# Multiple and Logistic Regression

## 2018/05/17

library(openintro)
library(broom)
library(dplyr)
library(ggplot2)
library(modelr)
library(plotly)
knitr::opts_chunk$set(cache = TRUE) ### Fitting a parallel slopes model We use the lm() function to fit linear models to data. In this case, we want to understand how the price of MarioKart games sold at auction varies as a function of not only the number of wheels included in the package, but also whether the item is new or used. Obviously, it is expected that you might have to pay a premium to buy these new. But how much is that premium? Can we estimate its value after controlling for the number of wheels? We will fit a parallel slopes model using lm(). In addition to the data argument, lm() needs to know which variables you want to include in your regression model, and how you want to include them. It accomplishes this using a formula argument. A simple linear regression formula looks like y ~ x, where y is the name of the response variable, and x is the name of the explanatory variable. Here, we will simply extend this formula to include multiple explanatory variables. A parallel slopes model has the form y ~ x + z, where z is a categorical explanatory variable, and x is a numerical explanatory variable. The output from lm() is a model object, which when printed, will show the fitted coefficients. • Explore the dataset mario_kart using glimpse() or str(). data("marioKart") mario_kart <- filter(marioKart, totalPr < 80) glimpse(mario_kart) ## Observations: 141 ## Variables: 12 ##$ 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, ...
## $cond <fct> new, used, new, new, new, new, used, new, used, use... ##$ 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...
## $shipSp <fct> standard, firstClass, firstClass, standard, media, ... ##$ sellerRate <int> 1580, 365, 998, 7, 820, 270144, 7284, 4858, 27, 201...
## $stockPhoto <fct> yes, yes, no, yes, yes, yes, yes, yes, yes, no, yes... ##$ wheels     <int> 1, 1, 1, 1, 2, 0, 0, 2, 1, 1, 2, 2, 2, 2, 1, 0, 1, ...
## $title <fct> ~~ Wii MARIO KART &amp; WHEEL ~ NINTENDO Wii ~ BRAN... • Use lm() to fit a parallel slopes model for total price as a function of the number of wheels and the condition of the item. Use the argument data to specify the dataset you’re using. (mod <- lm(formula = totalPr ~ wheels + cond, data = mario_kart)) ## ## Call: ## lm(formula = totalPr ~ wheels + cond, data = mario_kart) ## ## Coefficients: ## (Intercept) wheels condused ## 42.370 7.233 -5.585 ### Using geom_line() and augment() Parallel slopes models are so-named because we can visualize these models in the data space as not one line, but two parallel lines. To do this, we’ll draw two things: • a scatterplot showing the data, with color separating the points into groups • a line for each value of the categorical variable Our plotting strategy is to compute the fitted values, plot these, and connect the points to form a line. The augment() function from the broom package provides an easy way to add the fittted values to our data frame, and the geom_line() function can then use that data frame to plot the points and connect them. Note that this approach has the added benefit of automatically coloring the lines apropriately to match the data. You already know how to use ggplot() and geom_point() to make the scatterplot. The only twist is that now you’ll pass your augment()-ed model as the data argument in your ggplot() call. When you add your geom_line(), instead of letting the y aesthetic inherit its values from the ggplot() call, you can set it to the .fitted column of the augment()-ed model. This has the advantage of automatically coloring the lines for you. • augment() the model mod and explore the returned data frame using glimpse(). Notice the new variables that have been created. # Augment the model augmented_mod <- augment(mod) glimpse(augmented_mod) ## Observations: 141 ## Variables: 10 ##$ totalPr    <dbl> 51.55, 37.04, 45.50, 44.00, 71.00, 45.00, 37.02, 53...
## $wheels <int> 1, 1, 1, 1, 2, 0, 0, 2, 1, 1, 2, 2, 2, 2, 1, 0, 1, ... ##$ cond       <fct> new, used, new, new, new, new, used, new, used, use...
## $.fitted <dbl> 49.60260, 44.01777, 49.60260, 49.60260, 56.83544, 4... ##$ .se.fit    <dbl> 0.7087865, 0.5465195, 0.7087865, 0.7087865, 0.67645...
## $.resid <dbl> 1.9473995, -6.9777674, -4.1026005, -5.6026005, 14.1... ##$ .hat       <dbl> 0.02103158, 0.01250410, 0.02103158, 0.02103158, 0.0...
## $.sigma <dbl> 4.902339, 4.868399, 4.892414, 4.881308, 4.750591, 4... ##$ .cooksd    <dbl> 1.161354e-03, 8.712334e-03, 5.154337e-03, 9.612441e...

