Re: the standardized covariance is not equal to the correlation coefficient
Rönkkö, Mikko
Your understanding is correct. It's possible that you may have calculated the correlation incorrectly. It would be helpful to see the code and output to better understand the issue. Can you share the code and output that you used to calculate the correlation? That way we can help identify any potential errors or misunderstandings.
Best regards,
Mikko
From: lav...@googlegroups.com <lav...@googlegroups.com> on behalf of Serena <wsere...@gmail.com>
Date: Thursday, 23. May 2024 at 11.40
To: lavaan <lav...@googlegroups.com>
Subject: the standardized covariance is not equal to the correlation coefficient
Hi all,
To my understanding,
1) in simple linear regression, the standardized regression coefficient (beta coefficient) is equal to the correlation coefficient (Pearson's r) between the independent and dependent variables when both variables are standardized.
2) in bivariate scenarios (considering two variables), the standardized covariance and the correlation coefficient are numerically equivalent. This means that the value of the standardized covariance between two variables will be the same as the Pearson correlation
coefficient between those variables.
But in R, I found
1) the standardized regression coefficient is not equal to the correlation coefficient;
2) the standardized covariance and the correlation coefficient are also not equivalent.
(I extracted these parameters through CFA (multiple variables) analysis.)
Is this because the calculated functions are different or other reasons? Or is my understanding incorrect? Or is it because these are analyzing multiple variables simultaneously?
Thanks in advance,
Serena
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Serena
I run a CFA model that includes all constructs I will use in my SEM model.
model <- "
#factor1
fac1 =~ var1 + var2 + var3 + var4 + var5
#factor2
fac2 =~ var6 + var7 + var8 + var9
# factors3-10
fac3 =~ var10 + var11 + var12 + var13
...
fac8 =~ var29 + var30 + var31 + var32
fac9 =~ fac3 + fac4 + fac5
fac10 =~ fac6 + fac7 + fac8
"
Then, I use cor_matrix <- lavInspect(fit, "cor.all") to get the correlations.
I use para_estmt <- parameterEstimates(fit, standardized = TRUE) to get the standard regression coefficients and covariances.
I found the results are not the same, but most are similar.
My understanding is that the CFA is analyzing multiple variables simultaneously, not just bivariate. Then there some variances are correlated and impact each other. Thus the coefficient is not the same?
Rönkkö, Mikko
Hi,
HS.model <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9 '
fit <- cfa(HS.model, data = HolzingerSwineford1939)
parameterEstimates(fit, standardized = TRUE)
lavInspect(fit, "cor.lv")
Gives
…
19 visual ~~ visual 0.809 0.145 5.564 0 0.524 1.094 1.000 1.000 1.000
20 textual ~~ textual 0.979 0.112 8.737 0 0.760 1.199 1.000 1.000 1.000
21 speed ~~ speed 0.384 0.086 4.451 0 0.215 0.553 1.000 1.000 1.000
22 visual ~~ textual 0.408 0.074 5.552 0 0.264 0.552 0.459 0.459 0.459
23 visual ~~ speed 0.262 0.056 4.660 0 0.152 0.373 0.471 0.471 0.471
24 textual ~~ speed 0.173 0.049 3.518 0 0.077 0.270 0.283 0.283 0.283
> lavInspect(fit, "cor.lv")
visual textul speed
visual 1.000
textual 0.459 1.000
speed 0.471 0.283 1.000
The correlations, marked with colors, are the same as they should be. Without seeing your code or the output, it is difficult to say why that would be different in your case.
Best regards,
Mikko
To view this discussion on the web visit https://groups.google.com/d/msgid/lavaan/91398562-21ad-4e82-9ad1-52148935386an%40googlegroups.com.
Serena
var1 1
var2 -0.33 1
var3 -0.52 0.75 1
var1 ~~ var3 -0.856 0.059 -14.399 0 -0.973 -0.74 -0.544 -0.544 -0.544
var2 ~~ var3 1.595 0.065 24.642 0 1.468 1.722 0.803 0.803 0.803
Rönkkö, Mikko
Hi,
Seeing the full model specification would be helpful.
Best regards,
Mikko
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Serena
fac1 =~var1 + var2 + var3 + var4 + var5
fac2 =~ var6 + var7 + var8 + var9 + var10 + var11
fac3 =~ var12 + var13 + var14 + var15 + var16 + var17
fac5 =~ af1_t2 + af2_t2 + af3_t2 + af4_t2
fac6 =~ cs1_t2 + cs2_t2 + cs3_t2 + cs4_t2
fac7 =~ cf1_t2 + cf2_t2 + cf3_t2 + cf4_t2
fac8 =~ rs1_t2 + rs2_t2 + rs3_t2 + rs4_t2
fac9 =~ rf1_t2 + rf2_t2 + rf3_t2 + rf4_t2
higher_fac10 =~ fac4 + fac6 + fac8
higher_fac11 =~ fac5 + fac7 + fac9
fac12 =~ es + ts + tm + em + mc
#regress latent factors on the covariates
fac1 + fac2 + fac3 + higher_fac10 + higher_fac11 + fac12 + performance ~ Age + Gender + Language
"
fit_all <- cfa(model, data = dataset, estimator = "MLR", cluster = "id")
summary(fit_all, fit.measures = TRUE, standardized = TRUE, rsquare = TRUE)
cor_matrix <- lavInspect(fit_all, "cor.all")
para_estmt <- parameterEstimates(fit_all, standardized = TRUE)
Rönkkö, Mikko
Hi,
That clarifies a lot and if you had included the model in the first post, we could have resolved the issue right away.
Because you are regressing the factors on covariates, these parameters
parameterEstimates(fit, standardized = TRUE):
var1 ~~ var2 -0.592 0.045 -13.061 0 -0.681 -0.503 -0.352
-0.352 -0.352
var1 ~~ var3 -0.856 0.059 -14.399 0 -0.973 -0.74 -0.544
-0.544 -0.544
var2 ~~ var3 1.595 0.065 24.642 0 1.468 1.722 0.803
0.803 0.803
are not factor covariances but residual covariances. They differ from factor covariances because they are not the factors, but the residuals from the regressions of the factors on the observed covariates.
Best regards,
Mikko
To view this discussion on the web visit https://groups.google.com/d/msgid/lavaan/e49068db-f636-4b0d-ace2-4432efcda927n%40googlegroups.com.
Serena
parameterEstimates(fit, standardized = TRUE):
var1 ~~ var2 -0.592 0.045 -13.061 0 -0.681 -0.503 -0.352 -0.352 -0.352
var1 ~~ var3 -0.856 0.059 -14.399 0 -0.973 -0.74 -0.544 -0.544 -0.544
var2 ~~ var3 1.595 0.065 24.642 0 1.468 1.722 0.803 0.803 0.803
Rönkkö, Mikko
Hi,
You can get the model implied correlations with
lavInspect(fit_all, "cor.all")
If you want to get the correlations directly from the model, you need to make sure that the factors are modeled as exogenous variables without any incoming regression paths.
Best regards,
Mikko
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