Week 14 Chapter 16 – Partial Correlation and Multiple Regression and Correlation

Chapter 16 Partial Correlation and Multiple Regression and Correlation

In This Presentation Partial correlations Multiple regression Using the multiple regression line to predict Y Multiple correlation coefficient (R 2 ) Limitations of multiple regression and correlation

Introduction Multiple Regression and Correlation allow us to: 1. Disentangle and examine the separate effects of the independent variables. 2. Use all of the independent variables to predict Y. 3. Assess the combined effects of the independent variables on Y.

Partial Correlation Partial Correlation measures the correlation between X and Y controlling for Z Comparing the bivariate (“zero-order”) correlation to the partial (“first-order”) correlation allows us to determine if the relationship between X and Y is direct, spurious, or intervening Interaction cannot be determined with partial correlations

Partial Correlation Note the subscripts in the symbol for a partial correlation coefficient: r xy●z which indicates that the correlation coefficient is for X and Y controlling for Z

Partial Correlation Example The table below lists husbands’ hours of housework per week (Y), number of children (X), and husbands’ years of education (Z) for a sample of 12 dual-career households

Partial Correlation Example A correlation matrix appears below The bivariate (zero-order) correlation between husbands’ housework and number of children is This indicates a positive relationship

Partial Correlation Example Calculating the partial (first-order) correlation between husbands’ housework and number of children controlling for husbands’ years of education yields +0.43

Partial Correlation Example Comparing the bivariate correlation (+0.50) to the partial correlation (+0.43) finds little change The relationship between number of children and husbands’ housework controlling for husbands’ education has not changed Therefore, we have evidence of a direct relationship

Multiple Regression Previously, the bivariate regression equation was: In the multivariate case, the regression equation becomes:

Multiple Regression Y = a + b 1 X 1 + b 2 X 2 Notation a is the Y intercept, where the regression line crosses the Y axis b 1 is the partial slope for X1 on Y b 1 indicates the change in Y for one unit change in X 1, controlling for X 2 b 2 is the partial slope for X 2 on Y b 2 indicates the change in Y for one unit change in X 2, controlling for X 1

Multiple Regression using SPSS Suppose we are interested in the link between Daily Calorie Intake and Female Life Expectancy in a third world country Suppose further that we wish to look at other variables that might predict Female life expectancy One way to do this is to add additional variables to the equation and conduct a multiple regression analysis. E.g. literacy rates with the assumption that those who read can access health and medical information

Multiple Regression using SPSS: Steps to Set Up the Analysis In Data Editor go to Analyze/ Regression/ Linear and click Reset Put Average Female Life Expectancy into the Dependent box Put Daily Calorie Intake and People who Read % into the Independents box Under Statistics, select Estimates, Confidence Intervals, Model Fit, Descriptives, Part and Partial Correlation, R Square Change, Collinearity Diagnostics, and click Continue Under Options, check Include Constant in the Equation, click Continue and then OK Compare your output to the next several slides

Interpreting Your SPSS Multiple Regression Output First let’s look at the zero-order (pairwise) correlations between Average Female Life Expectancy (Y), Daily Calorie Intake (X1) and People who Read (X2). Note that these are.776 for Y with X 1,.869 for Y with X 2, and.682 for X 1 with X 2 r YX 1 r YX 2 r X 1 X 2

Examining the Regression Weights Above are the raw (unstandardized) and standardized regression weights for the regression of female life expectancy on daily calorie intake and percentage of people who read. The standardized regression coefficient (beta weight) for daily caloric intake is.342. The beta weight for percentage of people who read is much larger,.636. What this weight means is that for every unit change in percentage of people who read (that is, for every increase by a factor of one standard deviation on the people who read variable), Y (female life expectancy) will increase by a multiple of.636 standard deviations. Note that both the beta coefficients are significant at p <.001

R, R Square, and the SEE Above is the model summary, which has some important statistics. It gives us R and R square for the regression of Y (female life expectancy) on the two predictors. R is.905, which is a very high correlation. R square tells us what proportion of the variation in female life expectancy is explained by the two predictors, a very high.818. It gives us the standard error of estimate, which we can use to put confidence intervals around the unstandardized regression coefficients

F Test for the Significance of the Regression Equation Next we look at the F test of the significance of the Regression equation, Y =.342 X X 2. Is this so much better a predictor of female literacy (Y) than simply using the mean of Y that the difference is statistically significant? The F test is a ratio of the mean square for the regression equation to the mean square for the “residual” (the departures of the actual scores on Y from what the regression equation predicted). In this case we have a very large value of F, which is significant at p <.001. Thus it is reasonable to conclude that our regression equation is a significantly better predictor than the mean of Y.

Confidence Intervals around the Regression Weights Finally, your output provides confidence intervals around the unstandardized regression coefficients. Thus we can say with 95% confidence that the unstandardized weight to apply to daily calorie intake to predict female life expectancy ranges between.004 and.010, and that the undstandardized weight to apply to percentage of people who read ranges between.247 and.383

Limitations Multiple regression and correlation are among the most powerful techniques available to researchers. But powerful techniques have high demands. These techniques require: Every variable is measured at the interval-ratio level Each independent variable has a linear relationship with the dependent variable Independent variables do not interact with each other Independent variables are uncorrelated with each other When these requirements are violated (as they often are), these techniques will produce biased and/or inefficient estimates. There are more advanced techniques available to researchers that can correct for violations of these requirements. Such techniques are beyond the scope of this text.