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Linear regression
1.
Business Statistics: A
First Course (3rd Edition) Chapter 10 Simple Linear Regression © 2003 Prentice-Hall, Inc. Chap 10-1
2.
© 2003 Prentice-Hall,
Inc. Chap 10-2 Chapter Topics Types of Regression Models Determining the Simple Linear Regression Equation Measures of Variation Assumptions of Regression and Correlation Residual Analysis Measuring Autocorrelation Inferences about the Slope
3.
© 2003 Prentice-Hall,
Inc. (continued) Chap 10-3 Chapter Topics Correlation - Measuring the Strength of the Association Estimation of Mean Values and Prediction of Individual Values Pitfalls in Regression and Ethical Issues
4.
Purpose of Regression
Analysis Regression Analysis is Used Primarily to Model Causality and Provide Prediction Predict the values of a dependent (response) variable based on values of at least one independent (explanatory) variable Explain the effect of the independent variables on the dependent variable © 2003 Prentice-Hall, Inc. Chap 10-4
5.
Types of Regression
Models © 2003 Prentice-Hall, Inc. Chap 10-5 Positive Linear Relationship Negative Linear Relationship Relationship NOT Linear No Relationship
6.
Simple Linear Regression
Model Relationship Between Variables is Described by a Linear Function The Change of One Variable Causes the Other Variable to Change A Dependency of One Variable on the Other © 2003 Prentice-Hall, Inc. Chap 10-6
7.
Simple Linear Regression
Model Population regression line is a straight line that describes the dependence of the aavveerraaggee vvaalluuee ((ccoonnddiittiioonnaall mmeeaann)) of one variable on the other Population Y intercept i i i Y b b X e 0 1 = + + © 2003 Prentice-Hall, Inc. Chap 10-7 Population Regression Line (conditional mean) Population Slope Coefficient Random Error Dependent (Response) Variable Independent (Explanatory) Variable Y|X m (continued)
8.
Simple Linear Regression
Model © 2003 Prentice-Hall, Inc. (continued) i i i Y b b X e 0 1 = + + Chap 10-8 = Random Error Y X (Observed Value of Y) = Observed Value of Y Y|X i m b b X 0 1 = + i e b0 b1 (Conditional Mean)
9.
Linear Regression Equation
Sample regression line provides an eessttiimmaattee of the population regression line as well as a predicted value of Y Yˆ =b +b X = Simple Regression Equation © 2003 Prentice-Hall, Inc. Chap 10-9 Sample Y Intercept Sample Slope Coefficient i 0 1 i i Residual Y = b + b X + e 0 1 (Fitted Regression Line, Predicted Value)
10.
Linear Regression Equation
0 b 1 b and are obtained by finding the values of 0 b 1 b and that minimizes the sum of the ˆ n n å - =å Y Y e 0 b b0 1 b b1 © 2003 Prentice-Hall, Inc. Chap 10-10 squared residuals provides an eessttiimmaattee of provides and eessttiimmaattee of (continued) ( )2 2 i i i 1 1 i i = =
11.
Linear Regression Equation
i 0 1 i i Y = b + b X + e © 2003 Prentice-Hall, Inc. (continued) i i i Y b b X e 0 1 = + + Chap 10-11 Y X Observed Value Y|X i m b b X 0 1 = + i e b0 b1 Y ˆ= b + b X i 0 1 i i e 1 b 0 b
12.
Interpretation of the
Slope and Y |X 0 b m 0 = = | m © 2003 Prentice-Hall, Inc. Chap 10-12 Intercept is the average value of Y when the value of X is zero. 1 measures the change in the average value of Y as a result of a one-unit change in X. Y X X b D = D
13.
Interpretation of the
Slope and | 0 ˆY X b m 0 = = Dm ˆb = Y | X © 2003 Prentice-Hall, Inc. (continued) Chap 10-13 Intercept is the eessttiimmaatteedd average value of Y when the value of X is zero. 1 D X is the eessttiimmaatteedd change in the average value of Y as a result of a one-unit change in X.
14.
Simple Linear Regression:
© 2003 Prentice-Hall, Inc. Chap 10-14 Example You wish to examine the linear dependency of the annual sales of produce stores on their sizes in square footage. Sample data for 7 stores were obtained. Find the equation of the straight line that fits the data best. Annual Store Square Sales Feet ($1000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760
15.
