Ok, I conflated measurement and structural invariance. My error. Measurement invariance implies equality of (for continuous indicator items) factor loadings, intercepts and item variances. I am less confident about the requirements for ordinal indicators but definitely factor loadings and thresholds. Mplus has the delta and theta parameterization and for delta, the scale factors can be constrained and for theta the item variances can be constrained but there is quite a bit of complexity. They describe invariance for categorical indicators in the manual and, I think, on the demo videos/handouts and on the discussion list. I don’t completely understand but it seems that Muthen’s recommendations differ from Roger Millsap’s in his 2004 MBR article.
>>Strong longitudinal measurement invariance basically does not exist when factor means increase or decrease drastically over time (as is the case in my sample). Not even the number of factors is likely to be invariant across time, because lower factor means = lower variation in the items = lower intercorrelation of items = lower probability to detect multiple factors.
Given variables measured as real numbers (range +/- infinity, decimal values) from a multivariate normal distribution, there’s no reason for the variance or covariances to differ as factor means decrease. However, the typical 1-5 or 1-7 likert scale treated as continuous could well be a different story because of floor or ceiling effects. The same seems like it ought to be true for categorical variables because changing thresholds can only mean, I think, changing amounts of skew and, therefore, different correlations.
Given what I know about your analysis, I’d use mplus. You have a question about factor composition. Certainly, separate ESEMs would give you insight into factor composition stability and let you also look at residual covariances. Regardless of whether you declare items to be categorical or continuous, you’re also going to find out about factor variances, item intercepts/thresholds and residual variances. Those numbers have to be similar for measurement invariance to hold.
Gene
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of torvon
Sent: Tuesday, July 16, 2013 10:19 AM
To: [hidden email]
Subject: Re: Binomial multivariate repeated GLM
Gene ,
You grasped my question perfectly - these are indeed 9 items of a pre-existing questionnaire that is supposed to measure one underlying disorder. We want to show, however, that symptoms react very differently to what happens to the subjects between time 1 and time 2.
Unfortunately, I would not know what analysis to perform exactly to test whether symptoms change differentially over time. I can perform measurement invariance tests in MPLUS with ordered variables, ad know much more about MPLUS that SPSS, actually, but don't think it would help.
Strong longitudinal measurement invariance basically does not exist when factor means increase or decrease drastically over time (as is the case in my sample). Not even the number of factors is likely to be invariant across time, because lower factor means = lower variation in the items = lower intercorrelation of items = lower probability to detect multiple factors.
However, differential change of symptoms over time is just one of many explanations for lack of measurement invariance across time in samples with drastically increasing factor means, so measurement invariance wouldn't really directly tackle my question, or am I missing something?
Would you have a recommendation here as to what to test? Also, item intercepts wouldn't exist in ordered models, so I couldn't compare these.
Thank you for all the helpful comments so far
ta-ta
Eiko
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