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
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Response: Blood Alcohol Content (BAC)
Factors:
◦ Gender(GEN)
◦ Alcohol Consumption (ALC)
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Factor levels:
◦ GEN: 1=Male; 0=Female
◦ ALC: 1=1 drink; 2=2 drinks; 4=4 drinks
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Main effect of GEN, main effect of ALC, and
their interaction
Explore the difference in average response across both
factors.
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Table of marginal means (contingency table)
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In SPSS:
Analyze  Compare Means  Means
/Or General linear models  descriptive stat
e.g. Table of marginal means (contingency table)
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Response:
in Dependent list
Factors:
in independent list
of two seperate layers
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e.g. Table of marginal means (contingency table)
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Interpret
Explore the difference in average response across both
factors.
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Plots of marginal means (profile plot)
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In SPSS:
Graphs legacy dialogs  Line  multiple
/Or GLM  univariate  (plots)
e.g. Plot of marginal means (profile plot)
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Profile plot with lines representing gender
e.g. Plot of marginal means (profile plot)
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Interpretation:
◦ 6 points
◦ Effect of gender
◦ Effect of drink
consumption
◦ Interaction (parallel
or crossed)
Explore the difference in average response across both
factors.
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Clustered boxplot
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In SPSS:
Graphs  legacy dialogs  Boxplot
Clustered
e.g. Clustered boxplot
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Clustered boxplot with clusters defined by
gender
e.g. Clustered boxplot
General Linear Model (GLM)-Assumptions
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Independent samples
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Normality:
◦ Boxplot
◦ QQ plot
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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)
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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)
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GLM: Non-additive model output
GLM: Non-additive model output
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In columns “SS” and “df”
◦ Corrected total = corrected model + error
◦ Corrected model=GEN + ALC + GEN*ALC
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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
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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
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Model： define the model, Include intercept or not
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Contrasts: The linear combination of the level means
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Profile Plots: the plot of estimated means
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Post Hoc
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Save
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Options
GLM - Options
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Profile Plots: different from the one
obtained with line chart
This plots estimated means form the model
Equivalent regression models and tests
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Define dummy variables
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Write down estimated model equation
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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
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estimated regression model with interaction
term (full non-additive model)
Equivalent regression models and tests
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estimated regression model with NO
interaction term (additive model)
Equivalent regression models and tests
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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 )
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Note on Q3 (b)
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Linear combination of mean difference in BAC
for male V.S. female
m1  m 2  m 4
3
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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
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Thus, in Q3 (c)
 2  1
4  2
◦ rate (1to2) =
, rate (2to4) =
2 1
42
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In Q7 (c)
◦ rate (1to2) =  2 ,
rate (2to4) =
3   2
2