LAB SESSION 16

LINEAR REGRESSION ANALYSIS

 

 

INTRODUCTION: In an earlier lab, we looked at bivariate data, and used the linear correlation coefficient to see if there was a relationship between the two variables. You also looked at a method of developing a line of best fit. In this lab we will look at a method of deciding whether the equation of that line is of any use to us in making point predictions and developing confidence intervals.

 

Before beginning this lab, you should review the commands for performing a regression analysis in Lab 5. Use the data in Exercise 13.33 just to refresh your memory.

Enter x values in column A (independent variable)

Enter y values in column B (dependant variable)

 

Choose: Tools>Data Analysis>Regression OK

Enter: Input y range: B1: B11 (or select cells)

Input X range; A1:A11 ( or select cells)

Select: Labels (if you labeled your columns)

Confidence level: 95% (or desired level)

Select: Output Range: C3 (or upper left corner of where you

want output to appear)

 

Check the following boxes, depending on desired output:

 

Residuals: to obtain predicted values and their residuals

Residual Plots: scatterplot of residuals against their x values

Line Fit Plots: scatterplot of y against x

 

Note: You can make the output nicer looking if you choose

Format>Column>Autofit

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SUMMARY OUTPUT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

 

 

Multiple R

0.973362

 

 

 

 

 

 

 

R Square

0.947433

 

 

 

 

 

 

 

Adjusted R Square

0.940863

 

 

 

 

 

 

 

Standard Error

10.17383

 

 

 

 

 

 

 

Observations

10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

 

 

Regression

1

14924.45

14924.45

144.1882

2.13E-06

 

 

 

Residual

8

828.0539

103.5067

 

 

 

 

 

Total

9

15752.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

-13.4135

7.167862

-1.87134

0.098204

-29.9426

3.115613

-29.9426

3.115613

x

2.3028

0.191775

12.00784

2.13E-06

1.860566

2.745034

1.860566

2.745034

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RESIDUAL OUTPUT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Observation

Predicted y

Residuals

 

 

 

 

 

 

1

14.22008

0.779918

 

 

 

 

 

 

2

18.82568

6.174318

 

 

 

 

 

 

3

23.43128

6.568718

 

 

 

 

 

 

4

32.64248

-2.64248

 

 

 

 

 

 

5

39.55088

-9.55088

 

 

 

 

 

 

6

92.51528

-12.5153

 

 

 

 

 

 

7

101.7265

-11.7265

 

 

 

 

 

 

8

97.12088

-2.12088

 

 

 

 

 

 

9

101.7265

8.273522

 

 

 

 

 

 

10

113.2405

16.75952

 

 

 

 

 

 

 

For additional review, do exercise 3.38 in your text.

There are several steps in doing a linear regression analysis. First we obtain a least squares estimate for the model equation y = b0 +b1 x +e, using the REGRESSION command as practiced above. Next, we need to check our assumptions about the random error component, e. (The mean value of the experimental error is zero. We must also assume that the distribution of the ys is approximately normal and the variances s2 of the distribution of random errors is a constant.) Lastly, we can construct a confidence interval for our predictions.

 

 

 

We will use Exercise 13.59 to demonstrate the procedure. Enter the data into columns B and C

 

 

Median Income

% Immunized

 

 

 

 

 

 

 

17723

43

 

 

 

 

 

 

 

27005

51

 

 

 

 

 

 

 

33424

62

 

 

 

 

 

 

 

43337

66

 

 

 

 

 

 

 

19226

46

 

 

 

 

 

 

 

29775

59

 

 

 

 

 

 

 

40607

65

 

 

 

 

 

 

 

45496

62

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Correlation Coefficient

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.930250135

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Line of Best Fit:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SUMMARY OUTPUT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

 

 

Multiple R

0.930250135

 

 

 

 

 

 

 

R Square

0.865365314

 

 

 

 

 

 

 

Adjusted R Square

0.842926199

 

 

 

 

 

 

 

Standard Error

3.517836207

 

 

 

 

 

 

 

Observations

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

 

 

Regression

1

477.248971

477.2489705

38.56504

0.000805

 

 

 

Residual

6

74.2510295

12.37517158

 

 

 

 

 

Total

7

551.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

31.719566

4.21814734

7.519786161

0.000286

21.39812

42.04101

21.39812

42.04101

Median Income

0.000780393

0.00012567

6.210075577

0.000805

0.000473

0.001088

0.000473

0.001088

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RESIDUAL OUTPUT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Observation

Predicted % Immunized

Residuals

 

 

 

 

 

 

1

45.55047742

-2.5504774

 

 

 

 

 

 

2

52.79408854

-1.7940885

 

 

 

 

 

 

3

57.80343348

4.19656652

 

 

 

 

 

 

4

65.5394728

0.4605272

 

 

 

 

 

 

5

46.72340863

-0.7234086

 

 

 

 

 

 

6

54.95577813

4.04422187

 

 

 

 

 

 

7

63.40899894

1.59100106

 

 

 

 

 

 

8

67.22434205

-5.2243421

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Note the coefficients : The intercept is 31.72 and the slope is 0.00078 so we have the following equation:

Percent = 31.72 + 0.00078(Median Income)

 

We used the default of 95% for the confidence level. Note the upper and lower 95% confidence limits appear in blue on the above chart.

 

ASSIGNMENT: Do Exercises 13.58, 13.60, 13.75, 13.76 in your text.