Centering Predictors

Transcription

1 Centering Predictors Longitudinal Data Analysis Workshop Section 3 University of Georgia: Institute for Interdisciplinary Research in Education and Human Development Section 3: Centering

2 Covered this Section We will expand on this example to cover a few more important concepts in multilevel linear models The importance of centering of variables Provides a meaningful value for each parameter (fixed effect) in the model Interpreting conditional main effects (as they are conditional interaction variables being zero) Distinguishing within person/nesting unit from between person/nesting unit effects How total variation is partitioned by random effects Implications for how residuals are correlated Implications for hypothesis testing (type 1 and 2 errors) Implications for modeling dependencies Section 3: Centering 2

3 Data for this Section To help demonstrate the concepts of this section, we will be using a data set with three variables Female (Gender): Male (=0) or Female (=1) Height in inches Weight in pounds The end point of our lecture will be to build a linear model that predicts a person s weight Linear model: a statistical model for an outcome that uses a linear combination (a weighted sum; weighted by a slope) of one or more predictor variables In this section, we will revisit the centered/uncentered models to show how centering affects the parameter estimates and the standard errors (and likewise, the p values of the effects) We will lead off with our last model from the last section: Predicting weight from height, gender, and the interaction of height and gender Section 3: Centering 3

4 Model 5: Predicting Weight from Height and Gender (with Interaction); ( ANCOVAish ) Linear Model Using UNCENTERED Height: where 0, Parameter estimates: Intercept: Interpretation? The answer is under this box The intercept is the predicted (expected) weight when 0and 0 The weight of a male 0 who is zero inches 0 tall is Do we have any zero inch tall males in the workshop? Interaction term: Interpretation? The The answer interaction is under term this can box be looked at either by the height variable or by the gender variable as gender is categorical, we will focus on it alone For gender: when 1, then is combined with to make the slope for height In other words, the slope for females 1 is pounds per inch less than for males Section 3: Centering 4

5 Where Centering Matters: Conditional Main Effects Linear Model Using UNCENTERED Height: where 0, Main effect of height: Interpretation? The This answer is the is conditional under this main box effect of height (conditional on 0) For males 0, the weight of a person increases pounds per inch Main effect of gender: Interpretation? The This answer is the conditional is under this main box effect of gender, conditional on 0 This indicates that women 1 weigh pounds less than men when both are zero inches tall What would the gender difference be at 60 inches tall? These interpretations are all contingent on zero being a meaningful number! Section 3: Centering 5

6 Comparing Centered and Uncentered Results To show how centering affects the parameters of a model, here are the results for each relevant term, side by side: Parameter Centered Estimate Centered S.E. Uncentered Estimate Uncentered S.E. Intercept Height Main Effect Gender Main Effect Height*Gender Residual Variance Log Likelihood (Residual) The parameters that change are everything below the intercept But their interpretation does not change at all The log likelihoods are identical (we will come to know this as indicating the models are identical) Section 3: Centering 6

7 Centering Because our intercept and main effects are implausible, we centered our data so as to bring the intercept more into line with the data we collected To center the data, subtract a value from each of the predictor/independent variables Centering will alter the meaning of certain parameters The intercept Some slopes (depending on method of centering) Two methods of centering are popular: Grand mean centering/centering by a constant Cluster mean centering in longitudinal data this is called person mean centering We chose to center by using the grand mean of height a value of 67.9 inches Section 3: Centering 7

8 Cluster/Person Mean Centering Another popular method for centering is that of cluster mean centering Taking each person s independent variable(s) and subtracting the mean(s) from their cluster/sampling unit In our example we really do not have any clustered data gender is an IV The issue with clustered data is that variables thought to be collected at level 1 contain level 2 information In longitudinal: time varying covariates Therefore, we must put the level 2 information into the statistical model This is typically accomplished in MLM by adding the mean of the cluster In longitudinal: adding the person mean across all time points Then an issue becomes: what do we do with the level 2 effect? Would be zero if we cluster mean centered it We can leave it alone (what would happen to the intercept?) We can grand mean center it Section 3: Centering 8

9 Wrapping Up Centering Summary The scale of variables may lead to parameter values that are not plausible Sometimes interpretation changes (grand mean centering) Sometimes inference changes (cluster mean centering) Detailed shortly Centering helps to: Make parameter estimates understandable Help estimation of random effects in some types of models Disentangle types of effects (for cluster mean centering) Section 3: Centering 9