Multi trait multi method is a measurement model that accounts for correlations between traits measured by different methods. This type of analysis can be used to examine the effects that a particular measure has on other measures, such as the effect of one instrument on another. The goal of this analysis is to get a sense of how much variance in the data is due to the trait being measured, and how much is due to the particular measurement method being used.

There are several types of multitrait multi method mt mm models available. Two common models are the latent difference (LD) and the latent mean (LM) model. These models are based on classical test theory and can be used to estimate trait-method correlations. They can also be used to determine the degree to which a method factor increases or decreases as a function of trait level.

In the LD model, there are m standardized trait factors, and each method is associated with a standardized method factor. A linear relationship is modeled between the reference trait factor and the m method factors, with an intercept (b0m) and a regression slope (b1m). The covariance between the trait and method variables can then be estimated. The variance of the residual variable V ar(zm*) provides a measure of the magnitude of the method factor.

The LM model is a more sophisticated version of the LD model that is able to accommodate quadratic relationships between traits and methods. This allows researchers to ask more complex questions about convergent validity, such as whether method effects increase or decrease as a function of trait level. For example, a researcher might use the LM model to evaluate the effect of extraversion on the correlation between peer and self-ratings. This might show that as a person’s levels of extraversion increase, the discrepancy between self-ratings and peer reports will decrease.

It is important to note that the LD and LM models do not provide direct evidence about how well a particular measurement method relates to a specific trait. This is a limitation of these types of multitrait-multimethod analysis models. A more complete assessment of the validity of a measurement tool is possible using other methods, such as confirmatory factor analysis. For example, Campbell and Fiske (1959) outlined four criteria that can be used to evaluate convergent validity: the fact that measures of the same trait correlate reasonably highly, the degree to which correlations between measures vary across methods, the extent to which differences among methods are related to variation in trait level, and the presence or absence of discriminant validity. The correlations in the matrix that share neither trait nor method are called heterotrait-monomethod correlations, for example A1-B1 in the upper left of the triangle. These are not useful for evaluating construct validity, but may be useful in determining the likelihood that a measurement tool has been biased. For example, a bias could result in an error that makes it impossible to distinguish between depression and anxiety symptoms in a sample of children with severe symptoms.