For each additional wheel, the expected price of a MarioKart increases by $7.23 regardless of whether it is new or used. ### Syntax from math The babies data set contains observations about the birthweight and other characteristics of children born in the San Francisco Bay area from 1960–1967. We would like to build a model for birthweight as a function of the mother’s age and whether this child was her first (parity == 0). Use the mathematical specification below to code the model in R. $birthweight = \beta_0 + \beta_1 \cdot age + \beta_2 \cdot parity + \epsilon$ • Explore the dataset babies using glimpse() or str(). data("babies") glimpse(babies) ## Observations: 1,236 ## Variables: 8 ##$ case      <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $bwt <int> 120, 113, 128, 123, 108, 136, 138, 132, 120, 143, 14... ##$ gestation <int> 284, 282, 279, NA, 282, 286, 244, 245, 289, 299, 351...
## $parity <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0... ##$ age       <int> 27, 33, 28, 36, 23, 25, 33, 23, 25, 30, 27, 32, 23, ...
## $height <int> 62, 64, 64, 69, 67, 62, 62, 65, 62, 66, 68, 64, 63, ... ##$ weight    <int> 100, 135, 115, 190, 125, 93, 178, 140, 125, 136, 120...
## $smoke <int> 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1... • Use lm() to build the parallel slopes model specified above. It’s not necessary to use factor() in this case as the variable parity is coded using binary numeric values. # build model lm(bwt ~ age + parity, data = babies) ## ## Call: ## lm(formula = bwt ~ age + parity, data = babies) ## ## Coefficients: ## (Intercept) age parity ## 118.27782 0.06315 -1.65248 ### Syntax from plot This time, we’d like to build a model for birthweight as a function of the length of gestation and the mother’s smoking status. # build model mod <- lm(formula=bwt ~ gestation + smoke,data = babies) augment(mod) %>% ggplot(aes(y = bwt, x = gestation, color = factor(smoke))) + geom_point() ## Evaluating and extending parallel slopes model ### R-squared vs. adjusted R-squared Two common measures of how well a model fits to data are $$R^2$$ (the coefficient of determination) and the adjusted $$R^2$$. The former measures the percentage of the variability in the response variable that is explained by the model. To compute this, we define $R^2 = 1 - \frac{SSE}{SST} \,,$ where $$SSE$$ and $$SST$$ are the sum of the squared residuals, and the total sum of the squares, respectively. One issue with this measure is that the $$SSE$$ can only decrease as new variable are added to the model, while the $$SST$$ depends only on the response variable and therefore is not affected by changes to the model. This means that you can increase $$R^2$$ by adding any additional variable to your model—even random noise. The adjusted $$R^2$$ includes a term that penalizes a model for each additional explanatory variable (where p is the number of explanatory variables). $R^2_{adj} = 1 - \frac{SSE}{SST} \cdot \frac{n-1}{n-p-1} \,,$ We can see both measures in the output of the summary() function on our model object. (mod <- lm(formula = totalPr ~ wheels + cond, data = mario_kart)) ## ## Call: ## lm(formula = totalPr ~ wheels + cond, data = mario_kart) ## ## Coefficients: ## (Intercept) wheels condused ## 42.370 7.233 -5.585 • Use summary() to compute $$R^2$$ and adjusted $$R^2$$ on the model object called mod. # R^2 and adjusted R^2 summary(mod) ## ## Call: ## lm(formula = totalPr ~ wheels + cond, data = mario_kart) ## ## Residuals: ## Min 1Q Median 3Q Max ## -11.0078 -3.0754 -0.8254 2.9822 14.1646 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 42.3698 1.0651 39.780 < 2e-16 *** ## wheels 7.2328 0.5419 13.347 < 2e-16 *** ## condused -5.5848 0.9245 -6.041 1.35e-08 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 4.887 on 138 degrees of freedom ## Multiple R-squared: 0.7165, Adjusted R-squared: 0.7124 ## F-statistic: 174.4 on 2 and 138 DF, p-value: < 2.2e-16 • Use mutate() and rnorm() to add a new variable called noise to the mario_kart data set that consists of random noise. Save the new dataframe as mario_kart_noisy. # add random noise mario_kart_noisy <- mario_kart %>% mutate(noise = rnorm(nrow(mario_kart)) ) • Use lm() to fit a model that includes wheels, cond, and the random noise term. # compute new model mod2 <- lm( totalPr ~ wheels + cond + noise, data=mario_kart_noisy) • Use summary() to compute $$R^2$$ and adjusted $$R^2$$ on the new model object. Did the value of $$R^2$$ increase? What about adjusted $$R^2$$? # new R^2 and adjusted R^2 summary(mod2) ## ## Call: ## lm(formula = totalPr ~ wheels + cond + noise, data = mario_kart_noisy) ## ## Residuals: ## Min 1Q Median 3Q Max ## -10.5633 -3.0762 -0.6037 3.0086 14.7586 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 42.0659 1.0578 39.769 < 2e-16 *** ## wheels 7.5077 0.5473 13.717 < 2e-16 *** ## condused -5.4652 0.9123 -5.991 1.74e-08 *** ## noise 0.9528 0.4183 2.278 0.0243 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 4.815 on 137 degrees of freedom ## Multiple R-squared: 0.7269, Adjusted R-squared: 0.7209 ## F-statistic: 121.5 on 3 and 137 DF, p-value: < 2.2e-16 ### Prediction Once we have fit a regression model, we can use it to make predictions for unseen observations or retrieve the fitted values. Here, we explore two methods for doing the latter. A traditional way to return the fitted values (i.e. the $$\hat{y}$$’s) is to run the predict() function on the model object. This will return a vector of the fitted values. Note that predict() will take an optional newdata argument that will allow you to make predictions for observations that are not in the original data. A newer alternative is the augment() function from the broom package, which returns a data.frame with the response varible ($$y$$), the relevant explanatory variables (the $$x$$’s), the fitted value ($$\hat{y}$$) and some information about the residuals ($$e$$). augment() will also take a newdata argument that allows you to make predictions. • Compute the fitted values of the model as a vector using predict(). # return a vector predict(mod)[1:10] ## 1 2 3 4 5 6 7 8 ## 49.60260 44.01777 49.60260 49.60260 56.83544 42.36976 36.78493 56.83544 ## 9 10 ## 44.01777 44.01777 • Compute the fitted values of the model as a vector using predict(). augment(mod) ## totalPr wheels cond .fitted .se.fit .resid .hat ## 1 51.55 1 new 49.60260 0.7087865 1.94739955 0.02103158 ## 2 37.04 1 used 44.01777 0.5465195 -6.97776738 0.01250410 ## 3 45.50 1 new 49.60260 0.7087865 -4.10260045 0.02103158 ## 4 44.00 1 new 49.60260 0.7087865 -5.60260045 0.02103158 ## 5 71.00 2 new 56.83544 0.6764502 14.16455915 0.01915635 ## 6 45.00 0 new 42.36976 1.0651119 2.63023994 0.04749321 ## 7 37.02 0 used 36.78493 0.7065565 0.23507301 0.02089945 ## 8 53.99 2 new 56.83544 0.6764502 -2.84544085 0.01915635 ## 9 47.00 1 used 44.01777 0.5465195 2.98223262 0.01250410 ## 10 50.00 1 used 44.01777 0.5465195 5.98223262 0.01250410 ## 11 54.99 2 new 56.83544 0.6764502 -1.84544085 0.01915635 ## 12 56.01 2 new 56.83544 0.6764502 -0.82544085 0.01915635 ## 13 48.00 2 new 56.83544 0.6764502 -8.83544085 0.01915635 ## 14 56.00 2 new 56.83544 0.6764502 -0.83544085 0.01915635 ## 15 43.33 1 used 44.01777 0.5465195 -0.68776738 0.01250410 ## 16 46.00 0 used 36.78493 0.7065565 9.21507301 0.02089945 ## 17 46.71 1 new 49.60260 0.7087865 -2.89260045 0.02103158 ## 18 46.00 1 new 49.60260 0.7087865 -3.60260045 0.02103158 ## 19 55.99 2 new 56.83544 0.6764502 -0.84544085 0.01915635 ## 20 31.00 0 used 36.78493 0.7065565 -5.78492699 0.02089945 ## 21 53.98 2 new 56.83544 0.6764502 -2.85544085 0.01915635 ## 22 64.95 2 new 56.83544 0.6764502 8.11455915 0.01915635 ## 23 50.50 2 new 56.83544 0.6764502 -6.33544085 0.01915635 ## 24 46.50 1 used 44.01777 0.5465195 2.48223262 0.01250410 ## 25 55.00 2 new 56.83544 0.6764502 -1.83544085 0.01915635 ## 26 34.50 0 used 36.78493 0.7065565 -2.28492699 0.02089945 ## 27 36.00 0 used 36.78493 0.7065565 -0.78492699 0.02089945 ## 28 40.00 0 used 36.78493 0.7065565 3.21507301 0.02089945 ## 29 47.00 1 new 49.60260 0.7087865 -2.60260045 0.02103158 ## 30 43.00 0 used 36.78493 0.7065565 6.21507301 0.02089945 ## 31 31.00 0 used 36.78493 0.7065565 -5.78492699 0.02089945 ## 32 41.99 1 used 44.01777 0.5465195 -2.02776738 0.01250410 ## 33 49.49 2 used 51.25061 0.8279109 -1.76060777 0.02869514 ## 34 41.00 1 used 44.01777 0.5465195 -3.01776738 0.01250410 ## 35 44.78 1 used 44.01777 0.5465195 0.76223262 0.01250410 ## 36 47.00 0 used 36.78493 0.7065565 10.21507301 0.02089945 ## 37 44.00 1 used 44.01777 0.5465195 -0.01776738 0.01250410 ## 38 63.99 2 new 56.83544 0.6764502 7.15455915 0.01915635 ## 39 53.76 2 new 56.83544 0.6764502 -3.07544085 0.01915635 ## 40 46.03 1 new 49.60260 0.7087865 -3.57260045 0.02103158 ## 41 42.25 1 used 44.01777 0.5465195 -1.76776738 0.01250410 ## 42 46.00 2 used 51.25061 0.8279109 -5.25060777 0.02869514 ## 43 51.99 2 new 56.83544 0.6764502 -4.84544085 0.01915635 ## 44 55.99 2 new 56.83544 0.6764502 -0.84544085 0.01915635 ## 45 41.99 1 used 44.01777 0.5465195 -2.02776738 0.01250410 ## 46 53.99 2 new 56.83544 0.6764502 -2.84544085 0.01915635 ## 47 39.00 0 used 36.78493 0.7065565 2.21507301 0.02089945 ## 48 38.06 0 used 36.78493 0.7065565 1.27507301 0.02089945 ## 49 46.00 1 used 44.01777 0.5465195 1.98223262 0.01250410 ## 50 59.88 2 new 56.83544 0.6764502 3.04455915 0.01915635 ## 51 28.98 0 used 36.78493 0.7065565 -7.80492699 0.02089945 ## 52 36.00 1 used 44.01777 0.5465195 -8.01776738 0.01250410 ## 53 51.99 0 new 42.36976 1.0651119 9.62023994 0.04749321 ## 54 43.95 0 used 36.78493 0.7065565 7.16507301 0.02089945 ## 55 32.00 0 used 36.78493 0.7065565 -4.78492699 0.02089945 ## 56 40.06 1 used 44.01777 0.5465195 -3.95776738 0.01250410 ## 57 48.00 1 used 44.01777 0.5465195 3.98223262 0.01250410 ## 58 36.00 0 used 36.78493 0.7065565 -0.78492699 0.02089945 ## 59 31.00 0 used 36.78493 0.7065565 -5.78492699 0.02089945 ## 60 53.99 2 new 56.83544 0.6764502 -2.84544085 0.01915635 ## 61 30.00 0 used 36.78493 0.7065565 -6.78492699 0.02089945 ## 62 58.00 2 new 56.83544 0.6764502 1.16455915 0.01915635 ## 63 38.10 0 used 36.78493 0.7065565 1.31507301 0.02089945 ## 64 61.76 2 used 51.25061 0.8279109 10.50939223 0.02869514 ## 65 53.99 2 new 56.83544 0.6764502 -2.84544085 0.01915635 ## 66 40.00 1 used 44.01777 0.5465195 -4.01776738 0.01250410 ## 67 64.50 3 used 58.48345 1.2882085 6.01655183 0.06947257 ## 68 49.01 2 used 51.25061 0.8279109 -2.24060777 0.02869514 ## 69 47.00 1 new 49.60260 0.7087865 -2.60260045 0.02103158 ## 70 40.10 1 used 44.01777 0.5465195 -3.91776738 0.01250410 ## 71 41.50 1 new 49.60260 0.7087865 -8.10260045 0.02103158 ## 72 56.00 2 new 56.83544 0.6764502 -0.83544085 0.01915635 ## 73 64.95 2 new 56.83544 0.6764502 8.11455915 0.01915635 ## 74 49.00 2 used 51.25061 0.8279109 -2.25060777 0.02869514 ## 75 48.00 1 used 44.01777 0.5465195 3.98223262 0.01250410 ## 76 38.00 0 used 36.78493 0.7065565 1.21507301 0.02089945 ## 77 45.00 0 used 36.78493 0.7065565 8.21507301 0.02089945 ## 78 41.95 0 used 36.78493 0.7065565 5.16507301 0.02089945 ## 79 43.36 1 used 44.01777 0.5465195 -0.65776738 0.01250410 ## 80 54.99 2 new 56.83544 0.6764502 -1.84544085 0.01915635 ## 81 45.21 1 used 44.01777 0.5465195 1.19223262 0.01250410 ## 82 65.02 4 used 65.71629 1.7946635 -0.69628856 0.13483640 ## 83 45.75 1 used 44.01777 0.5465195 1.73223262 0.01250410 ## 84 64.00 2 new 56.83544 0.6764502 7.16455915 0.01915635 ## 85 36.00 0 used 36.78493 0.7065565 -0.78492699 0.02089945 ## 86 54.70 1 new 49.60260 0.7087865 5.09739955 0.02103158 ## 87 49.91 1 new 49.60260 0.7087865 0.30739955 0.02103158 ## 88 47.00 0 used 36.78493 0.7065565 10.21507301 0.02089945 ## 89 43.00 1 used 44.01777 0.5465195 -1.01776738 0.01250410 ## 90 35.99 0 used 36.78493 0.7065565 -0.79492699 0.02089945 ## 91 54.49 2 used 51.25061 0.8279109 3.23939223 0.02869514 ## 92 46.00 1 used 44.01777 0.5465195 1.98223262 0.01250410 ## 93 31.06 0 used 36.78493 0.7065565 -5.72492699 0.02089945 ## 94 55.60 2 used 51.25061 0.8279109 4.34939223 0.02869514 ## 95 40.10 0 new 42.36976 1.0651119 -2.26976006 0.04749321 ## 96 52.59 2 new 56.83544 0.6764502 -4.24544085 0.01915635 ## 97 44.00 2 used 51.25061 0.8279109 -7.25060777 0.02869514 ## 98 38.26 1 used 44.01777 0.5465195 -5.75776738 0.01250410 ## 99 51.00 2 used 51.25061 0.8279109 -0.25060777 0.02869514 ## 100 48.99 2 new 56.83544 0.6764502 -7.84544085 0.01915635 ## 101 66.44 2 new 56.83544 0.6764502 9.60455915 0.01915635 ## 102 63.50 2 new 56.83544 0.6764502 6.66455915 0.01915635 ## 103 42.00 0 used 36.78493 0.7065565 5.21507301 0.02089945 ## 104 47.00 1 new 49.60260 0.7087865 -2.60260045 0.02103158 ## 105 55.00 2 used 51.25061 0.8279109 3.74939223 0.02869514 ## 106 33.01 1 used 44.01777 0.5465195 -11.00776738 0.01250410 ## 107 53.76 2 new 56.83544 0.6764502 -3.07544085 0.01915635 ## 108 46.00 1 new 49.60260 0.7087865 -3.60260045 0.02103158 ## 109 43.00 0 used 36.78493 0.7065565 6.21507301 0.02089945 ## 110 42.55 1 used 44.01777 0.5465195 -1.46776738 0.01250410 ## 111 52.50 2 used 51.25061 0.8279109 1.24939223 0.02869514 ## 112 57.50 2 new 56.83544 0.6764502 0.66455915 0.01915635 ## 113 75.00 3 new 64.06828 1.0000415 10.93171876 0.04186751 ## 114 48.92 1 used 44.01777 0.5465195 4.90223262 0.01250410 ## 115 45.99 1 new 49.60260 0.7087865 -3.61260045 0.02103158 ## 116 40.05 1 used 44.01777 0.5465195 -3.96776738 0.01250410 ## 117 45.00 1 new 49.60260 0.7087865 -4.60260045 0.02103158 ## 118 50.00 2 used 51.25061 0.8279109 -1.25060777 0.02869514 ## 119 49.75 0 new 42.36976 1.0651119 7.38023994 0.04749321 ## 120 47.00 1 used 44.01777 0.5465195 2.98223262 0.01250410 ## 121 56.00 2 new 56.83544 0.6764502 -0.83544085 0.01915635 ## 122 41.00 1 used 44.01777 0.5465195 -3.01776738 0.01250410 ## 123 46.00 1 new 49.60260 0.7087865 -3.60260045 0.02103158 ## 124 34.99 1 used 44.01777 0.5465195 -9.02776738 0.01250410 ## 125 49.00 2 used 51.25061 0.8279109 -2.25060777 0.02869514 ## 126 61.00 2 new 56.83544 0.6764502 4.16455915 0.01915635 ## 127 62.89 2 new 56.83544 0.6764502 6.05455915 0.01915635 ## 128 46.00 1 new 49.60260 0.7087865 -3.60260045 0.02103158 ## 129 64.95 2 new 56.83544 0.6764502 8.11455915 0.01915635 ## 130 36.99 0 used 36.78493 0.7065565 0.20507301 0.02089945 ## 131 44.00 1 used 44.01777 0.5465195 -0.01776738 0.01250410 ## 132 41.35 1 used 44.01777 0.5465195 -2.66776738 0.01250410 ## 133 37.00 0 used 36.78493 0.7065565 0.21507301 0.02089945 ## 134 58.98 2 new 56.83544 0.6764502 2.14455915 0.01915635 ## 135 39.00 0 used 36.78493 0.7065565 2.21507301 0.02089945 ## 136 40.70 1 used 44.01777 0.5465195 -3.31776738 0.01250410 ## 137 39.51 0 used 36.78493 0.7065565 2.72507301 0.02089945 ## 138 52.00 2 used 51.25061 0.8279109 0.74939223 0.02869514 ## 139 47.70 1 new 49.60260 0.7087865 -1.90260045 0.02103158 ## 140 38.76 0 used 36.78493 0.7065565 1.97507301 0.02089945 ## 141 54.51 2 new 56.83544 0.6764502 -2.32544085 0.01915635 ## .sigma .cooksd .std.resid ## 1 4.902339 1.161354e-03 0.402708933 ## 2 4.868399 8.712334e-03 -1.436710863 ## 3 4.892414 5.154337e-03 -0.848389768 ## 4 4.881308 9.612441e-03 -1.158579529 ## 5 4.750591 5.574926e-02 2.926332759 ## 6 4.899816 5.053659e-03 0.551419180 ## 7 4.905181 1.681147e-05 0.048608215 ## 8 4.899077 2.249739e-03 -0.587854989 ## 9 4.898517 1.591419e-03 0.614036807 ## 10 4.878184 6.403658e-03 1.231731888 ## 11 4.902639 9.463096e-04 -0.381259589 ## 12 4.904706 1.893237e-04 -0.170532281 ## 13 4.845644 2.169149e-02 -1.825361432 ## 14 4.904693 1.939387e-04 -0.172598235 ## 15 4.904866 8.464177e-05 -0.141610176 ## 16 4.840263 2.583436e-02 1.905485609 ## 17 4.898859 2.562308e-03 -0.598170028 ## 18 4.895349 3.974537e-03 -0.744993181 ## 19 4.904680 1.986092e-04 -0.174664189 ## 20 4.879726 1.018113e-02 -1.196202689 ## 21 4.899034 2.265580e-03 -0.589920943 ## 22 4.855017 1.829628e-02 1.676430592 ## 23 4.874681 1.115287e-02 -1.308872933 ## 24 4.900578 1.102520e-03 0.511087627 ## 25 4.902666 9.360817e-04 -0.379193635 ## 26 4.901254 1.588345e-03 -0.472475419 ## 27 4.904754 1.874384e-04 -0.162306590 ## 28 4.897361 3.144719e-03 0.664810290 ## 29 4.900072 2.074290e-03 -0.538200007 ## 30 4.875781 1.175148e-02 1.285147949 ## 31 4.879726 1.018113e-02 -1.196202689 ## 32 4.902124 7.357628e-04 -0.417513979 ## 33 4.902848 1.315656e-03 -0.365515142 ## 34 4.898356 1.629570e-03 -0.621353356 ## 35 4.904785 1.039625e-04 0.156942447 ## 36 4.825276 3.174557e-02 2.112264829 ## 37 4.905222 5.648698e-08 -0.003658274 ## 38 4.866239 1.422325e-02 1.478099008 ## 39 4.898043 2.628135e-03 -0.635371930 ## 40 4.895513 3.908619e-03 -0.738789386 ## 41 4.902867 5.591802e-04 -0.363980405 ## 42 4.884059 1.170136e-02 -1.090064849 ## 43 4.887380 6.523784e-03 -1.001045788 ## 44 4.904680 1.986092e-04 -0.174664189 ## 45 4.902124 7.357628e-04 -0.417513979 ## 46 4.899077 2.249739e-03 -0.587854989 ## 47 4.901493 1.492713e-03 0.458031070 ## 48 4.903987 4.946184e-04 0.263658603 ## 49 4.902261 7.030898e-04 0.408138446 ## 50 4.898186 2.575620e-03 0.628991915 ## 51 4.858711 1.853266e-02 -1.613896713 ## 52 4.856546 1.150293e-02 -1.650845158 ## 53 4.832389 6.760627e-02 2.016844445 ## 54 4.866054 1.561857e-02 1.481588208 ## 55 4.887793 6.965475e-03 -0.989423469 ## 56 4.893406 2.802864e-03 -0.814897814 ## 57 4.893260 2.837624e-03 0.819935167 ## 58 4.904754 1.874384e-04 -0.162306590 ## 59 4.879726 1.018113e-02 -1.196202689 ## 60 4.899077 2.249739e-03 -0.587854989 ## 61 4.870114 1.400524e-02 -1.402981909 ## 62 4.904194 3.768392e-04 0.240592564 ## 63 4.903908 5.261382e-04 0.271929772 ## 64 4.819876 4.687834e-02 2.181827236 ## 65 4.899077 2.249739e-03 -0.587854989 ## 66 4.893045 2.888492e-03 -0.827251716 ## 67 4.876193 4.052940e-02 1.276155547 ## 68 4.901375 2.130829e-03 -0.465166678 ## 69 4.900072 2.074290e-03 -0.538200007 ## 70 4.893644 2.746495e-03 -0.806661880 ## 71 4.855070 2.010496e-02 -1.675562463 ## 72 4.904693 1.939387e-04 -0.172598235 ## 73 4.855017 1.829628e-02 1.676430592 ## 74 4.901341 2.149892e-03 -0.467242751 ## 75 4.893260 2.837624e-03 0.819935167 ## 76 4.904101 4.491639e-04 0.251251850 ## 77 4.853667 2.053161e-02 1.698706389 ## 78 4.884908 8.116206e-03 1.068029769 ## 79 4.904897 7.741876e-05 -0.135433225 ## 80 4.902639 9.463096e-04 -0.381259589 ## 81 4.904152 2.543452e-04 0.245478742 ## 82 4.904806 1.218734e-03 -0.153165404 ## 83 4.902961 5.369254e-04 0.356663856 ## 84 4.866129 1.426303e-02 1.480164962 ## 85 4.904754 1.874384e-04 -0.162306590 ## 86 4.885435 7.957044e-03 1.054107431 ## 87 4.905151 2.893749e-05 0.063568128 ## 88 4.825276 3.174557e-02 2.112264829 ## 89 4.904442 1.853526e-04 -0.209556635 ## 90 4.904742 1.922448e-04 -0.164374382 ## 91 4.897178 4.453937e-03 0.672521687 ## 92 4.902261 7.030898e-04 0.408138446 ## 93 4.880253 9.971030e-03 -1.183795936 ## 94 4.890710 8.029235e-03 0.902965863 ## 95 4.901197 3.763354e-03 -0.475846028 ## 96 4.891531 5.008164e-03 -0.877088548 ## 97 4.864786 2.231340e-02 -1.505279580 ## 98 4.880180 5.932118e-03 -1.185514863 ## 99 4.905174 2.665668e-05 -0.052028020 ## 100 4.858308 1.710282e-02 -1.620831987 ## 101 4.834741 2.563232e-02 1.984257737 ## 102 4.871414 1.234172e-02 1.376867262 ## 103 4.884512 8.274103e-03 1.078368730 ## 104 4.900072 2.074290e-03 -0.538200007 ## 105 4.894442 5.966763e-03 0.778401443 ## 106 4.813060 2.168204e-02 -2.266481255 ## 107 4.898043 2.628135e-03 -0.635371930 ## 108 4.895349 3.974537e-03 -0.744993181 ## 109 4.875781 1.175148e-02 1.285147949 ## 110 4.903599 3.854926e-04 -0.302210897 ## 111 4.904027 6.625438e-04 0.259383029 ## 112 4.904888 1.227157e-04 0.137294864 ## 113 4.811529 7.605411e-02 2.285052621 ## 114 4.887082 4.300207e-03 1.009361659 ## 115 4.895294 3.996633e-03 -0.747061113 ## 116 4.893346 2.817046e-03 -0.816956798 ## 117 4.889096 6.487255e-03 -0.951786355 ## 118 4.904024 6.638336e-04 -0.259635386 ## 119 4.862490 3.978837e-02 1.547237493 ## 120 4.898517 1.591419e-03 0.614036807 ## 121 4.904693 1.939387e-04 -0.172598235 ## 122 4.898356 1.629570e-03 -0.621353356 ## 123 4.895349 3.974537e-03 -0.744993181 ## 124 4.843427 1.458352e-02 -1.858802502 ## 125 4.901341 2.149892e-03 -0.467242751 ## 126 4.892049 4.819157e-03 0.860378763 ## 127 4.877336 1.018587e-02 1.250844068 ## 128 4.895349 3.974537e-03 -0.744993181 ## 129 4.855017 1.829628e-02 1.676430592 ## 130 4.905191 1.279432e-05 0.042404838 ## 131 4.905222 5.648698e-08 -0.003658274 ## 132 4.899857 1.273496e-03 -0.549288929 ## 133 4.905187 1.407252e-05 0.044472630 ## 134 4.901733 1.277936e-03 0.443056056 ## 135 4.901493 1.492713e-03 0.458031070 ## 136 4.896922 1.969670e-03 -0.683122864 ## 137 4.899576 2.259209e-03 0.563488472 ## 138 4.904792 2.383611e-04 0.155579346 ## 139 4.902471 1.108535e-03 -0.393444786 ## 140 4.902257 1.186770e-03 0.408404057 ## 141 4.901119 1.502601e-03 -0.480425381 ### Fitting a model with interaction Including an interaction term in a model is easy—we just have to tell lm() that we want to include that new variable. An expression of the form lm(y ~ x + z + x:z, data = mydata) will do the trick. The use of the colon (:) here means that the interaction between x and z will be a third term in the model. • Use lm() to fit a model for the price of a MarioKart as a function of its condition and the duration of the auction, with interaction. lm(formula = totalPr ~ cond + duration + cond:duration, data = mario_kart) ## ## Call: ## lm(formula = totalPr ~ cond + duration + cond:duration, data = mario_kart) ## ## Coefficients: ## (Intercept) condused duration ## 58.268 -17.122 -1.966 ## condused:duration ## 2.325 ### Visualizing interaction models Interaction allows the slope of the regression line in each group to vary. In this case, this means that the relationship between the final price and the length of the auction is moderated by the condition of each item. Interaction models are easy to visualize in the data space with ggplot2 because they have the same coefficients as if the models were fit independently to each group defined by the level of the categorical variable. In this case, new and used MarioKarts each get their own regression line. To see this, we can set an aesthetic (e.g. color) to the categorical variable, and then add a geom_smooth() layer to overlay the regression line for each color. • Use the color aesthetic and the geom_smooth() function to plot the interaction model between duration and condition in the data space. Make sure you set the method and se arguments of geom_smooth(). # interaction plot ggplot(mario_kart, aes(y= totalPr, x= duration, color=cond)) + geom_point() + geom_smooth(method = "lm",se=0) ### Consequences of Simpson’s paradox In the simple linear regression model for average SAT score, (total) as a function of average teacher salary (salary), the fitted coefficient was -5.02 points per thousand dollars. This suggests that for every additional thousand dollars of salary for teachers in a particular state, the expected SAT score for a student from that state is about 5 points lower. In the model that includes the percentage of students taking the SAT, the coefficient on salary becomes 1.84 points per thousand dollars. Choose the correct interpretation of this slope coefficient. For every additional thousand dollars of salary for teachers in a particular state, the expected SAT score for a student from that state is about 2 points higher, after controlling for the percentage of students taking the SAT. ### Simpson’s paradox in action A mild version of (Simpson’s paradox)[https://en.wikipedia.org/wiki/Simpson%27s_paradox] can be observed in the MarioKart auction data. Consider the relationship between the final auction price and the length of the auction. It seems reasonable to assume that longer auctions would result in higher prices, since—other things being equal—a longer auction gives more bidders more time to see the auction and bid on the item. However, a simple linear regression model reveals the opposite: longer auctions are associated with lower final prices. The problem is that all other things are not equal. In this case, the new MarioKarts—which people pay a premium for—were mostly sold in one-day auctions, while a plurality of the used MarioKarts were sold in the standard seven-day auctions. Our simple linear regression model is misleading, in that it suggests a negative relationship between final auction price and duration. However, for the used MarioKarts, the relationship is positive. • Fit a simple linear regression model for final auction price (totalPr) as a function of duration (duration). slr <- ggplot(mario_kart, aes(y = totalPr, x = duration)) + geom_point() + geom_smooth(method = "lm", se = 0) # model with one slope lm(totalPr ~ duration, data = mario_kart) ## ## Call: ## lm(formula = totalPr ~ duration, data = mario_kart) ## ## Coefficients: ## (Intercept) duration ## 52.374 -1.317 • Use aes() to add a color aesthetic that’s mapped to the condition variable to the slr object. slr + aes(color = cond) ## Multiple Regression ### Fitting a MLR model In terms of the R code, fitting a multiple linear regression model is easy: simply add variables to the model formula you specify in the lm() command. In a parallel slopes model, we had two explanatory variables: one was numeric and one was categorical. Here, we will allow both explanatory variables to be numeric. • Fit a multiple linear regression model for total price as a function of the duration of the auction and the starting price # Fit the model using duration and startPr (mod <- lm(totalPr ~ duration + startPr, mario_kart)) ## ## Call: ## lm(formula = totalPr ~ duration + startPr, data = mario_kart) ## ## Coefficients: ## (Intercept) duration startPr ## 51.030 -1.508 0.233 ### Tiling the plane One method for visualizing a multiple linear regression model is to create a heatmap of the