MATH 376 -- Mathematical Statistics
Confidence and Prediction Bands in Simple Linear Regression
April 30, 2012
#
The following data set is supposed to represent
peak loads on an electric power plant measured
on a random sample of days of different maximum
temperature (in degrees F)
#
> temp <- c(95,82,90,81,99,100,93,95,93,87)
> load <- c(214,152,156,129,254,266,210,204,213,150)
#
We fit a simple regression model:
#
> model <- lm(load~temp)
> summary(model)

Call:
lm(formula = load ~ temp)

Residuals:
    Min      1Q  Median      3Q     Max 
-28.724 -11.811   4.929   8.645  21.016 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -419.8491    76.0578   -5.52  0.00056 ***
temp           6.7175     0.8294    8.10 3.99e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 16.18 on 8 degrees of freedom
Multiple R-squared: 0.8913,     Adjusted R-squared: 0.8777 
F-statistic:  65.6 on 1 and 8 DF,  p-value: 3.994e-05 

#
The goal of fitting this kind of model is usually to 
allow us to draw conclusions about the response (y = peak load)
for other values of x = temperature.  There are two things
one can do here -- "confidence" and "prediction" intervals.
#
For instance, say we wanted to understand possible values
of the peak load for a day with max temperature 96:
# 
> predict(model,newdata=data.frame(temp=96),interval="conf");
       fit     lwr      upr
1 225.0286 210.426 239.6313
> predict(model,newdata=data.frame(temp=96),interval="pred");
       fit      lwr      upr
1 225.0286 184.9668 265.0905
#
#
We can also construct confidence and prediction "bands" about
the regression line:
#
These are computed by the following R commands:
#
> temp.frame <- data.frame(temp=seq(80,110,1))
> pc <- predict(model,newdata=temp.frame,interval="conf")
> pp <- predict(model,newdata=temp.frame,interval="pred")
#
Next, we generate a plot showing all of the things computed:
#
> plot(temp,load,xlim=c(85,110),ylim=c(0,350))
> ablines(model)
> matlines(temp.frame$temp,pc,lty=c(1,3,3),col="black")
> matlines(temp.frame$temp,pp,lty=c(1,2,2),col="black")