348 0.04825670 -0.329348160 0.008331375 ## 195 3.5 female 6.500 4.229348 0.04825670 -0.729348160 0.008331375 ## 196 4.0 female 6.500 4.229348 0.04825670 -0.229348160 0.008331375 ## 197 4.0 female 6.500 4.229348 0.04825670 -0.229348160 0.008331375 ## 198 3.9 male 2.333 4.092732 0.04478818 -0.192732271 0.007176756 ## 199 3.3 male 2.333 4.092732 0.04478818 -0.792732271 0.007176756 ## 200 4.0 male 2.333 4.092732 0.04478818 -0.092732271 0.007176756 ## 201 3.8 male 2.333 4.092732 0.04478818 -0.292732271 0.007176756 ## 202 4.2 male 2.333 4.092732 0.04478818 0.107267729 0.007176756 ## 203 4.0 male 2.333 4.092732 0.04478818 -0.092732271 0.007176756 ## 204 3.8 male 2.333 4.092732 0.04478818 -0.292732271 0.007176756 ## 205 3.3 male 2.333 4.092732 0.04478818 -0.792732271 0.007176756 ## 206 4.1 male 3.000 4.142194 0.03808791 -0.042193904 0.005190100 ## 207 4.7 male 3.000 4.142194 0.03808791 0.557806096 0.005190100 ## 208 4.4 male 3.000 4.142194 0.03808791 0.257806096 0.005190100 ## 209 4.8 male 3.667 4.191656 0.03362167 0.608344462 0.004044269 ## 210 4.8 male 3.667 4.191656 0.03362167 0.608344462 0.004044269 ## 211 4.6 male 3.667 4.191656 0.03362167 0.408344462 0.004044269 ## 212 4.6 male 3.667 4.191656 0.03362167 0.408344462 0.004044269 ## 213 4.8 male 3.667 4.191656 0.03362167 0.608344462 0.004044269 ## 214 4.4 male 3.667 4.191656 0.03362167 0.208344462 0.004044269 ## 215 4.7 male 3.667 4.191656 0.03362167 0.508344462 0.004044269 ## 216 4.7 male 3.667 4.191656 0.03362167 0.508344462 0.004044269 ## 217 3.3 male 6.167 4.377044 0.04495853 -1.077043971 0.007231454 ## 218 4.4 male 4.000 4.216349 0.03253424 0.183650723 0.003786892 ## 219 4.3 male 4.000 4.216349 0.03253424 0.083650723 0.003786892 ## 220 4.9 male 4.000 4.216349 0.03253424 0.683650723 0.003786892 ## 221 4.4 male 4.000 4.216349 0.03253424 0.183650723 0.003786892 ## 222 4.7 male 4.000 4.216349 0.03253424 0.483650723 0.003786892 ## 223 4.3 male 4.833 4.278121 0.03369075 0.021879297 0.004060904 ## 224 4.8 male 4.833 4.278121 0.03369075 0.521879297 0.004060904 ## 225 4.5 male 4.833 4.278121 0.03369075 0.221879297 0.004060904 ## 226 4.7 male 4.833 4.278121 0.03369075 0.421879297 0.004060904 ## 227 3.3 female 8.167 4.352965 0.06843987 -1.052965167 0.016757907 ## 228 4.7 female 8.167 4.352965 0.06843987 0.347034833 0.016757907 ## 229 4.6 female 8.167 4.352965 0.06843987 0.247034833 0.016757907 ## 230 3.6 female 8.167 4.352965 0.06843987 -0.752965167 0.016757907 ## 231 4.0 male 6.500 4.401738 0.04887853 -0.401737710 0.008547473 ## 232 4.1 male 6.500 4.401738 0.04887853 -0.301737710 0.008547473 ## 233 4.0 male 6.500 4.401738 0.04887853 -0.401737710 0.008547473 ## 234 4.5 male 4.833 4.278121 0.03369075 0.221879297 0.004060904 ## 235 4.6 male 4.833 4.278121 0.03369075 0.321879297 0.004060904 ## 236 4.8 male 4.833 4.278121 0.03369075 0.521879297 0.004060904 ## 237 4.6 male 4.833 4.278121 0.03369075 0.321879297 0.004060904 ## 238 4.9 male 7.000 4.438815 0.05523981 0.461184604 0.010917064 ## 239 3.1 male 7.000 4.438815 0.05523981 -1.338815396 0.010917064 ## 240 3.7 male 7.000 4.438815 0.05523981 -0.738815396 0.010917064 ## 241 3.7 female 4.667 4.093421 0.03786035 -0.393421361 0.005128267 ## 242 3.9 female 3.833 4.031576 0.04016909 -0.131575780 0.005772784 ## 243 3.9 female 3.833 4.031576 0.04016909 -0.131575780 0.005772784 ## 244 3.2 female 3.833 4.031576 0.04016909 -0.831575780 0.005772784 ## 245 4.4 female 3.167 3.982188 0.04495870 0.417811698 0.007231509 ## 246 4.2 female 3.167 3.982188 0.04495870 0.217811698 0.007231509 ## 247 4.7 female 3.167 3.982188 0.04495870 0.717811698 0.007231509 ## 248 3.9 female 3.167 3.982188 0.04495870 -0.082188302 0.007231509 ## 249 3.6 female 3.167 3.982188 0.04495870 -0.382188302 0.007231509 ## 250 3.4 female 3.167 3.982188 0.04495870 -0.582188302 0.007231509 ## 251 4.4 female 3.167 3.982188 0.04495870 0.417811698 0.007231509 ## 252 4.4 male 3.167 4.154578 0.03672119 0.245422148 0.004824307 ## 253 4.1 male 3.167 4.154578 0.03672119 -0.054577852 0.004824307 ## 254 3.6 male 3.167 4.154578 0.03672119 -0.554577852 0.004824307 ## 255 3.5 male 3.167 4.154578 0.03672119 -0.654577852 0.004824307 ## 256 4.1 male 3.167 4.154578 0.03672119 -0.054577852 0.004824307 ## 257 3.8 male 3.167 4.154578 0.03672119 -0.354577852 0.004824307 ## 258 4.0 male 3.167 4.154578 0.03672119 -0.154577852 0.004824307 ## 259 4.8 male 3.167 4.154578 0.03672119 0.645422148 0.004824307 ## 260 4.2 male 3.167 4.154578 0.03672119 0.045422148 0.004824307 ## 261 4.6 male 3.167 4.154578 0.03672119 0.445422148 0.004824307 ## 262 4.3 male 3.167 4.154578 0.03672119 0.145422148 0.004824307 ## 263 4.8 male 3.167 4.154578 0.03672119 0.645422148 0.004824307 ## 264 3.8 male 3.167 4.154578 0.03672119 -0.354577852 0.004824307 ## 265 4.5 female 5.833 4.179887 0.04239694 0.320113474 0.006430880 ## 266 4.9 female 5.833 4.179887 0.04239694 0.720113474 0.006430880 ## 267 4.9 female 5.833 4.179887 0.04239694 0.720113474 0.006430880 ## 268 4.8 female 5.833 4.179887 0.04239694 0.620113474 0.006430880 ## 269 4.7 female 5.833 4.179887 0.04239694 0.520113474 0.006430880 ## 270 4.6 female 5.833 4.179887 0.04239694 0.420113474 0.006430880 ## 271 4.3 male 5.667 4.339966 0.03973620 -0.039966284 0.005649034 ## 272 4.4 male 5.667 4.339966 0.03973620 0.060033716 0.005649034 ## 273 4.5 male 5.667 4.339966 0.03973620 0.160033716 0.005649034 ## 274 4.2 male 5.667 4.339966 0.03973620 -0.139966284 0.005649034 ## 275 4.8 female 6.500 4.229348 0.04825670 0.570651840 0.008331375 ## 276 4.6 female 6.500 4.229348 0.04825670 0.370651840 