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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