HISTORY OF RELATIONSHIPS 7
Problems that arise in estimation tend to manifest themselves in different
guises for the two models. In particular, the asymptotic theory for each is
very different. In Section 2 we pointed out that a difficulty found by Lindley
for the MLE of the slope of a functional relationship was attributable to the
likelihood not having a maximum. Even when further assumptions are made
that ensure a maximum for the likelihood function, inconsistencies in esti-
mation of error variance persist. These can be attributed to the number of
parameters being a monotonic function of the number of observations. In-
deed, consistency problems may persist for slope estimators unless the mean
square deviation for true values converges to a finite limit as the number of
observations increases.
The parallel in the bivariate linear structural relationship model to that of
no maximum for the likelihood for the functional model is the unidentified
case for (1) with x, ul' e1 being NI[(,U, 0, 0), diag(axx'
auu' aee)].
With-
out further assumptions, there are six parameters to estimate, i.e.,
Po, P
1
,
.u,
axx, auu, aee,
but only five estimating equations. The situation is widely
discussed in the literature; see, e.g., Moran (1971) and Fuller (1987, Section
1.1.3).
To resolve these rather different problems in either functional or structural
relationships, some assumptions must be made about the error variances.
Generally speaking, similar assumptions in either case will lead to the same
point estimators of slope and intercept. The classic "minimum" assumption
in the bivariate case is that the ratio of the error variances is known. Con-
fidence intervals and questions of consistency will still be model dependent.
Anderson ( 1984) compared the two models to fixed and random effect mod-
els in the analysis of variance. Depending on the observational set-up, there
are also parallels between the structural model and random sampling and
between the functional model and designed experiments.
Following a suggestion from Kendall and Stuart in the original ( 1961) edi-
tion of their
Advanced Theory of Statistics,
Vol. II, Chapter 29, of a model
that included both the functional and structural relationship, Dolby ( 1976)
considered what he termed the
ultra-structural relationship.
One way of look-
ing at this model is that there are groupings of data having
within groups
structural relationships (the same relationship in each group) and a
between
groups
functional relationship with the same slope and intercept as that within
groups. Not surprisingly, certain minimum assumptions are needed to en-
sure identifiability for the structural components and existence of maximum
likelihood estimates and consistency for the functional component. Some dif-
ficulties with Dolby's original estimators were considered by Patefield ( 1978).
Gieser ( 1985) showed that, without replication within groups, the model may
be treated as a functional model for which there is an equivalent structural
model. Using this example, Gieser pointed out that some difficulties of esti-
mating parameters in functional relationships may be avoided by considering
a corresponding structural model.
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