0.008331375 ## 277 4.9 female 6.500 4.229348 0.04825670 0.670651840 0.008331375 ## 278 4.8 female 6.500 4.229348 0.04825670 0.570651840 0.008331375 ## 279 4.8 female 6.500 4.229348 0.04825670 0.570651840 0.008331375 ## 280 4.6 female 6.500 4.229348 0.04825670 0.370651840 0.008331375 ## 281 4.7 female 6.500 4.229348 0.04825670 0.470651840 0.008331375 ## 282 4.1 female 1.667 3.870955 0.06162598 0.229044757 0.013587174 ## 283 3.8 female 1.667 3.870955 0.06162598 -0.070955243 0.013587174 ## 284 4.0 female 1.667 3.870955 0.06162598 0.129044757 0.013587174 ## 285 4.1 female 1.667 3.870955 0.06162598 0.229044757 0.013587174 ## 286 4.0 female 1.667 3.870955 0.06162598 0.129044757 0.013587174 ## 287 4.1 female 1.667 3.870955 0.06162598 0.229044757 0.013587174 ## 288 3.5 male 6.667 4.414122 0.05094741 -0.914121657 0.009286365 ## 289 4.1 male 6.667 4.414122 0.05094741 -0.314121657 0.009286365 ## 290 3.6 male 6.667 4.414122 0.05094741 -0.814121657 0.009286365 ## 291 4.0 male 3.667 4.191656 0.03362167 -0.191655538 0.004044269 ## 292 3.9 male 3.667 4.191656 0.03362167 -0.291655538 0.004044269 ## 293 3.8 male 3.667 4.191656 0.03362167 -0.391655538 0.004044269 ## 294 4.4 male 3.833 4.203965 0.03297321 0.196034670 0.003889770 ## 295 4.7 male 3.833 4.203965 0.03297321 0.496034670 0.003889770 ## 296 3.8 female 6.167 4.204654 0.04510128 -0.404654421 0.007277449 ## 297 4.1 female 6.167 4.204654 0.04510128 -0.104654421 0.007277449 ## 298 4.1 female 3.333 3.994498 0.04356300 0.105501906 0.006789489 ## 299 4.7 female 3.333 3.994498 0.04356300 0.705501906 0.006789489 ## 300 4.3 female 3.333 3.994498 0.04356300 0.305501906 0.006789489 ## 301 4.4 female 3.333 3.994498 0.04356300 0.405501906 0.006789489 ## 302 4.5 female 3.333 3.994498 0.04356300 0.505501906 0.006789489 ## 303 3.1 female 3.333 3.994498 0.04356300 -0.894498094 0.006789489 ## 304 3.7 female 3.333 3.994498 0.04356300 -0.294498094 0.006789489 ## 305 4.5 female 3.333 3.994498 0.04356300 0.505501906 0.006789489 ## 306 3.0 female 3.333 3.994498 0.04356300 -0.994498094 0.006789489 ## 307 4.6 female 3.333 3.994498 0.04356300 0.605501906 0.006789489 ## 308 3.7 male 3.667 4.191656 0.03362167 -0.491655538 0.004044269 ## 309 3.6 male 3.667 4.191656 0.03362167 -0.591655538 0.004044269 ## 310 3.2 female 3.500 4.006882 0.04228629 -0.806882041 0.006397358 ## 311 3.3 female 3.500 4.006882 0.04228629 -0.706882041 0.006397358 ## 312 2.9 female 3.500 4.006882 0.04228629 -1.106882041 0.006397358 ## 313 4.2 male 2.667 4.117500 0.04121336 0.082499835 0.006076836 ## 314 4.5 male 5.667 4.339966 0.03973620 0.160033716 0.005649034 ## 315 3.8 female 6.000 4.192270 0.04368577 -0.392270474 0.006827810 ## 316 3.7 female 6.000 4.192270 0.04368577 -0.492270474 0.006827810 ## 317 3.7 female 6.500 4.229348 0.04825670 -0.529348160 0.008331375 ## 318 4.0 female 6.500 4.229348 0.04825670 -0.229348160 0.008331375 ## 319 3.7 female 6.500 4.229348 0.04825670 -0.529348160 0.008331375 ## 320 4.5 female 2.333 3.920343 0.05350042 0.579657279 0.010240373 ## 321 3.8 female 2.333 3.920343 0.05350042 -0.120342721 0.010240373 ## 322 3.9 female 2.333 3.920343 0.05350042 -0.020342721 0.010240373 ## 323 4.6 female 2.333 3.920343 0.05350042 0.679657279 0.010240373 ## 324 4.5 female 2.333 3.920343 0.05350042 0.579657279 0.010240373 ## 325 4.2 female 2.333 3.920343 0.05350042 0.279657279 0.010240373 ## 326 4.0 female 2.333 3.920343 0.05350042 0.079657279 0.010240373 ## 327 3.8 male 2.333 4.092732 0.04478818 -0.292732271 0.007176756 ## 328 3.5 male 2.333 4.092732 0.04478818 -0.592732271 0.007176756 ## 329 2.7 male 2.333 4.092732 0.04478818 -1.392732271 0.007176756 ## 330 4.0 male 2.333 4.092732 0.04478818 -0.092732271 0.007176756 ## 331 4.6 male 2.333 4.092732 0.04478818 0.507267729 0.007176756 ## 332 3.9 male 2.333 4.092732 0.04478818 -0.192732271 0.007176756 ## 333 4.5 male 7.167 4.451199 0.05746369 0.048800656 0.011813768 ## 334 3.7 male 7.167 4.451199 0.05746369 -0.751199344 0.011813768 ## 335 2.4 male 1.667 4.043345 0.05286711 -1.643344792 0.009999369 ## 336 3.1 male 1.667 4.043345 0.05286711 -0.943344792 0.009999369 ## 337 2.5 male 1.667 4.043345 0.05286711 -1.543344792 0.009999369 ## 338 3.0 female 5.167 4.130499 0.03875022 -1.130499048 0.005372170 ## 339 4.5 male 3.500 4.179272 0.03447535 0.320728409 0.004252250 ## 340 4.8 male 3.500 4.179272 0.03447535 0.620728409 0.004252250 ## 341 4.9 male 3.500 4.179272 0.03447535 0.720728409 0.004252250 ## 342 4.5 male 3.500 4.179272 0.03447535 0.320728409 0.004252250 ## 343 4.6 male 3.500 4.179272 0.03447535 0.420728409 0.004252250 ## 344 4.5 male 3.500 4.179272 0.03447535 0.320728409 0.004252250 ## 345 4.9 male 3.500 4.179272 0.03447535 0.720728409 0.004252250 ## 346 4.4 male 3.500 4.179272 0.03447535 0.220728409 0.004252250 ## 347 4.6 male 3.500 4.179272 0.03447535 0.420728409 0.004252250 ## 348 4.6 male 3.333 4.166888 0.03551641 0.433112356 0.004512941 ## 349 5.0 male 3.333 4.166888 0.03551641 0.833112356 0.004512941 ## 350 4.9 male 3.333 4.166888 0.03551641 0.733112356 0.004512941 ## 351 4.6 male 3.333 4.166888 0.03551641 0.433112356 0.004512941 ## 352 4.8 male 3.333 4.166888 0.03551641 0.633112356 0.004512941 ## 353 4.9 male 3.333 4.166888 0.03551641 0.733112356 0.004512941 ## 354 4.9 male 3.333 4.166888 0.03551641 0.733112356 0.004512941 ## 355 4.9 male 3.333 4.166888 0.03551641 0.733112356 0.004512941 ## 356 5.0 male 3.333 4.166888 0.03551641 0.833112356 0.004512941 ## 357 4.5 male 3.333 4.166888 0.03551641 0.333112356 0.004512941 ## 358 3.5 male 5.833 4.352276 0.04136624 -0.852276076 