Path analysis is a type of statistical method to investigate the direct and indirect relationship among a set of exogenous (independent, predictor, input) and endogenous (dependent, output) variables. Path analysis can be viewed as generalization of regression and mediation analysis where multiple input, mediators, and output can be used. The purpose of path analysis is to study relationships among a set of observed variables, e.g., estimate and test direct and indirect effects in a system of regression equations and estimate and test theories about the absence of relationships

Path analysis is often conducted based on path diagrams. Path diagram represents a model using shapes and paths. For example, the diagram below portrays the multiple regression model $Y=\beta_0 + \beta_X X + \beta_W W + \beta_Z Z + e$.

In a path diagram, different shapes and paths have different meanings:

- Squares or rectangular boxes: observed or manifest variables
- Circles or ovals: errors, factors, latent variables
- Single-headed arrows: linear relationship between two variables. Starts from an independent variable and ends on a dependent variable.
- Double-headed arrows: variance of a variable or covariance between two variables
- Triangle: a constant variable, usually a vector of ones

A simplified path diagram is often used in practice in which the intercept term is removed and the residual variances are directly put on the outcome variables. For example, for the regression example, the path diagram is shown below.

In R, path analysis can be conducted using R package `lavaan`

. We now show how to conduct path analysis using several examples.

The NLSY data include three variables – mother's education (ME), home environment (HE), and child's math score. Assume we want to test whether home environment is a mediator between mother’s education and child's math score. The path diagram for the mediation model is:

To estimate the paths in the model, we use the R package `lavaan`

. To specify the mediation model, we follow the rules below. First, a model is put into a pair of quotation marks. Second, to specify the regression relationship, we use a symbol `~`

. The variable on the left is the outcome and the ones on the right are predictors or covariates. Third, parameter names can be used for paths in model specification such as `a`

, `b`

and `cp`

. Fourth, we can define new parameters using the notation `:=`

. On the left is the name of the new parameter and on the right is the formula to define the new parameter such as `a*b`

that defines the mediation effect and `a*b + cp`

that defines the total effect.

To estimate the model, the `sem()`

function from `lavaan`

can be used. To view the results, the `summary()`

function is used. For example, for the mediation example, the output is given below. From the output, we can see

> library(lavaan) This is lavaan 0.5-23.1097 lavaan is BETA software! Please report any bugs. > usedata('nlsy') > > mediation<-' + math ~ b*HE + cp*ME + HE ~ a*ME + ab := a*b + total := a*b + cp + ' > > mediation.res<-sem(mediation, data=nlsy) > summary(mediation.res) lavaan (0.5-23.1097) converged normally after 21 iterations Number of observations 371 Estimator ML Minimum Function Test Statistic 0.000 Degrees of freedom 0 Minimum Function Value 0.0000000000000 Parameter Estimates: Information Expected Standard Errors Standard Regressions: Estimate Std.Err z-value P(>|z|) math ~ HE (b) 0.465 0.143 3.252 0.001 ME (cp) 0.463 0.120 3.869 0.000 HE ~ ME (a) 0.139 0.043 3.249 0.001 Variances: Estimate Std.Err z-value P(>|z|) .math 20.621 1.514 13.620 0.000 .HE 2.724 0.200 13.620 0.000 Defined Parameters: Estimate Std.Err z-value P(>|z|) ab 0.065 0.028 2.298 0.022 total 0.528 0.120 4.410 0.000 >

- An individual path can be tested. For example, the coefficient from ME to HE is 0.139, which is significant based on the z-test.
- The residual variance parameters are also automatically estimated.
- The mediation effect is estimated and tested using the defined parameter. For example, the mediation effect here is 0.065 with the standard error 0.028. It is significant based on a z-test (Sobel test). Note that the result is the same as the mediation analysis before.

Assume we hypothesize that there is no direct effect from ME to math. To test the hypothesis, we can fit a model illustrated below.

The input and output of the analysis are given below. To evaluate the hypothesis, we can check the model fit. The null hypothesis is “\(H_{0}\): The model fits the data well or the model is supported”. The alternative hypothesis is “\(H_{1}\): The model does not fit the data or the model is rejected”. The model with the direct effect fits the data perfectly. Therefore, if the current model also fits the data well, we fail to reject the null hypothesis. Otherwise, we reject it. The test of the model can be conducted based on a chi-squared test. From the output, the Chi-square is 14.676 with 1 degree of freedom. The p-value is about 0. Therefore, the null hypothesis is rejected. This indicates that the model without direct effect is not a good model.

