1. Quote of the Day
"Not everything that counts can be counted and not everything that can be counted counts".
Albert Einstein

2. Using Statistics to Evaluate Cause -The Case for Path Analysis-
Professor J. Schutte
Psychology 524
April 11, 2017

3. Two Paradigms for Investigating Cause
Internal Validation – Experimental Control
Specifying the Conditions (Experimental & Control Groups)
Testing Differences (Simple and Complex ANOVA)
External Validation – Statistical Control
Specifying the People/Variables (Sampling Frames)
Testing Relationships (Partial and Multiple Correlation)

4. What is Cause in Non-Experimental Settings?
Cause is a Philosophical not a Statistical Concept
In Non-Experimental Settings, it’s Based on:
Covariation
Over a valid Time frame
Of a non-spurious nature
Related through theory or logic

5. Topics in Using Correlation for Causal Analysis:
I. Statistical Covariation – Pearson’s r
II. Third Variable Effects – Partial Correlation
III. The Logic of Multivariate Relationships
IV. Multiple Correlation and Regression
V. Path Analysis – The Essentials
VI. Using AMOS to Automatic Path Analysis

6. I. Statistical Covariation
Pearson’s r - The Bivariate Correlation Coefficient

7. The Graphical View– A Scatter Diagram
Y (WT) X’
A HIGH POSITIVE CORRELATION
. .
. . . Y’
X (HT)

8. The Graphical View– A Scatter Diagram
Y (Prejudice) X’ Y’
A HIGH NEGATIVE CORRLEATION
X (Education)

9. The Graphical View– A Scatter Diagram
Y (Births in Brazil) X’
NO CORRELATION Y’
X (Rainfall in NYC)

10. 1) 2) 3) 4) The Algebraic View – Shared Variance
Take the Variance in X = S2x
and the Variance in Y = S2y 2) 3) 4)
The correlation is simply 3) divided by 2):

11. 1) Find the raw scores, means, squared deviations and cross-products:
Manually Calculating a Correlation
1) Find the raw scores, means, squared deviations and cross-products:
3) Square r to determine variation explained r2 = .75

12. An Example of Calculating a Correlation from SPSS
INPUT

13. An Example of Calculating a Correlation from SPSS
OUTPUT

14. II. Partial Relationships
Looking for the effects of a third variable

15. The Partial Correlation Coefficient
Step 1 – Determine the zero order correlations (r).
Assume our previous correlation (rxy=.86) between Mother’s Education (x) and Daughter’s Education (y). Now assume you now want to control for the effects of a third variable, Father’s Education (Z), as it relates to the original Mother-Daughter correlation (rxy). You would first calculate the zero order correlation Mother’s and Father’s education (rxz), finding it to be .88; then calculate the same for Daughter and Father (ryz), finding it to be How much does the Father’s Education account for the original Mother-Daughter correlation.
.86 – (.88) (.88)
.091
Step 2 – Calculate the partial correlation (rxy.z)
___
.227 = = =
.40
Step 3 – Draw conclusions
Before z (rxy)2 = .75
Therefore, Z accounts for (.59/.75) or 79% of the covaration between X&Y
After z (rxy.z)2 =

16. Using SPSS for finding Partial Correlation Coefficients
INPUT

17. Using SPSS for finding Partial Correlation Coefficients
OUTPUT

18. III. The Logic of Multivariate Partial Relationships
Multiple Correlation and Multiple Regression

19. Causal Systems I. The Logic of Multiple Relationships X1 Y X2 NR R NR
One Dependent Variable, Multiple Independent Variables NR X1 Y R NR X2
In 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.

20. Causal Systems R2y.x1x2 = r2yx1 + r2yx2 R2y.x1x2 = r2yx1 + r2yx2.x1
II. Multiple Correlation and Coefficient of Determination
r2yx1
r2yx2 X1 Y X1 NR Y NR X2 X2
R2y.x1x2 = r2yx1 + r2yx2
R2y.x1x2 = r2yx1 + r2yx2.x1
Notice that when the Independent Variables are independent of each other, the multiple correlation coefficient, here squared and called the coefficient of determination (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.

