Measuring difference between a non-positive-definite correlation matrix and its nearly positive definite equivalent

[Simon Harmel]

Hello all,

When encountering a non-positive definite (NPD) correlation matrix, I'm often tempted to use "Matrix::nearPD(NPD, corr=TRUE)$mat" to convert my NPD correlation matrix to a nearly positive definite correlation matrix (see below).

Question 1: Is there a way to measure how much change will this introduce to the original NPD correlation matrix and hence determine the extent to which this strategy is justified?

Thanks,

Simon

```

NPD_MATRIX <- structure(c(0.58, 0.55, 0.52, 0.34, 0.56, 0.45, 0.52, 0.42, 0.55,
0.64, 0.36, 0.2, 0.4, 0.29, 0.35, 0.29, 0.52, 0.36, 0.58, 0.53,
0.58, 0.36, 0.55, 0.57, 0.34, 0.2, 0.53, 0.77, 0.52, 0.33, 0.63,
0.57, 0.56, 0.4, 0.58, 0.52, 0.32, 0.42, 0.52, 0.5, 0.45, 0.29,
0.36, 0.33, 0.42, 1, 0.4, 0.33, 0.52, 0.35, 0.55, 0.63, 0.52,
0.4, 0.62, 0.52, 0.42, 0.29, 0.57, 0.57, 0.5, 0.33, 0.52, 0.73
), dim = c(8L, 8L), dimnames = list(c("L2DA", "L2DF", "L2G",
"L2L", "L2M", "L2P", "L2R", "L2V"), c("L2DA", "L2DF", "L2G",
"L2L", "L2M", "L2P", "L2R", "L2V")))

as.matrix(Matrix::nearPD(NPD_MATRIX, corr=TRUE)$mat)

```

[Jeremy Miles]

Can you just find the difference?

Tweak to your code:

pdm <- as.matrix(Matrix::nearPD(NPD_MATRIX, corr=TRUE)$mat)

round(pdm - NPD_MATRIX, 3)

L2DA L2DF L2G L2L L2M L2P L2R L2V
L2DA 0.42 0.00 0.00 0.00 0.00 0 0.00 0.00
L2DF 0.00 0.36 0.00 0.00 0.00 0 0.00 0.00
L2G 0.00 0.00 0.42 0.00 0.00 0 0.00 0.00
L2L 0.00 0.00 0.00 0.23 0.00 0 0.00 0.00
L2M 0.00 0.00 0.00 0.00 0.68 0 0.00 0.00
L2P 0.00 0.00 0.00 0.00 0.00 0 0.00 0.00
L2R 0.00 0.00 0.00 0.00 0.00 0 0.38 0.00
L2V 0.00 0.00 0.00 0.00 0.00 0 0.00 0.27

It looks like it just increases the variances. 

This is probably a bad idea - it is a bit like adding measurement error to your data - that will make your indicators less reliable which will decrease chi-square and RMSEA (hooray!) and decrease CFI (boo!).

Jeremy

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[Simon Harmel]

Jeremy (or others),

Thanks for your input. I have no problem trying something better. But I wonder what other options are there to use when dealing with a non-positive definite (NPD) correlation matrix as input in SEM?

Thanks,

Simon

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[Terrence Jorgensen]

what other options are there to use when dealing with a non-positive definite (NPD) correlation matrix as input in SEM?

See the ridge= description on the ?lavOptions help page.

Terrence D. Jorgensen    (he, him, his)
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam
http://www.uva.nl/profile/t.d.jorgensen

[Jeremy Miles]

My preference would be to work out why I've got an NPD matrix and fix that.

In my experience it's one of: 

 - pairwise deletion (which is a bad idea anyway)

 - using polychoric correlations and something has gone wrong (usually associated with small sample size)

 - I've made some sort of mistake with the data (put the same variable in twice, put in a variable that's a sum of others). 

Jeremy

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