## Structural Equation Models

Structural Equation Models have been an area of interest for me for a very long time. In my PhD thesis, many years ago, I was preoccupied with analyzing errors-in-variables models. The errors-in-variables model can be interpreted as a specific Structural Equation Model. I have also developed a numerical procedure to fit non-linear errors-in-variables models. Later, in collaboration with my colleague Tenko Raykov, I have worked on different aspects of model choice and on model equivalence issues in Structural Equation Models. These are important practically relevant issues- it turns out that model equivalence (with respect to fit) appears much more often in Structural Equation Models than in other statistical models. Equivalent models represent, in a sense, a threat to the validity of inferences and need a careful investigation.

I have also been interested in estimation and confidence interval construction for latent correlations and for scale reliability. Testing for significance of latent regression in a Structural Equation Models is also an important issue and investigating the asymptotic power of such test draws a lot of attention. In a recent paper, we have shown some explicit formulae for this power in relation to the maximal reliability of the latent variable measurement indicators. I also have dealt with a detailed investigation of the relationship between the important psychometric concepts of maximal reliability and maximal validity. In the case of congeneric measurements, the maximal reliability and maximal validity are attained with the same weights but this is no longer true when the measures are not congeneric. We have established some useful inequalities in the latter case. The concept of reliability and maximal reliability of measurement indicators is meaningful also when these indicators are discrete but the treatment is more specific in this case. We have studied the asymptotic bias of the resulting estimators in order to improve the estimation via bias correction.

Structural Equation Models are only a part of the larger class of latent variable models. These models enjoy great popularity recently. Within this setting, the GLLAMM (generalized linear latent and mixed models) are general enough yet with a sufficient structure to allow flexible modeling of multivariate responses of mixed type, including continuous and categorical as a function of latent factors or random effects. Many challenges emerge in inference for these models and I am having a closer look at them.