0.006122003 ## 359 3.8 male 5.833 4.352276 0.04136624 -0.552276076 0.006122003 ## 360 3.9 male 5.833 4.352276 0.04136624 -0.452276076 0.006122003 ## 361 3.9 male 5.833 4.352276 0.04136624 -0.452276076 0.006122003 ## 362 4.2 male 5.833 4.352276 0.04136624 -0.152276076 0.006122003 ## 363 4.1 male 5.833 4.352276 0.04136624 -0.252276076 0.006122003 ## 364 4.8 male 6.167 4.377044 0.04495853 0.422956029 0.007231454 ## 365 4.8 male 6.167 4.377044 0.04495853 0.422956029 0.007231454 ## 366 4.8 male 6.167 4.377044 0.04495853 0.422956029 0.007231454 ## 367 4.8 male 6.167 4.377044 0.04495853 0.422956029 0.007231454 ## 368 4.9 male 6.167 4.377044 0.04495853 0.522956029 0.007231454 ## 369 4.2 male 3.333 4.166888 0.03551641 0.033112356 0.004512941 ## 370 4.5 male 3.333 4.166888 0.03551641 0.333112356 0.004512941 ## 371 3.9 male 3.333 4.166888 0.03551641 -0.266887644 0.004512941 ## 372 4.4 male 3.333 4.166888 0.03551641 0.233112356 0.004512941 ## 373 4.0 female 5.167 4.130499 0.03875022 -0.130499048 0.005372170 ## 374 3.6 female 5.167 4.130499 0.03875022 -0.530499048 0.005372170 ## 375 3.7 female 4.167 4.056344 0.03869495 -0.356343675 0.005356856 ## 376 2.7 female 4.167 4.056344 0.03869495 -1.356343675 0.005356856 ## 377 4.5 female 2.500 3.932727 0.05161850 0.567273332 0.009532618 ## 378 4.4 female 2.500 3.932727 0.05161850 0.467273332 0.009532618 ## 379 3.9 female 2.500 3.932727 0.05161850 -0.032726668 0.009532618 ## 380 3.6 female 2.500 3.932727 0.05161850 -0.332726668 0.009532618 ## 381 4.4 female 2.500 3.932727 0.05161850 0.467273332 0.009532618 ## 382 4.4 female 2.500 3.932727 0.05161850 0.467273332 0.009532618 ## 383 4.7 male 4.333 4.241043 0.03232826 0.458956983 0.003739091 ## 384 4.5 male 4.333 4.241043 0.03232826 0.258956983 0.003739091 ## 385 4.1 male 4.333 4.241043 0.03232826 -0.141043017 0.003739091 ## 386 3.7 male 4.333 4.241043 0.03232826 -0.541043017 0.003739091 ## 387 4.3 male 3.000 4.142194 0.03808791 0.157806096 0.005190100 ## 388 3.5 male 3.000 4.142194 0.03808791 -0.642193904 0.005190100 ## 389 3.7 male 3.000 4.142194 0.03808791 -0.442193904 0.005190100 ## 390 4.0 female 6.333 4.216964 0.04662245 -0.216964213 0.007776633 ## 391 4.0 female 6.333 4.216964 0.04662245 -0.216964213 0.007776633 ## 392 3.1 female 6.333 4.216964 0.04662245 -1.116964213 0.007776633 ## 393 4.5 female 6.333 4.216964 0.04662245 0.283035787 0.007776633 ## 394 4.8 male 3.333 4.166888 0.03551641 0.633112356 0.004512941 ## 395 4.2 male 3.333 4.166888 0.03551641 0.033112356 0.004512941 ## 396 4.9 male 3.333 4.166888 0.03551641 0.733112356 0.004512941 ## 397 4.8 male 3.333 4.166888 0.03551641 0.633112356 0.004512941 ## 398 3.5 male 2.833 4.129810 0.03959375 -0.629809957 0.005608603 ## 399 3.6 male 2.833 4.129810 0.03959375 -0.529809957 0.005608603 ## 400 4.4 male 2.833 4.129810 0.03959375 0.270190043 0.005608603 ## 401 3.4 male 2.833 4.129810 0.03959375 -0.729809957 0.005608603 ## 402 3.9 male 2.833 4.129810 0.03959375 -0.229809957 0.005608603 ## 403 3.8 male 2.833 4.129810 0.03959375 -0.329809957 0.005608603 ## 404 4.8 male 2.833 4.129810 0.03959375 0.670190043 0.005608603 ## 405 4.6 male 2.833 4.129810 0.03959375 0.470190043 0.005608603 ## 406 5.0 male 2.833 4.129810 0.03959375 0.870190043 0.005608603 ## 407 3.8 male 2.833 4.129810 0.03959375 -0.329809957 0.005608603 ## 408 4.2 male 2.833 4.129810 0.03959375 0.070190043 0.005608603 ## 409 3.3 female 6.667 4.241732 0.04998498 -0.941732107 0.008938826 ## 410 4.7 female 6.667 4.241732 0.04998498 0.458267893 0.008938826 ## 411 4.6 female 6.667 4.241732 0.04998498 0.358267893 0.008938826 ## 412 4.6 female 6.667 4.241732 0.04998498 0.358267893 0.008938826 ## 413 4.0 female 6.667 4.241732 0.04998498 -0.241732107 0.008938826 ## 414 4.2 female 6.833 4.254042 0.05178679 -0.054041899 0.009594877 ## 415 4.9 female 6.833 4.254042 0.05178679 0.645958101 0.009594877 ## 416 4.5 female 6.833 4.254042 0.05178679 0.245958101 0.009594877 ## 417 4.8 female 6.833 4.254042 0.05178679 0.545958101 0.009594877 ## 418 3.8 female 6.833 4.254042 0.05178679 -0.454041899 0.009594877 ## 419 4.8 male 7.833 4.500587 0.06669455 0.299413178 0.015914101 ## 420 5.0 male 7.833 4.500587 0.06669455 0.499413178 0.015914101 ## 421 5.0 male 7.833 4.500587 0.06669455 0.499413178 0.015914101 ## 422 4.9 male 7.833 4.500587 0.06669455 0.399413178 0.015914101 ## 423 4.6 male 7.833 4.500587 0.06669455 0.099413178 0.015914101 ## 424 5.0 male 7.833 4.500587 0.06669455 0.499413178 0.015914101 ## 425 4.8 male 7.833 4.500587 0.06669455 0.299413178 0.015914101 ## 426 4.9 male 7.833 4.500587 0.06669455 0.399413178 0.015914101 ## 427 4.9 male 7.833 4.500587 0.06669455 0.399413178 0.015914101 ## 428 3.9 male 7.833 4.500587 0.06669455 -0.600586822 0.015914101 ## 429 3.9 male 7.833 4.500587 0.06669455 -0.600586822 0.015914101 ## 430 4.5 male 5.833 4.352276 0.04136624 0.147723924 0.006122003 ## 431 4.5 male 5.833 4.352276 0.04136624 0.147723924 0.006122003 ## 432 3.3 male 2.000 4.068039 0.04869463 -0.768038531 0.008483274 ## 433 3.1 male 2.000 4.068039 0.04869463 -0.968038531 0.008483274 ## 434 2.8 male 2.000 4.068039 0.04869463 -1.268038531 0.008483274 ## 435 3.1 male 2.000 4.068039 0.04869463 -0.968038531 0.008483274 ## 436 4.2 male 2.000 4.068039 0.04869463 0.131961469 0.008483274 ## 437 3.4 male 2.000 4.068039 0.04869463 -0.668038531 0.008483274 ## 438 3.0 male 2.000 4.068039 0.04869463 -1.068038531 0.008483274 ## 439 3.3 female 7.833 4.328197 0.06398835 -1.028197272 0.014648840 ## 440 3.6 female 7.833 4.328197 0.06398835 -0.728197272 0.014648840 ## 441 3.7 