Dummy and Interactions
Tengyuan Liang
11/29/2018
# install.packages("readr")
library(readr)
MidCity <- read_csv("~/Github/Business_Statistics_41000/Lecture/Section5/MidCity.csv",col_types = cols(Nbhd = col_factor(levels = c("1", "2", "3"))))
# View(MidCity)
MidCity## # A tibble: 128 x 8
## Home Nbhd Offers SqFt Brick Bedrooms Bathrooms Price
## <dbl> <fct> <dbl> <dbl> <chr> <dbl> <dbl> <dbl>
## 1 1 2 2 1790 No 2 2 114300
## 2 2 2 3 2030 No 4 2 114200
## 3 3 2 1 1740 No 3 2 114800
## 4 4 2 3 1980 No 3 2 94700
## 5 5 2 3 2130 No 3 3 119800
## 6 6 1 2 1780 No 3 2 114600
## 7 7 3 3 1830 Yes 3 3 151600
## 8 8 3 2 2160 No 4 2 150700
## 9 9 2 3 2110 No 4 2 119200
## 10 10 2 3 1730 No 3 3 104000
## # … with 118 more rowsDummies for Neighbourhood
reg1 = lm(Price~Nbhd+SqFt, data=MidCity)
summary(reg1)##
## Call:
## lm(formula = Price ~ Nbhd + SqFt, data = MidCity)
##
## Residuals:
## Min 1Q Median 3Q Max
## -38107 -10924 -305 9643 38506
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 21241.174 13133.642 1.617 0.10835
## Nbhd2 10568.698 3301.096 3.202 0.00174 **
## Nbhd3 41535.306 3533.668 11.754 < 2e-16 ***
## SqFt 46.386 6.746 6.876 2.67e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15260 on 124 degrees of freedom
## Multiple R-squared: 0.6851, Adjusted R-squared: 0.6774
## F-statistic: 89.91 on 3 and 124 DF, p-value: < 2.2e-16MidCity = cbind(MidCity, pred1 = predict(reg1))library(ggplot2)
coeff = coefficients(lm(Price~SqFt, data=MidCity))
summary(lm(Price~SqFt, data=MidCity))##
## Call:
## lm(formula = Price ~ SqFt, data = MidCity)
##
## Residuals:
## Min 1Q Median 3Q Max
## -46593 -16644 -1610 15124 54829
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -10091.130 18966.104 -0.532 0.596
## SqFt 70.226 9.426 7.450 1.3e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 22480 on 126 degrees of freedom
## Multiple R-squared: 0.3058, Adjusted R-squared: 0.3003
## F-statistic: 55.5 on 1 and 126 DF, p-value: 1.302e-11ggplot(MidCity, aes(x = SqFt, y = Price, color = Nbhd)) + geom_point() + geom_line(mapping = aes(y = MidCity$pred1)) + geom_abline(intercept = coeff[1], slope = coeff[2], linetype="dashed")
Dummies with Interaction
reg2 = lm(Price~Nbhd+SqFt+Nbhd*SqFt, data=MidCity)
summary(reg2)##
## Call:
## lm(formula = Price ~ Nbhd + SqFt + Nbhd * SqFt, data = MidCity)
##
## Residuals:
## Min 1Q Median 3Q Max
## -37791 -10287 217 8989 38708
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 32906.423 22784.778 1.444 0.151238
## Nbhd2 -7224.312 32569.556 -0.222 0.824831
## Nbhd3 23752.725 33848.749 0.702 0.484183
## SqFt 40.300 11.825 3.408 0.000887 ***
## Nbhd2:SqFt 9.128 16.495 0.553 0.580996
## Nbhd3:SqFt 9.026 16.827 0.536 0.592681
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15360 on 122 degrees of freedom
## Multiple R-squared: 0.6861, Adjusted R-squared: 0.6732
## F-statistic: 53.32 on 5 and 122 DF, p-value: < 2.2e-16MidCity = cbind(MidCity, pred2 = predict(reg2))library(ggplot2)
ggplot(MidCity, aes(x = SqFt, y = Price, color = Nbhd)) + geom_point() + geom_line(mapping = aes(y = MidCity$pred2)) + geom_abline(intercept = coeff[1], slope = coeff[2], linetype="dashed")
Dummies for Brick
reg4 = lm(Price~SqFt + Brick, data=MidCity)
summary(reg4)##
## Call:
## lm(formula = Price ~ SqFt + Brick, data = MidCity)
##
## Residuals:
## Min 1Q Median 3Q Max
## -38412 -14665 -1772 13912 45016
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9444.289 16577.134 -0.570 0.57
## SqFt 66.058 8.265 7.992 7.54e-13 ***
## BrickYes 23445.096 3709.805 6.320 4.21e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 19640 on 125 degrees of freedom
## Multiple R-squared: 0.4739, Adjusted R-squared: 0.4655
## F-statistic: 56.3 on 2 and 125 DF, p-value: < 2.2e-16MidCity = cbind(MidCity, pred4 = predict(reg4))
ggplot(MidCity, aes(x = SqFt, y = Price, color = Brick)) + geom_point() + geom_line(mapping = aes(y = MidCity$pred4)) + geom_abline(intercept = coeff[1], slope = coeff[2], linetype="dashed")
Crazy interaction
Now let’s look at a crazy interaction \(Brick*Nbhd\). How many categories? Answer \(2*3 = 6\).
