Statistics for the Social Sciences Psychology 340 Spring 2005 Prediction cont.

Statistics for the Social Sciences Outline (for week) Simple bi-variate regression, least-squares fit line –The general linear model –Residual plots –Using SPSS Multiple regression –Comparing models, (?? Delta r 2 ) –Using SPSS

Statistics for the Social Sciences From last time Review of last time Y = intercept + slope(X) + error Y X

Statistics for the Social Sciences From last time Y X 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.

Statistics for the Social Sciences Seeing patterns in the error –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) Residual plots

Statistics for the Social Sciences Seeing patterns in the error 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. Residual plot Scatter plot The scatter plot shows a nice linear relationship.

Statistics for the Social Sciences Seeing patterns in the error Residual plot The scatter plot also shows a nice linear relationship. 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. Scatter plot

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

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

Statistics for the Social Sciences Regression in SPSS Analyze: Regression, Linear

Statistics for the Social Sciences Regression in SPSS –Predicted (criterion) variable into Dependent Variable field –Predictor variable into the Independent Variable field Enter :

Statistics for the Social Sciences Regression in SPSS The variables in the model r r 2 We’ll get back to these numbers in a few weeks Slope (indep var name) Intercept (constant) Unstandardized coefficients

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

Statistics for the Social Sciences 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.

Statistics for the Social Sciences Multiple Regression Y = intercept + slope (X) + error Bi-variate regression prediction models

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

Statistics for the Social Sciences Multiple Regression Multiple regression prediction models First Explanatory Variable Second Explanatory Variable Fourth Explanatory Variable whatever variability is left over Third Explanatory Variable

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

Statistics for the Social Sciences 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 r 2 s of the models) versus For example:

Statistics for the Social Sciences Multiple Regression Response variable Total variability it test performance Total study time r =.6 Model #1: Some co-variance between the two variables R 2 for Model =.36 64% variance unexplained If we know the total study time, we can predict 36% of the variance in test performance

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

Statistics for the Social Sciences Multiple Regression Response variable Total variability it test performance breakfast r =.0 Model #3: No co-variance between these test performance and breakfast food Total study time r =.6 Test time r =.1 R 2 for Model =.49 51% variance unexplained Not related, so we can NOT explain more the of variance in test performance

Statistics for the Social Sciences Multiple Regression Response variable Total variability it test performance breakfast r =.0 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 r 2 ’s Total study time r =.6 Test time r =.1 Hrs of sleep r =.45 R 2 for Model =.60 40% variance unexplained Model #4: Some co-variance between these test performance and hours of sleep

Statistics for the Social Sciences 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

Statistics for the Social Sciences Regression in SPSS Analyze: Regression, Linear

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

Statistics for the Social Sciences Multiple Regression in SPSS The variables in the model r for the entire model r 2 for the entire model Unstandardized coefficients Coefficient for var1 (var name) Coefficient for var2 (var name)

Statistics for the Social Sciences Multiple Regression in SPSS The variables in the model r for the entire model r 2 for the entire model Standardized coefficients Coefficient for var1 (var name)Coefficient for var2 (var name)

Statistics for the Social Sciences 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

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

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

Statistics for the Social Sciences Multiple Regression in SPSS –Click the ‘R squared change’ box

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

Statistics for the Social Sciences Multiple Regression in SPSS The variables in the first model (math SAT) r 2 for the first model Coefficients for var1 (var name) Shows the results of two models The variables in the second model (math and verbal SAT) Model 1

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

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

Statistics for the Social Sciences 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).