V1004637 Errors of Measurement in Statistics - Carnegie Mellon University
TECHNOMETRICS
Errors
VOL. 10, No. 4
of Measurement
NOVEMBER 1968
in Statistics*
W. G. COCHRAN
Harvard
University
In this review of some of the recent work in the study of errora of measurement,
attention is centered on the type of mathematical model used to represent errom of
measurement, on the extent to which standard techniques of analysis become
erroneous and misleading if certain types of errors are present (and the possible
remedial procedures), and the techniques that are available for the numerical study
of errors of measurement.
1. INTRODUCTION.
This paper reviews some of the recent work in the study of errors of measurement as they affect data analysis. This is a difficult field, as evidenced by the
prolonged struggles of the econometricians to find satisfactory methods for
coping with such errors in their investigations of relationships between variables,
and by the slow rate of progress that has rewarded major efforts to study errors
of measurement in sample surveys. It is also a field of growing importance.
For example, the recent rapid increase in data-gathering projects in the social
and medical sciences is producing large bodies of data containing variables
obviously difficult to measure, such as people¡¯s behavior, opinions, feelings,
and motivations. Concurrently, there are signs of a rise in research interest
in addition to that stimulated by problems in econometrics and sample surveys,
and one of the objectives of this review is to encourage research in this area.
Attention will be centered on four aspects of the problem.
(1) The types of mathematical model that have been used to represent
errors of measurement.
(2) The extent to which errors of measurement are automatically taken into
account in the standard techniques of analysis, and the extent to which these
methods become erroneous and misleading if certain types of errors are present.
(3) The amount of harm done by errors of measurement, from the viewpoint
of the investigator, in producing unsuspected biases or reduced precision, and
the remedial procedures that are available to avoid these undesirable consequences.
(4) Techniques for the study of errors of measurement.
Before looking at mathematical models that attempt to describe the statistical properties of errors of measurement, a reminder about some of the
Received June, 1968.
* This work wa+~facilitated by a grant from the National Science Foundation (C&341),
and by the Office of Naval Research, Contract Nonr 1866(37). Thii paper is hmed on the
Rieta lecture of the Institute of Mathematical Statistics, Washington, D. C., December, 1967.
637
638
WILLIAM
G. COCHRAN
different situations in which measurements are made may be useful. The quantity
being measured may be fixed and static through time. It may be fixed at a
point in time, but vary with time. Sometimes we deliberately measure the
wrong quantity, using a substitute measurement that is cheaper or more convenient, as for example in the standard manometer used for measuring blood
pressure by a sleeve wrapped round the arm instead of a direct measurement
of intra-arterial
blood pressure. Sometimes we measure the wrong quantity
because we don¡¯t know how to measure the right quantity. I once spent some
time trying to dev se a practical method of measuring the strength of maternal
affection in the dairy cow, but did not get far, When the quantity being measured
varies with time, we may in effect be using the wrong measurement if we treat
measurements made some months earlier as if they applied to the situation
today. Sometimes the error of measurement is actually a sampling error. The
amount of insect infestation on a plot in a field experiment may be estimated
by taking from the plot a small sample of plants and recording the mean number
of insects per plant in the sample. As Mahalanobis, 1946, has pointed out,
there may be some margin of physical uncertainty in what is being measured
even at a fixed point in time; the yield of wheat may increase by 4-570 owing
to moisture when the rainy season begins in the parts of India.
To turn to the measuring instrument, measurement and recording may be
completely automatic. Some human action may be involved, but of a simple
and easily checkable type, as in reading a clear, stationary dial. The human
action may involve complex subjective judgment to a greater or less degree.
Several different types of human action may enter. In many sample surveys
it is recognized that errors of measurement can arise from the person being
interviewed, from the interviewer, from the supervisor or leader of a team
of interviewers, and from the processor who transmits the information from
the recorded interview on to the punched cards or tapes that will be analysed.