fitted values in the plane defined by the two explanatory variables. This heatmap will illustrate how the model output changes over different combinations of the explanatory variables. This is a multistep process: • First, create a grid of the possible pairs of values of the explanatory variables. The grid should be over the actual range of the data present in each variable. Store the result as a data frame called grid. grid <- mario_kart %>% data_grid(duration = seq_range(duration, by = 1), startPr = seq_range(startPr, by = 1) ) • Use augment() with the newdata argument to find the $$\hat{y}$$’s corresponding to the values in grid. (price_hats <- augment(mod, newdata = grid)) ## duration startPr .fitted .se.fit ## 1 1 0.01 49.52371 0.9965480 ## 2 1 1.01 49.75666 0.9848633 ## 3 1 2.01 49.98962 0.9749931 ## 4 1 3.01 50.22257 0.9669932 ## 5 1 4.01 50.45553 0.9609101 ## 6 1 5.01 50.68848 0.9567805 ## 7 1 6.01 50.92144 0.9546298 ## 8 1 7.01 51.15439 0.9544713 ## 9 1 8.01 51.38734 0.9563059 ## 10 1 9.01 51.62030 0.9601223 ## 11 1 10.01 51.85325 0.9658970 ## 12 1 11.01 52.08621 0.9735951 ## 13 1 12.01 52.31916 0.9831715 ## 14 1 13.01 52.55212 0.9945718 ## 15 1 14.01 52.78507 1.0077343 ## 16 1 15.01 53.01802 1.0225908 ## 17 1 16.01 53.25098 1.0390687 ## 18 1 17.01 53.48393 1.0570921 ## 19 1 18.01 53.71689 1.0765836 ## 20 1 19.01 53.94984 1.0974647 ## 21 1 20.01 54.18279 1.1196579 ## 22 1 21.01 54.41575 1.1430866 ## 23 1 22.01 54.64870 1.1676765 ## 24 1 23.01 54.88166 1.1933558 ## 25 1 24.01 55.11461 1.2200558 ## 26 1 25.01 55.34757 1.2477109 ## 27 1 26.01 55.58052 1.2762590 ## 28 1 27.01 55.81347 1.3056416 ## 29 1 28.01 56.04643 1.3358035 ## 30 1 29.01 56.27938 1.3666933 ## 31 1 30.01 56.51234 1.3982626 ## 32 1 31.01 56.74529 1.4304664 ## 33 1 32.01 56.97825 1.4632629 ## 34 1 33.01 57.21120 1.4966132 ## 35 1 34.01 57.44415 1.5304809 ## 36 1 35.01 57.67711 1.5648325 ## 37 1 36.01 57.91006 1.5996369 ## 38 1 37.01 58.14302 1.6348651 ## 39 1 38.01 58.37597 1.6704903 ## 40 1 39.01 58.60892 1.7064876 ## 41 1 40.01 58.84188 1.7428340 ## 42 1 41.01 59.07483 1.7795081 ## 43 1 42.01 59.30779 1.8164901 ## 44 1 43.01 59.54074 1.8537614 ## 45 1 44.01 59.77370 1.8913051 ## 46 1 45.01 60.00665 1.9291052 ## 47 1 46.01 60.23960 1.9671469 ## 48 1 47.01 60.47256 2.0054165 ## 49 1 48.01 60.70551 2.0439012 ## 50 1 49.01 60.93847 2.0825890 ## 51 1 50.01 61.17142 2.1214689 ## 52 1 51.01 61.40438 2.1605305 ## 53 1 52.01 61.63733 2.1997640 ## 54 1 53.01 61.87028 2.2391605 ## 55 1 54.01 62.10324 2.2787115 ## 56 1 55.01 62.33619 2.3184091 ## 57 1 56.01 62.56915 2.3582459 ## 58 1 57.01 62.80210 2.3982149 ## 59 1 58.01 63.03505 2.4383097 ## 60 1 59.01 63.26801 2.4785242 ## 61 1 60.01 63.50096 2.5188525 ## 62 1 61.01 63.73392 2.5592894 ## 63 1 62.01 63.96687 2.5998298 ## 64 1 63.01 64.19983 2.6404688 ## 65 1 64.01 64.43278 2.6812021 ## 66 1 65.01 64.66573 2.7220254 ## 67 1 66.01 64.89869 2.7629347 ## 68 1 67.01 65.13164 2.8039261 ## 69 1 68.01 65.36460 2.8449963 ## 70 1 69.01 65.59755 2.8861418 ## 71 2 0.01 48.01558 0.8524393 ## 72 2 1.01 48.24854 0.8368861 ## 73 2 2.01 48.48149 0.8233544 ## 74 2 3.01 48.71445 0.8119450 ## 75 2 4.01 48.94740 0.8027485 ## 76 2 5.01 49.18036 0.7958417 ## 77 2 6.01 49.41331 0.7912845 ## 78 2 7.01 49.64626 0.7891176 ## 79 2 8.01 49.87922 0.7893606 ## 80 2 9.01 50.11217 0.7920114 ## 81 2 10.01 50.34513 0.7970460 ## 82 2 11.01 50.57808 0.8044196 ## 83 2 12.01 50.81104 0.8140685 ## 84 2 13.01 51.04399 0.8259132 ## 85 2 14.01 51.27694 0.8398607 ## 86 2 15.01 51.50990 0.8558081 ## 87 2 16.01 51.74285 0.8736461 ## 88 2 17.01 51.97581 0.8932612 ## 89 2 18.01 52.20876 0.9145392 ## 90 2 19.01 52.44171 0.9373669 ## 91 2 20.01 52.67467 0.9616339 ## 92 2 21.01 52.90762 0.9872340 ## 93 2 22.01 53.14058 1.0140663 ## 94 2 23.01 53.37353 1.0420357 ## 95 2 24.01 53.60649 1.0710530 ## 96 2 25.01 53.83944 1.1010353 ## 97 2 26.01 54.07239 1.1319061 ## 98 2 27.01 54.30535 1.1635946 ## 99 2 28.01 54.53830 1.1960358 ## 100 2 29.01 54.77126 1.2291701 ## 101 2 30.01 55.00421 1.2629429 ## 102 2 31.01 55.23717 1.2973045 ## 103 2 32.01 55.47012 1.3322091 ## 104 2 33.01 55.70307 1.3676154 ## 105 2 34.01 55.93603 1.4034852 ## 106 2 35.01 56.16898 1.4397839 ## 107 2 36.01 56.40194 1.4764800 ## 108 2 37.01 56.63489 1.5135445 ## 109 2 38.01 56.86784 1.5509510 ## 110 2 39.01 57.10080 1.5886753 ## 111 2 40.01 57.33375 1.6266953 ## 112 2 41.01 57.56671 1.6649908 ## 113 2 42.01 57.79966 1.7035432 ## 114 2 43.01 58.03262 1.7423355 ## 115 2 44.01 58.26557 1.7813519 ## 116 2 45.01 58.49852 1.8205781 ## 117 2 46.01 58.73148 1.8600007 ## 118 2 47.01 58.96443 1.8996076 ## 119 2 48.01 59.19739 1.9393875 ## 120 2 49.01 59.43034 1.9793299 ## 121 2 50.01 59.66330 2.0194252 ## 122 2 51.01 59.89625 2.0596645 ## 123 2 52.01 60.12920 2.1000394 ## 124 2 53.01 60.36216 2.1405424 ## 125 2 54.01 60.59511 2.1811663 ## 126 2 55.01 60.82807 2.2219043 ## 127 2 56.01 61.06102 2.2627505 ## 128 2 57.01 61.29397 2.3036990 ## 129 2 58.01 61.52693 2.3447445 ## 130 2 59.01 61.75988 2.3858819 ## 131 2 60.01 61.99284 2.4271066 ## 132 2 61.01 62.22579 2.4684142 ## 133 2 62.01 62.45875 2.5098007 ## 134 2 63.01 62.69170 2.5512621 ## 135 2 64.01 62.92465 2.5927949 ## 136 2 65.01 63.15761 2.6343958 ## 137 2 66.01 63.39056 2.6760614 ## 138 2 67.01 63.62352 2.7177890 ## 139 2 68.01 63.85647 2.7595755 ## 140 2 69.01 64.08943 2.8014185 ## 141 3 0.01 46.50746 0.7686211 ## 142 3 1.01 46.74041 0.7492544 ## 143 3 2.01 46.97337 0.7319796 ## 144 3 3.01 47.20632 0.7169480 ## 145 3 4.01 47.43928 0.7043033 ## 146 3 5.01 47.67223 0.6941758 ## 147 3 6.01 47.90518 0.6866769 ## 148 3 7.01 48.13814 0.6818934 ## 149 3 8.01 48.37109 0.6798826 ## 150 3 9.01 48.60405 0.6806690 ## 151 3 10.01 48.83700 0.6842431 ## 152 3 11.01 49.06995 0.6905615 ## 153 3 12.01 49.30291 0.6995499 ## 154 3 13.01 49.53586 0.7111070 ## 155 3 14.01 49.76882 0.7251100 ## 156 3 15.01 50.00177 0.7414204 ## 157 3 16.01 50.23473 0.7598895 ## 158 3 17.01 50.46768 0.7803642 ## 159 3 18.01 50.70063 0.8026910 ## 160 3 19.01 50.93359 0.8267198 ## 161 3 20.01 51.16654 0.8523067 ## 162 3 21.01 51.39950 0.8793156 ## 163 3 22.01 51.63245 0.9076197 ## 164 3 23.01 51.86541 0.9371016 ## 165 3 24.01 52.09836 0.9676537 ## 166 3 25.01 52.33131 0.9991777 ## 167 3 26.01 52.56427 1.0315846 ## 168 3 27.01 52.79722 1.0647938 ## 169 3 28.01 53.03018 1.0987325 ## 170 3 29.01 53.26313 1.1333351 ## 171 3 30.01 53.49608 1.1685428 ## 172 3 31.01 53.72904 1.2043025 ## 173 3 32.01 53.96199 1.2405663 ## 174 3 33.01 54.19495 1.2772915 ## 175 3 34.01 54.42790 1.3144392 ## 176 3 35.01 54.66086 1.3519747 ## 177 3 36.01 54.89381 1.3898666 ## 178 3 37.01 55.12676 1.4280864 ## 179 3 38.01 55.35972 1.4666087 ## 180 3 39.01 55.59267 1.5054100 ## 181 3 40.01 55.82563 1.5444694 ## 182 3 41.01 56.05858 1.5837679 ## 183 3 42.01 56.29154 1.6232880 ## 184 3 43.01 56.52449 1.6630139 ## 185 3 44.01 56.75744 1.7029313 ## 186 3 45.01 56.99040 1.7430270 ## 187 3 46.01 57.22335 1.7832889 ## 188 3 47.01 57.45631 1.8237061 ## 189 3 48.01 57.68926 1.8642684 ## 190 3 49.01 57.92221 1.9049666 ## 191 3 50.01 58.15517 1.9457921 ## 192 3 51.01 58.38812 1.9867372 ## 193 3 52.01 58.62108 2.0277945 ## 194 3 53.01 58.85403 2.0689574 ## 195 3 54.01 59.08699 2.1102197 ## 196 3 55.01 59.31994 2.1515757 ## 197 3 56.01 59.55289 2.1930201 ## 198 3 57.01 59.78585 2.2345479 ## 199 3 58.01 60.01880 2.2761546 ## 200 3 59.01 60.25176 2.3178360 ## 201 3 60.01 60.48471 2.3595881 ## 202 3 61.01 60.71767 2.4014072 ## 203 3 62.01 60.95062 2.4432898 ## 204 3 63.01 61.18357 2.4852328 ## 205 3 64.01 61.41653 2.5272331 ## 206 3 65.01 61.64948 2.5692880 ## 207 3 66.01 61.88244 2.6113948 ## 208 3 67.01 62.11539 2.6535510 ## 209 3 68.01 62.34834 2.6957543 ## 210 3 69.01 62.58130 2.7380025 ## 211 4 0.01 44.99933 0.7651698 ## 212 4 1.01 45.23229 0.7436173 ## 213 4 2.01 45.46524 0.7240558 ## 214 4 3.01 45.69820 0.7066507 ## 215 4 4.01 45.93115 0.6915648 ## 216 4 5.01 46.16410 0.6789528 ## 217 4 6.01 46.39706 0.6689545 ## 218 4 7.01 46.63001 0.6616885 ## 219 4 8.01 46.86297 0.6572453 ## 220 4 9.01 47.09592 0.6556824 ## 221 4 10.01 47.32887 0.6570204 ## 222 4 11.01 47.56183 0.6612416 ## 223 4 12.01 47.79478 0.6682913 ## 224 4 13.01 48.02774 0.6780815 ## 225 4 14.01 48.26069 0.6904954 ## 226 4 15.01 48.49365 0.7053947 ## 227 4 16.01 48.72660 0.7226255 ## 228 4 17.01 48.95955 0.7420256 ## 229 4 18.01 49.19251 0.7634294 ## 230 4 19.01 49.42546 0.7866736 ## 231 4 20.01 49.65842 0.8115999 ## 232 4 21.01 49.89137 0.8380584 ## 233 4 22.01 50.12433 0.8659085 ## 234 4 23.01 50.35728 0.8950204 ## 235 4 24.01 50.59023 0.9252749 ## 236 4 25.01 50.82319 0.9565637 ## 237 4 26.01 51.05614 0.9887886 ## 238 4 27.01 51.28910 1.0218611 ## 239 4 28.01 51.52205 1.0557014 ## 240 4 29.01 51.75500 1.0902380 ## 241 4 30.01 51.98796 1.1254070 ## 242 4 31.01 52.22091 1.1611507 ## 243 4 32.01 52.45387 1.1974178 ## 244 4 33.01 52.68682 1.2341621 ## 245 4 34.01 52.91978 1.2713423 ## 246 4 35.01 53.15273 1.3089211 ## 247 4 36.01 53.38568 1.3468652 ## 248 4 37.01 53.61864 1.3851447 ## 249 4 38.01 53.85159 1.4237324 ## 250 4 39.01 54.08455 1.4626039 ## 251 4 40.01 54.31750 1.5017373 ## 252 4 41.01 54.55046 1.5411125 ## 253 4 42.01 54.78341 1.5807116 ## 254 4 43.01 55.01636 1.6205180 ## 255 4 44.01 55.24932 1.6605169 ## 256 4 45.01 55.48227 1.7006947 ## 257 4 46.01 55.71523 1.7410390 ## 258 4 47.01 55.94818 1.7815385 ## 259 4 48.01 56.18113 1.8221828 ## 260 4 49.01 56.41409 1.8629625 ## 261 4 50.01 56.64704 1.9038689 ## 262 4 51.01 56.88000 1.9448940 ## 263 4 52.01 57.11295 1.9860303 ## 264 4 53.01 57.34591 2.0272713 ## 265 4 54.01 57.57886 2.0686105 ## 266 4 55.01 57.81181 2.1100422 ## 267 4 56.01 58.04477 2.1515611 ## 268 4 57.01 58.27772 2.1931622 ## 269 4 58.01 58.51068 2.2348410 ## 270 4 59.01 58.74363 2.2765931 ## 271 4 60.01 58.97659 2.3184146 ## 272 4 61.01 59.20954 2.3603019 ## 273 4 62.01 59.44249 2.4022514 ## 274 4 63.01 59.67545 2.4442599 ## 275 4 64.01 59.90840 2.4863246 ## 276 4 65.01 60.14136 2.5284425 ## 277 4 66.01 60.37431 2.5706111 ## 278 4 67.01 60.60726 2.6128279 ## 279 4 68.01 60.84022 2.6550906 ## 280 4 69.01 61.07317 2.6973970 ## 281 5 0.01 43.49121 0.8430732 ## 282 5 1.01 43.72416 0.8216643 ## 283 5 2.01 43.95712 0.8020601 ## 284 5 3.01 44.19007 0.7843959 ## 285 5 4.01 44.42302 0.7688053 ## 286 5 5.01 44.65598 0.7554169 ## 287 5 6.01 44.88893 0.7443493 ## 288 5 7.01 45.12189 0.7357075 ## 289 5 8.01 45.35484 0.7295775 ## 290 5 9.01 45.58779 0.7260231 ## 291 5 10.01 45.82075 0.7250820 ## 292 5 11.01 46.05370 0.7267645 ## 293 5 12.01 46.28666 0.7310524 ## 294 5 13.01 46.51961 0.7379002 ## 295 5 14.01 46.75257 0.7472378 ## 296 5 15.01 46.98552 0.7589730 ## 297 5 16.01 47.21847 0.7729969 ## 298 5 17.01 47.45143 0.7891872 ## 299 5 18.01 47.68438 0.8074138 ## 300 5 19.01 47.91734 0.8275422 ## 301 5 20.01 48.15029 0.8494370 ## 302 5 21.01 48.38325 0.8729654 ## 303 5 22.01 48.61620 0.8979991 ## 304 5 23.01 48.84915 0.9244157 ## 305 5 24.01 49.08211 0.9521001 ## 306 5 25.01 49.31506 0.9809451 ## 307 5 26.01 49.54802 1.0108512 ## 308 5 27.01 49.78097 1.0417270 ## 309 5 28.01 50.01392 1.0734890 ## 310 5 29.01 50.24688 1.1060607 ## 311 5 30.01 50.47983 1.1393727 ## 312 5 31.01 50.71279 1.1733620 ## 313 5 32.01 50.94574 1.2079714 ## 314 5 33.01 51.17870 1.2431491 ## 315 5 34.01 51.41165 1.2788482 ## 316 5 35.01 51.64460 1.3150263 ## 317 5 36.01 51.87756 1.3516448 ## 318 5 37.01 52.11051 1.3886690 ## 319 5 38.01 52.34347 1.4260672 ## 320 5 39.01 52.57642 1.4638109 ## 321 5 40.01 52.80937 1.5018738 ## 322 5 41.01 53.04233 1.5402324 ## 323 5 42.01 53.27528 1.5788652 ## 324 5 43.01 53.50824 1.6177524 ## 325 5 44.01 53.74119 1.6568762 ## 326 5 45.01 53.97415 1.6962201 ## 327 5 46.01 54.20710 1.7357693 ## 328 5 47.01 54.44005 1.7755100 ## 329 5 48.01 54.67301 1.8154296 ## 330 5 49.01 54.90596 1.8555166 ## 331 5 50.01 55.13892 1.8957604 ## 332 5 51.01 55.37187 1.9361511 ## 333 5 52.01 55.60483 1.9766799 ## 334 5 53.01 55.83778 2.0173382 ## 335 5 54.01 56.07073 2.0581186 ## 336 5 55.01 56.30369 2.0990138 ## 337 5 56.01 56.53664 2.1400173 ## 338 5 57.01 56.76960 2.1811230 ## 339 5 58.01 57.00255 2.2223252 ## 340 5 59.01 57.23550 2.2636186 ## 341 5 60.01 57.46846 2.3049984 ## 342 5 61.01 57.70141 2.3464599 ## 343 5 62.01 57.93437 2.3879990 ## 344 5 63.01 58.16732 2.4296115 ## 345 5 64.01 58.40028 2.4712939 ## 346 5 65.01 58.63323 2.5130426 ## 347 5 66.01 58.86618 2.5548544 ## 348 5 67.01 59.09914 2.5967263 ## 349 5 68.01 59.33209 2.6386553 ## 350 5 69.01 59.56505 2.6806388 ## 351 6 0.01 41.98308 0.9831789 ## 352 6 1.01 42.21603 0.9632646 ## 353 6 2.01 42.44899 0.9449471 ## 354 6 3.01 42.68194 0.9283208 ## 355 6 4.01 42.91490 0.9134781 ## 356 6 5.01 43.14785 0.9005073 ## 357 6 6.01 43.38081 0.8894901 ## 358 6 7.01 43.61376 0.8805001 ## 359 6 8.01 43.84671 0.8735996 ## 360 6 9.01 44.07967 0.8688386 ## 361 6 10.01 44.31262 0.8662522 ## 362 6 11.01 44.54558 0.8658601 ## 363 6 12.01 44.77853 0.8676651 ## 364 6 13.01 45.01149 0.8716536 ## 365 6 14.01 45.24444 0.8777959 ## 366 6 15.01 45.47739 0.8860471 ## 367 6 16.01 45.71035 0.8963490 ## 368 6 17.01 45.94330 0.9086319 ## 369 6 18.01 46.17626 0.9228167 ## 370 6 19.01 46.40921 0.9388171 ## 371 6 20.01 46.64216 0.9565421 ## 372 6 21.01 46.87512 0.9758977 ## 373 6 22.01 47.10807 0.9967889 ## 374 6 23.01 47.34103 1.0191213 ## 375 6 24.01 47.57398 1.0428023 ## 376 6 25.01 47.80694 1.0677421 ## 377 6 26.01 48.03989 1.0938547 ## 378 6 27.01 48.27284 1.1210581 ## 379 6 28.01 48.50580 1.1492749 ## 380 6 29.01 48.73875 1.1784322 ## 381 6 30.01 48.97171 1.2084620 ## 382 6 31.01 49.20466 1.2393009 ## 383 6 32.01 49.43762 1.2708900 ## 384 6 33.01 49.67057 1.3031747 ## 385 6 34.01 49.90352 1.3361045 ## 386 6 35.01 50.13648 1.3696331 ## 387 6 36.01 50.36943 1.4037174 ## 388 6 37.01 50.60239 1.4383179 ## 389 6 38.01 50.83534 1.4733983 ## 390 6 39.01 51.06829 1.5089251 ## 391 6 40.01 51.30125 1.5448676 ## 392 6 41.01 51.53420 1.5811973 ## 393 6 42.01 51.76716 1.6178882 ## 394 6 43.01 52.00011 1.6549163 ## 395 6 44.01 52.23307 1.6922594 ## 396 6 45.01 52.46602 1.7298971 ## 397 6 46.01 52.69897 1.7678106 ## 398 6 47.01 52.93193 1.8059826 ## 399 6 48.01 53.16488 1.8443970 ## 400 6 49.01 53.39784 1.8830389 ## 401 6 50.01 53.63079 1.9218946 ## 402 6 51.01 53.86375 1.9609515 ## 403 6 52.01 54.09670 2.0001977 ## 404 6 53.01 54.32965 2.0396224 ## 405 6 54.01 54.56261 2.0792152 ## 406 6 55.01 54.79556 2.1189670 ## 407 6 56.01 55.02852 2.1588687 ## 408 6 57.01 55.26147 2.1989124 ## 409 6 58.01 55.49442 2.2390903 ## 410 6 59.01 55.72738 2.2793954 ## 411 6 60.01 55.96033 2.3198211 ## 412 6 61.01 56.19329 2.3603611 ## 413 6 62.01 56.42624 2.4010097 ## 414 6 63.01 56.65920 2.4417614 ## 415 6 64.01 56.89215 2.4826111 ## 416 6 65.01 57.12510 2.5235542 ## 417 6 66.01 57.35806 2.5645861 ## 418 6 67.01 57.59101 2.6057026 ## 419 6 68.01 57.82397 2.6468998 ## 420 6 69.01 58.05692 2.6881740 ## 421 7 0.01 40.47495 1.1632240 ## 422 7 1.01 40.70791 1.1450791 ## 423 7 2.01 40.94086 1.1283308 ## 424 7 3.01 41.17382 1.1130422 ## 425 7 4.01 41.40677 1.0992742 ## 426 7 5.01 41.63973 1.0870846 ## 427 7 6.01 41.87268 1.0765270 ## 428 7 7.01 42.10563 1.0676498 ## 429 7 8.01 42.33859 1.0604952 ## 430 7 9.01 42.57154 1.0550982 ## 431 7 10.01 42.80450 1.0514860 ## 432 7 11.01 43.03745 1.0496769 ## 433 7 12.01 43.27041 1.0496803 ## 434 7 13.01 43.50336 1.0514961 ## 435 7 14.01 43.73631 1.0551150 ## 436 7 15.01 43.96927 1.0605186 ## 437 7 16.01 44.20222 1.0676797 ## 438 7 17.01 44.43518 1.0765632 ## 439 7 18.01 44.66813 1.0871270 ## 440 7 19.01 44.90108 1.0993226 ## 441 7 20.01 45.13404 1.1130963 ## 442 7 21.01 45.36699 1.1283904 ## 443 7 22.01 45.59995 1.1451440 ## 444 7 23.01 45.83290 1.1632940 ## 445 7 24.01 46.06586 1.1827761 ## 446 7 25.01 46.29881 1.2035257 ## 447 7 26.01 46.53176 1.2254783 ## 448 7 27.01 46.76472 1.2485705 ## 449 7 28.01 46.99767 1.2727403 ## 450 7 29.01 47.23063 1.2979274 ## 451 7 30.01 47.46358 1.3240739 ## 452 7 31.01 47.69654 1.3511240 ## 453 7 32.01 47.92949 1.3790245 ## 454 7 33.01 48.16244 1.4077250 ## 455 7 34.01 48.39540 1.4371774 ## 456 7 35.01 48.62835 1.4673365 ## 457 7 36.01 48.86131 1.4981595 ## 458 7 37.01 49.09426 1.5296065 ## 459 7 38.01 49.32721 1.5616396 ## 460 7 39.01 49.56017 1.5942236 ## 461 7 40.01 49.79312 1.6273253 ## 462 7 41.01 50.02608 1.6609139 ## 463 7 42.01 50.25903 1.6949603 ## 464 7 43.01 50.49199 1.7294375 ## 465 7 44.01 50.72494 1.7643202 ## 466 7 45.01 50.95789 1.7995850 ## 467 7 46.01 51.19085 1.8352097 ## 468 7 47.01 51.42380 1.8711738 ## 469 7 48.01 51.65676 1.9074581 ## 470 7 49.01 51.88971 1.9440447 ## 471 7 50.01 52.12267 1.9809168 ## 472 7 51.01 52.35562 2.0180588 ## 473 7 52.01 52.58857 2.0554560 ## 474 7 53.01 52.82153 2.0930948 ## 475 7 54.01 53.05448 2.1309624 ## 476 7 55.01 53.28744 2.1690467 ## 477 7 56.01 53.52039 2.2073366 ## 478 7 57.01 53.75334 2.2458216 ## 479 7 58.01 53.98630 2.2844917 ## 480 7 59.01 54.21925 2.3233378 ## 481 7 60.01 54.45221 2.3623512 ## 482 7 61.01 54.68516 2.4015236 ## 483 7 62.01 54.91812 2.4408475 ## 484 7 63.01 55.15107 2.4803157 ## 485 7 64.01 55.38402 2.5199213 ## 486 7 65.01 55.61698 2.5596580 ## 487 7 66.01 55.84993 2.5995198 ## 488 7 67.01 56.08289 