# What is the precise definition of a "Heywood Case"?

I had been using the term "Heywood Case" somewhat informally to refer to situations where an online, 'finite response' iteratively updated estimate of the variance became negative due to numerical precision issues. (I am using a variant of Welford's method to add data and to remove older data.) I was under the impression that it applied to any situation where a variance estimate became negative, either due to numerical error or modeling error, but a colleague was confused by my usage of the term. A google search doesn't turn up much, other than that it is used in Factor Analysis, and seems to refer to the consequences of a negative variance estimate. What is the precise definition? And who was the original Heywood?

Googling "Heywood negative variance" quickly answers these questions. Looking at a recent (2008) paper by Kolenikov & Bollen, for example, indicates that:

• " “Heywood cases” [are] negative estimates of variances or correlation estimates greater than one in absolute value..."

• "The original paper (Heywood 1931) considers specific parameterizations of factor analytic models, in which some parameters necessary to describe the correlation matrices were greater than 1."

Reference

"Heywood, H. B. (1931), ‘On finite sequences of real numbers’, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 134(824), 486–501."

• (+1) A good paper, thank you. I also found this article interesting, Conditions for factor (in)determinacy in factor analysis (Krijnen et al.), j.mp/dwo7c8.
– chl
Oct 31, 2010 at 9:39
• my google search for 'Heywood case definition' had been rather unsatisfactory. I am glad to see that such a google search now links to this question. Nov 1, 2010 at 4:31
• So it looks like this phrase is not used for the case of numerical issues (loss of precision) causing a negative variance estimate, but is not ambiguous. Nov 1, 2010 at 16:39
• @shabbychef The trick with searches is to focus them with appropriate keywords. "Case" and, to some extent, "definition" don't accomplish much. Including "negative" and "variance" encouraged Google to cough up more relevant material ;-).
– whuber
Nov 1, 2010 at 17:05

From the APA dictionary of psychology:

[...] any correlation coefficient, regression coefficient, factor loading, or similar parameter estimate having a value that is impossible or very rare (e.g., a negative error variance estimate). Heywood cases may indicate any of the following: a sample that is too small to adequately estimate the parameters; data that do not have a normal distribution or that contain outliers; a misspecified model that is not appropriate for the data; or a parameter whose true value is so close to a boundary (e.g., 1 or 0) in the population that its estimate exceeded this limit due to sampling fluctuation.

I found this definition more complete and useful.