© McGraw-Hill Higher Education. All Rights Reserved. Chapter 2F Statistical Tools in Evaluation.

© McGraw-Hill Higher Education. All Rights Reserved.

Chapter 2FChapter 2F

Statistical Tools in EvaluationStatistical Tools in Evaluation


Linear RegressionLinear Regression Predict one variable from others If measurement on one variable is difficult Prediction is not perfect but contains error Error (SEE) is low if r is high Equation is Y=(bX)+C Y is the predicted value b is the slope of the line and X is the value of

the other C is the Y intercept (constant)


Prediction-Regression AnalysisPrediction-Regression Analysis

• Regression – statistical model used to predict performance on one variable from another.

• Simple regression – predicting a score on one variable (Y) from one other variable (X).

• Multiple regression – predicting a score on one variable (Y) from two or more other variables (X1, X1, etc.)


General Prediction EquationGeneral Prediction Equation

Y = (bX) + C

b = slope of regression line (rate of change in Y per unit change in X)

c = Y-intercept or constant (Y when X=zero)


Standard Error of Estimate Standard Error of Estimate (SEE)(SEE)

• R=regression while r=correlation

• Predicted Score = Y

• Y will not be perfect unless r = 1

• When r 1 there is prediction error

• The standard deviation of this error = SEE

• SEE = Sy 1 - r2


Standard Error of Estimate Standard Error of Estimate (SEE)(SEE)

• Expect to find the subjects’ real score in the boundaries:

Y ± 2 (SEE) 95% of the time• The equation with the lowest SEE is the

most accurate.


Other important measuresOther important measures

• R = correlation between predicted and real score Ranges between 0 and 1.00

An index of prediction accuracy

• R2 = coefficient of determination Proportion of variance in criterion (Y scores)

explained by the predictor (X scores)

An index of prediction accuracy


62.00 64.00 66.00 68.00 70.00 72.00 74.00 76.00

height

130.00

140.00

150.00

160.00

170.00

180.00

190.00

200.00

wei

gh

t

R Sq Linear = 0.871


Regression EquationRegression Equation

Y=(bX)+C Y is the predicted value b is the slope of the line and X is the value C is the Y intercept (constant)


Confidence Intervals Confidence Intervals (CI)(CI)

SEE x 2 Determines error around the predicted score Multiply the SEE x 2 to get 95% confidence


Simple RegressionSimple Regression

Subject Height Weight Rgrip1 115 182 71 185 1003 73 326.6 97.54 68 160 1005 69 200 1306 74 195 1307 60 125 37.488 71 180 999 70 175 90

Trying to predict height from weight. Run SPSS regression and choose linear.


Model Summary

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .761a .580 .527 3.49808

a. Predictors: (Constant), Wt

Coefficientsa

Model

Unstandardized Coefficients

Standardized

Coefficients

B Std. Error Beta t Sig.

1 (Constant) 56.812 3.517 16.155 .000

Wt .063 .019 .761 3.322 .011

a. Dependent Variable: HT


Simple Regression SolvedSimple Regression Solved Subject #1, Y=(bx)+C Y=(.06x115)+56.81 Y=6.9+56.81 Y=63.71 Predicted score = 63.71+(SEEx2) Answer = 63.71+6.98 95% of the time the real score will fall between 56.73 - 70.69

Coefficientsa

Model


Standardized

Coefficients


1 (Constant) 56.812 3.517 16.155 .000

Wt .063 .019 .761 3.322 .011



Multiple RegressionMultiple Regression

• Predict criterion (Y) using several predictors (X1, X2, X3, etc)

• Basic multiple regression equation has one intercept (c) and several bs (one for each predictor variable).

Y = (bX1 + bX2 + bX3) + c

• Important measures: R, R2, SEE


Multiple RegressionMultiple Regression

Subject Height Weight Rgrip1 115 182 71 185 1003 73 326.6 97.54 68 160 1005 69 200 1306 74 195 1307 60 125 37.488 71 180 999 70 175 90

Trying to predict height from weight and Rgrip. Run SPSS regression and choose linear.


Model Summary

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .930a .864 .826 2.12476

a. Predictors: (Constant), Rgrip, Wt

Coefficientsa

Model


Standardized

Coefficients


1 (Constant) 56.149 2.143 26.200 .000

Wt .028 .015 .337 1.893 .100

Rgrip .083 .022 .682 3.832 .006



Multiple Regression SolvedMultiple Regression Solved Subject #1, Y= (bX1) + (bX2) + C Y=(.02x115) + (.08x18) + 56.14 Y=2.3+1.44+56.14 Y=59.88 Predicted score = 59.88+(SEEx2) Answer = 59.88+4.24 95% of the time the real score will fall between 55.64 – 64.12

Coefficientsa

Model


Standardized

Coefficients


1 (Constant) 56.149 2.143 26.200 .000

Wt .028 .015 .337 1.893 .100

Rgrip .083 .022 .682 3.832 .006



OutcomesOutcomes Simple vs. Multiple Regression

– R increased– SEE decreased

THEREFORE

– 95% confidence intervals decreased– Prediction accuracy increased


SPSSSPSS

Analyze– Regression– Linear

Dependent variable (what to predict) Independent variable (used to predict) Constant, B value(s) and SEE

© McGraw-Hill Higher Education. All Rights Reserved. Chapter 2F Statistical Tools in Evaluation.

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