reg5 = lm(Price~SqFt+Brick*Nbhd, data=MidCity)
summary(reg5)##
## Call:
## lm(formula = Price ~ SqFt + Brick * Nbhd, data = MidCity)
##
## Residuals:
## Min 1Q Median 3Q Max
## -31279 -7405 -847 6889 35775
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20735.558 10766.923 1.926 0.0565 .
## SqFt 45.562 5.484 8.308 1.64e-13 ***
## BrickYes 13106.669 5106.897 2.566 0.0115 *
## Nbhd2 5820.591 3187.082 1.826 0.0703 .
## Nbhd3 33023.314 3375.878 9.782 < 2e-16 ***
## BrickYes:Nbhd2 3267.031 6335.286 0.516 0.6070
## BrickYes:Nbhd3 13053.182 6506.989 2.006 0.0471 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12350 on 121 degrees of freedom
## Multiple R-squared: 0.7986, Adjusted R-squared: 0.7886
## F-statistic: 79.95 on 6 and 121 DF, p-value: < 2.2e-16MidCity = cbind(MidCity, pred5 = predict(reg5))
ggplot(MidCity, aes(x = SqFt, y = Price, color = interaction(Brick, Nbhd))) + geom_point() + geom_line(mapping = aes(y = MidCity$pred5)) + geom_abline(intercept = coeff[1], slope = coeff[2], linetype="dashed")
reg6 = lm(Price~SqFt+ Brick*Nbhd + SqFt*Brick*Nbhd, data=MidCity)
summary(reg6)##
## Call:
## lm(formula = Price ~ SqFt + Brick * Nbhd + SqFt * Brick * Nbhd,
## data = MidCity)
##
## Residuals:
## Min 1Q Median 3Q Max
## -31359 -7173 -781 6906 35843
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18969.783 20764.749 0.914 0.3628
## SqFt 46.478 10.717 4.337 3.1e-05 ***
## BrickYes 42464.160 46497.577 0.913 0.3630
## Nbhd2 9323.775 30588.472 0.305 0.7611
## Nbhd3 53901.413 31594.024 1.706 0.0907 .
## BrickYes:Nbhd2 -38015.319 62223.215 -0.611 0.5424
## BrickYes:Nbhd3 -93694.197 64939.291 -1.443 0.1518
## SqFt:BrickYes -15.773 24.704 -0.638 0.5244
## SqFt:Nbhd2 -1.784 15.469 -0.115 0.9084
## SqFt:Nbhd3 -10.133 15.658 -0.647 0.5188
## SqFt:BrickYes:Nbhd2 21.657 32.015 0.676 0.5001
## SqFt:BrickYes:Nbhd3 52.858 32.843 1.609 0.1102
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12430 on 116 degrees of freedom
## Multiple R-squared: 0.8045, Adjusted R-squared: 0.7859
## F-statistic: 43.39 on 11 and 116 DF, p-value: < 2.2e-16MidCity = cbind(MidCity, pred6 = predict(reg6))
ggplot(MidCity, aes(x = SqFt, y = Price, color = interaction(Brick, Nbhd))) + geom_point() + geom_line(mapping = aes(y = MidCity$pred6)) + geom_abline(intercept = coeff[1], slope = coeff[2], linetype="dashed")
Merging Neibhorhood 1 and 2
MidCity <- read_csv("~/Github/Business_Statistics_41000/Lecture/Section5/MidCity.csv", col_types = cols(Nbhd = col_factor(levels = c("1", "2", "3"))))
# View(MidCity)
# library(GGally)
# ggpairs(MidCity[,2:8], aes(colour = interaction(Brick, Nbhd), alpha = 0.4))
# Merge Nbhd 1&2
MidCity = cbind(MidCity, NbhdNew = MidCity$Nbhd)
levels(MidCity$NbhdNew) <- c("1&2", "1&2", "3")
summary(lm(Price~SqFt+NbhdNew+Brick+Bedrooms+Bathrooms, data = MidCity))##
## Call:
## lm(formula = Price ~ SqFt + NbhdNew + Brick + Bedrooms + Bathrooms,
## data = MidCity)
##
## Residuals:
## Min 1Q Median 3Q Max
## -34382 -7364 -53 7789 35778
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16374.106 10531.829 1.555 0.12260