2.6395010 ## 489 7 68.01 56.31584 2.6795963 ## 490 7 69.01 56.54880 2.7198005 ## 491 8 0.01 38.96683 1.3675234 ## 492 8 1.01 39.19978 1.3509679 ## 493 8 2.01 39.43274 1.3356334 ## 494 8 3.01 39.66569 1.3215625 ## 495 8 4.01 39.89865 1.3087957 ## 496 8 5.01 40.13160 1.2973718 ## 497 8 6.01 40.36455 1.2873263 ## 498 8 7.01 40.59751 1.2786918 ## 499 8 8.01 40.83046 1.2714971 ## 500 8 9.01 41.06342 1.2657666 ## 501 8 10.01 41.29637 1.2615204 ## 502 8 11.01 41.52932 1.2587734 ## 503 8 12.01 41.76228 1.2575356 ## 504 8 13.01 41.99523 1.2578113 ## 505 8 14.01 42.22819 1.2595995 ## 506 8 15.01 42.46114 1.2628938 ## 507 8 16.01 42.69410 1.2676826 ## 508 8 17.01 42.92705 1.2739488 ## 509 8 18.01 43.16000 1.2816710 ## 510 8 19.01 43.39296 1.2908228 ## 511 8 20.01 43.62591 1.3013742 ## 512 8 21.01 43.85887 1.3132915 ## 513 8 22.01 44.09182 1.3265378 ## 514 8 23.01 44.32478 1.3410737 ## 515 8 24.01 44.55773 1.3568578 ## 516 8 25.01 44.79068 1.3738471 ## 517 8 26.01 45.02364 1.3919975 ## 518 8 27.01 45.25659 1.4112641 ## 519 8 28.01 45.48955 1.4316018 ## 520 8 29.01 45.72250 1.4529658 ## 521 8 30.01 45.95545 1.4753114 ## 522 8 31.01 46.18841 1.4985948 ## 523 8 32.01 46.42136 1.5227728 ## 524 8 33.01 46.65432 1.5478036 ## 525 8 34.01 46.88727 1.5736465 ## 526 8 35.01 47.12023 1.6002621 ## 527 8 36.01 47.35318 1.6276125 ## 528 8 37.01 47.58613 1.6556614 ## 529 8 38.01 47.81909 1.6843738 ## 530 8 39.01 48.05204 1.7137163 ## 531 8 40.01 48.28500 1.7436572 ## 532 8 41.01 48.51795 1.7741662 ## 533 8 42.01 48.75091 1.8052144 ## 534 8 43.01 48.98386 1.8367746 ## 535 8 44.01 49.21681 1.8688207 ## 536 8 45.01 49.44977 1.9013283 ## 537 8 46.01 49.68272 1.9342740 ## 538 8 47.01 49.91568 1.9676358 ## 539 8 48.01 50.14863 2.0013930 ## 540 8 49.01 50.38158 2.0355258 ## 541 8 50.01 50.61454 2.0700157 ## 542 8 51.01 50.84749 2.1048451 ## 543 8 52.01 51.08045 2.1399975 ## 544 8 53.01 51.31340 2.1754571 ## 545 8 54.01 51.54636 2.2112093 ## 546 8 55.01 51.77931 2.2472400 ## 547 8 56.01 52.01226 2.2835360 ## 548 8 57.01 52.24522 2.3200850 ## 549 8 58.01 52.47817 2.3568751 ## 550 8 59.01 52.71113 2.3938953 ## 551 8 60.01 52.94408 2.4311349 ## 552 8 61.01 53.17704 2.4685842 ## 553 8 62.01 53.40999 2.5062336 ## 554 8 63.01 53.64294 2.5440743 ## 555 8 64.01 53.87590 2.5820979 ## 556 8 65.01 54.10885 2.6202964 ## 557 8 66.01 54.34181 2.6586624 ## 558 8 67.01 54.57476 2.6971885 ## 559 8 68.01 54.80771 2.7358682 ## 560 8 69.01 55.04067 2.7746949 ## 561 9 0.01 37.45870 1.5867361 ## 562 9 1.01 37.69166 1.5714972 ## 563 9 2.01 37.92461 1.5573323 ## 564 9 3.01 38.15757 1.5442709 ## 565 9 4.01 38.39052 1.5323412 ## 566 9 5.01 38.62347 1.5215699 ## 567 9 6.01 38.85643 1.5119816 ## 568 9 7.01 39.08938 1.5035991 ## 569 9 8.01 39.32234 1.4964426 ## 570 9 9.01 39.55529 1.4905297 ## 571 9 10.01 39.78824 1.4858753 ## 572 9 11.01 40.02120 1.4824913 ## 573 9 12.01 40.25415 1.4803863 ## 574 9 13.01 40.48711 1.4795658 ## 575 9 14.01 40.72006 1.4800320 ## 576 9 15.01 40.95302 1.4817837 ## 577 9 16.01 41.18597 1.4848162 ## 578 9 17.01 41.41892 1.4891218 ## 579 9 18.01 41.65188 1.4946895 ## 580 9 19.01 41.88483 1.5015052 ## 581 9 20.01 42.11779 1.5095520 ## 582 9 21.01 42.35074 1.5188104 ## 583 9 22.01 42.58370 1.5292583 ## 584 9 23.01 42.81665 1.5408716 ## 585 9 24.01 43.04960 1.5536241 ## 586 9 25.01 43.28256 1.5674880 ## 587 9 26.01 43.51551 1.5824341 ## 588 9 27.01 43.74847 1.5984321 ## 589 9 28.01 43.98142 1.6154506 ## 590 9 29.01 44.21437 1.6334579 ## 591 9 30.01 44.44733 1.6524215 ## 592 9 31.01 44.68028 1.6723090 ## 593 9 32.01 44.91324 1.6930877 ## 594 9 33.01 45.14619 1.7147254 ## 595 9 34.01 45.37915 1.7371898 ## 596 9 35.01 45.61210 1.7604494 ## 597 9 36.01 45.84505 1.7844730 ## 598 9 37.01 46.07801 1.8092302 ## 599 9 38.01 46.31096 1.8346914 ## 600 9 39.01 46.54392 1.8608276 ## 601 9 40.01 46.77687 1.8876107 ## 602 9 41.01 47.00983 1.9150137 ## 603 9 42.01 47.24278 1.9430103 ## 604 9 43.01 47.47573 1.9715752 ## 605 9 44.01 47.70869 2.0006841 ## 606 9 45.01 47.94164 2.0303136 ## 607 9 46.01 48.17460 2.0604412 ## 608 9 47.01 48.40755 2.0910454 ## 609 9 48.01 48.64050 2.1221056 ## 610 9 49.01 48.87346 2.1536020 ## 611 9 50.01 49.10641 2.1855158 ## 612 9 51.01 49.33937 2.2178289 ## 613 9 52.01 49.57232 2.2505243 ## 614 9 53.01 49.80528 2.2835854 ## 615 9 54.01 50.03823 2.3169965 ## 616 9 55.01 50.27118 2.3507429 ## 617 9 56.01 50.50414 2.3848102 ## 618 9 57.01 50.73709 2.4191848 ## 619 9 58.01 50.97005 2.4538539 ## 620 9 59.01 51.20300 2.4888052 ## 621 9 60.01 51.43596 2.5240269 ## 622 9 61.01 51.66891 2.5595079 ## 623 9 62.01 51.90186 2.5952375 ## 624 9 63.01 52.13482 2.6312057 ## 625 9 64.01 52.36777 2.6674026 ## 626 9 65.01 52.60073 2.7038193 ## 627 9 66.01 52.83368 2.7404468 ## 628 9 67.01 53.06663 2.7772770 ## 629 9 68.01 53.29959 2.8143017 ## 630 9 69.01 53.53254 2.8515135 ## 631 10 0.01 35.95058 1.8154678 ## 632 10 1.01 36.18353 1.8012978 ## 633 10 2.01 36.41649 1.7880806 ## 634 10 3.01 36.64944 1.7758375 ## 635 10 4.01 36.88239 1.7645889 ## 636 10 5.01 37.11535 1.7543538 ## 637 10 6.01 37.34830 1.7451501 ## 638 10 7.01 37.58126 1.7369942 ## 639 10 8.01 37.81421 1.7299009 ## 640 10 9.01 38.04716 1.7238832 ## 641 10 10.01 38.28012 1.7189526 ## 642 10 11.01 38.51307 1.7151184 ## 643 10 12.01 38.74603 1.7123879 ## 644 10 13.01 38.97898 1.7107665 ## 645 10 14.01 39.21194 1.7102572 ## 646 10 15.01 39.44489 1.7108612 ## 647 10 16.01 39.67784 1.7125771 ## 648 10 17.01 39.91080 1.7154017 ## 649 10 18.01 40.14375 1.7193295 ## 650 10 19.01 40.37671 1.7243530 ## 651 10 20.01 40.60966 1.7304625 ## 652 10 21.01 40.84262 1.7376468 ## 653 10 22.01 41.07557 1.7458924 ## 654 10 23.01 41.30852 1.7551845 ## 655 10 24.01 41.54148 1.7655065 ## 656 10 25.01 41.77443 1.7768405 ## 657 10 26.01 42.00739 1.7891672 ## 658 10 27.01 42.24034 1.8024663 ## 659 10 28.01 42.47329 1.8167164 ## 660 10 29.01 42.70625 1.8318953 ## 661 10 30.01 42.93920 1.8479802 ## 662 10 31.01 43.17216 1.8649476 ## 663 10 32.01 43.40511 1.8827735 ## 664 10 33.01 43.63807 1.9014340 ## 665 10 34.01 43.87102 1.9209046 ## 666 10 35.01 44.10397 1.9411611 ## 667 10 36.01 44.33693 1.9621790 ## 668 10 37.01 44.56988 1.9839341 ## 669 10 38.01 44.80284 2.0064025 ## 670 10 39.01 45.03579 2.0295605 ## 671 10 40.01 45.26875 2.0533847 ## 672 10 41.01 45.50170 2.0778523 ## 673 10 42.01 45.73465 2.1029407 ## 674 10 43.01 45.96761 2.1286281 ## 675 10 44.01 46.20056 2.1548930 ## 676 10 45.01 46.43352 2.1817145 ## 677 10 46.01 46.66647 2.2090724 ## 678 10 47.01 46.89942 2.2369470 ## 679 10 48.01 47.13238 2.2653192 ## 680 10 49.01 47.36533 2.2941706 ## 681 10 50.01 47.59829 2.3234833 ## 682 10 51.01 47.83124 2.3532400 ## 683 10 52.01 48.06420 2.3834242 ## 684 10 53.01 48.29715 2.4140198 ## 685 10 54.01 48.53010 2.4450113 ## 686 10 55.01 48.76306 2.4763840 ## 687 10 56.01 48.99601 2.5081234 ## 688 10 57.01 49.22897 2.5402159 ## 689 10 58.01 49.46192 2.5726482 ## 690 10 59.01 49.69488 2.6054077 ## 691 10 60.01 49.92783 2.6384821 ## 692 10 61.01 50.16078 2.6718598 ## 693 10 62.01 50.39374 2.7055295 ## 694 10 63.01 50.62669 2.7394805 ## 695 10 64.01 50.85965 2.7737024 ## 696 10 65.01 51.09260 2.8081853 ## 697 10 66.01 51.32555 2.8429198 ## 698 10 67.01 51.55851 2.8778968 ## 699 10 68.01 51.79146 2.9131074 ## 700 10 69.01 52.02442 2.9485433 • Add these to the data_space plot by using the fill aesthetic and geom_tile(). (data_space <- mario_kart %>% ggplot(aes(x = duration, y = startPr)) + geom_point(aes(color = totalPr))) # tile the plane data_space + geom_tile(data = price_hats, aes(fill = .fitted), alpha = 0.5 ) ### Models in 3D An alternative way to visualize a multiple regression model with two numeric explanatory variables is as a plane in three dimensions. This is possible in R using the plotly package. We have created three objects that you will need: • x: a vector of unique values of duration • y: a vector of unique values of startPr • plane: a matrix of the fitted values across all combinations of x and y Much like ggplot(), the plot_ly() function will allow you to create a plot object with variables mapped to x, y, and z aesthetics. The add_markers() function is similar to geom_point() in that it allows you to add points to your 3D plot. Note that plot_ly uses the pipe (%>%) operator to chain commands together. # draw the 3D scatterplot p <- plot_ly(data = mario_kart, z = ~totalPr, x = ~duration, y = ~startPr, opacity = 0.6) %>% add_markers()  plane <- matrix(seq_range(price_hats$.fitted, n=70))
# draw the plane
p %>%
y = ~y,
z = ~plane,
showscale = FALSE)
# api_create(p, filename = "mario_kart_3D_scatter")

