Statistical Concepts for Disease Detectives
Division C
Descriptive Epidemiology
deals with the frequency and the distribution of illness and risk factors in populations –
generally in terms of person, place and time. It enables epidemiologists to determine the
extent of a disease - the epidemiologist collects information to characterize and
summarize the health event or problem
 Used as a first step to look at health-related outcomes
 Examine numbers of cases to identify an increase
 Examine patterns of cases to see who gets sick and where and when they get sick
(person, place and time)
Mean – average – sum of all the given elements divided by the number of
elements. Means are typically reported when data are normally
distributed.
where sum of x scores and n = number of units in a sample
Median - The median of a series of numbers is the number that appears in the middle of
the list when arranged from smallest to largest. Medians are typically reported when
data are skewed.
For a list with an odd number of members, the way to find the middle number is to
take the number of members and add one. Then divide that value by two. In our case,
there are 9 numbers in the series. 9+1 = 10 and half of 10 is 5. The fifth number in the
series is the median or 14.
If the number of members of the series was even, the average of the two middle
numbers would be the median.
Mode - The mode is the number in the series that appears the most often. If there is no
single number that appears more than any other number in the series, there is no value
for the mode.
Variance - measures how far the results are from the expected
results
Standard deviation - expresses how close the results are to each other - it is the square
root of the variance
Standard error of the mean - standard deviation (sample standard
deviation) divided by the square root of the sample size
Confidence intervals of means - specifies a range of values within which the
mean may lie.
Z Scores for Commonly Used Confidence Intervals
Desired Confidence
Interval
90%
95%
99%
Z Score
1.645
1.96
2.576
Analytic Epidemiology
the epidemiologist relies on comparisons between groups to determine the role of various risk
factors in causing the problem. This is generally expressed as an association between an illness
and various risk factors. These associations have two components – strength and statistical
significance. Strength is measured by the relative risk or odds ratio while statistical significance
is determined by using a chi square, McNemar test or Fishers exact test to calculate a p-value.
Z-test - compares sample and population means to determine if there is a significant difference.
It requires a simple random sample from a population with a Normal distribution and where
the mean is known. The Z-test is preferable when the sample number n is greater than 30.
The Z value indicates the number of standard deviation units of the sample from the
population mean.
The Z measure is calculated as: z = (x - m) / SE
where x is the mean sample to be standardized, m (mu) is the
population mean and SE is the standard error of the mean.
where s is the population standard deviation and n is the sample
size.
The z value is then looked up in a z-table. A negative z value means it is below the
population mean (the sign is ignored in the lookup table).
T-test - An independent one-sample t-test is used to test whether the average of a sample
differ significantly from a population mean, a specified value μ0
Paired T-test -The dependent t-test for paired samples is used when the samples are paired.
This implies that each individual observation of one sample has a unique corresponding
member in the other sample
Note: The Z test, T-test and paired t-tests are used with continuous data such as age, weight
or height. Epidemiologists might use these tests to determine if there is a significant
difference between the ages or weights of a group of cases and a control group. They might
also used these tests to show that case and control groups are comparable. Paired t-tests are
used in settings where the observations are matched as in a before and after type of study.
One might use a paired t-test to show if there was a statistically significant reduction in body
mass index after a weight control program. Each participant would contribute two values – a
before BMI and an after BMI. These would be analyzed as paired data.
Chi-square - Any statistical test that uses the chi square distribution used to decide whether
there is any difference between the observed (experimental) value and the expected
(theoretical) value - Chi square test for independence of two attributes
where O is the observed frequency and E is the expected frequency.
Fischers exact test - a statistical test used to determine if there are nonrandom associations
between two categorical variables - two such variables and , with and observed states,
respectively.
