Mixed Model Variance Analysis with Normal Error and Possibly Non-Normal Other Random Effects: Part I: The Univariate Case

Abstract

The mixed model with one factor represented by fixed effects, one factor by random effects, and a normal error, has often stipulated that these random effects be a sample drawn from a normally distributed population. In the case of a single response (or univariate) experiment, the variance of this normal distribution is a natural measure of the dispersion of these random effects, and confidence bounds on the ratio of this variance to the error variance [13], [14], [9] and simultaneous confidence bounds on both variances (in the latter case with a confidence coefficient $\geqq$ a specified value) [9] have already been found for certain classes of experimental designs. But when a distribution is not normal--or not assumed at the outset to be normal--the variance may not reveal as much about the distribution as some other measure such as interquartile range. In the present paper we seek confidence bounds on what, in a sense to be explained presently, might be called representations of the interquartile range and of analogous differences between higher order quantiles of the population from which the random effects are drawn. The method of obtaining these bounds involves an element of approximation comparable to grouping continuous data into $k$ classes, since it replaces the actual random-effects variate by a "substitute variate" having $k$ equally probable discrete values. The main idea is this. Let us assume, for simplicity of discussion, that we have a real valued stochastic variate. One comment here might be helpful. If the stochastic variate is observable, it seems natural to attempt to approximate its unknown distribution by introducing unknown probabilities over a finite set of preassigned class intervals, then trying to estimate these probabilities and then (especially for a continuous distribution) increasing the number of class intervals. On the other hand, if the variate is unobservable, as in the present set-up, it seems natural to try to approximate the distribution by replacing the stochastic variate by a "substitute" variate which is supposed to take, as a first approximation, two (unknown) values with equal probabilities, or as a second approximation, three (unknown) values with equal probabilities, or in general $k$ (unknown) values with equal probabilities. We then try to estimate, in terms of our observations, these unknown values, which may be regarded as approximations to the 1st, 3rd, $\cdots, (2k - 1)$th quantiles of the unknown distribution. The random effects variate postulated in our model may have either a continuous or a discrete distribution, provided in the latter case there are enough distinct values to make these $k$ quantiles meaningful parameters of the distribution. From now on, for brevity, we shall refer to these unknown values as the quantiles of the unknown distribution. It will be seen later that it is the differences between these unknown values rather than the unknown values themselves which we can estimate or make inferences about, and the number, $k - 1,$ of such differences which can be estimated is restricted by the experiment. Turning now to the confidence bounds, we observe that in the derivation of these bounds use is made of the same kind of sums of squares as in the normal variance components analysis. Unlike the more familiar confidence statements where the confidence coefficient may be specified at will, here except for the case of two blocks, only a lower bound on the confidence coefficient is specifiable, and this includes as a factor a decreasing function of $k$, the number of discrete values of the substitute variable. For $k = 2, 3, 4, 5$ the geometric shape of a $(k - 1)$-dimensional confidence region has been found. It is also shown how the usual inference about the fixed effects can be made from this model and then how the above type of confidence bounds can be found for each of several random-effects factors in an experiment with orthogonal design. In the case of a multiresponse--usually called multivariate--experiment, the model frequently stipulates that the random effects be samples from a multivariate normal population. Although the variance matrix of this distribution has a readily available estimator, confidence bounds have presented many difficulties. For the extremely restricted model in which the variance matrix of each random-effects factor is proportional to the variance matrix of the error, Roy and Gnanadesikan [10] obtained simultaneous confidence bounds on the characteristic roots of this latter matrix and on the proportionality constants. In Part II of this paper the authors present (with a confidence coefficient greater than or equal to a preassigned value) confidence bounds on the characteristic roots of the variance-covariance matrix of a random-effects variate without assuming any such relation to the error matrix. The second part will also consider the $p$-variate extension of the univariate substitute variate and the associated confidence bounds for the case where the $p$-dimensional distribution of the random effects is not necessarily normal. This development will be only indicated in principle for $p > 2$ but will be discussed in some detail for the case $p = 2$.