Two-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 usedconsider “main effect” and “interaction effect”

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

ExampleExample

Table of marginal means (contingency table)

In SPSS: Analyze Compare Means Means

/Or General linear models descriptive stat

Explore the difference in average response across both factors.

Response: in Dependent list

Factors: in independent list of two seperate layers

e.g. Table of marginal means (contingency table)

e.g. Table of marginal means (contingency table)

Interpret

Plots of marginal means (profile plot)

In SPSS:

Graphs legacy dialogs Line multiple

/Or GLM univariate (plots)

Explore the difference in average response across both factors.

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)

e.g. Plot of marginal means (profile plot)

Clustered boxplot

In SPSS:

Graphs legacy dialogs Boxplot Clustered

Explore the difference in average response across both factors.

Clustered boxplot with clusters defined by gender

e.g. Clustered boxplot

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

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

GLM: Non-additive model output

Hypothesis test : no interaction between GEN and ALC

H0

Equivalent to the regression Equivalent to the regression models:models:

H0 :4 5 0

HA : 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

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3210

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alcalcgenALCGENBAC

ALCGENBAC

GLM: Additive model output & equivalent regression model

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320

alcalcgenALCGENBAC

alcalcALCGENBAC

GLM: Additive model output & equivalent regression model

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alcalcgenALCGENBAC

genALCGENBAC

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

Profile Plots: different from the one obtained with line chart

This plots estimated means form the model

GLM - Options

Define dummy variables

Write down estimated model equation

Use extra-sum-of-squares F-test for testing interaction term

Interpret meaning of estimated coefficient, t-test 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 , and its CI.

Is “GEN” effective in predicting BAC? (t-test on )

Equivalent regression models and tests

1

1

Linear combination of mean difference in BAC for male V.S. female

Substitute ’s by corresponding ’s

Note on Q3 (b)

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Y

Rate of increase in mean BAC for an increase in drinks consumption

Thus, in Q3 (c)

◦ rate (1to2) = , rate (2to4) =

In Q7 (c)◦ rate (1to2) = , rate (2to4) =

Note on Q3 (c) and Q7 (c)

2

3 2

2

1212

2424