David A. Kenny
December 18, 2002
(thanks to Jim Conway)
Multitrait Multimethod Matrix
Definition A set
of t traits are each measured by m methods. The resulting
data are tm measures, and the correlation matrix is called a multitrait-multimethod
matrix. The matrix was originally proposed by
Donald T. Campbell and Donald Fiske. The matrix is commonly abbreviated as MTMM.
Standard MTMM Evaluation Convergent
validity is demonstrated by strong correlations between two methods measuring
the same trait. Discriminant validity
is demonstrated by weak correlations between two different traits measured
by two different
methods relative to the same trait different method correlations.
The lack of method variance is demonstrated by weak correlations between
two traits measured by the same method, relative to same two traits measured
by different methods.
Standard CFA Estimation The standard
confirmatory factor analysis model of the matrix is to have each measure
load on its trait and method factors. The traits factors are correlated,
as well as the method factors. Usually, the trait and method
factors are assumed to be independent. There must be at least three
traits and methods for this approach to be identified. The loading
structure is as follows:
Factor
Trait Method
Measures 1 2 3
1 2 3
T1M1
x x
T2M1
x x
T3M1
x x
T1M2
x
x
T2M2
x x
T3M2
x x
T1M3
x
x
T2M3
x
x
T3M3
x x
where an "x" means that
the measure loads on the relevant trait or method factor.
Difficulties The standard
CFA model for the MTMM is not identified for two very important cases:
when the loadings for each factor are exactly equal or when there is no
discriminant validity between two or more
factors. Although actual data never exactly satisfy these conditions,
they usually approach one of the cases, and so the standard CFA model is
empirically underidentified. Heywood
cases, impossible values (correlations larger than one and negative
variances), and convergence problems are quite commonly found during estimation.
Thus given these problems, the "standard" model should not be estimated.
Correlated Uniqueness Model In this model,
there are no method factors, but measures that share a common method have
correlated errors or uniquenesses. The error variance-covariance
matrix would be as follows:
T1M1 x
T2M1 x
x
T3M1 x
x x
T1M2
x
T2M2
x x
T3M2
x x
x
T1M3
x
T2M3
x x
T3M3
x x
x
T1M1 T2M1 T3M1
T1M2 T2M2 T3M2
T1M3 T2M3 T3M3
where an "x" means
that a free error variance or covariance.
For this model to be identified there must be at least two traits and three
methods. This model does not have the difficulties that the standard
CFA model has.
Direct Product Model This model
can be very difficult to set up and to interpret. It is sometimes
referred to as the "multiplicative model." Within this model, the
correlation between two measures equals the product of the similarity between
traits being measured, the similarity of two methods being used,
and how much variance of each measure is meaningful (communalities).
Within this model, the same trait measured by dissimilar methods should
correlate zero because the correlation between measures is the product
of the similarity between the methods as well as traits.
One of the advantages of this
method is that it estimates a correlation matrix for the methods.
With this matrix, one can determine the similarity of the different methods.
It is possible that the similarity between methods might be one which would
mean that the methods that were nominally different were in fact the same.
In essence, the methods would have no discriminant validity.
There have been a few comparisons
between the empirical utility of the standard additive and this newer multiplicative
model. The additive model appears to work better. Nonetheless,
the direct product model deserves
attention.