female 7.833 4.328197 0.06398835 -0.628197272 0.014648840 ## 442 3.6 male 3.333 4.166888 0.03551641 -0.566887644 0.004512941 ## 443 4.3 male 3.333 4.166888 0.03551641 0.133112356 0.004512941 ## 444 4.1 female 4.500 4.081037 0.03794809 0.018962586 0.005152064 ## 445 4.9 female 4.500 4.081037 0.03794809 0.818962586 0.005152064 ## 446 4.8 female 4.500 4.081037 0.03794809 0.718962586 0.005152064 ## 447 3.7 female 4.333 4.068653 0.03822881 -0.368653467 0.005228571 ## 448 3.9 female 4.333 4.068653 0.03822881 -0.168653467 0.005228571 ## 449 4.5 female 4.333 4.068653 0.03822881 0.431346533 0.005228571 ## 450 3.6 female 4.333 4.068653 0.03822881 -0.468653467 0.005228571 ## 451 4.4 female 4.333 4.068653 0.03822881 0.331346533 0.005228571 ## 452 3.4 female 4.333 4.068653 0.03822881 -0.668653467 0.005228571 ## 453 4.4 female 4.333 4.068653 0.03822881 0.331346533 0.005228571 ## 454 4.5 male 6.833 4.426431 0.05306158 0.073568551 0.010073069 ## 455 4.5 male 6.833 4.426431 0.05306158 0.073568551 0.010073069 ## 456 4.5 male 6.833 4.426431 0.05306158 0.073568551 0.010073069 ## 457 4.6 male 6.833 4.426431 0.05306158 0.173568551 0.010073069 ## 458 4.1 male 6.833 4.426431 0.05306158 -0.326431449 0.010073069 ## 459 4.5 male 6.833 4.426431 0.05306158 0.073568551 0.010073069 ## 460 3.5 female 5.333 4.142809 0.03941338 -0.642808840 0.005557619 ## 461 4.4 female 5.333 4.142809 0.03941338 0.257191160 0.005557619 ## 462 4.4 female 5.333 4.142809 0.03941338 0.257191160 0.005557619 ## 463 4.1 female 5.333 4.142809 0.03941338 -0.042808840 0.005557619 ## .sigma .cooksd .std.resid ## 1 0.5285624 2.137443e-03 1.10351503 ## 2 0.5292627 2.071580e-06 -0.03435436 ## 3 0.5291649 3.003252e-04 -0.41364416 ## 4 0.5283005 2.935232e-03 1.29315993 ## 5 0.5288296 1.310807e-03 0.86818494 ## 6 0.5292119 1.557483e-04 0.29926398 ## 7 0.5255230 1.126695e-02 -2.54534079 ## 8 0.5292541 2.429743e-05 -0.12680279 ## 9 0.5280460 3.193979e-03 -1.45383345 ## 10 0.5287072 2.346156e-03 0.98298885 ## 11 0.5291946 2.904385e-04 -0.34585752 ## 12 0.5287072 2.346156e-03 0.98298885 ## 13 0.5284715 3.339838e-03 1.17282404 ## 14 0.5292494 5.910622e-05 -0.15602232 ## 15 0.5292494 5.910622e-05 -0.15602232 ## 16 0.5290539 8.837978e-04 0.60331846 ## 17 0.5287072 2.346156e-03 0.98298885 ## 18 0.5287237 3.759402e-03 0.96830651 ## 19 0.5290646 1.385058e-03 0.58774303 ## 20 0.5290646 1.385058e-03 0.58774303 ## 21 0.5284906 5.382095e-03 1.15858824 ## 22 0.5290646 1.385058e-03 0.58774303 ## 23 0.5291725 6.334069e-04 0.39746129 ## 24 0.5292525 3.302708e-05 0.13733555 ## 25 0.5291098 4.673601e-04 0.51662298 ## 26 0.5289763 8.734572e-04 0.70626670 ## 27 0.5292019 1.872167e-04 0.32697927 ## 28 0.5288014 1.405508e-03 0.89591042 ## 29 0.5285850 2.063513e-03 1.08555414 ## 30 0.5292019 1.872167e-04 0.32697927 ## 31 0.5290189 7.626131e-04 0.65176558 ## 32 0.5291405 3.833637e-04 0.46210936 ## 33 0.5292594 1.230694e-05 0.08279693 ## 34 0.5292207 1.332616e-04 0.27245315 ## 35 0.5286225 1.998674e-03 -1.05514035 ## 36 0.5284044 2.769562e-03 -1.22142116 ## 37 0.5288328 1.388993e-03 0.86498769 ## 38 0.5290010 8.466266e-04 0.67531416 ## 39 0.5290010 8.466266e-04 0.67531416 ## 40 0.5243065 1.592083e-02 -2.92848294 ## 41 0.5291277 4.378348e-04 0.48564063 ## 42 0.5288328 1.388993e-03 0.86498769 ## 43 0.5282295 3.084876e-03 1.33991634 ## 44 0.5282295 3.084876e-03 1.33991634 ## 45 0.5290689 5.807650e-04 0.58137864 ## 46 0.5285016 2.273479e-03 1.15028191 ## 47 0.5290689 5.807650e-04 0.58137864 ## 48 0.5285016 2.273479e-03 1.15028191 ## 49 0.5289213 1.021423e-03 0.77101306 ## 50 0.5290413 6.758060e-04 -0.62123932 ## 51 0.5292618 4.791188e-06 -0.05230817 ## 52 0.5292525 3.302708e-05 0.13733555 ## 53 0.5292019 1.872167e-04 0.32697927 ## 54 0.5283271 2.847471e-03 1.27519786 ## 55 0.5285850 2.063513e-03 1.08555414 ## 56 0.5291098 4.673601e-04 0.51662298 ## 57 0.5283271 2.847471e-03 1.27519786 ## 58 0.5289763 8.734572e-04 0.70626670 ## 59 0.5283271 2.847471e-03 1.27519786 ## 60 0.5283122 2.245181e-03 -1.28526102 ## 61 0.5285722 1.631828e-03 -1.09572824 ## 62 0.5292624 2.337244e-06 0.04146844 ## 63 0.5292223 8.937237e-05 -0.26727957 ## 64 0.5292599 7.567956e-06 -0.07777741 ## 65 0.5288280 9.463331e-04 0.86973343 ## 66 0.5288280 9.463331e-04 0.86973343 ## 67 0.5281277 2.467101e-03 -1.40429257 ## 68 0.5291978 1.549052e-04 -0.33759712 ## 69 0.5292508 2.979680e-05 -0.14806434 ## 70 0.5291035 3.776626e-04 -0.52712990 ## 71 0.5291035 3.776626e-04 -0.52712990 ## 72 0.5289678 6.980688e-04 -0.71666268 ## 73 0.5292327 7.252658e-05 0.23100122 ## 74 0.5287908 1.116124e-03 -0.90619546 ## 75 0.5282021 2.335115e-03 -1.35752997 ## 76 0.5292628 1.216342e-06 -0.03098298 ## 77 0.5282021 2.335115e-03 -1.35752997 ## 78 0.5284779 1.728671e-03 -1.16802326 ## 79 0.5259917 1.101880e-02 -2.38107770 ## 80 0.5277472 5.114951e-03 -1.62228504 ## 81 0.5277472 5.114951e-03 -1.62228504 ## 82 0.5273712 6.381099e-03 -1.81198320 ## 83 0.5289785 6.188637e-04 0.70355775 ## 84 0.5292617 3.706735e-06 -0.05445002 ## 85 0.5292529 2.280327e-05 0.13505192 ## 86 0.5292028 1.316949e-04 0.32455386 ## 87 0.5292291 7.440535e-05 -0.24395196 ## 88 0.5289785 6.188637e-04 0.70355775 ## 89 0.5288095 2.530195e-03 0.88807906 ## 90 0.5276981 8.716136e-03 1.64830140 ## 91 0.5289830 1.563115e-03 0.69802348 ## 92 0.5290333 1.282884e-03 -0.63236562 ## 93 0.5289830 1.563115e-03 0.69802348 ## 94 0.5291433 2.610293e-04 -0.45678174 ## 95 0.5292599 7.567956e-06 -0.07777741 ## 96 0.5292223 8.937237e-05 -0.26727957 ## 97 0.5292223 8.937237e-05 -0.26727957 ## 98 0.5292112 1.135166e-04 0.30122693 ## 