As in the scalar case, the weighted mean of multiple estimates can provide a maximum likelihood estimate. For independent estimates we simply replace the variance by the covariance matrix and the arithmetic inverse by the matrix inverse (both denoted in the same way, via superscripts); the weight matrix then reads (see https://en.wikipedia.org/wiki/Weight...lued_estimates)

,

where stands for the covariance matrix of the vector-valued quantity .

The weighted mean in this case is:

(where the order of the matrix-vector product is not commutative).

The covariance of the weighted mean is:

For example, consider the weighted mean of the point with high variance in the second component and with high variance in the first component. Then

then the weighted mean is:

On the other hand, for scalar quantities it is well known that correlations between estimates can be easily accounted. In the general case (see https://en.wikipedia.org/wiki/Weight...r_correlations), suppose that , is the covariance matrix relating the quantities , is the common mean to be estimated, and is the design matrix (of length ). The Gauss–Markov theorem states that the estimate of the mean having minimum variance is given by:

with

The question is, how can correlated vector-valued estimates be combined?

In our case, how to proceed if and are not independent and all the terms in the covariance matrix are known?

In other words, are there analogous expressions to the last two for vector-valued estimates?

Any suggestion or reference, please? ]]>

I am looking to assess a 15-item measure to fit on to a one-factor structure. After eliminating poorly-fitting items (based on poor factor loadings and corrected-item total correlations), I have been have been left with two possible new models, a 9 or 8-item model.

The 9-item model shows better model fit; X2= 47.672, p = 0.002, CMIN/DF = 2.073, CFI = 0.949, TLI = 0.92, RMSEA = 0.067, PCLOSE= 0.142, however, the corresponding Chronbach’s is unacceptably low (α = 0.497).

Alternatively, the 8-item model presents slightly lower goodness of fit; X2 = 49.995, p ≥ 0.001, CMIN/DF = 3.125, CFI = 0.931, TLI = 0.87, RMSEA = 0.94, PCLOSE = 0.008, but with much higher internal consistency (α = 0.778).

My question is which model would be more appropriate to report in my results. I recognise that the problems I’ve had with finding appropriate model-fit seem to confirm the original measure to be poor. This is not an issue as it is going to represent a key critique of the surrounding literature in my discussion. However, I still want to report the best, most appropriate model I have been able to find, even if this is with the acknowledged caveat of either poor internal consistency or slightly lower model fit.

Any advice would be greatly appreciated! :)

Thanks ]]>