Negative variance error in CFA model (with addressing measurement error/repeatability))

[Magali Frauendorf]

Hi,

 

Similar to this approach described by James Grace (https://prd-wret.s3-us-west-2.amazonaws.com/assets/palladium/production/s3fs-public/atoms/files/SEM_09_1_Modeling_with_latent_variables.pdf), I want to address measurement errors in my model.

However, my “eta” is a latent variable described by 3 indicators instead of 1 (in this example). I want to model the following:

BCol (bill colour of an individual), being the latent variable defined by nLS, nHS and nCS (measured and standardized lightness, hue and chroma (saturation); the three axes/parameters that define a colour). The variables themselves are correlated:

nCS & nLS =0.44

nCS & nHS = -0.74

nHS & nLS =0.92

m1, m2 and m3 are the measurement errors on the observed variables (calculated with ICC package; repeatability of measurements). Because I want to address the measurement error I added the latent variables lLS, lHS and lCS between the latent BCol and the measured variables.

My syntax is the following:

lLS=~nLS

lHS=~nHS

lCS=~nCS

BCol=~lLS+lHS+lCS

nLS ~~ 0.05* nLS

nHS~~0.05*nHS

nCS~~0.06*nCS

 

My output looks like this:

lavaan 0.6-5 ended normally after 58 iterations

 

  Estimator                                         ML

  Optimization method                           NLMINB

  Number of free parameters                          9

                                                     

  Number of observations                           598

                                                      

Model Test User Model:

                                                     

  Test statistic                                 0.000

  Degrees of freedom                                 0

 

Parameter Estimates:

 

  Information                                 Expected

  Information saturated (h1) model          Structured

  Standard errors                             Standard

 

Latent Variables:

                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all

  lLS =~                                                                

    nLS               1.000                               0.974    0.975

  lHS =~                                                               

    nHS               1.000                               0.974    0.975

  lCS =~                                                               

    nCS               1.000                               0.969    0.969

  BCol =~                                                              

    lLS               1.000                               0.758    0.758

    lHS               1.680    0.080   21.076    0.000    1.273    1.273

    lCS              -0.810    0.043  -18.724    0.000   -0.617   -0.617

 

Intercepts:

                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all

   .nLS              -0.000    0.041   -0.000    1.000   -0.000   -0.000

   .nHS              -0.000    0.041   -0.000    1.000   -0.000   -0.000

   .nCS               0.000    0.041    0.000    1.000    0.000    0.000

   .lLS               0.000                               0.000    0.000

   .lHS               0.000                               0.000    0.000

   .lCS               0.000                               0.000    0.000

    BCol              0.000                               0.000    0.000

 

Variances:

                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all

   .nLS               0.050                               0.050    0.050

   .nHS               0.050                               0.050    0.050

   .nCS               0.060                               0.060    0.060

   .lLS               0.404    0.031   12.943    0.000    0.426    0.426

   .lHS              -0.589    0.057  -10.359    0.000   -0.621   -0.621

   .lCS               0.581    0.039   15.020    0.000    0.619    0.619

    BCol              0.545    0.054   10.066    0.000    1.000    1.000

 

R-Square:

                   Estimate

    nLS               0.950

    nHS               0.950

    nCS               0.940

    lLS               0.574

    lHS                  NA

    lCS               0.381

 

I understand that the warning comes from the negative residual covariance in lHS (-0.589) and also results in the standardized estimate being larger than 1 and an NA in the R-square. But what can be the reason for it and what can I do about it. I understand that misspecification of the model can be a reason. The sample size seems ok to me for such a simple model. Can the reason lie in weakness covariance among the indicators? I does not look to me like this because nHS is quite strongly related to the other two variables.

I know that I could fix the variance to zero, but I think that is not what I am interested in when I want to address measurement errors.  Maybe I did not specify the model correctly?

The covariance matrix of the latent variables look like this:

     lLS    lHS    lCS    BCol 

lLS   0.948                    

lHS   0.915  0.948             

lCS  -0.441 -0.741  0.938      

BCol  0.545  0.915 -0.441  0.545

 

 

I also have doubts because the model fit seems perfect:

gfi  agfi   cfi   rni  srmr rmsea 

    1     1     1     1     0     0 

 

Thanks in advance for your help!

[Edward Rigdon]

Magali--

     1. Your approach is needlessly complicated. The individual factors (ILS, IHS, ICS) add nothing. You could remove them and let the observed variables load on the common factor (BCol) and nothing would change.