#### Coefficient magnitude

The coefficients from our model for the total auction price of MarioKarts as a function of auction duration and starting price are shown below.

mod
##
## Call:
## lm(formula = totalPr ~ duration + startPr, data = mario_kart)
##
## Coefficients:
## (Intercept)     duration      startPr
##      51.030       -1.508        0.233

A colleague claims that these results imply that the duration of the auction is a more important determinant of final price than starting price, because the coefficient is larger. This interpretation is false because:

The coefficients have different units (dollars per day and dollars per dollar, respectively) and so they are not directly comparable.

### Practicing interpretation

Fit a multiple regression model for the total auction price of an item in the mario_kart data set as a function of the starting price and the duration of the auction. Compute the coefficients and choose the correct interpretation of the duration variable.

mod <- lm(formula = totalPr ~ startPr + duration, data = mario_kart)
coef(mod)
## (Intercept)     startPr    duration
##  51.0295070   0.2329542  -1.5081260

For each additional day the auction lasts, the expected final price declines by $1.51, after controlling for starting price. ### Visualizing parallel planes By including the duration, starting price, and condition variables in our model, we now have two explanatory variables and one categorical variable. Our model now takes the geometric form of two parallel planes! The first plane corresponds to the model output when the condition of the item is new, while the second plane corresponds to the model output when the condition of the item is used. The planes have the same slopes along both the duration and starting price axes—it is the $$z$$-intercept that is different. Once again we have stored the x and y vectors for you. Since we now have two planes, there are matrix objects plane0 and plane1 stored for you as well. • Use plot_ly to draw 3D scatterplot for totalPr as a function of duration, startPr, and cond by mapping the z variable to the response and the x and y variables to the explanatory variables. Duration should be on the x-axis and starting price should be on the y-axis. Use color to represent cond. # draw the 3D scatterplot p <- plot_ly(data = mario_kart, z = ~totalPr, x = ~duration, y = ~startPr, opacity = 0.6) %>% add_markers(color = ~cond)  • Use add_surface() (twice) to draw two planes through the cloud of points, one for new MarioKarts and another for used ones. Use the objects plane0 and plane1. x0 <- mario_kart %>% filter(cond == "new") %>% select(duration, startPr) plane0 <- as.matrix(table(x0$duration, x0\$startPr))
dim(plane0)
## [1]  4 13
# draw two planes
p %>%
y = ~y,
z = ~plane0,
showscale = FALSE) %>%
y = ~y,
z = ~plane0,
showscale = FALSE)
sessionInfo()
## R version 3.4.4 (2018-03-15)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 17134)
##
## Matrix products: default
##
## locale:
##
## attached base packages:
## [1] methods   stats     graphics  grDevices utils     datasets  base
##
## other attached packages:
## [1] bindrcpp_0.2.2  plotly_4.7.1    modelr_0.1.2    ggplot2_2.2.1
## [5] dplyr_0.7.5     broom_0.4.4     openintro_1.7.1
##
## loaded via a namespace (and not attached):
##  [1] tidyselect_0.2.4  xfun_0.1          reshape2_1.4.3
##  [4] purrr_0.2.5       lattice_0.20-35   colorspace_1.3-2
##  [7] htmltools_0.3.6   viridisLite_0.3.0 yaml_2.1.19
## [10] rlang_0.2.1       pillar_1.2.3      later_0.7.2
## [13] foreign_0.8-70    glue_1.2.0        bindr_0.1.1
## [16] plyr_1.8.4        stringr_1.3.1     munsell_0.4.3
## [19] blogdown_0.6      gtable_0.2.0      htmlwidgets_1.2
## [22] codetools_0.2-15  psych_1.8.4       evaluate_0.10.1
## [25] knitr_1.20        httpuv_1.4.3      crosstalk_1.0.0
## [28] parallel_3.4.4    Rcpp_0.12.17      xtable_1.8-2
## [31] backports_1.1.2   scales_0.5.0      promises_1.0.1
## [34] jsonlite_1.5      mime_0.5          mnormt_1.5-5
## [37] digest_0.6.15     stringi_1.1.7     bookdown_0.7
## [40] shiny_1.0.5       grid_3.4.4        rprojroot_1.3-2
## [43] tools_3.4.4       magrittr_1.5      lazyeval_0.2.1
## [46] tibble_1.4.2      tidyr_0.8.1       pkgconfig_2.0.1
## [49] data.table_1.11.4 assertthat_0.2.0  rmarkdown_1.9
## [52] httr_1.3.1        R6_2.2.2          nlme_3.1-137
## [55] compiler_3.4.4
## Adding cites for R packages using knitr
knitr::write_bib(.packages(), "packages.bib")

# References

R Core Team. 2018. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

Wickham, Hadley, Romain François, Lionel Henry, and Kirill Müller. 2018. Dplyr: A Grammar of Data Manipulation. https://CRAN.R-project.org/package=dplyr.