1. Inference for Least Squares Lines
Lecture Unit
Inference for Least Squares Lines

2. The Estimated Coefficients
Bivariate data: (x1 , y1 ), (x2 , y2 ), (x3 , y3 ), … , (xn , yn )
To calculate the estimates of the slope and intercept of the least squares line , use the formulas:
The least squares prediction equation that estimates the mean value of y for a particular value of x is:

3. The Simple Linear Regression Line
Example:
A car dealer wants to find the relationship between the odometer reading and the selling price of used cars.
A random sample of 100 cars is selected, and the data recorded.
Find the regression line.
Explanatory variable x
Response variable y

4. The Simple Linear Regression Line
Solving by hand: Calculate a number of statistics
where n = 100.

5. The Simple Linear Regression Line
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
100
ANOVA df SS MS F
Significance F
Regression 1
E-24
Residual 98
Total 99
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Intercept
3.57E-98
Odometer
5.75E-24

6. Interpreting the Least Squares Line
No data
The intercept is b0 = $
This is the slope of the line.
For each additional mile on the odometer,
the price decreases by an average of $0.0669
Do not interpret the intercept
as the “Price of cars that have
not been driven”

7. Error Variable: Required Conditions
The error e is a critical part of the regression model.
Four requirements involving the distribution of e must be satisfied.
The distribution of the e’s can be described by a normal model
The mean of e is zero: E(e) = 0.
The standard deviation of e is se for all values of x.
The errors associated with different values of y are all independent.

8. but the mean value changes with x
The Normality of 
The distribution of the e’s can be described by a normal model
The mean of e is zero: E(e) = 0.
The standard deviation of e is se for all values of x.
E(y|x3)
The standard deviation remains constant, m3
b0 + b1x3
E(y|x2)
b0 + b1x2 m2
E(y|x1)
but the mean value changes with x m1
b0 + b1x1
From the first three assumptions we have:
y is normally distributed with mean
E(y) = b0 + b1x, and a constant standard deviation se x1 x2 x3

9. Assessing the Model
The least squares method will produces a regression line whether or not there is a linear relationship between x and y.
Consequently, it is important to assess how well the linear model fits the data.
Several methods are used to assess the model. All are based on the sum of squares for errors, SSE.

10. Sum of Squares for Errors SSE
This is the sum of differences between the observed y’s and the predicted y’s.
It can serve as a measure of how well the line fits the data. SSE is defined by
A shortcut formula

11. Standard Error of Estimate
The mean error is equal to zero.
If the standard deviation e of the residuals is small the errors tend to be close to zero (close to the mean error). Then, the model fits the data well.
Therefore we can use e as a measure of the suitability of using a linear model.
An estimator of e is given by se

12. Standard Error of Estimate, Example
Calculate the standard error of estimate for the previous example and describe what it tells you about the model fit.
Solution

13. The slope is not equal to zero
Testing the slope
When no linear relationship exists between two variables, the regression line should be horizontal. q q q q q q q q q q q q
Linear relationship.
Linear relationship.
Linear relationship.
Linear relationship.
No linear relationship.
Different inputs (x) yield
the same output (y).
Different inputs (x) yield
different outputs (y).
The slope is not equal to zero
The slope is equal to zero

14. Testing the Slope We can draw inference about 1 from b1 by testing
H1: 1 = 0 (or < 0,or > 0)
The test statistic is
If the error variable is normally distributed, the statistic is Student t distribution with d.f. = n-2.
where
The standard error of b1.

15. Testing the Slope, Example
Test to determine whether there is enough evidence to infer that there is a linear relationship between the car auction price and the odometer reading for all three-year-old Tauruses in the previous example.

16. Testing the Slope, Example
Bivariate data: (x1 , y1 ), (x2 , y2 ), (x3 , y3 ), … , (x100 , y100 )
H0: 1 = 0
HA: 1  0
Solving by hand
To compute “t” we need the values of b1 and sb1.
P-value = 2P(t98 > |-13.44|) ~ 0

17. Testing the Slope (Example)
Using the computer
Odometer
Price
37400
14600
44800
14100
Regression Statistics
45800
14000
Multiple R
30900
15600
R Square
31700
Adjusted R Square
34000
14700
Standard Error
45900
14500
Observations
100
19100
15700
40100
15100
ANOVA
40200
14800 df SS MS F
Significance F
32400
15200
Regression 1
E-24
43500
Residual 98
32700
Total 99
34500
37700
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
41400
Intercept
3.57E-98
24500
5.75E-24
35800
15000
48600
24200
15400
H0: 1 = 0
HA: 1  0
Reject H0:1=0. There is strong
evidence to infer that the odometer
reading affects the auction selling price.

18. Confidence Interval for the the Slope, Example
Bivariate data: (x1 , y1 ), (x2 , y2 ), (x3 , y3 ), … , (x100 , y100 )

19. Confidence Interval for the Slope (Example)
Using the computer
Odometer
Price
37400
14600
44800
14100
Regression Statistics
45800
14000
Multiple R
30900
15600
R Square
31700
Adjusted R Square
34000
14700
Standard Error
45900
14500
Observations
100
19100
15700
40100
15100
ANOVA
40200
14800 df SS MS F
Significance F
32400
15200
Regression 1
E-24
43500
Residual 98
32700
Total 99
34500
37700
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
41400
Intercept
3.57E-98
24500
5.75E-24
35800
15000
48600
24200
15400
95% confidence interval for the slope 1

20. Coefficient of determination
Reduction in prediction error when use x:
TSS-SSE = SSR

21. Coefficient of determination
Reduction in prediction error when use x:TSS-SSE = SSR or TSS = SSR + SSE
The regression model
SSR
Overall variability in y
TSS
Explained in part by
Remains, in part, unexplained
The error
SSE

22. Coefficient of determination: graphically
Two data points (x1,y1) and (x2,y2)
of a certain sample are shown. y
Variation in y = SSR + SSE
(TSS) y1 x1 x2
Total variation in y =
Variation explained by the regression line
+ Unexplained variation (error)

23. Coefficient of determination
R2 (=r2 ) measures the proportion of the variation in y that is explained by the variation in x.
r2 takes on any value between zero and one since the correlation r satisfies -1r1).
r2 = 1: Perfect match between the line and the data points.
r2 = 0: There are no linear relationship between x and y.

