The most influential critic in the world today happens to be a critic of wine. His name is Robert Parker.
Meet Robert Parker
Meet Clive Coates
Clive Coates, Master of Wine, is one of the world's leading wine authorities.
Parker is the wine writer who matters. Clives Coates is very serious and well-respected, but in terms of commercial impact his influence is zero.
The data
wine <- read.csv("http://andrewpbray.github.io/data/Bordeaux.csv")
dim(wine)
## [1] 72 9
names(wine)
## [1] "Wine" "Price" "ParkerPoints" ## [4] "CoatesPoints" "P95andAbove" "FirstGrowth" ## [7] "CultWine" "Pomerol" "VintageSuperstar"
The mission
Develop a regression model to estimate the percentage effect on price of a 1% increase in ParkerPoints and a 1% increase in CoatesPoins using a subset of or all seven predictors.
Exploratory Data Analysis
library(car) scatterplotMatrix(wine[, 2:4])
EDA
Fitting the simplest model.
\[ \widehat{Price} \sim ParkerPoints + CoatesPoints + P95andAbove \\+ FirstGrowth + CultWine + Pomerol + VintageSuperstar \]
m1 <- lm(Price ~ . - Wine, data = wine)
\(Y\) versus \(\hat{Y}\)
plot(wine$Price ~ m1$fit, data = wine) abline(0, 1)
A second model
wine <- transform(wine, logPrice = log(Price),
logParkerPoints = log(ParkerPoints),
logCoatesPoints = log(CoatesPoints))
m2 <- lm(logPrice ~ . - Price - Wine - ParkerPoints - CoatesPoints,
data = wine)
Recall: logging the predictors of interest as well as the response allows us to interpret the estimates slopes as the percentage increase in the \(Y\) associated with a 1% increase in the \(x\).
SLR residual plots
MLR standardized res plots
MLR standardized res plots
\(Y\) versus \(\hat{Y}\)
MMPs
AVPs
Multicollinearity
vif(m2)
## P95andAbove FirstGrowth CultWine Pomerol ## 4.013 1.625 1.188 1.124 ## VintageSuperstar logParkerPoints logCoatesPoints ## 1.139 5.825 1.410
One value exceeds the traditional cutoff of 5, but not by too much. Multicollinearity is a minor issue here.
Model validity
We're confident we're working with a valid model.
- Linearity: the conditional mean of the response is a linear function of the predictors. (See \(Y\) vs \(\hat{Y}\) and mmps)
- The errors have constant variance and are uncorrelated. (See standardized residual plots vs predictors)
- The errors are normally distributed with mean zero. (see QQ plot)
It's also sensible to build a model with:
- No highly influential points. (see Cook's dist plot)
- Low multicollinearity. (see VIF)
Model summary
## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -51.1416 8.98557 -5.6915 3.389e-07 ## P95andAbove 0.1006 0.13697 0.7341 4.656e-01 ## FirstGrowth 0.8697 0.12524 6.9441 2.332e-09 ## CultWine 1.3532 0.14569 9.2878 1.784e-13 ## Pomerol 0.5364 0.09366 5.7275 2.946e-07 ## VintageSuperstar 0.6159 0.22067 2.7910 6.918e-03 ## logParkerPoints 11.5886 2.06763 5.6048 4.744e-07 ## logCoatesPoints 1.6205 0.61154 2.6499 1.013e-02
Note that all of the predictors are signficant at the .05 level (another reason to not worry too much about the borderline VIF), with the exception of P95andAbove.
We could consider dropping it from the model to remove redudancy with logParkerPrice.
Updated model summary
m3 <- update(m2, . ~ . - P95andAbove) summary(m3)$coef
## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -56.4755 5.26798 -10.721 5.205e-16 ## FirstGrowth 0.8615 0.12430 6.931 2.299e-09 ## CultWine 1.3360 0.14330 9.323 1.339e-13 ## Pomerol 0.5362 0.09333 5.745 2.641e-07 ## VintageSuperstar 0.5947 0.21800 2.728 8.186e-03 ## logParkerPoints 12.7843 1.26915 10.073 6.662e-15 ## logCoatesPoints 1.6045 0.60898 2.635 1.052e-02
Since the parameter estimates barely budged, this is essentially the same model as the previous one: no need to re-check diagnostic plots.