99 0.5292562 1.561603e-05 0.11172476 ## 100 0.5292112 1.135166e-04 0.30122693 ## 101 0.5292112 1.135166e-04 0.30122693 ## 102 0.5278315 4.354926e-03 1.57659870 ## 103 0.5274657 5.465617e-03 1.76624268 ## 104 0.5290362 6.918014e-04 0.62837877 ## 105 0.5281556 3.370257e-03 1.38695471 ## 106 0.5278315 4.354926e-03 1.57659870 ## 107 0.5291526 3.372427e-04 0.43873479 ## 108 0.5274657 5.465617e-03 1.76624268 ## 109 0.5284380 2.511610e-03 1.19731073 ## 110 0.5288783 1.172382e-03 0.81802276 ## 111 0.5285939 2.037567e-03 -1.07841710 ## 112 0.5292045 1.792285e-04 -0.31984116 ## 113 0.5292536 2.969898e-05 -0.13019717 ## 114 0.5292536 2.969898e-05 -0.13019717 ## 115 0.5289821 8.563538e-04 -0.69912913 ## 116 0.5283375 2.817207e-03 -1.26806108 ## 117 0.5280395 3.722869e-03 -1.45770507 ## 118 0.5291140 4.547801e-04 -0.50948514 ## 119 0.5292045 1.792285e-04 -0.31984116 ## 120 0.5283375 2.817207e-03 -1.26806108 ## 121 0.5289227 1.022897e-03 -0.76941699 ## 122 0.5292633 2.040983e-07 -0.01086840 ## 123 0.5289227 1.022897e-03 -0.76941699 ## 124 0.5285038 2.279897e-03 -1.14869129 ## 125 0.5285038 2.279897e-03 -1.14869129 ## 126 0.5290842 5.380764e-04 0.55804305 ## 127 0.5286181 3.119091e-03 -1.05873865 ## 128 0.5292119 1.557483e-04 0.29926398 ## 129 0.5288587 1.222929e-03 -0.83857793 ## 130 0.5286192 1.945997e-03 1.05782526 ## 131 0.5291420 3.668615e-04 -0.45929729 ## 132 0.5286548 1.838590e-03 -1.02821825 ## 133 0.5291113 3.303818e-04 0.51405581 ## 134 0.5291113 3.303818e-04 0.51405581 ## 135 0.5285888 1.465213e-03 1.08256163 ## 136 0.5285888 1.465213e-03 1.08256163 ## 137 0.5288044 9.971408e-04 0.89305969 ## 138 0.5291113 3.303818e-04 0.51405581 ## 139 0.5292529 2.280327e-05 0.13505192 ## 140 0.5287982 4.005182e-03 0.89901235 ## 141 0.5291546 9.369626e-04 -0.43482615 ## 142 0.5291840 1.796615e-04 0.37151804 ## 143 0.5292443 4.311669e-05 0.18200154 ## 144 0.5285091 1.705349e-03 -1.14461395 ## 145 0.5290823 4.097080e-04 0.56103454 ## 146 0.5292443 4.311669e-05 0.18200154 ## 147 0.5292327 7.252658e-05 0.23100122 ## 148 0.5288954 8.689880e-04 0.79959956 ## 149 0.5287002 1.329773e-03 0.98913234 ## 150 0.5280106 2.956183e-03 -1.47479381 ## 151 0.5287908 1.116124e-03 -0.90619546 ## 152 0.5283122 2.245181e-03 -1.28526102 ## 153 0.5292508 2.979680e-05 -0.14806434 ## 154 0.5286548 1.838590e-03 -1.02821825 ## 155 0.5291259 4.156828e-04 0.48890430 ## 156 0.5288587 1.222929e-03 -0.83857793 ## 157 0.5292119 1.557483e-04 0.29926398 ## 158 0.5288296 1.310807e-03 0.86818494 ## 159 0.5288296 1.310807e-03 0.86818494 ## 160 0.5292597 1.113461e-05 -0.08001665 ## 161 0.5286548 1.838590e-03 -1.02821825 ## 162 0.5222837 2.169940e-02 -3.47168193 ## 163 0.5292562 1.561603e-05 0.11172476 ## 164 0.5292112 1.135166e-04 0.30122693 ## 165 0.5284137 1.846182e-03 -1.21479041 ## 166 0.5292112 1.135166e-04 0.30122693 ## 167 0.5290230 5.225386e-04 -0.64628391 ## 168 0.5288614 8.739005e-04 -0.83578607 ## 169 0.5278000 3.177873e-03 -1.59379474 ## 170 0.5279939 4.491802e-03 1.48459698 ## 171 0.5292627 2.246655e-06 -0.03320217 ## 172 0.5275173 6.175390e-03 -1.74072622 ## 173 0.5292297 1.025091e-04 -0.24195189 ## 174 0.5288850 1.151385e-03 -0.81088304 ## 175 0.5285850 2.063513e-03 1.08555414 ## 176 0.5289763 8.734572e-04 0.70626670 ## 177 0.5292525 3.302708e-05 0.13733555 ## 178 0.5292277 1.087061e-04 0.24909080 ## 179 0.5292536 2.969898e-05 -0.13019717 ## 180 0.5290362 6.918014e-04 0.62837877 ## 181 0.5292045 1.792285e-04 -0.31984116 ## 182 0.5290362 6.918014e-04 0.62837877 ## 183 0.5268952 7.195987e-03 -2.02663702 ## 184 0.5285939 2.037567e-03 -1.07841710 ## 185 0.5259228 1.014151e-02 -2.40592499 ## 186 0.5286789 1.778985e-03 1.00766674 ## 187 0.5291526 3.372427e-04 0.43873479 ## 188 0.5283375 2.817207e-03 -1.26806108 ## 189 0.5268952 7.195987e-03 -2.02663702 ## 190 0.5292277 1.087061e-04 0.24909080 ## 191 0.5291743 3.730164e-04 0.39345625 ## 192 0.5292633 4.586295e-07 0.01379633 ## 193 0.5287297 2.234287e-03 0.96294613 ## 194 0.5290382 1.095908e-03 -0.62556534 ## 195 0.5281582 5.374444e-03 -1.38532710 ## 196 0.5291542 5.314400e-04 -0.43562490 ## 197 0.5291542 5.314400e-04 -0.43562490 ## 198 0.5291864 3.225324e-04 -0.36586359 ## 199 0.5279590 5.456543e-03 -1.50484335 ## 200 0.5292456 7.466659e-05 -0.17603363 ## 201 0.5290857 7.440562e-04 -0.55569355 ## 202 0.5292395 9.990856e-05 0.20362629 ## 203 0.5292456 7.466659e-05 -0.17603363 ## 204 0.5290857 7.440562e-04 -0.55569355 ## 205 0.5279590 5.456543e-03 -1.50484335 ## 206 0.5292597 1.113461e-05 -0.08001665 ## 207 0.5286192 1.945997e-03 1.05782526 ## 208 0.5291259 4.156828e-04 0.48890430 ## 209 0.5284980 1.799448e-03 1.15300255 ## 210 0.5284980 1.799448e-03 1.15300255 ## 211 0.5289187 8.107622e-04 0.77394015 ## 212 0.5289187 8.107622e-04 0.77394015 ## 213 0.5284980 1.799448e-03 1.15300255 ## 214 0.5291737 2.110591e-04 0.39487775 ## 215 0.5287291 1.256482e-03 0.96347135 ## 216 0.5287291 1.256482e-03 0.96347135 ## 217 0.5268530 1.015025e-02 -2.04460846 ## 218 0.5291937 1.534772e-04 0.34803045 ## 219 0.5292489 3.184183e-05 0.15852373 ## 220 0.5282969 2.126802e-03 1.29556401 ## 221 0.5291937 1.534772e-04 0.34803045 ## 222 0.5287799 1.064443e-03 0.91655059 ## 223 0.5292624 2.337244e-06 0.04146844 ## 224 0.5287002 1.329773e-03 0.98913234 ## 225 0.5291616 2.403648e-04 0.42053400 ## 226 0.5288954 8.689880e-04 0.79959956 ## 227 0.5269374 2.291960e-02 -2.00855856 ## 228 0.5290112 2.489575e-03 0.66197801 ## 229 0.5291356 1.261524e-03 0.47122540 ## 230 0.5280753 1.172003e-02 -1.43630072 ## 231 0.5289282 1.673629e-03 -0.76314553 ## 232 0.5290743 