Shift parameter of scaled and shifted estimators (MLMV and WLSMV)

[Tobias Krieger]

Dear all,

I try to understand what the scale and shift parameters do in MLMV and WLSMV, but I could not find much (except for example for the paper from Asparouhov & Muthén, 2010, Simple Second Order Chi-Square Correction).

If I understood it correctly, scale and shift parameters influence the chi-square value of the robust estimators to follow a chi-square distribution:

The shift parameter influences the location of the distribution and the scale parameter the size and shape. Is that correct?

Why I am writing is the following question:

Can a shift parameter also be negative? For models with 1 degree of freedom I have some results with cfa in lavaan where the shift parameter is near zero, but with a negative sign. This happens in some models with WLSMV and in some others with MLMV. I don't get an error message from Lavaan.

So are the results from those models correct and can I trust them or is a negative shift parameter problematic and indicating some problems with the model?

Thank you very much for your help and best regards,

Tobias 

[Yves Rosseel]

On 12/21/22 12:06, Tobias Krieger wrote:
> If I understood it correctly, scale and shift parameters influence the
> chi-square value of the robust estimators to follow a chi-square
> distribution:
> The shift parameter influences the location of the distribution and the
> scale parameter the size and shape. Is that correct?

That sounds about right. The idea is that after scaling/shifting, the
'expected' value of the modified test statistic equals df (the degrees
of freedom) again, and its variance is 2*df. The 'b' parameter is only
needed for the expectation, while 'a' is used for both the expectation
and the variance.

> Why I am writing is the following question:
> Can a shift parameter also be negative?

I see no reason why not. As long as it gets the job done.

Yves.

[Tobias Krieger]

Dear Yves,

Thank you very much for your fast answer which I highly appreciate!

This helps me a lot!

Just to better understand it: What do you mean by "expectation"?

And the 'a' and 'b' parameters are both part of the shift parameter if I understood it correctly from Asparouhov & Muthén (2010). Simple second order chi-square correction, is that correct? Or is one the shift and the other the scale parameter?

Thank you again very much and best regards,

Tobias

[Yves Rosseel]

On 1/2/23 15:32, Tobias Krieger wrote:
> Just to better understand it: What do you mean by "expectation"?

The expectation operator E(X), see
https://en.wikipedia.org/wiki/Expected_value. Think of it as the average
of its argument (X) across an infinite number of replications.

> And the 'a' and 'b' parameters are both part of the shift parameter if I

> understood it correctly from /Asparouhov & Muthén (2010). Simple second
> order chi-square correction/, is that correct? Or is one the shift and

> the other the scale parameter?

I think 'a' is called the 'scaling' parameter, and 'b' is called the
'shift' parameter, because the scaled test statistic is written as

a*T + b

where T is the original test statistic.

Yves.

[Tobias Krieger]

Dear Yves,

Thank you again for all your valuable answers and your time.

All the best

Tobias