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Statistics One
Lecture 7
Introduction to Regression
1
Three segments
•  Overview
•  Calculation of regression coefficients
•  Assumptions

2
Lecture 7 ~ Segment 1
Regression: Overview

3
Regression: Overview
•  Important concepts & topics
–  Simple regression vs. multiple regression
–  Regression equation
–  Regression model

4
Regression: Overview
•  Regression: a statistical analysis used to
predict scores on an outcome variable,
based on scores on one or multiple predictor
variables
–  Simple regression: one predictor variable
–  Multiple regression: multiple predictors
5
Regression: Overview
•  Example: IMPACT (see Lab 2)
–  An online assessment tool to investigate the
effects of sports-related concussion
•  http://www.impacttest.com

6
IMPACT example
•  IMPACT provides data on 6 variables
–  Verbal memory
–  Visual memory
–  Visual motor speed
–  Reaction time
–  Impulse control
–  Symptom score
7
IMPACT: Correlations pre-injury

8
IMPACT: Correlations pre-injury

9
IMPACT: Correlations post-injury

10
IMPACT: Correlations post-injury

11
IMPACT example
•  For this example, assume:
–  Symptom Score is the outcome variable
–  Simple regression example:
•  Predict Symptom Score from just one variable

–  Multiple regression example:
•  Predict Symptom Score from two variables
12
Regression equation
•  Y = m + bX + e
–  Y is a linear function of X
–  m = intercept
–  b = slope
–  e = error (residual)
13
Regression equation
•  Y = B0 + B1X1 + e
–  Y is a linear function of X1
–  B0 = intercept = regression constant
–  B1 = slope = regression coefficient
–  e = error (residual)
14
Model R and

2
R

•  R = multiple correlation coefficient

• 

–  R = rÝY
–  The correlation between the predicted scores
and the observed scores

2
R

–  The percentage of variance in Y explained by
the model
15
IMPACT example
•  Y = B0 + B1X1 + e

–  Let Y = Symptom Score
–  Let X1 = Impulse Control
–  Solve for B0 and B1

– In R, function lm
16
IMPACT example
Ŷ = 20.48 + 1.43(X)	

	

r = .40	

	

R2 = 16%	

	

17
IMPACT example

18
Regression model
•  The regression model is used to model or
predict future behavior
–  The model is just the regression equation
–  Later in the course we will discuss more
complex models that consist of a set of
regression equations
19
Regression: It gets better
•  The goal is to produce better models so we
can generate more accurate predictions
–  Add more predictor variables, and/or…
–  Develop better predictor variables

20
IMPACT example
•  Y = B0 + B1X1 + B2X2 + e

–  Let Y = Symptom Score
–  Let X1 = Impulse Control
–  Let X2 = Verbal Memory
–  Solve for B0 and B1 and B2

– In R, function lm
21
IMPACT example
Ŷ = 4.13 + 1.48(X1) + 0.22(X2)	


	

R2 = 22%	

	


22
IMPACT example

23
Model R and

2
R

•  R = multiple correlation coefficient

• 

–  R = rÝY
–  The correlation between the predicted scores
and the observed scores

2
R

–  The percentage of variance in Y explained by
the model
24
IMPACT example
R2 = 22%	

	

rÝY = .47	


25
Segment summary
•  Important concepts & topics
–  Simple regression vs. multiple regression
–  Regression equation
–  Regression model

26
END SEGMENT

27
Lecture 7 ~ Segment 2
Calculation of regression coefficients

28
Estimation of coefficients
•  Regression equation:
–  Y = B0 + B1X1 + e
–  Ŷ = B0 + B1X1
•  (Y – Ŷ) = e (residual)

29
Estimation of coefficients
•  The values of the coefficients (e.g., B1) are
estimated such that the regression model
yields optimal predictions
–  Minimize the residuals!

30
Estimation of coefficients
•  Ordinary Least Squares estimation
–  Minimize the sum of the squared (SS) residuals
–  SS.RESIDUAL = Σ(Y –

2
Ŷ)

31
IMPACT example

32
Estimation of coefficients
•  Sum of Squared deviation scores (SS) in variable Y
–  SS.Y
SS.Y à	


33
Estimation of coefficients
•  Sum of Squared deviation scores (SS) in variable X
–  SS.X
SS.X à	


34
Estimation of coefficients
•  Sum of Cross Products
–  SP.XY
SS.Y à	

SP.XY	

SS.X à	

35
Estimation of coefficients
•  Sum of Cross Products = SS of the Model
–  SP.XY = SS.MODEL
SS.Y à	

SS.MODEL	

SS.X à	

36
Estimation of coefficients
•  SS.RESIDUAL = (SS.Y – SS.MODEL)

SS.Y à	


SS.RESIDUAL	

SS.MODEL	


SS.X à	

37
Estimation of coefficients
•  Formula for the unstandardized coefficient
–  B1 = r x (SDy/ SDx)

38
Estimation of coefficients
•  Formula for the standardized coefficient
–  If X and Y are standardized then
•  SDy = SDx = 1
•  B = r x (SDy/ SDx)
•  β = r

39
Segment summary
•  Important concepts
–  Regression equation and model
–  Ordinary least squares estimation
–  Unstandardized regression coefficients
–  Standardized regression coefficients

40
END SEGMENT

41
Lecture 7 ~ Segment 3
Assumptions

42
Assumptions
•  Assumptions of linear regression
–  Normal distribution for Y
–  Linear relationship between X and Y
–  Homoscedasticity

43
Assumptions
•  Assumptions of linear regression
–  Reliability of X and Y
–  Validity of X and Y
–  Random and representative sampling

44
Assumptions
•  Assumptions of linear regression
–  Normal distribution for Y
–  Linear relationship between X and Y
–  Homoscedasticity

45
Anscombe’s quartet

46
Anscombe’s quartet
•  Regression equation for all 4 examples:
–  Ŷ = 3.00 + 0.50(X1)

47
Anscombe’s quartet
•  To test assumptions, save residuals
–  Y = B0 + B1X1 + e
–  e = (Y – Ŷ)

48
Anscombe’s quartet
•  Then examine a scatterplot with
–  X on the X-axis
–  Residuals on the Y-axis

49
Anscombe’s quartet

50
Segment summary
•  Assumptions when interpreting r
–  Normal distributions for Y
–  Linear relationship between X and Y
–  Homoscedasticity
–  Examine residuals to evaluate assumptions

51
END SEGMENT

52
END LECTURE 7

53

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