Mplus version 8 was used for these examples. All the files for this portion of this seminar can be downloaded here.

Path analysis is used to estimate a system of equations in which all of the variables are observed. Unlike models that include latent variables, path models assume perfect measurement of the observed variables; only the structural relationships between the observed variables are modeled. This type of model is often used when one or more variables is thought to mediate the relationship between two others (mediation models). Similar models setups can be used to estimate models where the errors (residuals) of two otherwise unrelated dependent variables are allowed to correlated (seemingly unrelated regression), as well as models where the relationship between variables is thought to vary across groups (multiple group models).

## 1.0 A just identified model

The examples on this page use a dataset (path.dat) that contains four variables: the respondent’s high school gpa (**hs**), college gpa (**col**), GRE score (**gre**) and graduate school gpa (**grad**). We begin with the model illustrated below, where GRE scores are
predicted using high school and college gpa (**hs** and **col **respectively); and graduate school gpa (**grad**) is predicted using GRE, high school gpa and college gpa. This model is just identified, meaning that it has zero degrees of freedom.

In the **model** command, the keyword **on** is used to indicate that the model regresses **gre** on **hs** and **col**, and **grad** on **hs**, **col** and **gre**. The
**output** command with the **stdyx;** option was included to obtain standardized regression coefficients and R-squared values. (The **stdyx;**
option produces coefficients standardized on both y and x, but other types of standardization are available and can be requested using the **standardized;** option.)

title: Path analysis -- just identified model data: file is path.dat; variable: names are hs gre col grad; model: gre on hs col; grad on hs col gre; output: stdyx;

Here is the output from Mplus.

Path analysis -- just identified model SUMMARY OF ANALYSIS Number of groups 1 Number of observations 200 Number of dependent variables 2 Number of independent variables 2 Number of continuous latent variables 0 Observed dependent variables Continuous GRE GRAD Observed independent variables HS COL Estimator ML Information matrix OBSERVED Maximum number of iterations 1000 Convergence criterion 0.500D-04 Maximum number of steepest descent iterations 20 Input data file(s) D:/data/path.dat Input data format FREE THE MODEL ESTIMATION TERMINATED NORMALLY MODEL FIT INFORMATION Number of Free Parameters 9 Loglikelihood H0 Value -1367.594 H1 Value -1367.594 Information Criteria Akaike (AIC) 2753.189 Bayesian (BIC) 2782.874 Sample-Size Adjusted BIC 2754.361 (n* = (n + 2) / 24) Chi-Square Test of Model Fit Value 0.000 Degrees of Freedom 0 P-Value 0.0000 RMSEA (Root Mean Square Error Of Approximation) Estimate 0.000 90 Percent C.I. 0.000 0.000 Probability RMSEA <= .05 0.000 CFI/TLI CFI 1.000 TLI 1.000 Chi-Square Test of Model Fit for the Baseline Model Value 247.004 Degrees of Freedom 5 P-Value 0.0000 SRMR (Standardized Root Mean Square Residual) Value 0.000 MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value GRE ON HS 0.309 0.065 4.756 0.000 COL 0.400 0.071 5.625 0.000 GRAD ON HS 0.372 0.075 4.937 0.000 COL 0.123 0.084 1.465 0.143 GRE 0.369 0.078 4.754 0.000 Intercepts GRE 15.534 2.995 5.186 0.000 GRAD 6.971 3.506 1.989 0.047 Residual Variances GRE 49.694 4.969 10.000 0.000 GRAD 59.998 6.000 10.000 0.000 STANDARDIZED MODEL RESULTS STDYX Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value GRE ON HS 0.335 0.068 4.887 0.000 COL 0.396 0.068 5.859 0.000 GRAD ON HS 0.356 0.070 5.073 0.000 COL 0.108 0.073 1.467 0.142 GRE 0.326 0.067 4.869 0.000 Intercepts GRE 1.643 0.378 4.343 0.000 GRAD 0.651 0.350 1.859 0.063 Residual Variances GRE 0.556 0.052 10.611 0.000 GRAD 0.523 0.051 10.240 0.000 R-SQUARE Observed Two-Tailed Variable Estimate S.E. Est./S.E. P-Value GRE 0.444 0.052 8.477 0.000 GRAD 0.477 0.051 9.333 0.000 QUALITY OF NUMERICAL RESULTS Condition Number for the Information Matrix 0.348E-04 (ratio of smallest to largest eigenvalue) DIAGRAM INFORMATION Use View Diagram under the Diagram menu in the Mplus Editor to view the diagram. If running Mplus from the Mplus Diagrammer, the diagram opens automatically. Diagram output d:datapath1.dgm

