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Assessment of Assumptions in SLR

1.Linearity: There is a real linear relationship between the mean response andthe explanatory variable.

Assessment:Scatterplot ofYvs. X: Look for a linear pattern!

Non-Linearity Linearity

Mortality (Y) vs. Wine (X)

WINE

806040200

MORTAL

IT

12

10

8

6

4

2 Rsq = 0.5559

log(Mortality) vs. log(Wine)

LNWINE

4.54.03.53.02.52.01.51.0

LNMORT

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

.8

.6 Rsq = 0.7384

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2.Equal Variability: The spread of the responses about the line is the same atall levels of the explanatory variable.

Assessment:

Scatterplot ofYvs. X: Check for roughly the same spread about a line

across all values ofX.

Scatterplot of Residuals vs. Fitted Values: Look for an equal spread about

the horizontal line at zero.

Lack of Equal Variability Equal Variability seems SatisfiedScatterplot

Dependent Variable: Y

Regression Standardized Predicted Value

3210-1-2

RegressionStandardizedResidual

8

6

4

2

0

-2

Scatterplot

Dependent Variable: LNY

Regression Standardized Predicted Value

3210-1-2

RegressionStandardizedResidual

2

1

0

-1

-2

-3

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3. Normality:At any fixed X, the response variable is normally distributed

centered on the line.

Assessment:

Normal Probability Plots: Look for an increasing linear pattern.

4. Independence:All responses are independent on one another.

Assessment: Check study design.

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An Illustration

On Original Scale: In the scatterplot of Y vs. X the assumptions of linearity and equal variability appear to

be violated. The scatterplot of residuals confirms the lack of equal variability.

Y vs. X

X

403836343230282624

Y

3000

2000

1000

0

-1000

Scatterplot

Dependent Variable: Y

Regression Standardized Predicted Value

3210-1-2

RegressionStandardizedResidual

8

6

4

2

0

-2

After a log transformation in Y: In the scatterplot of Y vs. X the assumptions of linearity and equal

variability appear to be satisfied. The residual plot confirms the equal variability assumption.

lnY vs. X

X

403836343230282624

LNY

8

6

4

2

0

-2

-4

Scatterplot

Dependent Variable: LNY

Regression Standardized Predicted Value

3210-1-2

RegressionStandardizedResidua

l

2

1

0

-1

-2

-3

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Example 2: Heart Disease Mortality vs. Wine Consumption

Y vs. X: Non-linearity and Unequal log(Y) vs. log(X): The assumptions appear

Variability to be satisfied.

Mortality (Y) vs. Wine (X)

WINE

806040200

MORTALIT

12

10

8

6

4

2 Rsq = 0.5559

log(Mortality) vs. log(Wine)

LNWINE

4.54.03.53.02.52.01.51.0

LNMORT

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

.8

.6 Rsq = 0.7384

Scatterplot

Dependent Variable: MORTALIT

Regression Standardized Predicted Value

1.0.50.0-.5-1.0-1.5-2.0-2.5-3.0

RegressionStandardizedResidual

2.0

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

Scatterplot

Dependent Variable: LNMORT

Regression Standardized Predicted Value

1.51.0.50.0-.5-1.0-1.5-2.0-2.5

RegressionStandardizedResidual

2.0

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

-2.0

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Example 3

SLR Fit Separate Means (SM or I-Means) Fit

DOSE

.5.4.3.2.10.0

BPDRO

P

40

30

20

10

0

666N =

DOSE

.40.20.10

BPDR

OP

40

30

20

10

0