Trait, Occasion and Error

When the same measure is taken on several occasions, what it captures at any one of them is partly the person, partly the moment and partly error. A latent state–trait model separates the three, but the separation is left inside the parameters: the variance of the trait, the residual variance of each occasion and the error variance of each variable. This page reads them and writes out the three shares for every occasion and for every measured variable — consistency, occasion specificity and reliability — in the form Geiser (2021) sets out for the singletrait–multistate model.

What to request from Mplus. The model itself, with a factor for each occasion measured by the variables of that occasion and one trait factor measured by the occasion factors, and OUTPUT: STDYX;. The loadings, the variance of the trait, the residual variance of every occasion and the error variance of every variable are read from MODEL RESULTS; the R-square of the standardized solution is read too, and reported beside the reliability the page computes, which is the same quantity. Nothing on this page can be typed in.

What to do

  1. Fit the model with one factor per occasion and a trait factor over them, and add OUTPUT: STDYX;.
  2. Drop the .out file on the box below.
  3. Read how much of each occasion, and of each variable, is the person and how much is the moment.
Drop the Mplus .out file here, or press Enter to choose one.

The file is read in your browser and is not sent anywhere.

The occasions

The occasions appear here once an output has been read.

The measured variables

The variables appear here once an output has been read.

Methods and formulas

The model the page reads

The singletrait–multistate model has one latent state variable for every occasion, measured by the variables taken on that occasion, and one trait variable measured by the latent states. What a state does not owe to the trait is its state residual: the influence of the situation and of the interaction between the person and the situation at that occasion (Geiser, 2021, Section 5.4.2). The measurement equation of a variable is therefore

Yit=λit(γtξ+ζt)+εit

with ξ the trait, γt the loading of the latent state at occasion t on that trait, ζt the state residual of occasion t, and εit the measurement error of that variable at that occasion. The trait, the state residuals and the errors are uncorrelated with one another, and that is what makes the variance add up in parts.

The three parts

The variance of a latent state and of a measured variable decompose as

Var(τt)=γt2Var(ξ)+Var(ζt),Var(Yit)=λit2Var(τt)+Var(εit)

where γt is the loading of the state of occasion t on the trait, which is one when the trait is scaled on that occasion. From this the three coefficients follow (Geiser, 2021, Section 5.4.2.3):

Con(Yit)=λit2γt2Var(ξ)Var(Yit),Osp(Yit)=λit2Var(ζt)Var(Yit),Rel(Yit)=Con+Osp

and the same two ratios taken over the variance of the latent state rather than of the observed variable give the consistency and the occasion specificity of the occasion itself, Con(τt)=γt2Var(ξ)/Var(τt) and Osp(τt)=Var(ζt)/Var(τt). The factor γt2 is essential whenever a state loading is not one. Where the trait scale is set on one occasion, the page reports every state loading so that this weighting is visible.

What the numbers say

Consistency is the share of a measure that is the person; occasion specificity is the share that is the situation and the interaction between the person and the situation; reliability is the two together, and is what is left once measurement error is taken out. A measure whose latent state carries more than half its variance from the trait is closer to a trait than to a state, and one whose state residual carries more than half is the other way round; Geiser (2021, Box 5.3) puts it that way and adds that no fixed rule exists, which is why the page reports the shares and not a verdict.

The reliability the page computes is the same quantity as the R-square Mplus prints for that variable in the standardized solution, and where the file carries it the two are set side by side and the largest departure between them is reported. What the page adds is the split of that reliability into the part that belongs to the person and the part that belongs to the occasion, which the single number does not show.

Unsupported inputs

A model that does not have the shape described: exactly one factor measured by other factors, and at least two factors measured by observed variables. A file in which the variance of the trait, the residual variance of an occasion or the error variance of a variable is missing. The output of a multiple-group run and of a two-level run. A run whose messages put it among those whose results are not to be interpreted. The page reads the model as it is written and does not check that the occasions are occasions: if the factors below the trait are the facets of a scale rather than repeated measurements, the same three numbers are the general and specific parts of that scale, and the words consistency and occasion specificity do not apply to them.

Self-test

Append ?selftest=1 to run the in-browser checks. They cover parameter parsing, state and observed-score variance decompositions, the identity between reported reliability and standardized R², non-unit trait loadings, and unsupported model structures.

References