## Technote (troubleshooting)

## Problem(Abstract)

At times I get a message from Amos which states that I have negative variances in my model and that my solution is not admissible. What does this mean and why do these messages occur?

## Resolving the problem

You may at times see both of these errors in the Amos output:

The following variances are negative.

This solution is not admissible

Although variances cannot be negative, Amos can produce variance estimates that are negative. The solution is then called inadmissible.

Negative variances and R-squared values greater than 1 are not theoretically possible, so the solution is considered improper and the other estimates are not reliable. When such out-of-range variance estimates occur, it is known as a Heywood case. There is a helpful description of Heywood cases in Chapter 2 of:

McDonald, R.P. (1985). Factor Aanalysis and Related Methods. Hillsdale NJ: Erlbaum.

McDonald notes that a common cause of Heywood cases is a failure to represent each factor with a sufficient number of variables with large loadings and suggests that researchers insure that every common factor is defined by at least 3, and preferably 4, variables with large loadings on it. He notes that sometimes a Heywood case can be 'cured' by fitting fewer factors, but notes that this may give an unacceptably small fit to the data. The reference to large loadings reflects the exploratory factor analysis context in which McDonald was discussing Heywood cases. In the context of confirmatory factor analysis, the implication is that it is preferable to have more than 2 manifest variables defining a latent variable. It is not mandatory and there are several examples in the AMOS User's Guide of latent variables that are defined by only 2 observed variables. Sometimes you can work around a Heywood case by placing additional constraints that may not be part of the theory but are not inconsistent with the theory. For example, if the model is a multiple group model, error variances with negative estimates in one group can be constrained to be equal across groups, without constraining the actual value. There may be other constraints that can be tried, such as constraining error variances within the same factors to be equal. These tricks can become ad hoc if not supported by the theory behind your model. .

Sometimes you can get around the negative variance problem by changing the estimation method from the default of maximum-likelihood to either generalized least squares (GLS) or unweighted least squares (ULS), as the latter 2 methods seem to be less prone to Heywood cases. There is a discussion of this approach at

https://stat.utexas.edu/software-faqs/lisrel

The discussion there concerns LISREL, but is applicable to AMOS. In AMOS, you would choose GLS or ULS by opening the View Menu and choosing 'Analysis Properties'. Then click the Estimation tab in the 'Analysis Properties' dialog box and check the radio button for either 'Generalized least squares' or 'Unweighted least squares'.

The negative variance message may indicate that some exogenous variables have an estimated covariance matrix that is not positive definite. Joreskog and Sorbom (1984) suggest that the problem may indicate that either that your model is wrong or that the sample is too small.

Joreskog, K.G. and Sorbom, D. (1984). LISREL-VI user's guide (3rd ed.). Mooresville, IN: Scientific Software.

It is possible to prevent the occurrence of negative variance estimates, and it may even be possible to prevent the occurrence of inadmissible solutions in general, by restricting the search for a solution to admissible parameter values. If you are using Bayesian estimation (AMOS versions 6 and above), you can define the prior distribution for a particular variance estimate to be positive. There is also an 'admissibility test' option to force variance estimates to be non-negative. See Example 27 in the AMOS User's Guide for a discussion of this option.

## Historical Number

36816