9.441332e-04 -0.57318439 ## 233 0.5289282 1.673629e-03 -0.76314553 ## 234 0.5291616 2.403648e-04 0.42053400 ## 235 0.5290492 5.058519e-04 0.61006678 ## 236 0.5287002 1.329773e-03 0.98913234 ## 237 0.5290492 5.058519e-04 0.61006678 ## 238 0.5288206 2.830546e-03 0.87712032 ## 239 0.5255203 2.385404e-02 -2.54627361 ## 240 0.5281263 7.264279e-03 -1.40514230 ## 241 0.5289431 9.563845e-04 -0.74606234 ## 242 0.5292275 1.205719e-04 -0.24959384 ## 243 0.5292275 1.205719e-04 -0.24959384 ## 244 0.5278299 4.816127e-03 -1.57746500 ## 245 0.5289014 1.527476e-03 0.79315365 ## 246 0.5291650 4.151218e-04 0.41348326 ## 247 0.5281941 4.508522e-03 1.36265924 ## 248 0.5292494 5.910622e-05 -0.15602232 ## 249 0.5289605 1.278109e-03 -0.72552791 ## 250 0.5285602 2.965788e-03 -1.10519830 ## 251 0.5289014 1.527476e-03 0.79315365 ## 252 0.5291388 3.498993e-04 0.46533380 ## 253 0.5292572 1.730408e-05 -0.10348259 ## 254 0.5286269 1.786655e-03 -1.05150990 ## 255 0.5283765 2.489076e-03 -1.24111536 ## 256 0.5292572 1.730408e-05 -0.10348259 ## 257 0.5290033 7.303631e-04 -0.67229898 ## 258 0.5292140 1.388066e-04 -0.29308805 ## 259 0.5284011 2.419933e-03 1.22375565 ## 260 0.5292591 1.198536e-05 0.08612287 ## 261 0.5288529 1.152549e-03 0.84454472 ## 262 0.5292196 1.228504e-04 0.27572834 ## 263 0.5284011 2.419933e-03 1.22375565 ## 264 0.5290033 7.303631e-04 -0.67229898 ## 265 0.5290511 7.960903e-04 0.60744315 ## 266 0.5281881 4.028620e-03 1.36647792 ## 267 0.5281881 4.028620e-03 1.36647792 ## 268 0.5284662 2.987423e-03 1.17671923 ## 269 0.5287027 2.101603e-03 0.98696053 ## 270 0.5288976 1.371159e-03 0.79720184 ## 271 0.5292601 1.088335e-05 -0.07580968 ## 272 0.5292559 2.455644e-05 0.11387440 ## 273 0.5292104 1.745010e-04 0.30355848 ## 274 0.5292228 1.334817e-04 -0.26549375 ## 275 0.5285871 3.290078e-03 1.08389861 ## 276 0.5289782 1.388021e-03 0.70401773 ## 277 0.5283291 4.544206e-03 1.27383905 ## 278 0.5285871 3.290078e-03 1.08389861 ## 279 0.5285871 3.290078e-03 1.08389861 ## 280 0.5289782 1.388021e-03 0.70401773 ## 281 0.5288034 2.238016e-03 0.89395817 ## 282 0.5291539 8.736402e-04 0.43620609 ## 283 0.5292529 8.384186e-05 -0.13513127 ## 284 0.5292286 2.773145e-04 0.24576030 ## 285 0.5291539 8.736402e-04 0.43620609 ## 286 0.5292286 2.773145e-04 0.24576030 ## 287 0.5291539 8.736402e-04 0.43620609 ## 288 0.5275245 9.428393e-03 -1.73712333 ## 289 0.5290584 1.113334e-03 -0.59693155 ## 290 0.5278846 7.478393e-03 -1.54709137 ## 291 0.5291875 1.786006e-04 -0.36324704 ## 292 0.5290876 4.136001e-04 -0.55277824 ## 293 0.5289463 7.458452e-04 -0.74230943 ## 294 0.5291840 1.796615e-04 0.37151804 ## 295 0.5287547 1.150306e-03 0.94006754 ## 296 0.5289238 1.442023e-03 -0.76819428 ## 297 0.5292407 9.645367e-05 -0.19867552 ## 298 0.5292403 9.135983e-05 0.20023518 ## 299 0.5282309 4.085367e-03 1.33899287 ## 300 0.5290699 7.660595e-04 0.57982107 ## 301 0.5289225 1.349648e-03 0.76961402 ## 302 0.5287336 2.097395e-03 0.95940697 ## 303 0.5276027 6.567401e-03 -1.69769430 ## 304 0.5290836 7.118683e-04 -0.55893661 ## 305 0.5287336 2.097395e-03 0.95940697 ## 306 0.5272099 8.117879e-03 -1.88748725 ## 307 0.5285031 3.009302e-03 1.14919992 ## 308 0.5287636 1.175336e-03 -0.93184063 ## 309 0.5285395 1.702072e-03 -1.12137183 ## 310 0.5279130 5.031245e-03 -1.53110299 ## 311 0.5282273 3.861440e-03 -1.34134750 ## 312 0.5267194 9.467996e-03 -2.10036947 ## 313 0.5292493 4.992962e-05 0.15652272 ## 314 0.5292104 1.745010e-04 0.30355848 ## 315 0.5289444 1.270234e-03 -0.74451606 ## 316 0.5287610 2.000415e-03 -0.93431267 ## 317 0.5286815 2.831044e-03 -1.00544622 ## 318 0.5291542 5.314400e-04 -0.43562490 ## 319 0.5286815 2.831044e-03 -1.00544622 ## 320 0.5285642 4.188699e-03 1.10206485 ## 321 0.5292333 1.805411e-04 -0.22879982 ## 322 0.5292625 5.158870e-06 -0.03867630 ## 323 0.5283019 5.758594e-03 1.29218838 ## 324 0.5285642 4.188699e-03 1.10206485 ## 325 0.5291007 9.749644e-04 0.53169428 ## 326 0.5292502 7.910200e-05 0.15144723 ## 327 0.5290857 7.440562e-04 -0.55569355 ## 328 0.5285345 3.050575e-03 -1.12518343 ## 329 0.5252269 1.684224e-02 -2.64382312 ## 330 0.5292456 7.466659e-05 -0.17603363 ## 331 0.5287297 2.234287e-03 0.96294613 ## 332 0.5291864 3.225324e-04 -0.36586359 ## 333 0.5292584 3.435915e-05 0.09285536 ## 334 0.5280868 8.141444e-03 -1.42934322 ## 335 0.5236188 3.285786e-02 -3.12400472 ## 336 0.5274101 1.082736e-02 -1.79330205 ## 337 0.5242881 2.898063e-02 -2.93390434 ## 338 0.5266121 8.276546e-03 -2.14407821 ## 339 0.5290507 5.261084e-04 0.60794388 ## 340 0.5284664 1.970623e-03 1.17659685 ## 341 0.5281886 2.656707e-03 1.36614784 ## 342 0.5290507 5.261084e-04 0.60794388 ## 343 0.5288974 9.053240e-04 0.79749487 ## 344 0.5290507 5.261084e-04 0.60794388 ## 345 0.5281886 2.656707e-03 1.36614784 ## 346 0.5291627 2.491821e-04 0.41839289 ## 347 0.5288974 9.053240e-04 0.79749487 ## 348 0.5288754 1.018755e-03 0.82107625 ## 349 0.5278264 3.769430e-03 1.57937948 ## 350 0.5281510 2.918836e-03 1.38980367 ## 351 0.5288754 1.018755e-03 0.82107625 ## 352 0.5284340 2.176858e-03 1.20022786 ## 353 0.5281510 2.918836e-03 1.38980367 ## 354 0.5281510 2.918836e-03 1.38980367 ## 355 0.5281510 2.918836e-03 1.38980367 ## 356 0.5278264 3.769430e-03 1.57937948 ## 357 0.5290339 6.026291e-04 0.63150044 ## 358 0.5277570 5.368689e-03 -1.61701662 ## 359 0.5286314 2.254343e-03 -1.04782901 ## 360 0.5288396 1.511871e-03 -0.85809980 ## 361 0.5288396 1.511871e-03 -0.85809980 ## 362 0.5292154 1.713843e-04 -0.28891219 ## 363 0.5291316 4.703918e-04 -0.47864139 ## 364 0.5288924 1.565310e-03 0.80291938 ## 365 0.5288924 1.565310e-03 