General Linear Model

Instructional Materials

• http://core.ecu.edu/psyc/wuenschk/PP/PP-MultReg.htm

• aka, http://tinyurl.com/multreg4u

Introducing the General

Linear Models

• As noted by the General, the GLM can be used to relate one set of things (Ys) to another set of things (X).

• It can also be used with only one set of things.

Bivariate Linear Function

Y = a + bX + error• This is probably what you have in mind

when thinking of a linear model.• Spatially, it is represented in two-

dimensional (Cartesian) space.

Least Squares Criterion

• Linear models produce parameter estimates (intercepts and slopes) such that the sum of squared deviations between Y and predicted Y is minimized.

2ˆ YY

Univariate Regression

• The mean is a univariate least squares predictor.

• The prediction model is• The sum of the squared deviations

between Y and mean Y is smaller than that for any reference value of Y.

errorYY ˆ

Fixed and Random Variables

• A FIXED variable is one for which you have every possible value of interest in your sample.– Example: Subject sex, female or male.

• A RANDOM variable is one where the sample values are randomly obtained from the population of values.– Example: Height of subject.

Correlation & Regression

• If Y is random and X is fixed, the model is a regression model.

• If both Y and X are random, the model is a correlation model.

• Researchers generally think that– Correlation = compute the corr coeff, r– Regression = find an equation to predict Y

from X

Assumptions, Bivariate Correlation

1. Homoscedasticity across Y|X2. Normality of Y|X3. Normality of Y ignoring X 4. Homoscedasticity across X|Y5. Normality of X|Y6. Normality of X ignoring Y• The first three should look familiar, you

make them with the pooled variances t.

Bivariate Normal

When Do Assumptions Apply?

• Only when employing t or F.• That is, obtaining a p value• or constructing a confidence interval.• With regression analysis, only the first

three assumptions (regarding Y) are made.

Sources of Error

• Y = a + bX + error• Error in the measurement of X and or Y or

in the manipulation of X.• The influence upon Y of variables other

than X (extraneous variables), including variables that interact with X.

• Any nonlinear influence of X upon Y.

The Regression Line

• r2 < 1 Predicted Y regresses towards mean Y

• In univariate regression, it regresses all the way to the mean for every case.

bXaY ˆxy rZZ ˆ

Uses of Correlation/Regression Analysis

• Measure the degree of linear association• Correlation does imply causation

– Necessary but not sufficient– Third variable problems

• Reliability• Validity• Independent Samples t – point biserial r

– Y = a + b Group (Group is 0 or 1)

Uses of Correlation/Regression Analysis

• Contingency tables -- Rows = a + bColumns

• Multiple correlation/regression

pp XbXbXbaY 2211

HighSchoolpMathVerbalECU GPAbSATbSATbaGPA 21

Uses of Correlation/Regression Analysis

• Analysis of variance (ANOVA)

• PolitConserv = a + b1 Republican? + b2 Democrat?k = 3, the third group is all others

11?22?11 kk GroupbGroupbGroupbaY

Uses of Correlation/Regression Analysis

• Canonical correlation/regression

• (homophobia, homo-aggression) = (psychopathic deviance, masculinity, hypomania, clinical defensiveness)

• High homonegativity = hypomanic, unusually frank, stereotypically masculine, psychopathically deviant (antisocial)

)()( 22112211 YbYbXaXa