Scatter Diagram: Example
12000 10000 8000 6000 4000 2000 © 2003 Prentice-Hall, Inc. Chap 10-15 0 0 1000 2000 3000 4000 5000 6000 Square Feet Annual Sales ($000) Excel Output
16.
Simple Linear Regression
ˆ i i 1636.415 1.487 © 2003 Prentice-Hall, Inc. Chap 10-16 Equation: Example 0 1 i Y b b X X = + = + From Excel Printout: Coefficients Intercept 1636.414726 X Variable 1 1.486633657
17.
Graph of the
Simple Linear Regression Equation: Example 12000 10000 8000 6000 4000 2000 © 2003 Prentice-Hall, Inc. Chap 10-17 0 0 1000 2000 3000 4000 5000 6000 Square Feet Annual Sales ($000) Yi = 1636.415 +1.487Xi Ù
18.
Interpretation of Results:
Yˆi = 1636.415 +1.487Xi © 2003 Prentice-Hall, Inc. Chap 10-18 Example The slope of 1.487 means that each increase of one unit in X, we predict the average of Y to increase by an estimated 1.487 units. The equation estimates that for each increase of 1 square foot in the size of the store, the expected annual sales are predicted to increase by $1487.
19.
Simple Linear Regression
in © 2003 Prentice-Hall, Inc. Chap 10-19 PHStat In Excel, use PHStat | Regression | Simple Linear Regression … EXCEL Spreadsheet of Regression Sales on Footage Microsoft Excel Worksheet
20.
Measures of Variation:
The Sum of Squares SST = SSR + SSE Total Sample Variability © 2003 Prentice-Hall, Inc. Chap 10-20 = Explained Variability + Unexplained Variability
21.
Measures of Variation:
The Sum of Squares Y © 2003 Prentice-Hall, Inc. (continued) Chap 10-21 SST = Total Sum of Squares Measures the variation of the Yi values around their mean, SSR = Regression Sum of Squares Explained variation attributable to the relationship between X and Y SSE = Error Sum of Squares Variation attributable to factors other than the relationship between X and Y
22.
Measures of Variation:
The Sum of Squares © 2003 Prentice-Hall, Inc. (continued) Chap 10-22 Xi Y X Y SST = å(Yi - Y)2 Ù SSE =å(Yi - Yi )2 Ù SSR = å(Yi - Y)2 _ _ _
23.
The ANOVA Table
in Excel © 2003 Prentice-Hall, Inc. Chap 10-23 ANOVA df SS MS F Significanc e F Regressio n k SSR MSR =SSR/k MSR/MSE P-value of the F Test Residuals n-k-1 SSE MSE =SSE/(n-k-1) Total n-1 SST
24.
Measures of Variation
The Sum of Squares: Example Excel Output for Produce Stores Degrees of freedom © 2003 Prentice-Hall, Inc. Chap 10-24 ANOVA df SS MS F Significance F Regression 1 30380456.12 30380456 81.17909 0.000281201 Residual 5 1871199.595 374239.92 Total 6 32251655.71 SSR SSE Regression (explained) df Error (residual) df Total df SST
25.
The Coefficient of
Determination r SSR 2 Regression Sum of Squares = = SST © 2003 Prentice-Hall, Inc. Total Sum of Squares Chap 10-25 Measures the proportion of variation in Y that is explained by the independent variable X in the regression model
26.
Coefficients of Determination
(r 2) r = +1 r = -1 ^ ^ r = +0.9 r = 0 © 2003 Prentice-Hall, Inc. Chap 10-26 and Correlation (r) r2 = 1, r2 = 1, r2 = .81, Y r2 = 0, ^ Yi = b0 + b1XiX Y Yi = b0 + b1Xi X ^ Y Yi = b0 + b1XiX Y Yi = b0 + b1XiX
27.
Standard Error of
Estimate S SSE YX å = = = n n © 2003 Prentice-Hall, Inc. Chap 10-27 n ( ˆ )2 1 i Y - Y - - 2 2 i The standard deviation of the variation of observations around the regression equation
28.
Measures of Variation:
Produce Store Example Excel Output for Produce Stores © 2003 Prentice-Hall, Inc. Chap 10-28 Regression Statistics Multiple R 0.9705572 R Square 0.94198129 Adjusted R Square 0.93037754 Standard Error 611.751517 Observations 7 r2 = .94 94% of the variation in annual sales can be explained by the variability in the size of the store as measured by square footage Syx
29.