0.80291938 ## 366 0.5288924 1.565310e-03 0.80291938 ## 367 0.5288924 1.565310e-03 0.80291938 ## 368 0.5286961 2.392986e-03 0.99275457 ## 369 0.5292611 5.954547e-06 0.06277302 ## 370 0.5290339 6.026291e-04 0.63150044 ## 371 0.5291161 3.868348e-04 -0.50595441 ## 372 0.5291510 2.951204e-04 0.44192463 ## 373 0.5292281 1.102868e-04 -0.24750146 ## 374 0.5286807 1.822544e-03 -1.00613216 ## 375 0.5290006 8.199620e-04 -0.67582793 ## 376 0.5254428 1.187942e-02 -2.57239007 ## 377 0.5285943 3.729038e-03 1.07813465 ## 378 0.5288095 2.530195e-03 0.88807906 ## 379 0.5292612 1.241125e-05 -0.06219886 ## 380 0.5290333 1.282884e-03 -0.63236562 ## 381 0.5288095 2.530195e-03 0.88807906 ## 382 0.5288095 2.530195e-03 0.88807906 ## 383 0.5288280 9.463331e-04 0.86973343 ## 384 0.5291248 3.012696e-04 0.49072909 ## 385 0.5292223 8.937237e-05 -0.26727957 ## 386 0.5286583 1.315115e-03 -1.02528824 ## 387 0.5292119 1.557483e-04 0.29926398 ## 388 0.5284094 2.579337e-03 -1.21785856 ## 389 0.5288587 1.222929e-03 -0.83857793 ## 390 0.5291657 4.434341e-04 -0.41198756 ## 391 0.5291657 4.434341e-04 -0.41198756 ## 392 0.5266691 1.175253e-02 -2.12097358 ## 393 0.5290972 7.546327e-04 0.53744911 ## 394 0.5284340 2.176858e-03 1.20022786 ## 395 0.5292611 5.954547e-06 0.06277302 ## 396 0.5281510 2.918836e-03 1.38980367 ## 397 0.5284340 2.176858e-03 1.20022786 ## 398 0.5284417 2.683114e-03 -1.19462491 ## 399 0.5286821 1.898718e-03 -1.00494469 ## 400 0.5291123 4.938088e-04 0.51249707 ## 401 0.5281598 3.602796e-03 -1.38430514 ## 402 0.5291541 3.572382e-04 -0.43590403 ## 403 0.5290382 7.357795e-04 -0.62558425 ## 404 0.5283329 3.038198e-03 1.27121795 ## 405 0.5288056 1.495433e-03 0.89185751 ## 406 0.5276937 5.122104e-03 1.65057839 ## 407 0.5290382 7.357795e-04 -0.62558425 ## 408 0.5292532 3.332508e-05 0.13313663 ## 409 0.5274184 9.625306e-03 -1.78927820 ## 410 0.5288271 2.279285e-03 0.87070277 ## 411 0.5289968 1.393078e-03 0.68070413 ## 412 0.5289968 1.393078e-03 0.68070413 ## 413 0.5291420 6.342032e-04 -0.45928772 ## 414 0.5292573 3.406866e-05 -0.10271288 ## 415 0.5283955 4.867462e-03 1.22771803 ## 416 0.5291377 7.056935e-04 0.46747180 ## 417 0.5286436 3.477062e-03 1.03765647 ## 418 0.5288348 2.404837e-03 -0.86295911 ## 419 0.5290759 1.756864e-03 0.57089354 ## 420 0.5287415 4.887832e-03 0.95223517 ## 421 0.5287415 4.887832e-03 0.95223517 ## 422 0.5289296 3.126375e-03 0.76156436 ## 423 0.5292427 1.936797e-04 0.18955192 ## 424 0.5287415 4.887832e-03 0.95223517 ## 425 0.5290759 1.756864e-03 0.57089354 ## 426 0.5289296 3.126375e-03 0.76156436 ## 427 0.5289296 3.126375e-03 0.76156436 ## 428 0.5285084 7.068836e-03 -1.14514378 ## 429 0.5285084 7.068836e-03 -1.14514378 ## 430 0.5292182 1.612907e-04 0.28027543 ## 431 0.5292182 1.612907e-04 0.28027543 ## 432 0.5280375 6.070293e-03 -1.45892750 ## 433 0.5273146 9.643372e-03 -1.83883748 ## 434 0.5259151 1.654659e-02 -2.40870244 ## 435 0.5273146 9.643372e-03 -1.83883748 ## 436 0.5292272 1.792000e-04 0.25066739 ## 437 0.5283362 4.592473e-03 -1.26897251 ## 438 0.5268902 1.173863e-02 -2.02879247 ## 439 0.5270505 1.902192e-02 -1.95921301 ## 440 0.5281546 9.541123e-03 -1.38756794 ## 441 0.5284384 7.100574e-03 -1.19701958 ## 442 0.5285985 1.745270e-03 -1.07468183 ## 443 0.5292267 9.622886e-05 0.25234883 ## 444 0.5292626 2.232252e-06 0.03596002 ## 445 0.5278740 4.163669e-03 1.55305357 ## 446 0.5281929 3.208933e-03 1.36341687 ## 447 0.5289821 8.563538e-04 -0.69912913 ## 448 0.5292045 1.792285e-04 -0.31984116 ## 449 0.5288783 1.172382e-03 0.81802276 ## 450 0.5288088 1.383950e-03 -0.88877311 ## 451 0.5290362 6.918014e-04 0.62837877 ## 452 0.5283375 2.817207e-03 -1.26806108 ## 453 0.5290362 6.918014e-04 0.62837877 ## 454 0.5292521 6.634682e-05 0.13985930 ## 455 0.5292521 6.634682e-05 0.13985930 ## 456 0.5292521 6.634682e-05 0.13985930 ## 457 0.5292007 3.692987e-04 0.32996676 ## 458 0.5290418 1.306230e-03 -0.62057053 ## 459 0.5292521 6.634682e-05 0.13985930 ## 460 0.5284075 2.769322e-03 -1.21924997 ## 461 0.5291265 4.433246e-04 0.48782825 ## 462 0.5291265 4.433246e-04 0.48782825 ## 463 0.5292596 1.228221e-05 -0.08119782
Join up the fitted values to plot the parallel lines model
ggplot(evals, aes(x=bty_avg, y=score, colour=gender)) + geom_point(alpha=0.5) + theme_bw() + geom_line(data = augment(parallel_lines), aes(y=.fitted, colour=gender))
Add an independent variable to the model that is the product of gender_is_male and bty_avg. This is called an interaction term.
Model 2: \[score = \beta_0 + \beta_1 \, gender\_is\_male + \beta_2 \, bty\_avg + \beta_3 \, (gender\_is\_male \times bty\_avg) + \epsilon\]
Model 2 for male professors: \[score = \beta_0 + \beta_1 + \beta_2 \, bty\_avg + \beta_3 \, bty\_avg + \epsilon\] \[score = (\beta_0 + \beta_1) + (\beta_2 + \beta_3) \, bty\_avg + \epsilon\]
Model 2 for female professors: \[score = \beta_0 + \beta_2 \, bty\_avg + \epsilon\]
ggplot(evals, aes(x=bty_avg, y=score, colour=gender)) + geom_point(alpha=0.5) +
geom_smooth(method=lm, fill=NA) + theme_bw()
Including the term bty_avg*gender on the right-side of the model specification in lm includes the interaction term plus both of the variables in the model.
summary(lm(score ~ bty_avg*gender, data=evals))$coefficients
## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 3.95005984 0.11799986 33.475124 2.920267e-125 ## bty_avg 0.03064259 0.02400361 1.276582 2.023952e-01 ## gendermale -0.18350903 0.15349459 -1.195541 2.324931e-01 ## bty_avg:gendermale 0.07961855 0.03246948 2.452105 1.457376e-02
What are the fitted lines for male and for female professors?