In the MODEL RESULTS section, the path coefficients (slopes) for the regression of **gre** on **hs** and **col** are shown, followed by those for the
regression **grad** on **hs**. Along with the unstandardized coefficients (in the column labeled Estimate), the standard errors (S.E), coefficients divided by the standard errors, and a p-values are shown. From this we see that **hs **and **col** significantly predict **gre**, and that **gre**
and **hs** (but not **col**) significantly predict **grad**. Additional parameters from the model are listed below the path coefficients. Note that
the regression intercepts are listed under the heading Intercepts rather than with the path coefficients. This is different from some general-purpose statistical packages where all of the coefficients (intercepts and slopes) are listed together. Because we requested standardized coefficients using the **stdyx** option of the **output** command, the standardized results are also included in the output (after the unstandardized results). Under the heading STDYX Standardization all of the model parameters are listed, standardized so that a one unit change represents a standard deviation change in the original variable (just as in a standardized regression model). As part of the standardized output, the r-squared values are presented under the heading R-SQUARE. Here the estimated r-squared value for each of the dependent variables in our model is given, along with standard errors and hypothesis tests.

## 2.0 Indirect and total effects

One of the appealing aspects of path models is the ability to assess indirect, as well as total effects (i.e., relationships among variables). Note that the total effect is the combination of the direct effect and indirect effects. In this example we will request the estimated indirect effect of **hs** on **grad** (through **gre**). Below is the diagram corresponding to this model with the desired indirect effect shown in blue. We can obtain the estimate of the indirect effect by adding the **model indirect: **command to our input file, and specifying **grad ind hs;**.

Here is the entire program. Notice that the **model indirect** has been added.

title: Path analysis -- with indirect effects data: file is path.dat; variable: names are hs gre col grad; model: gre on hs col; grad on hs col gre; model indirect: grad ind hs; output: stdyx;

The output for this model is shown below. The output is the same as the output from the previous example because we have estimated the same model; adding the indirect effects requests additional output from Mplus, but that does not change the model itself. The breakdown of the total, indirect, and direct effects appears below the MODEL RESULTS and STANDARDIZED MODEL RESULTS in a section labeled TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS. Because standardized coefficients were requested, the standardized total, indirect, and direct effects appear below the unstandardized effects.

Path analysis -- with indirect effects SUMMARY OF ANALYSIS Number of groups 1 Number of observations 200 Number of dependent variables 2 Number of independent variables 2 Number of continuous latent variables 0 Observed dependent variables Continuous GRE GRAD Observed independent variables HS COL Estimator ML Information matrix OBSERVED Maximum number of iterations 1000 Convergence criterion 0.500D-04 Maximum number of steepest descent iterations 20 Input data file(s) path.dat Input data format FREE THE MODEL ESTIMATION TERMINATED NORMALLY MODEL FIT INFORMATION Number of Free Parameters 9 Loglikelihood H0 Value -1367.594 H1 Value -1367.594 Information Criteria Akaike (AIC) 2753.189 Bayesian (BIC) 2782.874 Sample-Size Adjusted BIC 2754.361 (n* = (n + 2) / 24) Chi-Square Test of Model Fit Value 0.000 Degrees of Freedom 0 P-Value 0.0000 RMSEA (Root Mean Square Error Of Approximation) Estimate 0.000 90 Percent C.I. 0.000 0.000 Probability RMSEA <= .05 0.000 CFI/TLI CFI 1.000 TLI 1.000 Chi-Square Test of Model Fit for the Baseline Model Value 247.004 Degrees of Freedom 5 P-Value 0.0000 SRMR (Standardized Root Mean Square Residual) Value 0.000 MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value GRE ON HS 0.309 0.065 4.756 0.000 COL 0.400 0.071 5.625 0.000 GRAD ON HS 0.372 0.075 4.937 0.000 COL 0.123 0.084 1.465 0.143 GRE 0.369 0.078 4.754 0.000 Intercepts GRE 15.534 2.995 5.186 0.000 GRAD 6.971 3.506 1.989 0.047 Residual Variances GRE 49.694 4.969 10.000 0.000 GRAD 59.998 6.000 10.000 0.000 STANDARDIZED MODEL RESULTS STDYX Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value GRE ON HS 0.335 0.068 4.887 0.000 COL 0.396 0.068 5.859 0.000 GRAD ON HS 0.356 0.070 5.073 0.000 COL 0.108 0.073 1.467 0.142 GRE 0.326 0.067 4.869 0.000 Intercepts GRE 1.643 0.378 4.343 0.000 GRAD 0.651 0.350 1.859 0.063 Residual Variances GRE 0.556 0.052 10.611 0.000 GRAD 0.523 0.051 10.240 0.000 R-SQUARE Observed Two-Tailed Variable Estimate S.E. Est./S.E. P-Value GRE 0.444 0.052 8.477 0.000 GRAD 0.477 0.051 9.333 0.000 QUALITY OF NUMERICAL RESULTS Condition Number for the Information Matrix 0.348E-04 (ratio of smallest to largest eigenvalue) TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS Two-Tailed Estimate S.E. Est./S.E. P-Value Effects from HS to GRAD Total 0.487 0.075 6.453 0.000 Total indirect 0.114 0.034 3.362 0.001 Specific indirect GRAD GRE HS 0.114 0.034 3.362 0.001 Direct GRAD HS 0.372 0.075 4.937 0.000 STANDARDIZED TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS STDYX Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value Effects from HS to GRAD Total 0.465 0.068 6.858 0.000 Total indirect 0.109 0.032 3.455 0.001 Specific indirect GRAD GRE HS 0.109 0.032 3.455 0.001 Direct GRAD HS 0.356 0.070 5.073 0.000 DIAGRAM INFORMATION Use View Diagram under the Diagram menu in the Mplus Editor to view the diagram. If running Mplus from the Mplus Diagrammer, the diagram opens automatically. Diagram output c:\temp\path2.dgm

