Chapter 5 The Normal Error Linear Regression Model

5.1 Section 5.1 pt() and q(t)

If we want to calculate a t-statistic ourselves, we can divide the estimate, minus the associated hypothesized parameter value, by its standard error, and then using the pt() to obtain the p-value. We need to multiply by 2 to get values as extreme as ours in either tail of the t-distribution. Also enter the degrees of freedom associated with the residual standard error in the R output.

5.1.1 Calculating t-Statistic

b1 <- summary(Cars_M_A060)$coeff[2,1] SEb1 <- summary(Cars_M_A060)$coeff[2,2]
ts <- (b1-0)/SEb1
2*pt(-abs(ts), df=108)
## [1] 1.485428e-20

If we want to calculate confidence intervals with a level of confidence other than 95%, we can get the appropriate multiplier using the qt() command. For a A% interval, use an argument of (1-A/100)/2 to get the appropriate multiplier.

5.1.2 Obtaining p-value using qt()

abs(qt((1-.99)/2, df=108))
## [1] 2.62212

5.2 Confidence and Prediction Intervals

To calculate a 95% confidence interval for the average response, given a specified value of the explanatory variable, use the predict() command and specify interval="confidence".

To calculate a 95% prediction interval for a new observation, given a specified value of the explanatory variable, use the predict() command and specify interval="predict".

5.2.1 Confidence and Prediction Interval Example:

95% confidence interval for mean price among all cars that can accelerate from 0 to 60 mph in 7 seconds.

predict(Cars_M_A060, newdata=data.frame(Acc060=7), interval="confidence", level=0.95)
##       fit      lwr      upr
## 1 39.5502 37.21856 41.88184

95% prediction interval for price of an individual car that can accelerate from 0 to 60 mph in 7 seconds.

predict(Cars_M_A060, newdata=data.frame(Acc060=7), interval="prediction", level=0.95)
##       fit      lwr      upr
## 1 39.5502 18.19826 60.90215

5.3 Plots for Checking Model Assumptions

To check model assumptions, we use a plot of residuals vs predicted values, a histogram of the residuals, and a normal quantile-quantile plot. Code to produce each of these is given below.

To put the graphs side-by-side, we use the grid.arrange() function in the gridExtra() package.

5.3.1 Residual Plot, Histogram, and Normal QQ Plot

library(gridExtra)
P1 <- ggplot(data=data.frame(Cars_M_A060$residuals), aes(y=Cars_M_A060$residuals, x=Cars_M_A060$fitted.values)) + geom_point() + ggtitle("Cars Model Residual Plot") + xlab("Predicted Values") + ylab("Residuals") P2 <- ggplot(data=data.frame(Cars_M_A060$residuals),
aes(x=Cars_M_A060$residuals)) + geom_histogram() + ggtitle("Histogram of Residuals") + xlab("Residual") P3 <- ggplot(data=data.frame(Cars_M_A060$residuals),
aes(sample = Cars_M_A060\$residuals)) +
stat_qq() + stat_qq_line() + xlab("Normal Quantiles") +
ylab("Residual Quantiles") + ggtitle("Cars Model QQ Plot")
grid.arrange(P1, P2, P3, ncol=3)

5.4 Section 5.4: Transformations

When residual plots reveal issues with model assumptions, we can try modeling a transformation of the response variable, such as log().

5.4.1 Model with Transformation

Cars_M_Log <- lm(data=Cars2015, log(LowPrice)~Acc060)
summary(Cars_M_Log)
##
## Call:
## lm(formula = log(LowPrice) ~ Acc060, data = Cars2015)
##
## Residuals:
##      Min       1Q   Median       3Q      Max
## -0.84587 -0.19396  0.00908  0.18615  0.53350
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  5.13682    0.13021   39.45   <2e-16 ***
## Acc060      -0.22064    0.01607  -13.73   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.276 on 108 degrees of freedom
## Multiple R-squared:  0.6359, Adjusted R-squared:  0.6325
## F-statistic: 188.6 on 1 and 108 DF,  p-value: < 2.2e-16

When making predictions, we need to exponentiate.

exp(predict(Cars_M_Log, newdata=data.frame(Acc060=c(7)), interval="prediction", level=0.95))
##        fit      lwr      upr
## 1 36.31908 20.94796 62.96917

Likewise, we are interested in $$e^{b_j}$$.

exp(confint(Cars_M_Log))
##                   2.5 %     97.5 %
## (Intercept) 131.4610408 220.284693
## Acc060        0.7768669   0.827959