Linear Regression Assumptions
© 2003 Prentice-Hall, Inc. Chap 10-29 Normality Y values are normally distributed for each X Probability distribution of error is normal 2. Homoscedasticity (Constant Variance) 3. Independence of Errors
30.
Variation of Errors
Around the © 2003 Prentice-Hall, Inc. • Y values are normally distributed around the regression line. • For each X value, the “spread” or variance around the regression line is the same. Chap 10-30 Regression Line X1 X2 X Y f(e ) Sample Regression Line
31.
© 2003 Prentice-Hall,
Inc. Chap 10-31 Residual Analysis Purposes Examine linearity Evaluate violations of assumptions Graphical Analysis of Residuals Plot residuals vs. X and time
32.
Residual Analysis for
Linearity e e © 2003 Prentice-Hall, Inc. Chap 10-32 X Not Linear Linear X Y X Y X
33.
Residual Analysis for
Homoscedasticity Y Heteroscedasticity Homoscedasticity © 2003 Prentice-Hall, Inc. Chap 10-33 SR X SR X Y X X
34.
Residual Analysis:Excel Output
for Produce Stores Example Excel Output © 2003 Prentice-Hall, Inc. Chap 10-34 Residual Plot 0 1000 2000 3000 4000 5000 6000 Square Feet Observation Predicted Y Residuals 1 4202.344417 -521.3444173 2 3928.803824 -533.8038245 3 5822.775103 830.2248971 4 9894.664688 -351.6646882 5 3557.14541 -239.1454103 6 4918.90184 644.0981603 7 3588.364717 171.6352829
35.
Residual Analysis for
e e ( ) © 2003 Prentice-Hall, Inc. Chap 10-35 Independence The Durbin-Watson Statistic Used when data is collected over time to detect autocorrelation (residuals in one time period are related to residuals in another period) Measures violation of independence assumption 2 1 2 2 1 n i i i n i i D e - = = - = å å Should be close to 2. If not, examine the model for autocorrelation.
36.
Durbin-Watson Statistic in
© 2003 Prentice-Hall, Inc. Chap 10-36 PHStat PHStat | Regression | Simple Linear Regression … Check the box for Durbin-Watson Statistic
37.
Obtaining the Critical
Values of Durbin-Watson Statistic Table 13.4 Finding critical values of Durbin-Watson Statistic n dL dU dL dU 15 1.08 1.36 .95 1.54 16 1.10 1.37 .98 1.54 © 2003 Prentice-Hall, Inc. a = .0 5 k=1 k=2 Chap 10-37
38.
Using the Durbin-Watson
: No autocorrelation (error terms are independent) : There is autocorrelation (error terms are not independent) Reject H0 (positive autocorrelation) © 2003 Prentice-Hall, Inc. Inconclusive Reject H0 (negative autocorrelation) Chap 10-38 Statistic Accept H0 (no autocorrelatin) 0 H 1 H 0 d 2 4 L 4-dL dU 4-dU
39.
Residual Analysis for
Graphical Approach Cyclical Pattern No Particular Pattern © 2003 Prentice-Hall, Inc. Chap 10-39 Independence Not Independent Independent e e Time Time Residual Is Plotted Against Time to Detect Any Autocorrelation
40.
Inference about the
Slope: t b S S 1 1 © 2003 Prentice-Hall, Inc. Chap 10-40 t Test t Test for a Population Slope Is there a linear dependency of Y on X ? Null and Alternative Hypotheses H0: b1 = 0 (No Linear Dependency) H1: b1 ¹ 0 (Linear Dependency) Test Statistic 1 1 2 1 where YX ( ) b n b i i S X X b = = - = å - d. f . = n - 2
41.
Example: Produce Store
© 2003 Prentice-Hall, Inc. Chap 10-41 Data for 7 Stores: Estimated Regression Equation: The slope of this model is 1.487. Does Square Footage Affect Annual Sales? Annual Store Square Sales Feet ($000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760 Yˆ =1636.415 +1.487Xi
42.
Inferences about the
Slope: : b1 ¹ 0 a = .05 df = 7 - 2 = 5 Critical Value(s): From Excel Printout 1 b b1 S t Intercept 1636.4147 451.4953 3.6244 0.01515 Footage 1.4866 0.1650 9.0099 0.00028 Reject Reject .025 .025 © 2003 Prentice-Hall, Inc. Coefficients Standard Error t Stat P-value Chap 10-42 t Test Example H0: b1 = 0 H1 Test Statistic: Decision: Conclusion: There is evidence that square footage affects annual sales. -2.5706 0 2.5706 t Reject H0
43.
