Sampling and measurement error, part 2: errors in vital rates

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Stile Ceater For BeaIth Statistics
Department of Human Resources • Division of Health Services. P.O. Box 2091 • Raleigh, NC 2760i-2091 .9191733-4728
No.6 July 1984
SAMPLINC AND MEASUREMENT ERROR
PART 2: ERRORS IN VITAL RATES
by
Paul A. Buescher
IntroductIOn
Part 1 of this statistical primer on sampling and mea­surement
error (May 1984) dealt primarily with the prob­lems
of sampling errors. In addition, some general con­cepts
were presented concerning random vs. nonrandom
errors and sampling vs. nonsampling errors. The reader
may want to refer to Part 1 for background in these areas.
This second part addresses the problem of errors in
measures based on a "complete count," and particularly
the problem of errors in vital rates. Formulas are pres­ented
for calculating random errors of measurement in
terms of the standard error of simple rates, and the
standard error of the difference between two rates. The
general concept of a standard error is explained in Part 1
of this statistical primer.
Random Measurement Error In a Complete
EnumeratIOn
The discussion in Part 1 considered random error
associated with measurements based on a sample, but
the basic concept of random variability in surveys also
applies to measurement error in general. Even a measure
based on a "complete count" of the population has a
random error component that can be assessed in terms
of standard errors. Such a measure may also contain a
very large nonrandom error or bias, due to such factors
as a poor measurement instrument or inadequate field
procedures, and it is of course very important to mini­mize
this type of error. But this paper addresses only the
problem of random error. A measure based on a com­plete
enumeration that is observed at a point in time may
be considered as one possible outcome of a chance
process, in addition to the problem of random errors
due to imperfect measurements.
Let us use the example of random error in a death rate
(number of deaths in a given year per 1000 population)
that is based on complete registration of deaths. The rate
measured in a given year can be considered as a sample
of one of a large number of possible measurements, all
of which cluster in a normal distribution around the
"true" death rate of the population. In fact, the larger the
numerator of the measured death rate, the smaller the
standard error associated with this "sampling distribu­tion,"
and therefore the better this death rate will esti­mate
the true or underlying rate of the population. This
idea of an "underlying" rate is an abstract concept, since
the death rate observed in one year did actually occur,
but it is this underlying rate that health policies should
seek to address rather than annual rates which may
fluctuate dramatically. Since death occurring to an indi­vidual
is subject to chance, the death rate for the popula­tion
may vary from one time to another even if the force
of mortality or underlying death rate remains constant.
The concepts presented below apply to the standard
error of any simple or "crude" rate (death rate, birth rate,
marriage rate, etc.), which is probably the most fre­quently
used type of health measure. Standard errors
may also be estimated for adjusted rates and other more
complex measures, but description of these is beyond
the scope of the present paper (1).
Any rate with a small number of events in the numera­tor
will be unstable, with possibly large random fluctua­tions
from year to year that do not comprise a significant
trend. It has been shown that events of a rare nature
follow a Poisson probability distribution and that the
following approximation may be used to estimate the
standard error Of a Simple rate. I n the case of a death rate,
where D is the number of deaths in the numerator and P