Cause (Part II) - Causal Systems I. The Logic of Multiple Relationships II. Multiple Correlation Topics: III. Multiple Regression IV. Path Analysis

Cause (Part II) - Causal Systems Y X2 X1 One Dependent Variable, Multiple Independent Variables In this diagram the overlap of any two circles can be thought of as the r 2 between the two variables. When we add a third variable, however, we must ‘partial out’ the redundant overlap of the additional independent variables. R NR I. The Logic of Multiple Relationships

Cause (Part II) - Causal Systems II. Multiple Correlation Y X2 X1 R NR R 2 y.x 1 x 2 = r 2 yx 1 + r 2 yx 2 YX2X1 NR R 2 y.x 1 x 2 = r 2 yx 1 + r 2 yx 2.x 1 Notice that when the Independent Variables are independent of each other, the multiple correlation coefficient (R 2 ) is simply the sum of the individual r 2, but if the independent variables are related, R 2 is the sum of one zero order r 2 of one plus the partial r 2 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 Systems II. Multiple Regression YX2X1 X2 Y Y’ = a + b yx 1 X 1 + b yx 2 X 2 Y’ = B yx 1 X 1 + B yx 2 X 2 or Standardized If 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 Regression Y X2 X1 X2 Y Y’ = a + b yx 1 X 1 + b yx 2.x 1 X 2 or Standardized Y’ = B yx 1 X 1 + B yx 2.x 1 X 2 Once 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 Equations 1. Input the data 5. Solve for the Path Coefficients (Betas) 6. Interpret the Findings Cause (Part II) - Causal Systems

Path Analysis – Steps and Example Step1 – Input the data Y = DV - income X3 = IV - educ X2 = IV - pedu X1 = IV - pinc Assume you have information from ten respondents as to their income, education, parent’s education and parent’s 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 respondent’s income (Y), using the three other independent variables (X1, X2, X3)

Step 2 – Calculate the Correlation Matrix X1 X2 X3 Y Path Analysis – Steps and Example These 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 IV’s must interrelate as the values of the r 2 would sum to an R 2 indicating more than 100% of the variance in the DV which, of course, is impossible.

Step 3 – Specify the Path Diagram Y X3 X1 X2 b c X3 = Offspring’s education X2 = Parent’s education X1 = Parent’s income Y = Offspring’s income Time a d e f Path Analysis – Steps and Example Therefore, 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. r yx1 = a + br x3x1 + cr x2x1 2. r yx2 = c + br x3x2 + ar x1x2 3. r yx3 = b + ar x1x3 + cr x2x3 4. r x3x2 = d + er x1x2 6. r x1x2 = f 5. r x3x1 = e + dr x1x2 b c a d e f X3 X1 X2 Y Path Analysis – Steps and Example Click here for solution to two equations in two unknownshere With 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 – a, b and c Path Analysis – Steps and Example The 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.

Step 5 – Solve for the Path Coefficients – d and e Path Analysis – Steps and Example The easiest way to calculate B is to use the Regression module in SPSS. By indicating offspring education as the dependent variable and Parents Inc and Parents Edu as the independent variables, we can solve for the Beta Weights or Path Coefficients for each of these Independent Variables on the DV Offspring Edu. These circled numbers correspond to Beta for paths d and e, respectively, in the previous path diagram.

The SPSS Regression module also calculate R 2. According to this statistic, for our data, 50% of the variation in the respondent’s income (Y) is accounted for by the respondent’s education (X3), parent’s education (X2) and parent’s income (X1) Path Analysis – Steps and Example Step 5a – Solving for R 2 R 2 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 Y X3 X1 X2 r =.57 B = = (.82) +.63(.68).52 = (.75) +.31(.82).69 = (.75) +.31(.68) Time r =.69 B =.63 r =.82 B =.58 r = B =.68 e =.50 r =.52 B = -.21 r =.75 B =.35 The 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 R 2. The value of 1-R 2 is e. Path Analysis – Steps and Example r yx1 = a + br x3x1 + cr x2x1 r yx2 = c + br x3x2 + ar x1x2 r yx3 = b + ar x1x3 + cr x2x3

Step 6 – Interpret the Findings Y X3 X1 X X3 = Offspring’s education X2 = Parent’s education X1 = Parent’s income Y = Offspring’s income Time e =.50 Specifying the Path Coefficients (Betas), several facts are apparent, among which are that Parent’s 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 offspring’s education, in these data, is actually negative! Path Analysis – Steps and Example End

Exercise - Solving Two Equations in Two Unknowns Back