Model: \[score = \beta_0 + \beta_1 \, gender\_is\_male + \beta_2 \, bty\_avg \\ + \beta_3 \, (gender\_is\_male \times bty\_avg) + \epsilon\]
What would be appropriate hypotheses to test?
What do you conclude?
marioKart dataset in the openintro package includes eBay sales of the game Mario Kart for Nintendo Wii in October 2009.duration, in days) result in higher prices (totalPr)?library(openintro) glimpse(marioKart)
## Observations: 143 ## Variables: 12 ## $ ID <dbl> 150377422259, 260483376854, 320432342985, 280405224... ## $ duration <int> 3, 7, 3, 3, 1, 3, 1, 1, 3, 7, 1, 1, 1, 1, 7, 7, 3, ... ## $ nBids <int> 20, 13, 16, 18, 20, 19, 13, 15, 29, 8, 15, 15, 13, ... ## $ cond <fctr> new, used, new, new, new, new, used, new, used, us... ## $ startPr <dbl> 0.99, 0.99, 0.99, 0.99, 0.01, 0.99, 0.01, 1.00, 0.9... ## $ shipPr <dbl> 4.00, 3.99, 3.50, 0.00, 0.00, 4.00, 0.00, 2.99, 4.0... ## $ totalPr <dbl> 51.55, 37.04, 45.50, 44.00, 71.00, 45.00, 37.02, 53... ## $ shipSp <fctr> standard, firstClass, firstClass, standard, media,... ## $ sellerRate <int> 1580, 365, 998, 7, 820, 270144, 7284, 4858, 27, 201... ## $ stockPhoto <fctr> yes, yes, no, yes, yes, yes, yes, yes, yes, no, ye... ## $ wheels <int> 1, 1, 1, 1, 2, 0, 0, 2, 1, 1, 2, 2, 2, 2, 1, 0, 1, ... ## $ title <fctr> ~~ Wii MARIO KART & WHEEL ~ NINTENDO Wii ~ BRA...
ggplot(marioKart, aes(x=duration, y=totalPr)) + geom_point() + theme_bw()
totalPr?# create a data set without the outliers marioKart2 <- marioKart %>% filter(totalPr < 100)
ggplot(marioKart2, aes(x=duration, y=totalPr)) + geom_point() + theme_bw()
ggplot(marioKart2, aes(x=duration, y=totalPr)) + geom_point() + theme_bw() +
geom_smooth(method = "lm")
There appears to be a negative relationship between totalPr and duration.
That is, the longer an item is on auction, the lower the price.
Does this make sense?
Maybe there actually isn't a relationship.
We can investigate if the data are consistent with a slope of 0.
summary(lm(totalPr ~ duration, data=marioKart2))$coefficients
## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 52.373584 1.2607560 41.541411 3.010309e-80 ## duration -1.317156 0.2769021 -4.756756 4.866701e-06
We have strong evidence that the slope is not 0.
There must be something else affecting the relationship …
Consider the role of cond.cond is a categorical variable for the game's condition, either new or used.
ggplot(marioKart2, aes(x=duration, y=totalPr, color=cond)) + geom_point() + theme_bw()
New games, which are more desirable, were mostly sold in one-day auctions.
ggplot(marioKart2, aes(x=duration, y=totalPr, color=cond)) + geom_point() + geom_smooth(method="lm", fill=NA) + theme_bw()
cond changes the nature of the relationship between totalPr and duration.summary(lm(totalPr ~ duration*cond, data=marioKart2))$coefficients
## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 58.268226 1.3664729 42.641332 5.832075e-81 ## duration -1.965595 0.4487799 -4.379865 2.341705e-05 ## condused -17.121924 2.1782581 -7.860374 1.013608e-12 ## duration:condused 2.324563 0.5483731 4.239016 4.101561e-05
Source of data: British Medical Journal (Clinical Research Edition) March 29, 1986
treatment is one of 2 treatments (A or B)outcome is success or failure of the treatmentkidney_stones %>% count(treatment, outcome)
## # A tibble: 4 x 3 ## treatment outcome n ## <chr> <chr> <int> ## 1 A failure 77 ## 2 A success 273 ## 3 B failure 61 ## 4 B success 289
Which treatment would you choose?
The table below shows counts of patients being treated for kidney stones.
Which treatment would you choose?
What would make it easier to decide which treatment is better?
Proportion of observations in each cell in the table:
Proportion of observations in each row and column:
Proportion of successes in each column:
kidney_stones %>% count(treatment, outcome) %>% group_by(treatment) %>% mutate(perc_success = n / sum(n)) %>% filter(outcome=="success")
## # A tibble: 2 x 4 ## # Groups: treatment [2] ## treatment outcome n perc_success ## <chr> <chr> <int> <dbl> ## 1 A success 273 0.7800000 ## 2 B success 289 0.8257143
Which treatment would you choose?
Recall: The distribution of a variable is the pattern of values in the data for that variable, showing the frequency or relative frequency (proportions) of the occurrence of the values relative to each other.
We can also look at the joint distribution of two variables. If both variables are categorical, we can see their joint distribution in a contingency table showing the counts of observations in each way the data can be cross-classifed.
Counts:
Proportions in the joint distributions:
A marginal distribution is the distribution of only one of the variables in a contingency table.
Counts:
Proportions:
A conditional distribution is the distribution of a variable within a fixed value of a second variable.
What percentage of successes were Treatment A?
What percentage of Treatment A surgeries resulted in a success?
\(P(E_1)\) is the probability of an event \(E_1\)
\(P(E_1 \, | \, E_2)\) is the probability of \(E_1\) given that event \(E_2\) has occurred. It is a conditional probability.
Example:
From the tables, we estimate: \[P(\mbox{success}) = 0.80\] \[P(\mbox{success} \, | \, \mbox{treatment A}) = 0.78\] \[P(\mbox{success} \, | \, \mbox{treatment B}) = 0.83\]
Does there appear to be a relationship between success and treatment?
Yes! Success is more likely with treatment B.
\(E_1\) and \(E_2\) are independent if \(P(E_1 \, | \, E_2) = P(E_1)\).
That is, the conditional distribution of one variable is the same for all values of the other variable.
It appears that success and treatment are not independent.
kidney_stones %>% count(size, treatment, outcome) %>% group_by(size, treatment) %>% mutate(per_success = n / sum(n)) %>% filter(outcome=="success")
## # A tibble: 4 x 5 ## # Groups: size, treatment [4] ## size treatment outcome n per_success ## <chr> <chr> <chr> <int> <dbl> ## 1 large A success 192 0.7300380 ## 2 large B success 55 0.6875000 ## 3 small A success 81 0.9310345 ## 4 small B success 234 0.8666667
Column percentages:
Which treatment is better?
This example is another case of Simpson's paradox.
Be careful drawing conclusions from data!
It's important to understand how the data were collected and what other factors might have an affect.
Visualizing the kidney stone data: treatment and outcome
ggplot(kidney_stones, aes(x=treatment, fill=outcome)) + geom_bar(position = "fill") + labs(y="Proportion") + theme_bw()
Visualizing the kidney stone data: treatment and outcome by size
ggplot(kidney_stones, aes(x=treatment, fill=outcome)) + geom_bar(position = "fill") + labs(y="Proportion") + facet_grid(. ~ size) + theme_bw()
Examples of situations that may have confounding variables:
Great care must be taken to deal with potential confounders in observational studies.