PowerPoint Presentation

Cause (Part II) - Causal SystemsI. The Logic of Multiple RelationshipsII. Multiple CorrelationTopics:III. Multiple RegressionIV. Path Analysis

1Cause (Part II) - Causal SystemsYX2X1One Dependent Variable, Multiple Independent VariablesIn this diagram the overlap of any two circles can be thought of as the r2 between the two variables. When we add a third variable, however, we must partial out the redundant overlap of the additional independent variables.RNRNRI. The Logic of Multiple Relationships

2Cause (Part II) - Causal SystemsII. Multiple CorrelationYX2X1RNRNRR2y.x1x2 = r2yx1 + r2yx2YX2X1NRNRR2y.x1x2 = r2yx1 + r2yx2.x1Notice that when the Independent Variables are independent of each other, the multiple correlation coefficient (R2) is simply the sum of the individual r2, but if the independent variables are related, R2 is the sum of one zero order r2 of one plus the partial r2 of the other(s). This is required to compensate for the fact that multiple independent variables being related to each other would be otherwise double counted in explaining the same portion of the dependent variable. Partially out this redundancy solves this problem.

Cause (Part II) - Causal SystemsII. Multiple RegressionYX2X1X1X2YY = a + byx1X1 + byx2X2Y = Byx1X1 + Byx2X2or StandardizedIf we were to translate this into the language of regression, multiple independent variables, that are themselves independent of each other would have their own regression slopes and would simply appear as an another term added in the regression equation.

Cause (Part II) - Causal Systems Multiple RegressionYX2X1X1X2YY = a + byx1X1 + byx2.x1X2or StandardizedY = Byx1X1 + Byx2.x1X2Once we assume the Independent Variables are themselves related with respect to the variance explained in the Dependent Variable, then we must distinguish between direct and indirect predictive effects. We do this using partial regression coefficients to find these direct effects. When standardized these B-values are called Path coefficients or Beta Weights

III. Path Analysis The Steps and an Example 2. Calculate the Correlation Matrix 3. Specify the Path Diagram 4. Enumerate the Equations1. Input the data 5. Solve for the Path Coefficients (Betas) 6. Interpret the FindingsCause (Part II) - Causal Systems

Path Analysis Steps and ExampleStep1 Input the data

Y = DV - incomeX3 = IV - educX2 = IV - peduX1 = IV - pincAssume you have information from ten respondents as to their income, education, parents education and parents income. We would input these ten cases and four variables into SPSS in the usual way, as here on the right. In this analysis we will be trying to explain respondents income (Y), using the three other independent variables (X1, X2, X3)

Step 2 Calculate the Correlation Matrix

X1 X2 X3 YPath Analysis Steps and ExampleThese correlations are calculated in the usual manner through the analyze, correlate, bivariate menu clicks. Notice the zero order correlations of each IV with the DV. Clearly these IVs must interrelate as the values of the r2 would sum to an R2 indicating more than 100% of the variance in the DV which, of course, is impossible.

Step 3 Specify the Path Diagram YX3X1X2bcX3 = Offsprings educationX2 = Parents educationX1 = Parents incomeY = Offsprings incomeTime adefPath Analysis Steps and ExampleTherefore, we must specify a model that explains the relationship among the variables across time We start with the dependent variable on the right most side of the diagram and form the independent variable relationship to the left, indicating their effect on subsequent variables.

Step 4 Enumerate the Path Equations 1. ryx1 = a + brx3x1 + crx2x1 2. ryx2 = c + brx3x2 + arx1x2 3. ryx3 = b + arx1x3 + crx2x3 4. rx3x2 = d + erx1x2 6. rx1x2 = f 5. rx3x1 = e + drx1x2 bcadefX3X1X2YPath Analysis Steps and ExampleClick here for solution to two equations in two unknownsWith the diagram specified, we need to articulate the formulae necessary to find the path coefficients (arbitrarily indicated here by letters on each path). Overall correlations between an independent and the dependent variable can be separated into its direct effect plus the sum of its indirect effects.

Step 5 Solve for the Path Coefficients

Path Analysis Steps and ExampleThe easiest way to calculate B is to use the Regression module in SPSS. By indicating income as the dependent variable and pinc, pedu and educ as the independent variables, we can solve for the Beta Weights or Path Coefficients for each of the Independent Variables. These circled numbers correspond to Beta for paths a, c and b, respectively, in the previous path diagram.

The SPSS Regression module also calculate R2. According to this statistic, for our data, 50% of the variation in the respondents income (Y) is accounted for by the respondents education (X3), parents education (X2) and parents income (X1)Path Analysis Steps and ExampleStep 5a Solving for R2R2 is calculated by multiplying the Path Coefficient (Beta) by its respective zero order correlation and summed across all of the independent variables (see spreadsheet at right).

Checking the Findings YX3X1X2r = .57B =.31 .57 = .31 + -.21(.82) + .63(.68) .52 = -.21 + .63(.75) + .31(.82) .69 = .63 + -.21(.75) +.31(.68)Time r = .69 B = .63r = .82B = .57r = B =.68e = .50r = .52B = -.21r = .75B = .36The values of r and B tells us three things: 1) the value of Beta is the direct effect; 2) dividing Beta by r gives the proportion of direct effect; and 3) the product of Beta and r summed across each of the variables with direct arrows into the dependent variable is R2 . The value of 1-R2 is e.Path Analysis Steps and Exampleryx1 = a + brx3x1 + crx2x1 ryx2 = c + brx3x2 + arx1x2ryx3 = b + arx1x3 + crx2x3

Step 6 Interpret the Findings YX3X1X2.31-.21X3 = Offsprings educationX2 = Parents educationX1 = Parents incomeY = Offsprings incomeTime .63.36.57.68e = .50Specifying the Path Coefficients (Betas), several facts are apparent, among which are that Parents income has the highest percentage of direct effect (i.e., .63/.69 = 92% of its correlation is a direct effect, 8% is an indirect effect). Moreover, although the overall correlation of educ with income is positive, the direct effect of offsprings education, in these data, is actually negative!Path Analysis Steps and ExampleEnd

Exercise - Solving Two Equations in Two UnknownsBack