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FOUNDATIONAL CONCEPTS Model Specification: What is SLR? y = β₀ + β₁x + ε Response vs. predictor variable What does the line represent? (conditional mean) Slope interpretation. Intercept interpretation.
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Professor
School of Industrial and Systems Engineering
Learning Objectives:
estimation and prediction
prediction intervals for the
regression mean response
mean response to a data example
using R
Interpretation of estimated mean response:
Ø If x* is one of the observations for the predicting variable, then we use
estimation. Estimated regression line for the value x* is interpreted
as the average estimated mean response for all settings under which
the predicting variable is equal to x*.
Ø If x* is a new observation of the predicting variables, the we use
prediction. Predicted regression line for the value x* is interpreted as
the estimated mean response for one setting under which the
predicting variable is equal to x*.
At some selected value of x (say x*), we estimate the “mean response” of y
(or the regression line) via
% $
Because the estimators of! !
and! "
are normally distributed, so is
#. That
means we can draw inference using
variance.
One of the primary motivations for regression is to use the
regression equation to predict future responses. The prediction is
the same as the estimator for the “mean response”, which is.
th observation
and! "
But the prediction contains two sources of uncertainty:
!!
!
!
(
The new observation is independent of the regression
data, so the total variation in predicting y | x* is
!! !!
!
!!
!
!
( ( (
A 100( 1 - α)% prediction interval for a future y* (at x*) is
$ #
" =! +!! is the same as the line estimate, but the
interval is wider than the confidence
interval for the mean response.
Regression Line
Confidence Interval
Prediction Interval
Observations
! !
= !̅ or close to that. Both intervals get wider
as we go away from! ̅
"
is
greater
A company, which sells medical supplies to hospitals, clinics, and
doctor's offices, had considered the effectiveness of a new advertising
program. Management wants to know if the advertisement is related to
sales.
This company intends to increase the sales with an effective advertising
program.
What inferences can be made on the prediction of the sales given a
targeted advertisement expenditure?
a. What sales would you predict for an advertisement expenditure of $30,000?
b. What is the variance estimate of the estimated predicted sales for an
advertisement expenditure of $30,000?
c. What are the lower and upper limits of predicted sales for an advertisement
expenditure of $30,000 at 99% confidence level? How will the limits change if
we lower the confidence level to 95%?
d. Compare the confidence intervals of the estimated regression line versus the
predicted regression line. Interpret.
summary(model)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) - 157.3301 145.1912 - 1.084 0.
adv 2.7721 0.2794 9.921 8.87e- 10
Residual standard error: 101.4 on 23 degrees of freedom
xbar = mean(ADV)
n = 23+
mse =101.4^
var.beta1 = 0.2794^
sxx = mse/var.beta
pred.var = mse*(1+1/n+(xbar-300)^2/sxx)
pred.var
[1] 14286.
a. For advertising expenditure of
$30,000, the predicted sales is:
= 673.67 thousand
b. The variance of the predicted
sales is
( '
1 +
1
(-
∗ − - ̄)
1 $$
= 14286. 16
new = data.frame(adv = 300)
predict.lm(model, new, interval = "predict", level = 0.99)
fit lwr upr
1 674.3047 338.712 1009.
predict.lm(model, new, interval = "predict", level = 0.95)
fit lwr upr
1 674.3047 427.0146 921.
predict.lm(model, new, interval = "confidence", level = 0.99)
fit lwr upr
1 674.3047 496.6497 851.
predict.lm(model, new, interval = "confidence", level = 0.95)
fit lwr upr
1 674.3047 543.395 805.
c. A 99% prediction interval
at an advertisement
expenditure of $30,000 is
(338.712, 1009.897). A
95% interval is (427.014,
921.594).
d. A 99% confidence interval
at an advertisement
expenditure of $30,000 is
(496.649, 851.959). A 95%
interval is (543.395,
805.214).
The confidence intervals are narrower than the prediction intervals because the
prediction intervals have additional variance from the variation of a new measurement.
