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 (http://stats.idre.ucla.edu/wp-content/uploads/2016/02/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 (hsandcolrespectively); 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 keywordonis used to indicate that the model regressesgreonhsandcol; andgradonhs,col, andgre. Theoutput:command with thestdyx;option was included to obtain standardized regression coefficients and R-squared values. (Thestdyx;option produces coefficients standardized on both y and x, but other types of standardization are available and can be requested using thestandardized;option.)

Title: Path analysis -- just identified model Data: file is http://stats.idre.ucla.edu/wp-content/uploads/2016/02/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.

INPUT READING TERMINATED NORMALLY 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) http://stats.idre.ucla.edu/wp-content/uploads/2016/02/path.dat Input data format FREE THE MODEL ESTIMATION TERMINATED NORMALLY TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 0.000 Degrees of Freedom 0 P-Value 0.0000 Chi-Square Test of Model Fit for the Baseline Model Value 247.004 Degrees of Freedom 5 P-Value 0.0000 CFI/TLI CFI 1.000 TLI 1.000 Loglikelihood H0 Value -2789.415 H1 Value -2789.415 Information Criteria Number of Free Parameters 9 Akaike (AIC) 5596.830 Bayesian (BIC) 5626.515 Sample-Size Adjusted BIC 5598.002 (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation) Estimate 0.000 90 Percent C.I. 0.000 0.000 Probability RMSEA <= .05 0.000 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)

Under MODEL RESULTS the path coefficients (slopes) for the regression of

greonhsandcolare shown, followed by those for the regressiongradonhs. 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 thathsandcolsignificantly predictgre, and thatgreandhs(but notcol) significantly predictgrad. 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 thestdyxoption of theoutput: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.

### 1.1 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

hsongrad(throughgre). 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 themodel indirect:command to our input file, and specifyinggrad ind hs;.Here is the entire program; except for the highlighted portion of the output (and the title) this model is identical to the previous model.

Title: Path analysis -- with indirect effects. Data: file is http://stats.idre.ucla.edu/wp-content/uploads/2016/02/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, and some the output has been omitted since the output for this model is the same as the previous model except for the addition of sections showing the total, indirect and direct effects. The output is the same 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.

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 <output omitted> 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

Under Specific indirect, the effect labeled GRAD GRE HS (note each appears on its own line and the final outcome is listed first), gives the estimated coefficient for the indirect effect of

hsongrad, throughGRE(the blue path above). The coefficient labeled Direct is the direct effect ofhsongrad. We can say that part of the total effect ofhsongradis mediated bygrescores, but the significant direct path fromhstogradsuggests only partial mediation.

### 1.2 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

hstocol, creating a model with multiple possible indirect effects. The diagram below shows the model, with the three indirect paths we wish to examine highlighted with colored lines.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 fromhstograd. We can also useindto request a specific indirect path, for example, below we usegrad ind col hs;, to specify that we want to estimate the indirect effect fromhstocoltograd(i.e. the dashed orange path shown in the diagram above). Finally, we can use via to request all indirect effects that go through a third variable, for example below we usegrad via gre hs;to request all indirect paths fromhstogradthat involvegre, this includeshstogretograd(i.e. the solid blue path), andhstocoltogretograd(i.e. the dotted pink path). The new directional path (col on hs;), as well as the specific indirect (grad ind col hs;) and via (grad via gre hs;) options of the model indirect are highlighted in the input shown below.

Title: Multiple indirect paths Data: file is http://stats.idre.ucla.edu/wp-content/uploads/2016/02/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;

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 Sum of indirect 0.075 0.051 1.455 0.146 Specific indirect GRAD COL HS 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

In the first set of indirect effects (labeled Effects from HS to GRAD) gives the indirect effect of

hsongradthroughcol. Although we estimated a direct effect ofhsongradin 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 fromhstograd, that includeGRE, in this case, there are two such effects. This portion of the output shows thathshas a significant indirect effect ongrad, overall (Sum of indirect), as well as the two specific indirect effects, that is throughgre, as well as throughcolandgre. Note that this output does not include the total effect ofgradonhs, for this output we would simply specifygrad ind hs;as we did in the previous model.

