Updated 10-14-02

WEB REFERENCES These are web-sites that I've found that have have material related to Path Analysis

• Path Analysis class notes at University of Exeter, Dept of Psychology
• Path Analysis class notes at North Carolina State University, Dept of Public Administration.
• Path Analysis class notes at University of South Florida, Dept of Psychology.

PATH ANALYSIS

Forces researcher to articulate the theoretical models that underlie their design, variables and thinking.

Takes correlations and breaks them apart into causal and non-causal components.

Allows researcher to compute the magnitudes of causal relationships from correlation measurements. PROVIDED THE PATH DIAGRAM CORRECTLY REPRESENTS THE CAUSAL PROCESS UNDERLYING THE DATA.

So we always say, ``Assuming that the path model we have written down represents the truth, the magnitude of the relationships is X.''

To infer that X is a cause of Y - from Moore and McCabe (section on ``The Question of Causation'')

• The association is consistent across studies
• The alleged cause precedes the effect in time
• The alleged cause is plausible
• The direction of relation is correctly specified (reciprocal causation?)
• The relationship does not disappear when external variables are held constant

Partial Correlation

Correlation between two variables after adjusting for another variable (or set of variables)

$$r_{xy \cdot w} = \frac{r_{xy} - r_{xw} r_{yw}}{\sqrt{(1-r_{xw}^2)(1-r_{yw}^2)}}$$

Example (from Kline p. 29)

	X	Y	W
X shoe-size	1
Y vocabulary breadth	.5	1
W age	.8	.6	1

$$r_{xy \cdot w} = \frac{.5 - .8*.6}{\sqrt{(1-.8^2)(1-.6^2)}} = \frac{.02}{.48} = .04$$

There is no association between shoe-size and vocabulary breadth when we adjust for age.

This can be represented by

PUT PICTURE HERE

Another way to calculate $r_{xy \cdot w}$ is

1.

obtain the residuals from the regression of X on W, call them e_xw

2.

obtain the residuals from the regression of Y on W, call them e_yw

3.

Calculate the correlation between e_xw and e_yw, this correlation is $r_{xy \cdot w}$

Multiple Regression- Standardized regression

• Regression for Prediction vs.
• Regression for Explanation

Given variables Y, X₁, and X₂, we can standardize each of these variables and get $Y^s = \frac{Y - mean{Y}}{stddev(Y)}$, $X_1^s = \frac{X_1 - mean{X_1}}{stddev(X_1)}$, $X_2^s = \frac{X_2 - mean{X_2}}{stddev(X_2)}$.

Then the standardized multiple regression is

$$Y^s = \beta_1 X^s_1 + \beta_2 X^s_2 + \epsilon$$

NOTE: This model implies that $E(Y^s \vert X^s_1 X^s_2) = \beta_1 X^s_1 + \beta_2 X^s_2$, i.e. the expected value (or mean value) of Y^s given that we know X^s₁ and X^s₂ is $\beta_1 X^s_1 + \beta_2 X^s_2$

The Ordinary Least Squares estimates are

$$\begin{eqnarray*}\hat{\beta_1} &=& \frac{r_{yx_1} - r_{yx_2} r{x_1 x_2}}{(1-r_{x... ..._2} &=& \frac{r_{yx_2} - r_{yx_1} r{x_1 x_2}}{(1-r_{x_1 x_2}^2)} \end{eqnarray*}$$

Interpretation of the standardized regression coefficient $\beta_1$:

• A one standard deviation increase in X₁ results in $\beta_1$ standard deviations increase in Y when the variable X₂ is held constant. (``increase in X<sub>1</sub>'' is a euphemism for an intervention and ``results in'' is a euphemism for causes). This is the most common way people learn to interpret regression coefficients.

• We can expect to see an average difference of $\beta_1$ standard deviations in the Y value for individuals who differ on the X₁value by 1 standard deviation but do not differ on the X₂ variable. (This interpretation does not hint at any causal relationship)

  
  

  $$E(Y^s\vert X_1^s+1, X_2^s) - E(Y^s\vert X_1^s, X_2^s) = \beta... ... X^s_2 - \left[ \beta_1X^s_1 + \beta_2 X^s_2 \right] = \beta_1$$
  
  

• If we could intervene in the population and increase the X₁variable for each individual by one standard deviation while making sure that each individual's X₂ variable does not change, then we would expect the average Y value after the intervention to be $\beta_1$ standard deviations higher than it was before the intervention. (Full blown causal interpretation)

• Interpretation of $\beta_2$ is analogous.

