General Linear Model. Instructional Materials MultReg.htmhttp://core.ecu.edu/psyc/wuenschk/PP/PP- MultReg.htm.

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Presentation transcript:

General Linear Model

Instructional Materials MultReg.htmhttp://core.ecu.edu/psyc/wuenschk/PP/PP- MultReg.htm aka,

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.

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.

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|X 2.Normality of Y|X 3.Normality of Y ignoring X 4.Homoscedasticity across X|Y 5.Normality of X|Y 6.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 r 2 < 1  Predicted Y regresses towards mean Y In univariate regression, it regresses all the way to the mean for every case.

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 + b  Columns Multiple correlation/regression

Uses of Correlation/Regression Analysis Analysis of variance (ANOVA) PolitConserv = a + b 1 Republican? + b 2 Democrat? k = 3, the third group is all others

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)