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Summary  Principles and practice of structural equation modeling

1 Introduction

Er bestaat SEMNET, een mailinglist voor SEMfanaten. En er is een SEMlied, maar dat is nogal lelijk.
Belangrijke kernpunten: om SEM te kunnen gebruiken moet je veel kennis hebben van het vakgebied waarop je aan het modeleren bent. 
1.2 a Priori does not Mean exclusively Confirmatory

SEM can be viewed as confirmatory research. This means: the model is given at the start of the analysis and answered is whether the model is confirmed by the data.
 When using a strictly confirmatory application only a single model is rejected or accepted but other options are possible as well. However, this scope of testing is very narrow and does not allow other applications. Only in a FEW situations the situation is this narrow.
 Alternative Models is an application with less constraints. It means that more than one model is a priori available. Therefore, it is needed to have in depth knowledge of the theoretical field. One of the alternative models could be confirmed and all the others will be rejected.
 Model generation. This is the most common used method. When the initial a priori model does not work, and the model is modified by the researcher you can speak of Model generation. The main goal of model generation is to 'discover' a model with three properties: (1) it makes theoretical sense, (2) it is reasonably parsimonious, what means that there are little assumptions needed and as less steps in the model generation process made as possible and (3) the correspondence to the original data is very close, which make it reasonable to not reject the model.
 When using a strictly confirmatory application only a single model is rejected or accepted but other options are possible as well. However, this scope of testing is very narrow and does not allow other applications. Only in a FEW situations the situation is this narrow.

1.3 explicit distinction between observed and latent variables

Disctinction between factors (latent) and indicators (observed) allows us to test hypothesis about the measurement! A model could be specified in which X1  X3 are indicators for a specific factor and X4  X6 are indicators for a different factor, but all combined in one model.
 If the fit of the factor to the data is poor than the measurement hypothesis will be rejected.
 If the fit of the factor to the data is significant than the measurement could be used in the model.
This is different when compared to Multiple Regression and ANOVA as this two methods only can use observed variables. 
SEM knows two classes of variables: observed and latent.
Observed variables: the variables on which you have collected data and exist in the data file.
 Observed variables can be : categorical, ordinal or continuous. However, SEM only deals with continuous variables!
 A special type of observed variables is the indicator. A indicator is an observed value used as an indirect measure of the construct.
Latent variables: the hypothetical constructs or factors which are the exploratory variables but not directly measurble. Different types of observed variables are used to measure a latent variable.
 Examples of Latent variables: intelligence, motivation, groups (higher levels of analysis) but also method effects as observational issues and selfreport.

What is the residual error of indicator?A residual represents variance unexplained by the factor that the corresponding indicator is supposed to measure.

What is the unexplained variance of the residual error of the indicator?The unexplained variance of the residual error could be due to random measurement error or score unreliability

Does SEM takes into account residual error?Yes, SEM does take into account explicity the residual error. This makes the model more realistic than for example Multiple Regression that assumes that the residual error is zero.

Substantive latent variables or observed variables (or a combination of both) can be used as outcome variables. The residual error term represent the variance unexplained by their predictors. However, substantive latent variables does not have to be in the model to be a SEM model at all!

1.4 Covariances always, but Means Can Be analyzed, too

What is covariance?COVxy=Rxy*SDx*SDyThe covariance exists of the pearson correlation (Rxy) and the Standard Deviations of the correlation; x and y. Covariance represents the strength of the association between X and Y and their variability with a single number.

What are the two main goals concerning the use of covariances in SEM?1. Analyse patterns of covariance among a set of observed variables.2. Explain as much of the variance possible with the researcher's model.

What is the covariance structure?The part of SEM that represents hypothesis about variance and covariance.

What is the mean structure?The mean structure represent the estimation of factor means.

The difference between SEM and ANOVA concerning the means is that ANOVA only can use the means of the observed variables and that SEM can estimate the means of latent variables. However, means are not estimated often in SEM, but it is possible.

What is the added value to have a mean structure possibility in SEM?The added value of mean structure is that this makes it possible to test whether latent variables differ for groups, for example boys and girls, within the model.
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