ALISON BOWLING THE GENERAL LINEAR MODEL

ALTERNATIVE EXPRESSION OF THE MODEL

PREDICTORS (IVS) Continuous (covariates) E.g. age, weight, time Dichotomous E.g. male vs female, employed vs unemployed, control vs experiment, low dose vs placebo, high dose vs placebo Dichotomous variables are usually Dummy coded. Reference group = 0e.g. male Comparison group = 1 e.g. female

GLM APPROACH TO ANOVA GroupLowHigh Placebo00 Low Dose Viagra 10 High Dose Viagra 01

REGRESSION ANALYSIS

INTERPRETING THE COEFFICIENTS

GLM

ANOVA OUTPUT The GLM Univariate procedure runs a General Linear Model, with the IV automatically dummy coded so that the group with the highest code is treated as the reference category. The intercept is the mean of the High dose group The dose = 1 (placebo) coefficient is the difference between the mean of the high dose and placebo group, and is significant (p =.008) The dose = 2 (low) coefficient is the difference between the mean of the high and low doses (NS).

WHY IS THIS IMPORTANT? Enables us to build sophisticated models including both continuous and categorical predictors Enhances understanding covariance Necessary to build regression models in which the assumptions are violated.

BRIEF REVISION OF MULTIPLE REGRESSION Multiple regression with two or more continuous predictors When the predictors are correlated, the coefficients represent the effect each predictor in the model, taking into account the other predictors. i.e. the effect of a predictor after the correlation(s) with the other predictors are partialled out. Field (2013) album sales.sav DV: Album sales Predictors: Advertising budget, No of plays on radio, Attractiveness of the band.

BIVARIATE SCATTERPLOTS

CORRELATIONS

HIERARCHICAL REGRESSION Order of entry: 1.Advertising budget 2.Airplay 3.Attractiveness

COEFFICIENTS

MODEL

COGNITION AND ANTI-SACCADE DATA (PETER LINDSAY) DV: Percentage of anti-saccade errors Predictors: Visuospatial/constructional, Delayed memory. Predictors are correlated

HIERARCHICAL REGRESSION Delayed memory does not improve the prediction of errors, after controlling for visuospatial

CHECKING RESIDUALS FOR NORMALITY Residuals are skewed – DV is a proportion, and therefore likely to violate assumptions.

ANALYSIS OF COVARIANCE

HIERARCHICAL REGRESSION

ANCOVA The effect of Dose is the effect of Viagra, after controlling for the covariate.

ASSUMPTIONS AND DATA ISSUES Main assumption of the model Residuals ( not DV ) are normally distributed Data issues (relate to the data or mathematics) Mulitcollinearity Outliers Non-linearity Etc. If the DV is continuous, unbounded and measured on an interval or ratio scale, the assumptions are most likely met. Not otherwise.

WHAT TO DO WHEN THE ASSUMPTIONS ARE VIOLATED

COMMON LINK FUNCTIONS Logit (log odds) for proportions Logistic regression Poisson (log) for count data Negative binomial for count data.

GENERALISED LINEAR MODELS Parameter estimation Linear models use Ordinary Least Squares (OLS) to estimate parameters. This is done by minimising the sum of the squared residuals R 2 is the measure of goodness-of-fit of the model Generalised linear models use Maximum Likelihood as the method of estimation. Negative Log Likelihood (-2LL) is one of several measures of goodness of fit.

MORE SOPHISTICATED MODELS Mixed models These may be used as an alternative method of analysing repeated measures data. Includes random effects in the model Models the variance of the intercept and/or the slope Generalised Linear mixed models More sophisticated still Applies mixed model methodology to Generalised linear models. All of these use Maximum Likelihood.