### 2.0 An Over Identified Model

This is an example of an overidentified model, that is a model with positive degrees of freedom (as opposed to the previous models which can be described as saturated or just identified). Having positive degrees of freedom allows us to examine the fit of the model using the chi-squared test of model fit, along with fit indices, for example, CFI and RMSEA. In the illustration below, paths that are included in the model are represented by solid lines; paths that could be estimated, but are not, are represented by dotted lines. Note that now

hsdoes not have a direct effect on eithergradorgre, its only influence is viacol. This corresponds to the hypothesis that high school gpa is only associated with GRE scores and graduate school grades through its relationship with college gpa.The input file for this model is shown below.

Title: Path analysis -- over identified model Data: file is http://stats.idre.ucla.edu/wp-content/uploads/2016/02/path.dat ; Variable: Names are hs gre col grad; Model: col on hs; gre on col; grad on col gre; Output: stdyx;

Below is the output for this model.

INPUT READING TERMINATED NORMALLY Path analysis -- over identified model SUMMARY OF ANALYSIS Number of groups 1 Number of observations 200 Number of dependent variables 3 Number of independent variables 1 Number of continuous latent variables 0 Observed dependent variables Continuous GRE COL GRAD Observed independent variables HS 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) http://stats.idre.ucla.edu/wp-content/uploads/2016/02/path.dat Input data format FREE THE MODEL ESTIMATION TERMINATED NORMALLY TESTS OF MODEL FIT Chi-Square Test of Model FitValue 44.429 Degrees of Freedom 2 P-Value 0.0000Chi-Square Test of Model Fit for the Baseline Model Value 362.474 Degrees of Freedom 6 P-Value 0.0000 CFI/TLICFI 0.881 TLI 0.643Loglikelihood H0 Value -2811.629 H1 Value -2789.415 Information Criteria Number of Free Parameters 10 Akaike (AIC) 5643.258 Bayesian (BIC) 5676.242 Sample-Size Adjusted BIC 5644.561 (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation)Estimate 0.3266 90 Percent C.I. 0.247 0.412 Probability RMSEA <= .05 0.000SRMR (Standardized Root Mean Square Residual)Value 0.086MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value COL ON HS 0.605 0.048 12.500 0.000 GRE ON COL 0.625 0.056 11.101 0.000 GRAD ON COL 0.317 0.079 4.014 0.000 GRE 0.492 0.078 6.303 0.000 Intercepts GRE 19.887 3.009 6.609 0.000 COL 21.038 2.576 8.165 0.000 GRAD 9.779 3.664 2.669 0.008 Residual Variances GRE 55.313 5.531 10.000 0.000 COL 49.025 4.903 10.000 0.000 GRAD 67.311 6.731 10.000 0.000 STANDARDIZED MODEL RESULTS STDYX Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value COL ON HS 0.662 0.040 16.684 0.000 GRE ON COL 0.617 0.044 14.112 0.000 GRAD ON COL 0.276 0.068 4.092 0.000 GRE 0.434 0.065 6.671 0.000 Intercepts GRE 2.103 0.397 5.298 0.000 COL 2.251 0.363 6.210 0.000 GRAD 0.913 0.375 2.436 0.015 Residual Variances GRE 0.619 0.054 11.452 0.000 COL 0.561 0.053 10.677 0.000 GRAD 0.587 0.053 11.002 0.000 R-SQUARE Observed Two-Tailed Variable Estimate S.E. Est./S.E. P-Value GRE 0.381 0.054 7.056 0.000 COL 0.439 0.053 8.342 0.000 GRAD 0.413 0.053 7.743 0.000 QUALITY OF NUMERICAL RESULTS Condition Number for the Information Matrix 0.104E-03 (ratio of smallest to largest eigenvalue)

The chi-squared value compares the current model to a saturated model. Since our model is not saturated (i.e., our model has positive degrees of freedom), the chi-squared value is no longer zero and may be used to evaluative model fit. Similarly, the CFI and TLI which were equal to one in the just identified model now take on informative values. Further down, the RMSEA and SRMR now take on informative values (in a just identified model, they are displayed as zero). Having positive degrees of freedom, and hence, informative values of the fit indices allows us to better evaluate how well our model fits the data. The specific coefficient estimates from this model are generally interpreted as they were in the just identified model.