> library(lavaan) This is lavaan 0.5-23.1097 lavaan is BETA software! Please report any bugs. > usedata('nlsy') > > model2<-' + math ~ b*HE + HE ~ a*ME + ' > > model2.res<-sem(model2, data=nlsy) > summary(model2.res) lavaan (0.5-23.1097) converged normally after 17 iterations Number of observations 371 Estimator ML Minimum Function Test Statistic 14.676 Degrees of freedom 1 P-value (Chi-square) 0.000 Parameter Estimates: Information Expected Standard Errors Standard Regressions: Estimate Std.Err z-value P(>|z|) math ~ HE (b) 0.556 0.144 3.873 0.000 HE ~ ME (a) 0.139 0.043 3.249 0.001 Variances: Estimate Std.Err z-value P(>|z|) .math 21.453 1.575 13.620 0.000 .HE 2.724 0.200 13.620 0.000 >

Path analysis can be used to test more complex theories. In this example, we look at how age and education influence EPT using the ACTIVE data. Both age and education may influence EPT directly or through memory and reasoning ability. Therefore, we can fit a model shown below.

Suppose we want to test the total effect of `age`

on `EPT`

and its indirect effect. The direct effect is the path from `age`

to `ept1`

directly, denoted by `p1`

. One indirect path goes through `hvltt1`

, that is `p2*p7`

. The second indirect effect through `ws1`

is `p3*p8`

. The third indirect effect through `ls1`

is `p4*p9`

. The last indirect effect through `lt1`

is `p5*p10`

. The total indirect effect is `p2*p7+p3*p8+p4*p9+p5*p10`

. The total effect is the sum of them `p1+p2*p7+p3*p8+p4*p9+p5*p10`

.

The output from such a model is given below. From it, we can see that the indirect effect `ind1=p2*p7`

is significant. The total indirect (`indirect`

) from age to EPT is also significant. Finally, the total effect (`total`

) from age to EPT is significant.

> library(lavaan) This is lavaan 0.5-23.1097 lavaan is BETA software! Please report any bugs. > usedata('active.full') > > active.model<-' + hvltt1 ~ p1*age + edu + ws1 ~ p2*age + edu + ls1 ~ p3*age + edu + lt1 ~ p4*age + edu + ept1 ~ p5*age + p6*edu + p7*hvltt1 + p8*ws1 + p9*ls1 + p10*lt1 + ws1~~ls1 + ws1~~lt1 + ls1~~lt1 + hvltt1~~ls1 + hvltt1~~ws1 + hvltt1~~lt1 + ind1 := p1*p7 + total := p5 + p1*p7 + p2*p8 + p3*p9 + p4*p10 + indirect := p1*p7 + p2*p8 + p3*p9 + p4*p10 + ' > > active.res<-sem(active.model, data=active.full) > summary(active.res) lavaan (0.5-23.1097) converged normally after 79 iterations Number of observations 1114 Estimator ML Minimum Function Test Statistic 0.000 Degrees of freedom 0 Parameter Estimates: Information Expected Standard Errors Standard Regressions: Estimate Std.Err z-value P(>|z|) hvltt1 ~ age (p1) -0.161 0.027 -6.074 0.000 edu 0.429 0.052 8.177 0.000 ws1 ~ age (p2) -0.226 0.026 -8.737 0.000 edu 0.704 0.051 13.772 0.000 ls1 ~ age (p3) -0.276 0.029 -9.658 0.000 edu 0.877 0.057 15.486 0.000 lt1 ~ age (p4) -0.085 0.015 -5.894 0.000 edu 0.394 0.029 13.723 0.000 ept1 ~ age (p5) 0.014 0.021 0.644 0.519 edu (p6) 0.448 0.045 9.913 0.000 hvltt1 (p7) 0.202 0.025 8.045 0.000 ws1 (p8) 0.196 0.038 5.179 0.000 ls1 (p9) 0.246 0.035 7.090 0.000 lt1 (p10) 0.151 0.051 2.953 0.003 Covariances: Estimate Std.Err z-value P(>|z|) .ws1 ~~ .ls1 16.606 0.819 20.287 0.000 .lt1 5.714 0.371 15.390 0.000 .ls1 ~~ .lt1 6.573 0.415 15.852 0.000 .hvltt1 ~~ .ls1 8.444 0.713 11.838 0.000 .ws1 7.572 0.643 11.769 0.000 .lt1 2.856 0.349 8.191 0.000 Variances: Estimate Std.Err z-value P(>|z|) .hvltt1 20.618 0.874 23.601 0.000 .ws1 19.588 0.830 23.601 0.000 .ls1 24.030 1.018 23.601 0.000 .lt1 6.174 0.262 23.601 0.000 .ept1 12.177 0.516 23.601 0.000 Defined Parameters: Estimate Std.Err z-value P(>|z|) ind1 -0.033 0.007 -4.847 0.000 total -0.144 0.026 -5.594 0.000 indirect -0.158 0.017 -9.340 0.000 >