24. Coefficient of Determination, Example
Find the coefficient of determination for the used car price –odometer example. What does this statistic tell you about the model?
Solution
Solving by hand;

25. Coefficient of Determination
Using the computer From the regression output we have
64.8% of the variation in the auction
selling price is explained by the
variation in odometer reading. The
rest (35.2%) remains unexplained by
this model.
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
100
ANOVA df SS MS F
Significance F
Regression 1
E-24
Residual 98
Total 99
Coefficients
t Stat
P-value
Intercept
3.57E-98
Odometer
5.75E-24

26. Using the Regression Equation
We can use the least squares line to predict the values of y.
To make a prediction we use
Point prediction, and
Interval prediction

27. Point Prediction Example
Predict the selling price of a three-year-old Taurus with 40,000 miles on the odometer.
A point prediction
It is predicted that a 40,000 miles car would sell for $14,574.
How close is this prediction to the real price?

28. The confidence interval
Interval Estimates
Two intervals can be used to discover how closely the predicted value will match the true value of y.
Prediction interval – predicts y for a given value of x (price prediction for a specific car with 40,000 miles on odometer)
Confidence interval – estimates the average y for a given x (estimate the average price of all cars with 40,000 miles on odometer).
The prediction interval
The confidence interval

29. Interval Estimates, Example
Example - continued
Provide an interval estimate for the bidding price on a Ford Taurus with 40,000 miles on the odometer.
Two types of predictions are required:
A prediction for a specific car
An estimate for the average price per car

30. Interval Estimates, Example
Solution
A 95% prediction interval provides the price estimate for a single car:
t.025,98

31. Interval Estimates, Example
Solution – continued
A 95% confidence interval provides the estimate of the mean price per car for all Ford Taurus’s with 40,000 miles on the odometer.
The confidence interval (95%) =

32. The effect of the given x on the length of the interval
As x moves away from x the interval becomes longer. That is, the shortest interval is found at x.

33. The effect of the given x on the length of the interval
As x moves away from x the interval becomes longer. That is, the shortest interval is found at x.
As x moves away from x the interval becomes longer. That is, the shortest interval is found at x.

34. The effect of the given x on the length of the interval
As x moves away from x the interval becomes longer. That is, the shortest interval is found at x.
As x moves away from x the interval becomes longer. That is, the shortest interval is found at x.

35. Regression Diagnostics - I
The three conditions required for the validity of the regression analysis are:
the error variable is normally distributed.
the error variance is constant for all values of x.
The errors are independent of each other.
How can we diagnose violations of these conditions?

36. Residual Analysis
Examining the residuals (or standardized residuals), help detect violations of the required conditions.
Example – continued:
Nonnormality.
Use Excel to obtain the standardized residual histogram.
Examine the histogram and look for a bell shaped. diagram with a mean close to zero.

37. Standardized residual ‘i’ =
Residual Analysis
A Partial list of
Standard residuals
For each residual we calculate
the standard deviation as follows:
Standardized residual ‘i’ =
Residual ‘i’
Standard deviation

38. It seems the residual are normally distributed with mean zero
Residual Analysis
It seems the residual are normally distributed with mean zero

39. The spread increases with y
Heteroscedasticity
When the requirement of a constant variance is violated we have a condition of heteroscedasticity.
Diagnose heteroscedasticity by plotting the residual against the predicted y. + ^ y
Residual + + + + + + + + + + + + + + ^ + + + y + + + + + + + + +
The spread increases with y ^

40. Homoscedasticity
When the requirement of a constant variance is not violated we have a condition of homoscedasticity.
Example - continued

41. Non Independence of Error Variables
A time series is constituted if data were collected over time.
Examining the residuals over time, no pattern should be observed if the errors are independent.
When a pattern is detected, the errors are said to be autocorrelated.
Autocorrelation can be detected by graphing the residuals against time.

42. Non Independence of Error Variables
Patterns in the appearance of the residuals over time indicates that autocorrelation exists.
Residual
Residual + + + + + + + + + + + + + + +
Time
Time + + + + + + + + + + + + +
Note the runs of positive residuals,
replaced by runs of negative residuals
Note the oscillating behavior of the
residuals around zero.

43. Outliers
An outlier is an observation that is unusually small or large.
Several possibilities need to be investigated when an outlier is observed:
There was an error in recording the value.
The point does not belong in the sample.
The observation is valid.
Identify outliers from the scatter diagram.
It is customary to suspect an observation is an outlier if its |standard residual| > 2

44. An influential observation
An outlier
An influential observation + + + + + + + + + + + + +
… but, some outliers
may be very influential + + + + + + + + + + + + + +
The outlier causes a shift
in the regression line

45. Procedure for Regression Diagnostics
Develop a model that has a theoretical basis.
Gather data for the two variables in the model.
Draw the scatter diagram to determine whether a linear model appears to be appropriate.
Determine the regression equation.
Check the required conditions for the errors.
Check the existence of outliers and influential observations
Assess the model fit.
If the model fits the data, use the regression equation.