21. Causal Systems Y’ = a + byx1X1 + byx2X2 or Standardized
II. Multiple Regression
Y’ = a + byx1X1 + byx2X2 X1 Y X2
or Standardized X1
Y’ = Byx1X1 + Byx2X2 Y X2
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.

22. Causal Systems Y’ = a + byx1X1 + byx2.x1X2 or Standardized
Multiple Regression X1 Y
Y’ = a + byx1X1 + byx2.x1X2
or Standardized X2
Y’ = Byx1X1 + Byx2.x1X2 X1 Y X2
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”

23. IV. Path Analysis
The Essentials

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

25. Path Analysis – Steps and Example
Step1 – Input the data
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)
Y = DV - income
X3 = IV - educ
X2 = IV - pedu
X1 = IV - pinc

26. Path Analysis – Steps and Example
Step 2 – Calculate the Correlation Matrix
These correlations are calculated in the usual manner through the “analyze”, “correlate”, bivariate menu clicks. X1 X2 X3 Y
Notice the zero order correlations of each IV with the DV. Clearly these IV’s 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.

27. Path Analysis – Steps and Example
Step 3 – Specify the Path Diagram
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. X1 a e Y f X3 b d
Y = Offspring’s income c
X1 = Parent’s income X2
X2 = Parent’s education
X3 = Offspring’s education
Time

28. Path Analysis – Steps and Example
Step 4 – Enumerate the Path Equations
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. X1 a e X3 Y f b d
1. ryx1 = a + brx3x1 + crx2x1 c
2. ryx2 = c + brx3x2 + arx1x2 X2
3. ryx3 = b + arx1x3 + crx2x3
4. rx3x2 = d + erx1x2
5. rx3x1 = e + drx1x2
6. rx1x2 = f

29. Path Analysis – Steps and Example
Step 5 – Solve for the Path Coefficients
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.

30. Path Analysis – Steps and Example
Step 5a – Solving for R2
The SPSS Regression module also calculate R2. 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)
R2 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).

31. Path Analysis – Steps and Example
Checking the Findings
ryx1 = a + brx3x1 + crx2x1
.69 = (.68) -.21(.75)
e = .50
ryx2 = c + brx3x2 + arx1x2 X1
r = .69
B = .63
.57 = (.68) - .21(.82)
r = .75
B = .36
ryx3 = b + arx1x3 + crx2x3
.52 = (.75) + .31(.82)
r = .52
B = -.21 Y X3
r = B =.68
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 R2 . The value of 1-R2 is e.
r = .82
B = .57
r = .57
B =.31 X2
Time

32. Path Analysis – Steps and Example
Step 6 – Interpret the Findings
Y = Offspring’s income
X3 = Offspring’s education
X2 = Parent’s education X1
e = .50
X1 = Parent’s income
.63
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!
.36 Y
.68 X3
-.21
.57
.31 X2
Time

33. Automating Path Analysis
V. Using AMOS
Automating Path Analysis

34. First we input the data into SPSS in the usual way

35. And save it in the usual way (an spss.sav file)

36. We then open the AMOS Program having saved the Data Set

37. Input and Label each Variable in the Model

38. Place all of the variables in time sequence order

39. Next specify the causal and non-causal connections

40. Then indicate the error terms of the endogenous variables

41. Identify the dataset from File Datafile menu

42. If a new diagram, save as an .amw file first before calculating

43. Specify the output parameters from the View Analysis menu

44. Finally, click on the Output Button on the right of the upper box
Numbers on the arrow lines are path coefficients

45. Assumptions Linearity Homoscedasticity Uncorrelated error terms
Residuals normally distributed

46. Example #1 – Age at Marriage

47. Example #2 – College GPA