250B Final Flashcards
What is the general formula for a total variance and how can it be converted to total sum of squares?
Variance is squared deviations from grand mean over degrees of freedom (N - 1). To get SStotal, just multiply the total variance by the degrees of freedom.
How can we conceptualize the variance of Y in model comparison? How is this related to the variance sum law?
Y = Model + Error
Y = Fit + Residual
Y = Systematic variance + Unsystematic variance
Variance of Y arises through variance sum law: Sy^2 = Smodel^2 + Serror^2 - 0
The covariance term goes to 0 because the two sources of variance are uncorrelated
Define the terms Sy^2 = Smodel^2 + Serror^2 for two-group independent t-test.
Comparing group mean vs grand mean.
For null model:
Sy^2 = sum(Ybar - Ybar)^2 + sum(Y - Ybar)^2
For full model:
Sy^2 = sum(Yhat - Ybar)^2 + sum(Y - Yhat)^2
Define the terms Sy^2 = Smodel^2 + Serror^2 for one-way ANOVA.
Comparing group means vs grand mean.
For null model:
Sy^2 = sum(Ybar - Ybar)^2 + sum(Y - Ybar)^2
For full model:
Sy^2 = sum(Yhat - Ybar)^2 + sum(Y - Yhat)^2
Define the terms Sy^2 = Smodel^2 + Serror^2 for two-way ANOVA.
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Define the terms Sy^2 = Smodel^2 + Serror^2 for simple regression.
Compare reduced model (fewer predictors) with full model (more predictors).
For null model:
Sy^2 = sum(Yhat - Ybar)^2 + sum(Y - Ybar)^2
For full model:
Sy^2 = sum(Yhat - Ybar)^2 + sum(Y - Ybar)^2
Model comparison typically involves the comparison of error terms. Which term has more degrees of freedom, the full or reduced model sum-of-squares?
Reduced model has more df because fewer parameters are estimated.
Explain the terms in F test for model comparison formula and when to use it.
It’s change in error variance over full model error variance. The terms used are error variance of reduced model, error variance of full model, error df for reduced model and error df for full model.
You use this formula any time you want to compare the error reduction properties of two models.
A researcher has a two-way ANOVA problem but the effects (a,b, and ab) are correlated. How would model comparison proceed if the researcher wanted to use a hierarchical approach entering b, a, and the interaction in that order? What is being tested at each step?
Hierarchical: Type I SS.
The first model would include just B, and tests the amount of variance accounted for by B, SS(B). The second model adds in A, and tests the drop in residual variance from adding A after B, SS(A | B). The third model adds in the interaction, and tests the drop in residual variance that’s left over after accounting for A and B, SS(AxB | A, B).
Research and statistical analysis is a form of modeling. What does that mean?
This means that our goal as statisticians is to create a model of the world (a population) and test it to see if it accurately represents the world. We compare different statistical models of the world to one another and evaluate which ones are the best fit for the data.
How are partial eta sq computed from a table of SPSS output with SS? Are they the same as the change in R squared from a full (all predictors) to reduced model?
Partial eta squared tells you: how much variance would factor A explain if it was the only variable in the model?
Formula for partial eta-squared = SSeffect / (SSeffect + SSerror)
Sum of partial eta-squareds is not the same as change in R2 because the denominator for partial eta squares.
In SPSS output, what are the SS terms for Corrected Model, Error, Total, and Corrected Total?
Corrected Model SS = Corrected Total SS - Error SS. Variance due to two main effects and interaction.
Corrected Total SS is the SStotal we’re interested in. It comes from adding up SS for each factor, the interaction, and the error. Basically Corrected Model SS + Error SS
Error SS is the sum squares for error (within cells error) –> MSE
Total SS is the total sum of squares for intercept, main effects, interaction, and error
In SPSS output, what is that F value for the corrected model testing? How is that R Squared calculated?
The F value for the corrected model is the F test for change in error variance between reduced model and full model with all predictors (aka test for change in R squared)
The R squared is the amount of variance in DV accounted for by the mode l= Corrected Model / Corrected Total
Howell states, “ANOVA tells us that three treatments have different means. Multiple regression tell us that means are related to treatments”. Explain in what ways are these the same thing?
These are the same because when we say means are related to treatments, we mean that group means are different depending on what treatment they receive – aka, the treatment groups have different means.
If there were two groups, treatment and control, one could do a t-test, a one-way between ANOVA, or compute the correlation between treatment/control and the dependent variable. Are there any important differences between these three approaches? In what ways would the results be exactly the same?
a t-test and the ANOVA are the most direct approaches, and rejecting those null hypotheses give you direct information about whether or not there are group mean differences.
