Transcript Random error

A short introduction to epidemiology
Chapter 6: Precision
Neil Pearce
Centre for Public Health Research
Massey University
Wellington, New Zealand
Chapter 6
Precision
• Random error
• Basic statistics
• Study size and power
Random Error
• Random error is not unique to epidemiologic
studies and also occurs in randomized trials
• Even if the disease under study is not
associated with an exposure, there may be a
“chance” association in a particular study
(e.g. disease may be more common in the
exposed than in the non-exposed group)
Random Error
• The precision (lack of random error) of an
effect estimate (e.g. an odds ratio) is
reflected in the 95% confidence interval
• Random error reduces, and precision
increases, as the study size increases
Chapter 6
Precision
• Random error
• Basic statistics
• Study size and power
Basic statistics
• Mean
• Standard deviation
– If we take a sample from a population, and the
data is normally distributed, then 95% of
individual values in the sample will lie within
+1.96 standard deviations of the population
mean
The normal distribution (insert figure
showing the bell curve)
Basic statistics
• Standard error
– If we take repeated samples, then the sample
means will vary and the standard deviation of
the sample means is known as the standard
error
– 95% of sample means will lie within +1.96 SE of
the population mean
The distribution of sample means (insert
figure showing the bell curve)
Categorical data
• Suppose we want to calculate a proportion
(p)
• Under the binomial distribution, if the sample
is sufficiently large, the sampling distribution
will approximate to the normal distribution
with mean (p) and standard deviation:
s = (p(1-p)/n)0.5
Testing and estimation
• The p-value is the probability that differences as large or
larger as those observed could have arisen by chance if the
null hypothesis (of no association between exposure and
disease) is correct
• The confidence interval provides a range of values in which
it is plausible that the true effect estimate may lie
• The principal aim of an individual study should be to
estimate the size of the effect (using the effect estimate and
confidence interval) rather than just to decide whether or not
an effect is present (using the p-value)
Chapter 6
Precision
• Random error
• Basic statistics
• Study size and power
Study size and power
The study power depends on:
• The cut-off value (e.g. p<0.05) below which the p-value
would be considered “statistically significant”
• The disease rate in the non-exposed group in a cohort
study or the exposure prevalence of the controls in a casecontrol study
• The expected relative risk (i.e. the specified value of the
relative risk under the alternative (non-null) hypothesis))
• The ratio of the sizes of the two groups being studied
• The total number of study participants
Study size and power
Zb = N00.5(P1 – P0)B0.5 – ZaB
------------------------------K0.5
where:
• Zb
• Za
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N0
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P1
P0
A
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B
C
K
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=
standard normal deviate corresponding to a given statistical power
standard normal deviate corresponding to an alpha level (the
largest p-value that would be considered "statistically significant")
number of persons in the reference group (i.e. the non-exposed
group in a cohort study, or the controls in a case-control study)
outcome proportion in study group
outcome proportion in the reference group
allocation ratio of referent to study group (i.e., the relative size of
the two groups)
(1-P0) (P1+ (A-1) P0) + P0 (1-P1)
(1-P0) (AP1 - (A-1) P0) + AP0 (1-P1)
BC - A (P1-P0)2
Study size and power: example
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5,000 exposed persons and 5,000 non-exposed
persons
The risk in the non-exposed group is 0.005
We expect that exposure will double the risk of
disease, so the risk will be 0.010 in the exposed
group
Zb = 0.994
Power = 83%
Study size and power
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The study power is not the probability that the
study will estimate the size of the association
correctly
Rather, it is the probability that the study will yield
a "statistically significant" finding when an
association of the postulated size exists
The observed association could be greater or less
than expected, but still be "statistically significant"
Study size and power
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Standard calculator and microcomputer
programmes incorporating procedures for power
calculations are widely available.
EPI-INFO (Dean et al, 1990) can be downloaded
for free from http://www.cdc.gov/epiinfo/
Rothman’s Episheet programme (Rothman, 2002)
can be downloaded for free from
http://www.oup-usa.org/epi/rothman/
Study size and power
• Insert example of power curve from
Rothman’s Episheet
A short introduction to epidemiology
Chapter 6: Precision
Neil Pearce
Centre for Public Health Research
Massey University
Wellington, New Zealand