a good illustration of the need for collaboration between the statistician and
research worker if, between them, they are to optimize the collection and
analysis of data.
10. Miscellanea
Clearly, measurement error models (however broadly one defines that term)
are but one aspect of functional and structural relationship problems. Their
importance in the wider context of errors-in-variables models lies in the fact
that measurement error is a component of error that is often estimable (as in
the soil nitrogen data in the example discussed in Section 9). More generally,
some form of replication gives the best hope of estimating error variances,
but, except in very special experimental situations, replication corresponding
to fixed but unknown true values is rather rare, although in certain biological
contexts it may exist (sometimes attainable through use of an instrumental
variable, especially time). Whether or not one interprets natural biological
variation in such a situation as a measurement error, the mathematical treat-
ment required is the same as that for a physical measurement error.
This brief survey has reflected my personal interests; to those steeped in
econometric models or who specialize in factor analysis, my choice of topics
for discussion may have come as a disappointment. I have, however, tried to
reflect the importance of (even simple) structural and functional relationship
models in stimulating research in methods both in statistical inference and in
practice in an interdisciplinary role. One might add that research in this area
has been very international, as evidenced by the attendance at the Dundee
meeting in 1983 on Functional and Structural Relations and Factor Analysis
(FASRAFA) and at this meeting. The one major school working in this area
that has not been represented at either meeting is that from the German
Democratic Republic, but fortunately there is free access to their published
I often regret the diversity of notation that has grown up in this subject
area, so I welcome the plea from the editors of these proceedings to stan-
dardize notation as far as possible to that used in Fuller ( 1987). Although
his notation would not be everybody's first choice and may need extension to
deal with some aspects of error models not confined to measurement errors,
it is high time we grabbed the notational nettle: since Fuller has provided
the first comprehensive treatise on the subject, adopting his notation would
provide the best hope for setting up a standard.
11. Acknowledgments
I am grateful to Wayne Fuller and Leon Gieser for inviting me to the
Summer Research Conference and to a referee who suggested useful additions
and clarifications.
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