Under Specific indirect, the effect labeled GRAD GRE HS (note that each appears on its own line and the final outcome is listed first), gives the estimated coefficient for the indirect effect of **hs** on **grad**, through **GRE **. The coefficient labeled Direct is the direct effect of **hs** on **grad**. We can say that part of the total effect of **hs** on **grad** is mediated by **gre** scores, but the significant direct path from **hs** to **grad **suggests only partial mediation.

## 3.0 Specific indirect effects

The above example was overly simple since there was only one indirect effect. Often models will have multiple indirect effects. In this example we place a directional path (i.e., regression) from **hs** to **col**, creating a model with multiple possible indirect effects. The diagram below shows the model.

There are several ways to request calculation of indirect effects. The first, shown in the previous example (i.e., **grad ind hs;**) requests all indirect paths from **hs** to **grad**. We can also use **ind** to request a specific indirect path. For example, below we use **grad ind col hs;** to specify that we want to estimate the indirect effect from **hs** to **col** to **grad**. Finally, we can use **via** to request all indirect effects that go through a third variable. In the example below, we use **grad via gre hs;** to request all indirect paths from **hs** to **grad** that involve **gre**. This includes **hs** to **gre** to **grad **and **hs** to **col** to **gre** to **grad**.

title: Multiple indirect paths data: file is path.dat; variable: names are hs gre col grad; model: gre on col hs; grad on hs col gre; col on hs; model indirect: grad ind col hs; grad via gre hs; output: stdyx;

The abridged output is shown below. Note that the output for this model is similar in structure to the output from earlier models, except for the addition of the section showing the indirect effects.

<output omitted> TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS Two-Tailed Estimate S.E. Est./S.E. P-Value Effects from HS to GRAD Indirect 0.075 0.051 1.455 0.146 Effects from HS to GRAD via GRE Sum of indirect 0.204 0.047 4.333 0.000 Specific indirect GRAD GRE HS 0.114 0.034 3.362 0.001 GRAD GRE COL HS 0.090 0.026 3.487 0.000 STANDARDIZED TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS STDYX Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value Effects from HS to GRAD Indirect 0.071 0.049 1.460 0.144 Effects from HS to GRAD via GRE Sum of indirect 0.195 0.043 4.533 0.000 Specific indirect GRAD GRE HS 0.109 0.032 3.455 0.001 GRAD GRE COL HS 0.086 0.024 3.587 0.000

The first set of indirect effects (labeled Effects from HS to GRAD) gives the indirect effect of **hs** on **grad** through **col**. Although we estimated a direct effect of **hs** on **grad** in the model, this is not shown in this portion of the output (it is shown above), because we requested the specific indirect effect. The second set of indirect effects (labeled Effects from HS to GRAD via GRE) shows all possible indirect effects from **hs** to **grad** that include **GRE**. In this example, there are two such effects. This portion of the output shows that **hs** has a significant indirect effect on **grad**, overall (Sum of indirect), as well as the two specific indirect effects, that is through **gre**, as well as through **col** and **gre**. Note that this output does not include the total effect of **grad** on **hs**; for this output we would simply specify **grad ind hs;** as we did in the previous model.