Inferences about the
Slope: Confidence Interval Example Confidence Interval Estimate of the Slope: Intercept 475.810926 2797.01853 X Variable 11.06249037 1.91077694 © 2003 Prentice-Hall, Inc. Chap 10-43 1 n 2 b1 b t S - ± Excel Printout for Produce Stores Lower 95% Upper 95% At 95% level of confidence the confidence interval for the slope is (1.062, 1.911). Does not include 0. Conclusion: There is a significant linear dependency of annual sales on the size of the store.
44.
Inferences about the
Slope: SSR 1 2 F SSE ( n ) = - © 2003 Prentice-Hall, Inc. Chap 10-44 F Test F Test for a Population Slope Is there a linear dependency of Y on X ? Null and Alternative Hypotheses H0: b1 = 0 (No Linear Dependency) H1: b1 ¹ 0 (Linear Dependency) Test Statistic Numerator d.f.=1, denominator d.f.=n-2
45.
Relationship between a
t Test © 2003 Prentice-Hall, Inc. Chap 10-45 and an F Test Null and Alternative Hypotheses H: b= 0 (No Linear Dependency) 01 H1: b1 ¹ 0 (Linear Dependency) ( t )2 = F n - 2 1,n - 2
46.
Inferences about the
Slope: F Test Example Reject © 2003 Prentice-Hall, Inc. From Excel Printout Reject H0 Chap 10-46 ANOVA Test Statistic: df SS MS F Significance F Regression 1 30380456.12 30380456.12 81.179 0.000281 Residual 5 1871199.595 374239.919 Total 6 32251655.71 Decision: Conclusion: H0: b1 = 0 H1: b1 ¹ 0 a = .05 numerator df = 1 denominator df = 7 - 2 = 5 There is evidence that square footage affects annual sales. 0 6.61 a = .05 1,n 2 F -
47.
Purpose of Correlation
Analysis Correlation Analysis is Used to Measure Strength of Association (Linear Relationship) Between 2 Numerical Variables Only Strength of the Relationship is Concerned No Causal Effect is Implied © 2003 Prentice-Hall, Inc. Chap 10-47
48.
Purpose of Correlation
Analysis (continued) Population Correlation Coefficient r (Rho) is Used to Measure the Strength between the Variables Sample Correlation Coefficient r is an Estimate of r and is Used to Measure the Strength of the Linear Relationship in the Sample Observations © 2003 Prentice-Hall, Inc. Chap 10-48
49.
Sample of Observations
from Various r Values r = -1 r = -.6 r = 0 © 2003 Prentice-Hall, Inc. r = .6 r = 1 Chap 10-49 Y X Y X Y X Y X Y X
50.
Features of r
and r Unit Free Range between -1 and 1 The Closer to -1, the Stronger the Negative Linear Relationship The Closer to 1, the Stronger the Positive Linear Relationship The Closer to 0, the Weaker the Linear Relationship © 2003 Prentice-Hall, Inc. Chap 10-50
51.
t Test for
Correlation © 2003 Prentice-Hall, Inc. Chap 10-51 Hypotheses H0: r = 0 (No Correlation) H1: r ¹ 0 (Correlation) Test Statistic where ( ) ( ) ( ) ( ) 2 å å å 2 1 2 2 1 1 2 n i i i n n i i i i t r r n X X Y Y r r X X Y Y r = = = = - 1 - - - - = = - -
52.
Example: Produce Stores
Is there any evidence of linear relationship between Annual Sales of a store and its Square Footage at .05 level of significance? H0: r = 0 (No association) © 2003 Prentice-Hall, Inc. From Excel Printout r Regression Statistics Multiple R 0.9705572 R Square 0.94198129 Adjusted R Square 0.93037754 Standard Error 611.751517 Observations 7 H1: r ¹ 0 (Association) a = .05 df = 7 - 2 = 5 Chap 10-52
53.
Example: Produce Stores
= - r = = 1- - - Critical Value(s): Reject Reject .025 .025 © 2003 Prentice-Hall, Inc. Chap 10-53 Solution -2.5706 0 2.5706 Decision: Reject H0 Conclusion: There is evidence of a linear relationship at 5% level of significance 2 .9706 9.0099 1 .9420 2 5 t r r n The value of the t statistic is exactly the same as the t statistic value for test on the slope coefficient
54.
