# How to interpret the model fit indices generated by lavaan (in R)? Something wrong with the model specifications?

I am trying to replicate a path analysis SEM model using Lavaan in R, and was very confused about the results that it gave regarding the model fit statistics.

The code is as follows:

#Import Package
library(lavaan)

#Input Correlation Matrix
sigma <- matrix(c(1.00, -0.03,  0.39, -0.05, -0.08,
-0.03,  1.00,  0.07, -0.23, -0.16,
0.39,  0.07,  1.00, -0.13, -0.29,
-0.05, -0.23, -0.13,  1.00,  0.34,
-0.08, -0.16 ,-0.29,  0.34,  1.00), nr=5, byrow=TRUE)
rownames(sigma) <-c("Exercise", "Hardiness", "Fitness", "Stress", "Illness")
colnames(sigma) <-c("Exercise", "Hardiness", "Fitness", "Stress", "Illness")

#Create Covariance Matrix
sdevs <-c(66.5, 3.8, 18.4, 6.7, 624.8)
covmax <- cor2cov(sigma, sdevs)
as.matrix(covmax)

#Specify Model
mymodel<-'Illness ~ Exercise + Fitness
Illness ~ Hardiness + Stress
Fitness ~ Exercise + Hardiness
Stress ~ Exercise + Hardiness + Fitness
Exercise ~~ Exercise
Hardiness ~~ Hardiness
Exercise ~~ Hardiness'

#Fit the model with the covariance matrix
N = 363
fit.path <-sem(mymodel,sample.cov=covmax, sample.nobs=N, fixed.x=FALSE)

#Summary of the model fit
summary(fit.path, fit.measures = TRUE)


And the output I get is as follows:

 lavaan (0.5-12) converged normally after  93 iterations

Number of observations                         37300

Estimator                                         ML
Minimum Function Test Statistic                0.000
Degrees of freedom                                 0
P-value (Chi-square)                           1.000

Model test baseline model:

Minimum Function Test Statistic            16594.387
Degrees of freedom                                10
P-value                                        0.000

Full model versus baseline model:

Comparative Fit Index (CFI)                    1.000
Tucker-Lewis Index (TLI)                       1.000

Loglikelihood and Information Criteria:

Loglikelihood user model (H0)             -882379.005
Loglikelihood unrestricted model (H1)     -882379.005

Number of free parameters                         15
Akaike (AIC)                              1764788.009
Bayesian (BIC)                            1764915.910

Root Mean Square Error of Approximation:

RMSEA                                          0.000
90 Percent Confidence Interval          0.000  0.000
P-value RMSEA <= 0.05                          1.000

Standardized Root Mean Square Residual:

SRMR                                           0.000

Parameter estimates:

Information                                 Expected
Standard Errors                             Standard

Estimate  Std.err  Z-value  P(>|z|)
Regressions:
Illness ~
Exercise          0.318    0.048    6.640    0.000
Fitness          -8.835    0.174  -50.737    0.000
Hardiness       -12.146    0.793  -15.321    0.000
Stress           27.125    0.451   60.079    0.000
Fitness ~
Exercise          0.109    0.001   82.602    0.000
Hardiness         0.396    0.023   17.211    0.000
Stress ~
Exercise         -0.001    0.001   -2.614    0.009
Hardiness        -0.393    0.009  -44.332    0.000
Fitness          -0.040    0.002  -19.953    0.000

Covariances:
Exercise ~~
Hardiness        -7.581    1.309   -5.791    0.000

Variances:
Exercise       4422.131   32.381
Hardiness        14.440    0.106
Illness       318744.406 2334.012
Fitness         284.796    2.085
Stress           41.921    0.307


These are my questions:

• Why does the chi-squared say that there are no degrees of freedom?
• Why are the p-values exactly 1? Why is the CFI and TLI exactly 1?
• Why is the RMSEA 0?

• What would I need to do to simulate a more realistic model that doesn't appear artificially "perfect"?

• Does it have to do with the model specification?
• I don't think we can answer this thoroughly without seeing mymodel, but if there are zero df, then chi-square will be zero, and p will be 1 (or undefined), and the other fit indices are also correct. So yes, it probably has to do with the model specification. Jun 16, 2013 at 21:35
• Also, your SDs are very variable, this can cause convergence problems. A rough rule of thumb is to keep the smallest and the largest within about an order of magnitude (i.e. largest var < 10 * smallest var). Jun 16, 2013 at 21:37
• My apologies! I thought I had included the model specifications when I copied the code. My mistake. Here it is: #Specify Model mymodel<-'Illness ~ Exercise + Fitness Illness ~ Hardiness + Stress Fitness ~ Exercise + Hardiness Stress ~ Exercise + Hardiness + Fitness Exercise ~~ Exercise Hardiness ~~ Hardiness Exercise ~~ Hardiness' Jun 17, 2013 at 0:34
• I also like what you said about the variability of the standard deviations. Maybe it would be a good idea to multiply them by scalars to make them more similar? Jun 17, 2013 at 0:36
• Yes, that's what I usually do. Multiply / divide by 10 or 100 or whatever, until they're all similar. I aim for between 1 and 10, usually. Jun 17, 2013 at 0:44

It appears that this is a model where (almost) everything is regressed on everything else.

You have 5 variables in your model. That means you have 10 covariances.

You have 10 parameters.

The df of the model is equal to (number of covariances) - (number of parameters). This is zero. The model is described as saturated, and it's not testing anything. Because it's not testing anything, the fit indices are all perfect. (This will make sense if you look at the formulas for the fit indices - a zero chi-square should give you these fit indices).

What do you mean by simulate a model? If you don't want the fit to be perfect, add some constraints. Typically, one constrains to zero.

So yes, it has to do with the model specification. It's a an unusual model to test with an SEM, but if that's your model you want to test, that's your model. If you want to make it more testable, you need to add a variable which is a possible cause of one variable, but not of the others. For example, social support might influence stress, but should not (directly) incfuence ilness, and perhaps not the others. If you add social support, and put an arrow from social support ONLY to stress, you will add 6 covariances to the model, but only add 1 df. Hence your model will have 5 df, and the fit will no longer be perfect.

• Thank you for your good explanation. Just to test it out, I deleted one of the model specifications and it did increase the degrees of freedom and lower the model fit statistics. Good lesson, thanks for your help! Jun 17, 2013 at 1:38