The preceding examples, though incomplete, suggest that a variety of mathematical models, some simple, some quite complex, may be needed to describe
realistically the types of measurement error relevant to different measurement
problems. Since the introduction of terms representing the presence of errors
of measurement nearly always seems to complicate the mathematical analysis,
two obvious maxims in this field are to use as simple a model as will reasonably
represent the facts and to keep oneself aware of other people¡¯s work. The same
model may turn up in very diverse applications and if the consequences of a
certain model have already been worked out, one might as well take advantage
of this. Concurrently, the investigator has a responsibility for producing evidence
that his model does fit adequately. From the literature my impression is that
investigators have needed no urging to use simple models, but are not so well
provided with data that justify the assumptions made in the model, and have
not always worked out the full consequences of the errors of measurement.
For instance, most textbooks on sample survey theory, including my own,
present all the standard theory on the assumption that there are no errors of
measurement, and merely indicate some of the disturbances caused by errors
of measurement briefly in a later chapter.
One common type of error of measurement-the
occasional gross error-will
ERRORS OF MEASUREMENT
639
IN STATISTICS
not be discussed in this paper. Since computers can print and examine residuals
from a fitted model, there is current interest in developing programs by which
any likely gross errors are automatically
signposted for the investigator to
examine and consider. In my opinion, much work of practical interest remains
to be done, for instance, more testing of the performance of such methods in
the presence of more than one gross error, and further study of the merits of
methods like censoring that are less affected by the presence of gross errors.
PART I. SOMESIMPLE
MODELS
2. Continuous variates.
The subscript u = 1, 2, . * * , n, will denote the member of the sample, while
the symbol Y, refers to the recorded measurement. The symbol y,, denotes the
correct or true measurement. For exposition of theory, the symbol yU is sometimes introduced even when we do not know how to make a correct measurement, though some workers are unhappy with this practice.
The error of the measurement on the uth unit is d, = Y, - yU . With some
measurement processes we can conceive of repeated measurements of the same
unit, and with some simple non-destructive processes we can actually carry
out such repeated measurements. The subscript t will refer to the tth trial or
repeated measurement. Thus we write
Y¡±, = Y¡± + CL
(1)
In this representation, both Y,, and dUf have a frequency distribution for each
member of the population (i.e. for fixed u), whereas yU is assumed fixed for any
specific member of the population.
The simplest model is one in which
(2-l)
E(d,, 1 u) = 0;
E(di, 1 u) = 2;
E(&t
E(d,, , CL,,) = 0
, 4, 1 u, v> = 0,
(t # t¡¯);
u # 21.
In this model the errors are unbiased and have constant variance. They are
uncorrelated with the correct values, with one another on different units, and
on different trials for the same unit. This model may be expected to apply
to some of the simplest measurement processes. It has been used even without
evidence that it really applies, because it is sometimes the only model for which
the consequences of the errors have been successfully worked out. The additional assumptions that the d,, are normal and independent in the probability
sense are often adopted when needed. Sometimes we write E(dZ, 1 u) = g: ,
because some units are more difficult to measure precisely than others.
The next stage in the model is to recognize that the measuring instrument
may have an overall bias of amount a, writing
(2)
Y?Lt = yu + a + d,, )
where assumptions (2.1) about the d,, still apply.
As an introduction to the third stage, I would like to jump back historically
to Karl Pearson¡¯s 1902 paper on the mathematical theory of errors of measure-
640
WILLIAM
G. COCHRAN
ment. Pearson was interested in the nature of errors of measurement when the
quantity being measured is fixed and definite, while the measuring instrument
is a human being. He conducted two experiments, each having three persons
as measurers. In the first experiment, each person was presented with the same
set of lines of unequal lengths drawn on paper, and was asked to bisect each
line freehand. The exact middle of each line and the error of measurement
was recorded. In the second experiment the task of the measurer was a little
harder. A bright line of light slowly traversed a white strip on a black screen.
When a bell sounded, each observer drew a mark across a line lying before
him on a sheet of paper, to mark the proportional distance of the beam across
its strip when the bell sounded. Again, the exact position of the bright line
and the error of measurement were carefully noted. The sample sizes were
500 in the bisection series and 520 in the bright line series.
The principal conclusions drawn by Pearson from his analysis of the results
were as follows.
(1) In 5 of the 6 cases (3 measurers, 2 experiments), the mean errors differed
significantly from zero, the sixth being almost significant at the 5yo level.
This is the a term in model (2). These overall biases varied in size and direction
from one person to another.