The causal path model

The path model representing the ``full blown causal interpretation'' given on the previous page is

PUT FIGURE HERE

 

• Notice that this model (as in multiple regression) assumes that Cov(X₁, e) = 0 and Cov(X₂, e)=0.
• The error represents all other variables not in the model that could directly cause Y. The model assumes that all of these other variables are uncorrelated with X₁ and X₂.
• Note there could be other variables that are correlated with X₁ or X₂ which indirectly cause Y or are spuriously correlated with Y but this model assumes all of those such variables are uncorrelated with Y given X₁ and X₂.

If the assumption Cov(X₁, e) = 0 and Cov(X₂, e)=0 is violated, the estimated regression coefficients can be biased.

PUT FIGURE HERE

If some such W exists and we do not include it in the model, the path estimate for X₁ will equal the direct effect of X₁ on Y holding X₂ constant PLUS the indirect or spurious effect of X₁ on Y through W.

Suppression Effect

The old saying that correlation does not prove causation should be complimented by saying that a lack of correlation does not disprove causation (Bollen, 1989, p.52)

	# of errors	boredom	intelligence
# of errors	1
boredom	.35	1
intelligence	0	.7	1

Consider the following multiple regression:

# of errors = $\beta_1*$ boredom + $\beta_2*$ intelligence

then

$$\hat{\beta_2} = \frac{0 - .35*.7}{1-.7^2} = -.48$$

Suppression effect: You don't see a bivariate relationship unless you adjust for some other variable.

In the example above boredom is suppressing the relationship between intelligence and the number of errors made.

PUT FIGURE HERE

Effect Decomposition of a Bivariate Relationship

In Path Analysis we distinguish 3 types of causal effects

1.

direct - the influence of one variable on another that is unmediated by any other variable, i.e. each single headed arrow represents a direct effect

2.

indirect - effect that is mediated by at least one intervening variable

3.

total causal effect - sum of the direct and indirect

Total effect = Total causal effect + spurious effect

Note the Total effect is estimated by the simple bivariate regression of Y on X.

Total causal effect = direct effect + indirect effect

Spurious effect is then Total effect - Total causal effect.

For extra reading about direct,indirect, and total effects, see the following article
J. Pearl, "Direct and Indirect Effects" UCLA Cognitive Systems Laboratory, Technical Report (R-273), June 2001. In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, San Francisco, CA: Morgan Kaufmann, 411-420,

Conversation about indirect effects with SEMNET members

Chalk Talk discussion about how Effect Decomposition works for the three examples 1. Age, Vocabulary, Shoe size 2. boredom, intelligence, number of errors, and 3. education, income, conservatism

Calculation of Indirect effects

Path Multiplication Rule - The value of the effect associated with a compound path is the product of its path coefficients (this works for standardized regression coefficient or unstandardized)

     education ----> income ------> conservatism

Unstandardized regression coefficient of Income on Education is $\beta_1 = 1000 \; \$/year$, and the regression of conservatism (a 5 point Likert scale) on income yields a regression slope of $\beta_2 = 0.0002 \; points/\$

What is the indirect effect of education on conservatism?

If education goes up 1 year, income goes up $\$1000$

If income goes up $\$1000$ then conservatism goes up $0.0002 \times 1000 = .2 \; points$

So, the indirect effect of a 1 year increase in education through income on conservatism is a .2 increase in the conservatism scale.

Equivalent models

Two models are equivalent if they are covariance equivalent, i.e. if every covariance matrix generated by on e model (through some choice of parameters) can also be generated by the others.

For a detailed study of equivalent models check out: MacCallum R.C., Wegener, D.T., Uchino, B.N. and Fabrigan, L.R. (1993) "The problem of equivalent models in applications of covariance structure analysis" Psychological Bulletin Vol 144, No. 1, 185-199.

Other references for this topic:
Lee, S., and Hershberger, S. (1990). A simple rule for generating equivalent models in covariance structure modeling. Multivariate Behavioral Research, 25, 313-334.

Raykov, and Penev (1999). On SEM equivalence. Multivariate Behavioral Research, 34, 199-244.

These three models are equivalent:

A: X1 and X2 are both causes of Y but the causal relationship between them is unspecified.

B: X1 is a common cause for both Y and X2, or X2 partially mediates the relationship between Y and X1.

C: X2 is a common cause for both Y and X1, or X1 partially mediates the relationship between Y and X2.

What is implied by each model above when beta1 = 0?

A1: There is no causal effect of X1 on Y, it is only spuriously related to Y through its correlation with X2

B1: X1 only has an indirect causal influence on Y through X2. In other words, X2 fully mediates the influence of X1 on Y.

C1: There is no causal influence of X1 on Y, they are related because they both have a common cause, i.e. X2.

Handout about "Counting the # of parameters"
Handout about "Standardized vs. unstandardized estimates"

MW: Convert file classnotes2.tex to html for this also add stress/illness AMOS graphic