Statistics for the Social Sciences Psychology 340 Fall 2006 Prediction cont.

Outline (for week) Simple bi-variate regression, least-squares fit line The general linear model Residual plots Using SPSS Multiple regression Comparing models, (?? Delta r2)

From last time Review of last time Y = intercept + slope(X) + error Y 1 2 3 4 5 6 Y = intercept + slope(X) + error

From last time Y X The sum of the residuals should always equal 0. 6 1 2 3 4 5 6 The sum of the residuals should always equal 0. The least squares regression line splits the data in half Additionally, the residuals to be randomly distributed. There should be no pattern to the residuals. If there is a pattern, it may suggest that there is more than a simple linear relationship between the two variables.

Seeing patterns in the error Residual plots Useful tools to examine the relationship even further. These are basically scatterplots of the Residuals (often transformed into z-scores) against the Explanatory (X) variable (or sometimes against the Response variable)

Seeing patterns in the error Scatter plot Residual plot The scatter plot shows a nice linear relationship. The residual plot shows that the residuals fall randomly above and below the line. Critically there doesn't seem to be a discernable pattern to the residuals.

Seeing patterns in the error Scatter plot Residual plot The residual plot shows that the residuals get larger as X increases. This suggests that the variability around the line is not constant across values of X. This is referred to as a violation of homogeniety of variance. The scatter plot also shows a nice linear relationship.

Seeing patterns in the error Scatter plot Residual plot The scatter plot shows what may be a linear relationship. The residual plot suggests that a non-linear relationship may be more appropriate (see how a curved pattern appears in the residual plot).

Regression in SPSS Using SPSS Variables (explanatory and response) are entered into columns Each row is an unit of analysis (e.g., a person)

Regression in SPSS Analyze: Regression, Linear

Regression in SPSS Enter: Predicted (criterion) variable into Dependent Variable field Predictor variable into the Independent Variable field

Regression in SPSS The variables in the model r r2 We’ll get back to these numbers in a few weeks Unstandardized coefficients Slope (indep var name) Intercept (constant)

Regression in SPSS Recall that r = standardized  in bi-variate regression Standardized coefficient  (indep var name)

Multiple Regression Typically researchers are interested in predicting with more than one explanatory variable In multiple regression, an additional predictor variable (or set of variables) is used to predict the residuals left over from the first predictor.

Multiple Regression Bi-variate regression prediction models Y = intercept + slope (X) + error

Multiple Regression “residual” “fit” Bi-variate regression prediction models Y = intercept + slope (X) + error Multiple regression prediction models “fit” “residual”

Multiple Regression Multiple regression prediction models whatever variability is left over First Explanatory Variable Second Explanatory Variable Third Explanatory Variable Fourth Explanatory Variable

Multiple Regression Predict test performance based on: Study time Test time What you eat for breakfast Hours of sleep whatever variability is left over First Explanatory Variable Second Explanatory Variable Third Explanatory Variable Fourth Explanatory Variable

Multiple Regression Predict test performance based on: Study time Test time What you eat for breakfast Hours of sleep Typically your analysis consists of testing multiple regression models to see which “fits” best (comparing r2s of the models) For example: versus versus

Total variability it test performance Multiple Regression Model #1: Some co-variance between the two variables If we know the total study time, we can predict 36% of the variance in test performance R2 for Model = .36 Response variable Total variability it test performance Total study time r = .6 64% variance unexplained

Total variability it test performance Multiple Regression Model #2: Add test time to the model Little co-variance between these test performance and test time We can explain more the of variance in test performance R2 for Model = .49 Response variable Total variability it test performance Total study time r = .6 51% variance unexplained Test time r = .1

Total variability it test performance Multiple Regression Model #3: No co-variance between these test performance and breakfast food Not related, so we can NOT explain more the of variance in test performance R2 for Model = .49 Response variable Total variability it test performance breakfast r = .0 Total study time r = .6 51% variance unexplained Test time r = .1

Total variability it test performance Multiple Regression Model #4: Some co-variance between these test performance and hours of sleep We can explain more the of variance But notice what happens with the overlap (covariation between explanatory variables), can’t just add r’s or r2’s R2 for Model = .60 Response variable Total variability it test performance breakfast r = .0 Total study time r = .6 40% variance unexplained Hrs of sleep r = .45 Test time r = .1

Multiple Regression in SPSS Setup as before: Variables (explanatory and response) are entered into columns A couple of different ways to use SPSS to compare different models

Regression in SPSS Analyze: Regression, Linear

Multiple Regression in SPSS Method 1:enter all the explanatory variables together Enter: Predicted (criterion) variable into Dependent Variable field All of the predictor variables into the Independent Variable field

Multiple Regression in SPSS The variables in the model r for the entire model r2 for the entire model Unstandardized coefficients Coefficient for var1 (var name) Coefficient for var2 (var name)

Multiple Regression in SPSS The variables in the model r for the entire model r2 for the entire model Standardized coefficients Coefficient for var1 (var name) Coefficient for var2 (var name)

Multiple Regression Which  to use, standardized or unstandardized? Unstandardized ’s are easier to use if you want to predict a raw score based on raw scores (no z-scores needed). Standardized ’s are nice to directly compare which variable is most “important” in the equation

Multiple Regression in SPSS Method 2: enter first model, then add another variable for second model, etc. Enter: Click the Next button First Predictor variable into the Independent Variable field Predicted (criterion) variable into Dependent Variable field

Multiple Regression in SPSS Method 2 cont: Enter: Second Predictor variable into the Independent Variable field Click Statistics

Multiple Regression in SPSS Click the ‘R squared change’ box

Multiple Regression in SPSS Shows the results of two models The variables in the first model (math SAT) The variables in the second model (math and verbal SAT)

Multiple Regression in SPSS Shows the results of two models The variables in the first model (math SAT) The variables in the second model (math and verbal SAT) r2 for the first model Model 1 Coefficients for var1 (var name)

Multiple Regression in SPSS Shows the results of two models The variables in the first model (math SAT) The variables in the second model (math and verbal SAT) r2 for the second model Model 2 Coefficients for var1 (var name) Coefficients for var2 (var name)

Multiple Regression in SPSS Shows the results of two models The variables in the first model (math SAT) The variables in the second model (math and verbal SAT) Change statistics: is the change in r2 from Model 1 to Model 2 statistically significant?

Cautions in Multiple Regression We can use as many predictors as we wish but we should be careful not to use more predictors than is warranted. Simpler models are more likely to generalize to other samples. If you use as many predictors as you have participants in your study, you can predict 100% of the variance. Although this may seem like a good thing, it is unlikely that your results would generalize to any other sample and thus they are not valid. You probably should have at least 10 participants per predictor variable (and probably should aim for about 30).