The t-test and ANOVA will give the same p-value, and the F statistic will be the square of the t statistic.
Computing a correlation is whether variability in DV is explained by treatment. This would entail computing a point biserial correlation, and will yield the same test statistic as the previous two tests.
If there were four treatment groups in a one-way ANOVA, and you had three effect codes (and thus three regression coefficients in the GLM), what would the effect be for the missing code, T4, and why?
T4 would be the reference group, which is indicated by being coded at -1 on all other regression coefficients.
A researcher has five levels of an independent variable and runs both an ANOVA and a regression. Will the eta squared from the ANOVA be equal to the R2 from the regression? Why?
Yes, because they both measure the amount of variance accounted for by the model. Because in an omnibus test, the regression model predicts the cell means perfectly. This is the same thing as the structural model in ANOVA, where each person’s score is predicted from his or her group mean.
Explain how the interpretation of regression coefficients would change depending on the whether the treatment levels are effects coded versus dummy coded.
In dummy coding, the reference group is coded as 0 and the intercept is the mean for the reference group. Coefficients are the difference between coded group and the grand mean.
In effects coding, the reference group is coded as -1 and the intercept is the grand mean. Coefficients are differences between the group coded one for that coef and the grand mean.
Given equal n in one-way or factorial ANOVA, what special properties will the design matrix have? What will the means and correlations of the design matrix look like with equal and unequal n with effects or dummy coding?
Special Properties: If balanced design, design matrix will have orthogonal matrix in it.
Effects coding with equal n: Different effects will be uncorrelated in the design matrix. Effect of A uncorrelated with the effect of B. By effect, we mean that Dummy codes for A uncorrelated with Dummy codes for B and same for the interaction. If balanced design, the means of each effects codes will be zero and they will be some fraction for dummy codes.
Without equal n: Different effects will be correlated in the design matrix. Effects codes will no longer have means of 0 and you cant uniquely partition it. Effects will be correlated with each other (i.e. A1 will be correlated with B1, etc.)
How does the design matrix allow us to transition from the ANOVA model to the regression framework of the GLM?
The design matrix relates the predictors to the dependent variable. If we code each group in ANOVA as a predictor and put those codes into the design matrix, we have a method of getting from ANOVA framework to GLM framework.
What is the degrees of freedom numerator and denominator for the F test in the GLM approach to ANOVA?
Numerator DF: difference in df between reduced and full model (i.e., difference in num parameters estimated)
Denominator DF: full model DF
What are these incremental SS? SS(AB | A, B) SS(A | B, AB) SS(B | A, AB) SS(A | B) SS(B | A)
These SS are all SSregression SS(AB | A, B) = SS(A, B, AB) - SS(A, B) SS(A | B, AB) = SS(A, B, AB) - SS(B, AB) SS(B | A, AB) = SS(A, B, AB) - SS(A, AB) SS(A | B) = SS(A, B) - SS(B) SS(B | A) = SS(A, B) - SS(A)
A researcher is conducting a two-way ANOVA in a general linear model framework, but the effects are confounded. How would this impact eta squared under a Type I sum-of-squares versus a Type III sum-of-squares?
If the effects are confounded, we cannot partition eta-sq uniquely into variance due to each variable and variance due to the interaction.
In Type I we adjust only for terms that were entered before so we test the first factor and see how much variance it takes up without controlling for the other factor. So, its eta-sq will be enlarged. Sum of semi-partial correlations (eta squared) should be smaller for Type III
In Type III, we adjust everything for everything else and test main effects after controlling for other main effect and interaction. In Type III SS you get only variance unique to each factor and the interaction so eta-sq decreases.
What is the order of incremental SS for Type I SS?
Enter A first, B second, AB third.
SS(A) for factor A, then SS(B | A) for factor B, then SS(AB | B, A) for interaction AB
A researcher is conducting a two-way ANOVA in a general linear model framework, but the effects are confounded. How would the eta squares be computed using a change in R squared?
Related: first define semi-partial eta squared
When each factor’s eta squared reflects unique effect for a given factor controlling for all other factors (like in Type III SS), this is called semi-partial
We compute these semi-partial correlations by comparing Rsq in full model vs Rsq in reduced model (literally just subtract: Rsq_full - Rsq_reduced)
In an unbalanced design in factorial ANOVA, the effect of factor A is confounded by the effects of factor B and the AB interaction. What does that really mean?
This means that you can’t uniquely partition variance accounted for (SS) uniquely to each factor. It’s impossible to separate what is attributable to factor A and factor B.