## SqFt 37.111 6.427 5.774 6.03e-08 ***
## NbhdNew3 31046.000 2698.846 11.503 < 2e-16 ***
## BrickYes 19486.156 2353.868 8.278 1.84e-13 ***
## Bedrooms 2280.483 1907.399 1.196 0.23417
## Bathrooms 6972.212 2584.471 2.698 0.00797 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12260 on 122 degrees of freedom
## Multiple R-squared: 0.7999, Adjusted R-squared: 0.7917
## F-statistic: 97.53 on 5 and 122 DF, p-value: < 2.2e-16coeff = coefficients(lm(Price~SqFt, data=MidCity))
reg2 = lm(Price~NbhdNew+SqFt, data=MidCity)
summary(reg2)##
## Call:
## lm(formula = Price ~ NbhdNew + SqFt, data = MidCity)
##
## Residuals:
## Min 1Q Median 3Q Max
## -35396 -9610 -1762 8778 38551
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18152.749 13574.154 1.337 0.184
## NbhdNew3 35699.135 3137.188 11.379 < 2e-16 ***
## SqFt 50.675 6.852 7.396 1.78e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15810 on 125 degrees of freedom
## Multiple R-squared: 0.659, Adjusted R-squared: 0.6536
## F-statistic: 120.8 on 2 and 125 DF, p-value: < 2.2e-16MidCity = cbind(MidCity, pred2 = predict(reg2))
ggplot(MidCity, aes(x = SqFt, y = Price, color = NbhdNew)) + geom_point() + geom_line(mapping = aes(y = MidCity$pred2)) + geom_abline(intercept = coeff[1], slope = coeff[2], linetype="dashed")
reg5 = lm(Price~SqFt+Brick+NbhdNew, data=MidCity)
summary(reg5)##
## Call:
## lm(formula = Price ~ SqFt + Brick + NbhdNew, data = MidCity)
##
## Residuals:
## Min 1Q Median 3Q Max
## -29415 -7450 47 8343 39744
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17039.80 10861.84 1.569 0.119
## SqFt 48.23 5.49 8.785 1.07e-14 ***
## BrickYes 20271.33 2401.53 8.441 6.96e-14 ***
## NbhdNew3 33585.50 2522.60 13.314 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12650 on 124 degrees of freedom
## Multiple R-squared: 0.7834, Adjusted R-squared: 0.7782
## F-statistic: 149.5 on 3 and 124 DF, p-value: < 2.2e-16MidCity = cbind(MidCity, pred5 = predict(reg5))
ggplot(MidCity, aes(x = SqFt, y = Price, color = interaction(Brick, NbhdNew))) + geom_point() + geom_line(mapping = aes(y = MidCity$pred5)) + geom_abline(intercept = coeff[1], slope = coeff[2], linetype="dashed")
reg6 = lm(Price~SqFt+ Brick*NbhdNew + SqFt*Brick*NbhdNew, data=MidCity)
summary(reg6)##
## Call:
## lm(formula = Price ~ SqFt + Brick * NbhdNew + SqFt * Brick *
## NbhdNew, data = MidCity)
##
## Residuals:
## Min 1Q Median 3Q Max
## -30285 -6983 -715 8294 38889
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18237.214 15123.806 1.206 0.2302
## SqFt 48.064 7.680 6.258 6.25e-09 ***
## BrickYes 10090.665 29245.328 0.345 0.7307
## NbhdNew3 54633.983 28398.755 1.924 0.0567 .
## BrickYes:NbhdNew3 -61320.701 54307.890 -1.129 0.2611
## SqFt:BrickYes 3.624 14.717 0.246 0.8059
## SqFt:NbhdNew3 -11.720 13.848 -0.846 0.3991
## SqFt:BrickYes:NbhdNew3 33.461 26.341 1.270 0.2064
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12550 on 120 degrees of freedom
## Multiple R-squared: 0.7939, Adjusted R-squared: 0.7819
## F-statistic: 66.03 on 7 and 120 DF, p-value: < 2.2e-16MidCity = cbind(MidCity, pred6 = predict(reg6))
ggplot(MidCity, aes(x = SqFt, y = Price, color = interaction(Brick, NbhdNew))) + geom_point() + geom_line(mapping = aes(y = MidCity$pred6)) + geom_abline(intercept = coeff[1], slope = coeff[2], linetype="dashed")