Appeal for advice: interpreting path coefficients where latent vars based on ordinal data

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Jack Bailey

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Jul 17, 2018, 6:53:01 AM7/17/18
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Hi all,

I have the following competing mediator model (errors & correlations omitted for clarity's sake):


These latent variables are based on observed ordinal data. Econ is latent economic perceptions, Cmp_g is latent government competence. Cmp_o is latent opposition party competence. Support is latent government support. Coefficients are standardised.


My problem is that I'm not sure how to interpret the coefficients in the structural model. Ordinarily, I would have treated the variables as continuous, but they are highly skewed and I'm trying to get out of bad habits. So, do the regression coefficients on the paths (e.g. .713 from Econ to Cmp_g) represent change in the assumed latent normal distribution? Or are they still in probits?


Any help would be greatly appreciated.

Terrence Jorgensen

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Jul 17, 2018, 7:24:56 AM7/17/18
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So, do the regression coefficients on the paths (e.g. .713 from Econ to Cmp_g) represent change in the assumed latent normal distribution?


Yes

Or are they still in probits?


That's what a probit is.  The probit function links the latent response to the observed response.  So any estimated effect on an ordered outcome is interpreted in probit units, not as changes in actual probability.

Terrence D. Jorgensen
Postdoctoral Researcher, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam

Jack Bailey

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Jul 17, 2018, 8:59:16 AM7/17/18
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Thanks, Terrence. In that case, am I simply able to interpret the latent variables as if they are continuous variables in their own right that just happen to be measured in probits?

Andy Supple

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Jul 17, 2018, 9:20:21 AM7/17/18
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I interpreted the question differently from how Terrence did. Wouldn't it be the case that the latent variables themselves are continuous variables, it is the relationship from the latents to the indicators where the ordinal nature of the items comes into play? As such, the associations between the latents are interepretable just as they would be were the indicators continuously scaled?

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Jack Bailey

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Jul 18, 2018, 3:34:22 AM7/18/18
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This was what I was thinking too, Andy. But perhaps we're all talking at cross purposes. Is it the case that the relationships between the latent variables are interpretable as if their indicators were continuous, but they just happen to be measured in probits?


On Tuesday, 17 July 2018 14:20:21 UTC+1, Andy Supple wrote:
I interpreted the question differently from how Terrence did. Wouldn't it be the case that the latent variables themselves are continuous variables, it is the relationship from the latents to the indicators where the ordinal nature of the items comes into play? As such, the associations between the latents are interepretable just as they would be were the indicators continuously scaled?
On Tue, Jul 17, 2018 at 7:24 AM, Terrence Jorgensen <tjorge...@gmail.com> wrote:

So, do the regression coefficients on the paths (e.g. .713 from Econ to Cmp_g) represent change in the assumed latent normal distribution?


Yes

Or are they still in probits?


That's what a probit is.  The probit function links the latent response to the observed response.  So any estimated effect on an ordered outcome is interpreted in probit units, not as changes in actual probability.

Terrence D. Jorgensen
Postdoctoral Researcher, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam

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Terrence Jorgensen

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Jul 18, 2018, 9:23:58 AM7/18/18
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I was interpreting "latent" merely as the latent normal response underlying the observed categorical outcome, such that the constructs in your path diagram each had a single indicator.  But yes, probit regression is a generalized linear model, so the effects on the latent, probit-transformed outcome are linear (1-unit increase in the predictor is associated with a beta-units change in the latent outcome), whereas the effects are the observed outcome are nonlinear.  The linear effect is in probit units (whatever those are***), analogous to the linear effect in logistic regression being in logit units (whereas the nonlinear effect on the untransformed outcome can be expressed as an odds ratio).  

If Jack's 4 constructs are common factors with multiple categorical indicators, that interpretation applies to the effects of factors on (the latent responses underlying) their indicators.  The effects among constructs are linear because the latent variables are continuous.  The (indirect) effect of a latent predictor on a categorical indicator of a latent outcome is interpreted in probit units, but that is not something people are usually interested in expressing when working with structural regression models (even if the indicators are normal).

*** Note that probit regression outside of CFA is typically consistent with the theta parameterization, in which the residual variance (not total latent-indicator variance, as in the delta parameterization) is fixed to 1 for identification.  I've never read the specific language "probit units" because my training is in psychology, where logistic regression is used more often.  I just bring this up in case Jack (or economists in general) thinks the delta parameterization would not yield "probit units".

Jack Bailey

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Jul 18, 2018, 9:32:25 AM7/18/18
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Excellent, Terrence. Very useful!
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