     2. Your model has 0 DF because it is just-identified. You are estimating 9 parameters (3 observed variable intercepts, 2 loadings, 4 variances or residual variances) from 9 moments of the data (3 observed variable means, 3 variances and 3 covariances). DF = moments of the data - parameters estimated. A model with 0 DF cannot help but fit perfectly in terms of the chi-square and related indicates.

     3. The large negative residual variance is a clue that your data are inconsistent with the model. Think about your observed variables--lightness, hue and saturation. Your mdoe says that these three variables all change together, in response to the variable Bill Color. That would mean that Bill Color is really one dimensional--that all Bill Colors can be arranged along one dimension, with lightness, hue and saturation all moving monotonically from less to more or from more to less as you go from one extreme of Bill Color to another.

     I don't think that this is what you believe. Rather, Bill Color is the outcome or consequence of lightness, hue and saturation. That is the opposite of a factor model. Instead, that says that Bill Color is a composite of lightness, hue and saturation--

Bill Color = f (lightness, hue, saturation)

I think that the negative residual variance is telling you that, despite the unavoidably perfect fit, the model is wrong for your data.

--Ed Rigdon

.

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[Magali Frauendorf]

Hi Edward,

Thanks a lot for your answer!

I thought that a latent variable would be appropriate to use as “Bill colour” since two out of the three correlations were pretty high (0.74 and 0.92). If the variables are highly correlated I would expect them being arranged on one dimension. When I run a PCA on the same data I get a PC1 of 80%, which I think is quiet high. But it is true that the third correlation is much lower (0.44) and that could indicate that a composite variable might be more useful?! Would you say that the correlation between the variables can help to give you an idea if a latent variable or composite variable is more appropriate to use in a specific case? Of course next to looking at the theoretical background and hypothesis.   

I understand from literature that a composite variable always needs a response variable in order to be able to sum the collective effect of all observed variables. Is that correct? So I could use “individual survival” for instance as a response variable?

Concerning your first point: I run it again with a different dataset where the model run smoothly (no warnings and good model fit). The latent variable is body size and I added three latents between the observed variable and the latent variable body size to address measurement errors (that are specified/fixed in the model; bold numbers). I run it with once measurement error (variance) fixed at 0.3 for each observed variable and once fixed at 0 (no error) for each observed variable. Even though the path coefficients changes and it is possible to disentangle the effect of the measurement error from the remaining unexplained variation, the latent variable bill size does not change at all in both example. 

The easier approach that you suggested, by removing the latent variables (lWL, lTA and lTH) resulted also in exactly the same values for the latent variable. In that model the variance is estimated by the model and not fixed as in the other two examples. 

So can I conclude from this that addressing measurement errors is possible and makes sense in regression analysis (path analysis) but not in a CFA (models with latent constructs)? Or would it make sense if I add a response variable (e.g. survival ~ latent body size)?

Thanks!

Magali

[Edward Rigdon]

Magali--

     Your first point represents a common misperception, well addressed by a chapter from Karl Joreskog (attached). The common factor model implies, not so much high correlations as proportional correlations. If two observed variables load only on the same common factor, then the ratio of their correlations to any third observed variable must be constant (within sampling variance), across all other variables in the model. With few observed variables in the model, one cannot observe whether or not such proportionality constraints hold.

     You could use another variable to  If you have other observed variables--whether the are believed to load on the same common factor or not--you could see whether the proportionality constraints implied by the factor model seem to hold.

     Whether or not you can "account for measurement error" in any circumstance depends on what you mean by "measurement error."

     The difference between the models with an without the extra layer of common factors is cosmetic. If you want your output to include these representations, then keep the extra layer. But you could always reconstruct by hand the same results from a model without the extra layer, or even use the lavaan package's ability to define novel parameters to get the package to calculate those for you, along with standard errors.

     I like simple models because they are easier to explain and diagnose.

     This list relates to the lavaan package for estimating common factor-based models, but i will answer your other questions briefly.

     If you wanted to build a composite-based model, and if you wanted to estimate "optimal" weights, then yes, you need some criterion against which to assess that optimality. If you have a priori estimates of the "measurement error" associated with each observed variable, you could assign weights to the observed variables relative to that, giving higher weight to the observed variables less affected by "measurement error." Exact values of the weights are not generally useful. Rather, it is the relative weights that define the composite.

--Ed Rigdon

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[Magali Frauendorf]

Hi Edward,

Thanks a lot for all your answers. It helped me a lot!

Best,

Magali