Do "intrinsic" negative variance components exist for population variances of stochastic physical parameters?
2016-09-20T20:47:09Z (GMT) by
<p><b><i>NOTE 1:</i> </b>Errata and corrections have been published for this paper at the reference provided at <a href="https://dx.doi.org/10.6084/m9.figshare.3978087.v1">https://dx.doi.org/10.6084/m9.figshare.3978087</a></p><p><b><i>NOTE 2:</i></b> Further work which includes an analysis of intrinsic negative variance components as a consequence of non-normal stochastic physical parameters has been published at <a href="https://dx.doi.org/10.6084/m9.figshare.3979128.v1">https://dx.doi.org/10.6084/m9.figshare.3979128</a></p><p><b>Brief Introduction </b></p> <p>In statistical problems involving the random effects analysis of variance (ANOVA) model, it is known that, in some cases, estimates of the components of variance can be negative. It has been suggested that any occurrence of negative variance component estimates is “embarrassing” because it is fundamentally assumed that variance components must, by definition, be non-negative . That is, since variance is a squared quantity, then this quantity could not be negative , unless there are certain statistical errors or if the model is incorrect or inadequate (e.g., if variables in the data are negatively correlated, then that would result in negative covariances ).</p> <p>However, I have done some thinking and analysis in this area, and I have found that if a function of random variables is expressed in terms of a second-order Taylor series, then its variance (i.e., variance resulting from a large sample of discrete values of those random variables) could indeed consist of at least one negative variance component, depending on the nature of the function. The attached technical communication presents a physical and mathematical basis for the existence of these "intrinsic" negative variance components. Further, I provide a practical example in my field of optical design and engineering where the variance of a large sample of refractive optical powers of lenses called "achromatic doublet lenses" produced from a hypothetical lens manufacturing plant would consist of at least one negative variance component. </p> <p><b>Origin and Background of this Study</b></p> <p>I was interested in this area from the perspective of optical design and tolerancing analysis. Most of the time, optical tolerancing analysis of lens parameters and variables such as focal length and refractive index are performed with the assumption that errors sum in quadrature without any so-called "interaction components of variation". Ordinarily, this assumption is a good one, because optical designs require lens components to be produced with very small magnitudes of tolerances. Under such conditions, the "interaction components" are negligible. Additionally, statistical independence is assumed (i.e., variables are not correlated), except for instances where certain optical system variables (such as the air spaces between lenses) are used as so-called "compensators" which would be adjusted to negate the effects of significant random lens parameter errors arising from random production errors that are difficult to control. These problems led me to think about whether or not there were relationships between the assignment of optical tolerances to a lens design and ANOVA and components of variance analysis. </p><p>In my search for these relationships, I had found that although the literature available on error analysis, experimental physics, ANOVA, and regression are plentiful, it was somewhat difficult to put the pieces together. Texts on experimental physics focus on error propagation and skillfully designed "one factor at a time" experiments (which have been very successful), while statistics texts discuss ANOVA which stress on statistical design of experiments (DOEs) with "interactions", but do not illustrate how such models have aided in the development of fundamental equations in physics. Optical tolerancing is discussed in optics literature, but they do not really state when interactions and ANOVA might play a role. Interestingly, in one fairly recent study, Prateek Jain  applies the method of principal components analysis to optical tolerancing. Admittedly, I have not fully studied Jain's paper, primarily because I am unfamiliar with principal components analysis, but I guess that it may be of some interest to look more into it. </p> <p>In any case, while performing my researches, I had gone somewhat off my main path, and had encountered the problem of negative variances in ANOVA which seemed interesting. I had then proceeded to find fundamental equations in geometric optics where nonlinearity could lead to negative variance components. While performing such analysis, I had arrived at the results presented in the attached technical communication, which I felt did not quite fit into any specific research journal, due to its breadth of coverage on many fields such as optics, physics, multivariable calculus, and statistics (though future work in this area might fit into the "interdisciplinary" section of The American Statistician.) Meanwhile, I have decided to make my preliminary studies available for open review here on Figshare. </p> <p><b>References and Notes</b></p> <p>1. H. Sahai and M. I Ageel, <i>The Analysis of Variance: Fixed, Random, and Mixed Models</i>, (Birkhauser, Boston, 2000), p. 28.</p><p>2. Well, I suppose that if the data being analyzed is comprised of a set of complex numbers (i.e., a + ib where i is the square root of -1), then its standard deviation is a complex number, and hence, the square of the standard deviation of a complex number (i.e., its variance) would be negative. But this is beyond the scope of the present study (though it would be rather interesting to speculate the consequences of this idea).</p> <p>3. D. W. Smith and L. W. Murray, “An alternative to Eisenhart’s Model II and Mixed Model in the Case of Negative Variance Estimates,” <i>J. Amer. Statist. Assoc.</i> <b>79</b> (1984), pp. 145 – 151.</p> <p>4. P. Jain, "Optical tolerancing and principal components analysis," <i>App. Opt.</i> <b>54</b> (2015), pp. 1439 - 1442.</p>