Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 71

germane to the stated scientific problem, it is best to avoid them alto-
gether in analysis. In this context, “avoidance” does not mean inclusion
of an arbitrarily chosen rotation and translation system for mathe-
matical convenience. “Avoidance” means that any inferences made from
the data should remain the same regardless of, or invariant with
respect to, such an arbitrary choice. The existence of translation and
rotation affects the estimation of the essential parameters of interest:
the mean form M , and the covariance structure . For this reason,
translation and rotation are referred to as “nuisance parameters” in
statistics and must be considered carefully.
The difficulty of statistical inference in the presence of nuisance
parameters is well-studied in statistics. Jerzy Neyman and Elizabeth
Scott (Neyman and Scott, 1948) used a simple one-dimensional exam-
ple to illustrate the effect of nuisance parameters on statistical
inference. We consider a similar example here to clarify the concept of
nuisance parameter.
Suppose that we have introduced a new drug to treat high blood
pressure. We are specifically interested in the intrinsic efficacy of the
drug and the variability of the effectiveness. Let Y ij denote the reduc-
tion in blood pressure for the j -th patient treated in the i -th clinic.
Suppose that we treat two patients per clinic and that there are I clin-
ics in the study. Let denote the average reduction in blood pressure
and

2 denote the variability in this effect due to patient variability.
Because each clinic also has a slightly different environment, the mean
response to the administration of the drug varies from clinic to clinic,
but we are not interested in this aspect. Our interest lies in the effect
of the drug, , and the variability in this effect,

2 . The model that
describes the relationship among the parameters is Y ij ~N (

2 ) ,

α

i ,

1,2...., where α denotes the “clinic effect.” Given the
research problem and specific research question posed above, the dif-
ference in the effect of the drug from clinic to clinic, or the clinic effect ,
α i , is a “nuisance” parameter. Not only are we not interested in the nui-
sance parameter, estimation of the nuisance parameter can influence
estimation of the parameters of interest.
The field of statistics is based on the idea that as the sample size
increases; the amount of information increases and estimates that
more closely approach the truth are obtained. An estimator is called a
consistent estimator if, as sample size increases, the estimator more
closely approximates the true value. Nuisance parameters, if not taken
into account properly, can lead to statistical anomalies such as non-
estimability and inconsistency. Neyman and Scott (1948) showed in the

i 1,2,…, I ; j