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So far
--> looked at the effect of a discrete variable on a
continuous variable
t-test, ANOVA, 2-way ANOVA
Height class
Weight (kg)
Short Medium Tall
Height
Weigth (kg)
Short
Medium
Tall
Height (cm)
Weigth (kg)
Often -
interested in the relationship between 2 or more
continuous variables
REGRESSION
Eg height and age, height and weight, dose and response
Regression allows us to ask:
Dose (mg)
Blood Pressure Decrease (mmHg)
Body Weight (kg)
Blood Pressure Decrease (mmHg)
Body Weight (kg)
Dose (mg)
BP Decrease (mmHg)
Height (cm)
Weight (kg)
Height (cm)
Weight (kg)
Y X 0 1
ˆ ˆ ˆ =β + β
Height (cm)
Weight (kg)
Y X 0 1
ˆ ˆ ˆ =β + β
Height (cm)
Weight (kg)
Y X 0 1
ˆ ˆ ˆ =β + β
Hypothesis Tests Regarding Regression
as
So, --> make inferences about the POPULATION
based on a sample.
Hypotheses:
Slope
H
o
1
H
A
1
Intercept
H
o
0
H
A
0
Height (cm)
Weight (kg)
Y
Height (cm)
Weight (kg)
Height (cm)
Weight (kg)
ANOVA Table
Source of
Variation SS DF MS F
Regression
Residual
Total
Y Y )
i
2
Y Y )
i
2
( Y Y )
i
2
n-
n-
Height (cm)
Weight (kg)
How much of the variation in Y is explained by
the relationship?
--> Coefficient of Determination = r
2
X Y
Observation Fiber Diameter Breaking Strength
Hypotheses:
H o : β = 0
H
A
What is the predicted strength of a fiber that is
26um diameter?
PPTV
LIFE
Model Summaryb
.606 a^ .367 .349 6.
Model 1
R R Square
Adjusted R Square
Std. Error of the Estimate
a.Predictors: (Constant), PPTV b.Dependent Variable: LIFE
ANOVAb
826.733 1 826.733 20.877 .000 a 1425.636 36 39. 2252.368 37
Regression Residual Total
Model 1
Sum of Squares df Mean Square F Sig.
a.Predictors: (Constant), PPTV b.Dependent Variable: LIFE
Coefficientsa
(Constant) PPTV
Model 1
B Std. Error
Unstandardized Coefficients Beta
Standardized Coefficients t Sig.
a.Dependent Variable: LIFE
Scatterplot
Dependent Variable: LIFE
Regression Standardized Predicted Value
Regression Standardized Residual
TPPT
LIFE
Model Summaryb
.855 a^ .731 .724 4.
Model 1
R R Square
Adjusted R Square
Std. Error of the Estimate
a.Predictors: (Constant), TPPT b.Dependent Variable: LIFE
ANOVAb
1646.972 1 1646.972 97.938 .000 a 605.396 36 16. 2252.368 37
Regression Residual Total
Model 1
Sum of Squares df Mean Square F Sig.
a.Predictors: (Constant), TPPT b.Dependent Variable: LIFE
Coefficientsa
(Constant) TPPT
Model 1
B Std. Error
Unstandardized Coefficients Beta
Standardized Coefficients t Sig.
a.Dependent Variable: LIFE
Scatterplot
Dependent Variable: LIFE
Regression Standardized Predicted Value
Regression Standardized Residual
Scatterplot (television.STA 8v*40c)
P_PHYS
LIFE
Scatterplot (television.STA 8v*40c)
LOGP_PHY
LIFE
5 6 7 8 9 10 11 THORAX
LONGDAY
SLEEP
THORAX
A^ A
A
A
A
A A
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A A
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AA
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A A
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A A A A
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A A A
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AA A
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A AA
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A A A A
A A A
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A A (^) A A
A AA
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Longevity = B o + B 1 X 1 + B 2 X 2