Estimation of Mean
Values Confidence Interval Estimate for : The Mean of Y given a particular Xi Y ± t S + X - X Y|X Xi m = ˆ 1 ( ) å - t value from table with df=n-2 © 2003 Prentice-Hall, Inc. 2 Chap 10-54 2 2 n X X 1 i ( ) i n YX n i i - = Standard error of the estimate Size of interval vary according to distance away from mean, X
55.
Prediction of Individual
Values Prediction Interval for Individual Y ± t S + + X - X ˆ 1 1 ( ) © 2003 Prentice-Hall, Inc. Chap 10-55 Response Yi at a Particular Xi Addition of 1 increases width of interval from that for the mean of Y 2 2 2 n X X å - 1 i ( ) i n YX n i i - =
56.
Interval Estimates for
Different © 2003 Prentice-Hall, Inc. Confidence Interval for the mean of Y Chap 10-56 Values of X Y X Prediction Interval for a individual Yi A given X Yi = b0 + b1Xi Ù X
57.
Example: Produce Stores
© 2003 Prentice-Hall, Inc. Consider a store with 2000 square feet. Chap 10-57 Data for 7 Stores: Regression Equation Obtained: Annual Store Square Sales Feet ($000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760 Yˆ =1636.415 +1.487Xi
58.
Estimation of Mean
Values: Confidence Interval Estimate for Y|X Xi m = Y t S X X ± + - = ± ˆ 1 ( ) 4610.45 612.66 n X X å - © 2003 Prentice-Hall, Inc. Chap 10-58 Example Find the 95% confidence interval for the average annual sales for stores of 2,000 square feet 2 2 2 1 i ( ) i n YX n i i - = Predicted Sales ˆ 1636.415 1.487 4610.45( $000) i Y = + X = 2 5 2350.29 611.75 2.5706 YX n X S t t - = = = =
59.
Prediction Interval for
Y : Prediction Interval for Individual Y t S X X ± + + - = ± ˆ 1 1 ( ) 4610.45 1687.68 n X X © 2003 Prentice-Hall, Inc. Chap 10-59 Example Find the 95% prediction interval for annual sales of one particular store of 2,000 square feet Predicted Sales) 2 2 2 å - 1 i ( ) i n YX n i i - = X Xi Y = ˆ 1636.415 1.487 4610.45( $000) i Y = + X = 2 5 2350.29 611.75 2.5706 YX n X S t t - = = = =
60.
Estimation of Mean
Values and Prediction of Individual Values in PHStat In Excel, use PHStat | Regression | Simple Linear Regression … Check the “Confidence and Prediction Interval for X=” box EXCEL Spreadsheet of Regression Sales on Footage © 2003 Prentice-Hall, Inc. Chap 10-60 Microsoft Excel Worksheet
61.
Pitfalls of Regression
Analysis Lacking an Awareness of the Assumptions Underlining Least-squares Regression Not Knowing How to Evaluate the Assumptions Not Knowing What the Alternatives to Least-squares © 2003 Prentice-Hall, Inc. Chap 10-61 Regression are if a Particular Assumption is Violated Using a Regression Model Without Knowledge of the Subject Matter
62.
Strategy for Avoiding
the Pitfalls © 2003 Prentice-Hall, Inc. Chap 10-62 of Regression Start with a scatter plot of X on Y to observe possible relationship Perform residual analysis to check the assumptions Use a histogram, stem-and-leaf display, box-and- whisker plot, or normal probability plot of the residuals to uncover possible non-normality
63.
Strategy for Avoiding
the Pitfalls © 2003 Prentice-Hall, Inc. (continued) Chap 10-63 of Regression If there is violation of any assumption, use alternative methods (e.g., least absolute deviation regression or least median of squares regression) to least-squares regression or alternative least-squares models (e.g., curvilinear or multiple regression) If there is no evidence of assumption violation, then test for the significance of the regression coefficients and construct confidence intervals and prediction intervals
64.
© 2003 Prentice-Hall,
Inc. Chap 10-64 Chapter Summary Introduced Types of Regression Models Discussed Determining the Simple Linear Regression Equation Described Measures of Variation Addressed Assumptions of Regression and Correlation Discussed Residual Analysis Addressed Measuring Autocorrelation
65.
© 2003 Prentice-Hall,
Inc. (continued) Chap 10-65 Chapter Summary Described Inference about the Slope Discussed Correlation - Measuring the Strength of the Association Addressed Estimation of Mean Values and Prediction of Individual Values Discussed Pitfalls in Regression and Ethical Issues
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