(2) For a given measurer, the size of t,he bias varied throughout the series
of trials when the errors were grouped in successive sets of 25. This suggests that
with a human measurer the model may need a term a, representing a bias that
varies from one sample member to another.
(3) The errors were not in general normally distributed, but exhibited both
some skewness and some kurtosis.
(4) To come to a result that startled Pearson, the errors of two apparently
independent observers in measuring the same quantity were positively correlated in 5 cases out of 6. This phenomenon is well known in interview surveys,
though the explanations given there do not apply to Pearson¡¯s experiments.
When two independent interviewers question the same person, the respondent
may give an erroneous answer to the first interviewer and simply repeat the
same erroneous answer to the second through a conscious or unconscious effect
of memory. Alternatively,
on a delicate question of opinion, both interviewers
may have the same point of view and may induce the same erroneous answer
because of the way in which they ask the questions. Pearson¡¯s result can have
a simple explanation even when the quantity being measured is fixed and
objective. If two measurers both tend to underestimate high values and overestimate low values, these negative correlations between dUt and yu induce a
positive correlation between the errors dUl , dUl, of the two measurers. Pearson
does not mention this possibility in connection with his results and does not
present the results in a way which enables one to examine whether an effect
of this type was present. Instead, he uses some uncharacteristically
vague
language, such as (p. 412) ¡°certain factors affected by the immediate atmosphere
seem to be common elements of two or more personalities, and there results
from this a tendency in each pair of observers to judge in the same manner¡±,
and later (p. 433), ¡°this psychological or organic correlation.¡±
Pearson¡¯s results suggest several possible elaborations of model (2). The
ERRORS OF MEASUREMENT
641
IN STATISTICS
simplest is to introduce a ¡°variable bias¡± term a, , and to make the additional
assumption that the a, are uncorrelated with the correct values yU ; that is,
(3)
Y,, = yy + a, + c-L ,
cov (a, ) yJ = 0.
With the same algebra, we may be unwilling
uncorrelated, leading to
(4)
Yut = yu + a, + d,, ,
to assume that a, and y, are
cov (a, ) y¡±) # 0.
Further, Pearson¡¯s results indicate that intercorrelations in errors of measurement within the sample may occur. In sample surveys the most prominent
effect of this kind, and the one most studied, arises from the personal biases
of the interviewers, which produce an intra-interviewer
covariance term that
is positive. Pearson¡¯s results also constitute a warning that when we come to
study errors of measurement by havin, v two different observers measure the
same unit, their errors may not be independent.
The introduction of these correlation terms into the mathematical model
complicates the subsequent analysis, and will be postponed until section 17.
Much of the work that has been done outside of sample surveys assumes,
rightly or wrongly, that errors on different members of the sample are uncorrelated. For the present, therefore, we revert to model (4), with the conditions
(2.1)
E(d,, 1 u) = 0;
E(d:, 1 u) = CT:;
E(d,, , d,, 1 u, v) = 0,
-WL
, d,,.) = 0,
t # t¡¯;
u # v.
For some applications it is convenient to amalgamate the terms y, and a,
by writing y: = yU + a, , particularly when no feasible method of measuring
yU itself is known. The symbol y; might be called the operationally correct
value for the uth unit. Model (4) then reduces to the simpler form of model
(l), i.e.
(4)¡¯
Yut = Y: + d,t ,
though with the above difference in interpretation.
One less general form of model (4) is worth mention, since it has already
proved useful. If the relation between the variable bias a, and the correct
value yU is expressible as a linear regression of a, on yU with regression coefficient
y, we may rewrite model (4) as
(44
Yut = a + PY~ + a, + dt., ,
E(G)
= 0,
cov (G , Y¡±) = 0
where P = 1 + y. In (4a) I have reused the symbol a, to represent the deviation
from this regression, so that the covariance of a, and yU is now zero. This model
is the basis of Mandel¡¯s, 1959, theory of errors of measurement as applied to
the analysis of interlaboratory
tests. In Mandel¡¯s application, equation (4a)
applies to a single laboratory, the values of a, 8, ui , and ~2 possibly varying
from laboratory to laboratory. This model may also be appropriate when
different judges are scaling the same set of subjects on a O-10 scale. If the
scaling involves a subjective element, some judges may be reluctant, while
others are willing, to assign very high or very low values. Thus @ will vary
from judge to judge.
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