ISLR Q3.10 - Multiple Regression/Carseats
Question
p123
This question should be answered using the Carseats data set.
Fit a multiple regression model to predict Sales using Price, Urban, and US
Provide an interpretation of each coefficient in the model. Be careful - some of the variables in the model are qualitative!
Write out the model in equation form, being careful to handle the qualitative variables properly
For which of the predictors can you reject the null hypothesis \(H_0 :\beta_j =0\)?
On the basis of your response to the previous question, fit a smaller model that only uses the predictors for which there is evidence of association with the outcome
How well do the models in (a) and (e) fit the data?
Using the model from (e), obtain 95% confidence intervals for the coefficient(s).
Is there evidence of outliers or high leverage observations in the model from (e)?
library(ISLR)10a \(Sales = \beta_0 + \beta_1 Price + \beta_2 Urban + \beta_3 US\)
names(Carseats)## [1] "Sales" "CompPrice" "Income" "Advertising" "Population"
## [6] "Price" "ShelveLoc" "Age" "Education" "Urban"
## [11] "US"carseat.fit = lm(Sales ~ Price + Urban + US, data=Carseats)10b Interpret Coefficients
Model Summary
summary(carseat.fit)##
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9206 -1.6220 -0.0564 1.5786 7.0581
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.043469 0.651012 20.036 < 2e-16 ***
## Price -0.054459 0.005242 -10.389 < 2e-16 ***
## UrbanYes -0.021916 0.271650 -0.081 0.936
## USYes 1.200573 0.259042 4.635 4.86e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2335
## F-statistic: 41.52 on 3 and 396 DF, p-value: < 2.2e-16contrasts(Carseats$US)## Yes
## No 0
## Yes 1USYes coefficient: If the store is in the US (predictor=1), the sales increase at a rate ofthe coefficient is 1.2.
sales = 1.2 * USYes (1.2 from coefficient from model) (ignoring other predictors for simplicity)
Price: Price is highly significant (p-value) when it comes to sales. There is a slight negative correlation to sales. As prices goes up, sales go down.
UrbanYes: Does not have a significant p-value. This means that it does not effect the sales. Consider removing from model
10c Model with Qualitative Variables
Sales = 13.04 - 0.05 * Price + 1.2 * US 1 if US = Yes ; 0 if US = No
10d Reject Predictors/Features
We can reject the null hypothesis for Price and USYes because its p-value is highly significant (<<.05)
10e Smaller Model
Model Summary
carseat.fit2 = lm(Sales ~ Price + US, data=Carseats)
summary(carseat.fit2)##
## Call:
## lm(formula = Sales ~ Price + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9269 -1.6286 -0.0574 1.5766 7.0515
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.03079 0.63098 20.652 < 2e-16 ***
## Price -0.05448 0.00523 -10.416 < 2e-16 ***
## USYes 1.19964 0.25846 4.641 4.71e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2354
## F-statistic: 62.43 on 2 and 397 DF, p-value: < 2.2e-16Diagostic Plots
par(mfrow = c(2,2))
plot(carseat.fit2)
10f Compare Models
The models for both (a) and (e) do NOT fit the data well. The \(R^2\) statistic for both models show that the model ONLY explains 23% of the variance.
10g 95% Confidence Interval
Getting the confidence intervals for each coefficient confint(carseat.fit2)
10h Outliers and High Leverage
Diagnostic Plots
par(mfrow = c(2,2))
plot(carseat.fit2)
Based on the Residuals vs. Leverage graph (bottom right):
- There is one observation that is far right of the graph. This means that its leverage is really high. Also there are few more that are few more that have high leverage.