# Residual Covariance

## Contents

# A Way to Look At Why Your SEM Model Has Poor Fit

If you’ve done any Structural Equation Modeling, you’ve run into a bad fitting model. Embarassingly, aside from estimating covariances among correlated variables, I didn’t know much about how to inspect poor model fit until very recently. Enter the residual covariance matrix. It’s a super simple way to examine poor model fit.

Here’s an example. You have 6 observed variables, and you want to create a latent factor. I’m going to use the lavaan and the simsem package in R to generate data.

```
{% raw %}
library(simsem)
library(lavaan)
loading <- matrix(0, 6, 2)
loading[1:3, 1] <- NA
loading[4:6, 2] <- NA
LY <- bind(loading, 0.7)
latent.cor <- matrix(NA, 2, 2)
diag(latent.cor) <- 1
RPS <- binds(latent.cor, 0.5)
RTE <- binds(diag(6))
VY <- bind(rep(NA,6),2)
CFA.Model <- model(LY = LY, RPS = RPS, RTE = RTE, modelType = "CFA")
dat <- generate(CFA.Model,200)
mod <-
'
L =~ y1 + y2 + y3 + y4 + y5 + y6
'
fit <- cfa(mod, data = dat)
{% endraw %}
```

If we examine the residual correlation matrix, with `resid(fit, type='cor')$cor`

, we see moderate correlations between y1, y2, and y3.

```
{% raw %}
y1 y2 y3 y4 y5 y6
y1 0.000
y2 0.303 0.000
y3 0.253 0.208 0.000
y4 -0.153 -0.097 -0.067 0.000
y5 -0.035 -0.089 -0.069 0.075 0.000
y6 -0.041 -0.069 -0.071 0.071 0.008 0.000
{% endraw %}
```

We also have really poor fit, RMSEA = .20. So, that means that we’re not accounting for the covariance between these indicators in our model. Or, a one factor solution doesn’t take into accout how y1, y2, and y3 vary toghether. Let’s see what happens if we make two factors.

```
{% raw %}
mod.2 <-
'
L1 =~ y1 + y2 + y3
L2 =~ y4 + y5 + y6
'
fit.2 <- cfa(mod.2, data = dat)
{% endraw %}
```

If we examine the residual correlation matrix, with {% raw %} resid(fit.2, type=‘cor’)$cor {% endraw %}, we see those moderate correlations have disappeared.

```
{% raw %}
y1 y2 y3 y4 y5 y6
y1 0.000
y2 0.012 0.000
y3 -0.009 -0.005 0.000
y4 -0.085 -0.049 0.032 0.000
y5 0.049 -0.028 0.043 0.008 0.000
y6 0.041 -0.009 0.039 0.011 -0.023 0.000
{% endraw %}
```

We also now have a much improved fit, RMSEA = .024.

I find this method super helpful for anything in SEM – path models, CFA, growth curves, etc.

Author Sam Portnow

LastMod 2016-07-01