Uji Kelinearan dan Keberartian Regresi

Pertemuan 02

Matakuliah : I0174 – Analisis RegresiTahun : Ganjil 2007/2008

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Uji Kelinieran dan Keberartian Regresi

• Anova pada regresi Sederhana

• Selang Kepercayaan Parameter Regresi

• Uji Independen Antar Peubah

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Measures of Variation: The Sum of Squares

SST = SSR + SSE

Total Sample

Variability

= Explained Variability

+ Unexplained Variability

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Measures of Variation: The Sum of Squares

• 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

(continued)

Y

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Measures of Variation: The Sum of Squares

(continued)

Xi

Y

X

Y

SST = (Yi - Y)2

SSE =(Yi - Yi )2

SSR = (Yi - Y)2

_

_

_

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Venn Diagrams and Explanatory Power of Regression

Sales

Sizes

Variations in Sales explained by Sizes or variations in Sizes used in explaining variation in Sales

Variations in Sales explained by the error term or unexplained by Sizes

Variations in store Sizes not used in explaining variation in Sales

SSE

SSR

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The ANOVA Table in Excel

ANOVA

df SS MS FSignificance F

Regression

kSSR

MSR=SSR/k

MSR/MSEP-value of the F Test

Residuals

n-k-1

SSE

MSE=SSE/(n-k-1)

Total n-1SST

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Measures of VariationThe Sum of Squares: Example

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

Excel Output for Produce Stores

SSR

SSERegression (explained) df

Degrees of freedom

Error (residual) df

Total df

SST

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The Coefficient of Determination

•

• Measures the proportion of variation in Y that is explained by the independent variable X in the regression model

2 Regression Sum of Squares

Total Sum of Squares

SSRr

SST

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Venn Diagrams and Explanatory Power of Regression

Sales

Sizes

2

SSR

SSR S

r

SE

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Coefficients of Determination (r 2) and Correlation (r)

r2 = 1, r2 = 1,

r2 = .81, r2 = 0,Y

Yi = b0 + b1Xi

X

^

YYi = b0 + b1Xi

X

^Y

Yi = b0 + b1Xi

X

^

Y

Yi = b0 + b1Xi

X

^

r = +1 r = -1

r = +0.9 r = 0

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Standard Error of Estimate

•

• Measures the standard deviation (variation) of the Y values around the regression equation

2

1

ˆ

2 2

n

ii

YX

Y YSSE

Sn n

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Measures of Variation: Produce Store Example

Regression StatisticsMultiple R 0.9705572R Square 0.94198129Adjusted R Square 0.93037754Standard Error 611.751517Observations 7

Excel Output for Produce Stores

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.

Syxn

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Linear Regression Assumptions

• Normality– Y values are normally distributed for each

X– Probability distribution of error is normal

• Homoscedasticity (Constant Variance)• Independence of Errors

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Consequences of Violationof the Assumptions

• Violation of the Assumptions– Non-normality (error not normally distributed)– Heteroscedasticity (variance not constant)

• Usually happens in cross-sectional data– Autocorrelation (errors are not independent)

• Usually happens in time-series data• Consequences of Any Violation of the Assumptions

– Predictions and estimations obtained from the sample regression line will not be accurate

– Hypothesis testing results will not be reliable• It is Important to Verify the Assumptions

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• Y values are normally distributed around the regression line.

• For each X value, the “spread” or variance around the regression line is the same.

Variation of Errors Aroundthe Regression Line

X1

X2

X

Y

f(e)

Sample Regression Line

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Residual Analysis• Purposes

– Examine linearity – Evaluate violations of assumptions

• Graphical Analysis of Residuals– Plot residuals vs. X and time