Multitrait-multimethod matrix
The multitrait-multimethod (MTMM) matrix is an approach to examining construct validity developed by Campbell and Fiske (1959). It organizes convergent and discriminant validity evidence for comparison of how a measure relates to other measures. Definitions and key components. Multiple traits are used in this approach to examine (a) similar or (b) dissimilar traits ( constructs), as to establish convergent and discriminant validity between traits. Similarly, multiple methods are used in this approach to examine the differential effects (or lack thereof) caused by method specific variance. There are six major considerations when examining a construct's validity through  the MTMM matrix, which are as follows: Example. The example below provides a prototypical matrix and what the correlations between measures mean. The diagonal line is typically filled in with a reliability coefficient of the measure (e.g. alpha coefficient). Descriptions in brackets  indicate what is expected when the validity of the construct (e.g., depression or anxiety) and the validities of the measures are all high. In this example, the first row and the first column display the trait being assessed (i.e. anxiety or depression) as well as the method of assessing this trait (i.e. interview or survey as measured by  fictitious measures). The term heteromethod indicates that in this cell the correlation between "two separate methods" is being reported. Monomethod indicates the opposite, in that the same method is being used (e.g. interview, interview). Heterotrait indicates that the cell is reporting two supposedly different traits. Monotrait indicates the opposite- that the same trait is being used. In evaluating an actual matrix one wishes to examine the proportion of variance shared amongst traits and methods as to establish a sense of how much method-specific variance is induced by the measurement method, as well as provide a look at how unique  the trait is, as compared to another trait. That is, for example, the trait should matter more than the specific method of measuring. For example, if a person is measured as being highly depressed by one measure, then another type of measure should also indicate that the person is highly depressed. On the other hand, people who appear highly depressed on the Beck Depression Inventory should not necessarily get high anxiety scores on Beck's Anxiety Inventory. Since the inventories were written by the same person, and are similar in style, there might be some correlation, but this similarity in  method should not affect the scores much, so the correlations between these measures of different traits should be low. Analysis. A variety of statistical approaches have been used to analyze the data from the MTMM matrix. The standard method from Campbell and Fiske can be implemented using the MTMM.EXE program available at: https://web.archive.org/web/20160304173400/http://gim.med.ucla.edu/FacultyPages/Hays/utils/ One can also use confirmatory factor analysis due to the complexities in considering all of the data in the matrix. The Sawilowsky I test, however, considers all of the data in the matrix with a distribution-free statistical test for trend. The test is conducted by reducing  the heterotrait-heteromethod and heterotrait-monomethod triangles, and the validity and reliability diagonals, into a matrix of four levels. Each level consists of the minimum, median, and maximum value. The null hypothesis is these values are unordered, which is tested against the alternative hypothesis of an increasing ordered trend. The test statistic is found by counting the number of inversions (I). The critical value for alpha = 0.05 is 10, and for alpha = .01 is 14. One of the most used models to analyze MTMM data is the True Score model proposed by Saris and Andrews (). The True Score  model can be expressed using the following standardized equations: The assumptions are the following: Typically, the respondent must answer at least three different questions (i.e. traits) measured using at least three different methods. This model has been used to estimate the quality of thousands of survey questions, in particular in the frame of the European Social Survey.