Chapter 13 General Linear Model

Chapter 13General Linear Model

Part III: Additional Hypothesis Tests

Renee R. Ha, Ph.D.James C. Ha, Ph.DIntegrative Statistics for the Social & Behavioral Sciences

ANOVA/t test

Score = + IV + E

Linear Regression

Least Squares Regression Line: Y’ = byX + ay where: Y’ = predicted value of Y by = slope of the line that minimizes the errors in predicting Y from X ay = y-intercept of the line that minimizes the errors in predicting Y from X

Linear Equation for Regression

Y = a + b(X) + SEE

Linear or Additive Equation

Score = + IV1 + IV2 + (IV1)(IV2) + E Where: = The grand mean starting point IV1 = Effect of the first independent variable (sex) IV2 = Effect of the second independent variable (ethnic group) (IV1)(IV2) = Interaction effect of the independent variables (sex * group) E = Unexplained error

Multifactorial ANOVA

Also called MANOVA

Score = + IV1 + IV2 + IV3 + (IV1)(IV2) + (IV1)(IV3) +(IV2)(IV3) + (IV1)(IV2)(IV3) + E Where = The grand mean IV1 = Effect of the first independent variable IV2 = Effect of the second independent variable IV3 = Effect of the third independent variable (IV1)(IV2) = Interaction effect of the first two independent variables (IV1)(IV3) = Interaction effect of the first and third independent variables (IV2)(IV3) = Interaction effect of the second and third independent variables (IV1)(IV2)(IV3) = Interaction effect of all three independent variables E = Unexplained error

Repeated-Measures ANOVA

Linear or additive equation Score = + IV1 + Subj + E

Linear Equation for Multiple Regression

Y = a + b1(X1) + b2(X2) + E

Assumptions of General Linear Model

All of these formulas share in common similar assumptions (i.e., normal distribution) and a linear form where the effects of each variable on the score are additive.

All of these equations are fundamentally the same form of linear equation or model, and they can all be solved using the same process (matrix algebra).

Advantages of Understanding GLM

1. You know why there is an F-obtained value in you’re linear regression output.

2. You can mix and match the measurement scales of your independent variables.

3. You can address the problem of predictor variables that are correlated with one another.

Raw Data Files for SPSS

Figure 13.1

Choosing the General Linear Model Option in SPSS

Selecting your Independent and Dependent Variables in SPSS

Choosing the Model in SPSS

a R Squared = .926 (Adjusted R Squared = .347)

Table 13.1

Tests of Between-Subjects EffectsDependent Variable: DAYS

ANCOVA

ANCOVA: a specialized form of ANOVA that is used when an investigator wishes to remove the effects of a variable that is known to influence the dependent variable but is not the subject of the current experiment and analysis.