One-Way ANOVA
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Transcript One-Way ANOVA
Lab 5 instruction
a collection of statistical methods to compare
several groups according to their means on a
quantitative response variable
Two-Way ANOVA
two factors are used
consider “main effect” and “interaction effect”
Example
Response: Blood Alcohol Content (BAC)
Factors:
◦ Gender(GEN)
◦ Alcohol Consumption (ALC)
Factor levels:
◦ GEN: 1=Male; 0=Female
◦ ALC: 1=1 drink; 2=2 drinks; 4=4 drinks
Main effect of GEN, main effect of ALC, and
their interaction
Explore the difference in average response across both
factors.
Table of marginal means (contingency table)
In SPSS:
Analyze Compare Means Means
/Or General linear models descriptive stat
e.g. Table of marginal means (contingency table)
Response:
in Dependent list
Factors:
in independent list
of two seperate layers
e.g. Table of marginal means (contingency table)
Interpret
Explore the difference in average response across both
factors.
Plots of marginal means (profile plot)
In SPSS:
Graphs legacy dialogs Line multiple
/Or GLM univariate (plots)
e.g. Plot of marginal means (profile plot)
Profile plot with lines representing gender
e.g. Plot of marginal means (profile plot)
Interpretation:
◦ 6 points
◦ Effect of gender
◦ Effect of drink
consumption
◦ Interaction (parallel
or crossed)
Explore the difference in average response across both
factors.
Clustered boxplot
In SPSS:
Graphs legacy dialogs Boxplot
Clustered
e.g. Clustered boxplot
Clustered boxplot with clusters defined by
gender
e.g. Clustered boxplot
General Linear Model (GLM)-Assumptions
Independent samples
Normality:
◦ Boxplot
◦ QQ plot
Equal standard deviation (variance):
◦ Boxplot
◦ Summary statistics (rule of thumb: the ratio of the largest
s.d. over the smallest s.d. is less than 2)
General Linear Model (GLM)
Univariate GLM is the general linear model now
often used to implement such essential statistical
procedures as regression and ANOVA. In particular,
the procedure can be used to carry out two-way
analysis of variance.
Procedure:
Analyze
General Linear Model
Univariate
(defaulted: model with
Interactions)
GLM: Non-additive model output
GLM: Non-additive model output
In columns “SS” and “df”
◦ Corrected total = corrected model + error
◦ Corrected model=GEN + ALC + GEN*ALC
The column “F” contains the value of the f
statistic for each effect. The value of F is
computed as follows:
GLM: Non-additive model output
Hypothesis test
H 0 : no interaction between GEN and ALC
Equivalent to the regression models:
H0 : 4 5 0
H A : not both are zeros
Since the interaction term is not significant, we tend to
modify the model.
the model without the interaction term: additive model
GLM: Define additive model
GLM: Additive model output
& equivalent regression model
( BAC | GEN, ALC) 0
( BAC | GEN, ALC) 0 1 gen 2 alc2 3alc4
GLM: Additive model output
& equivalent regression model
( BAC | GEN, ALC) 0 2 alc2 3alc4
( BAC | GEN, ALC) 0 1 gen 2 alc2 3alc4
GLM: Additive model output
& equivalent regression model
( BAC | GEN, ALC) 0 1 gen
( BAC | GEN, ALC) 0 1 gen 2 alc2 3alc4
GLM - Options
Model: define the model, Include intercept or not
Contrasts: The linear combination of the level means
Profile Plots: the plot of estimated means
Post Hoc
Save
Options
GLM - Options
Profile Plots: different from the one
obtained with line chart
This plots estimated means form the model
Equivalent regression models and tests
Define dummy variables
Write down estimated model equation
Use extra-sum-of-squares F-test for testing
interaction term
Interpret meaning of estimated coefficient, ttest on coefficients and their 95% CI
Equivalent regression models and tests
estimated regression model with interaction
term (full non-additive model)
Equivalent regression models and tests
estimated regression model with NO
interaction term (additive model)
Equivalent regression models and tests
Construct extra-of-sum F-test use the above
two ANOVA tables,
In the regression model with NO interaction
term, interpret meaning of 1 , and its CI.
Is “GEN” effective in predicting BAC? (t-test on
1 )
Note on Q3 (b)
Linear combination of mean difference in BAC
for male V.S. female
m1 m 2 m 4
3
f1 f 2 f 4
3
Substitute ’s by corresponding Y ’s
Note on Q3 (c) and Q7 (c)
Rate of increase in mean
BAC for an increase in
drinks consumption
Thus, in Q3 (c)
2 1
4 2
◦ rate (1to2) =
, rate (2to4) =
2 1
42
In Q7 (c)
◦ rate (1to2) = 2 ,
rate (2to4) =
3 2
2