Let there exist two such variables and , with and observed states, respectively. Now
form an
matrix in which the entries
represent the number of observations in which
and
. Calculate the row and column sums and , respectively, and the total sum
(1)
of the matrix. Then calculate the conditional probability of getting the actual matrix given the
particular row and column sums, given by
(2)
which is a multivariate generalization of the hypergeometric probability function. Now find
all possible matrices of nonnegative integers consistent with the row and column sums and
. For each one, calculate the associated conditional probability using (2), where the sum of
these probabilities must be 1.
To compute the P-value of the test, the tables must then be ordered by some criterion that
measures dependence, and those tables that represent equal or greater deviation from
independence than the observed table are the ones whose probabilities are added together.
There are a variety of criteria that can be used to measure dependence. In the
case,
which is the one Fisher looked at when he developed the exact test, either the Pearson chisquare or the difference in proportions (which are equivalent) is typically used. Other
measures of association, such as the likelihood-ratio-test, -squared, or any of the other
measures typically used for association in contingency tables, can also be used.
The test is most commonly applied to
matrices, and is computationally unwieldy for
large or . For tables larger than
, the difference in proportion can no longer be used,
but the other measures mentioned above remain applicable (and in practice, the Pearson
statistic is most often used to order the tables). In the case of the
matrix, the P-value of
the test can be simply computed by the sum of all -values which are
.
For an example application of the
test, let be a journal, say either Mathematics
Magazine or Science, and let be the number of articles on the topics of mathematics and
biology appearing in a given issue of one of these journals. If Mathematics Magazine has five
articles on math and one on biology, and Science has none on math and four on biology, then
the relevant matrix would be
(3)
Computing
gives
(4)
and the other possible matrices and their
s are
(5)
(6)
(7)
(8)
which indeed sum to 1, as required. The sum of -values less than or equal to
is then 0.0476 which, because it is less than 0.05, is significant.
The following data were obtained from investigating a foodborne outbreak at a small
company outing. Only 20 people were in attendance and 12 became ill. You are looking at
exposure to tuna salad.
Table : Illness among participants of company A outing who ate and did not eat tuna salad
Ilness
Ate Tuna Salad
Yes
No
Total
Yes
10
5
15
No
2
3
5
Total
12
8
20
The relative risk is (8/13)/(4/7) = 1.67. These are small numbers and 3 out of the 4 cells
have fewer than 5 observations. The Fishers exact test (also known as the Fisher-Irwin
exact test) is an appropriate measure of the statistical significance of these differences. The
p-value for a one-tailed Fishers exact test is 0.296. While that for a two-tailed test is 0.603
McNemar test for paired data - McNemar's test is basically a paired version of Chi-square test
– a form of chi-square (Χ2) test for matched paired data which is used to compare paired
proportions. It can be used to analyze retrospective case-control studies, where each case is
matched to a particular control. Or it can be used to analyze experimental studies, where the two
treatments are given to matched subjects.
McNemar's test calculates a P value. This test uses only the number of discordant pairs, that is,
the number of pairs for which the control was exposed to the risk factor but the case was not (4
as an example) and the number of pairs where the case was exposed to risk factor but the control
was not (25 as an example). Call these two numbers R and S. Calculate chi-square using this
equation:
For this example, chi-square=13.79, which has one degree of freedom. The two-tailed P value is
0.0002. If there were really no association between risk factor and disease, there is a 0.02 percent
chance that the observed odds ratio would be so far from 1.0 (no association).
The below example is modified from Statistical Methods for Rates and Proportions (JL Fless, 2nd
edition). An investigator is interested in determining if there is a significant difference between
proportions of patients diagnosed as having schizophrenia by two different physicians (physician
A and physician B). Each of 100 patients is seen by both physicians who give a diagnosis –
each patient has two diagnoses, one from each position.
Table 8.7. Two-by-two table for comparing rates of schizophrenia by physician A and B
Physician A
Schizophrenia
Not schizophrenia
Total
Schizophrenia
35
5
40
Not schizophrenia
Total
25
60
35
40
60
100
Physician B
Χ2 = (|𝐵 − 𝐶| - 1)2 /(B+C) = (|5 − 25| - 1)2 /(5+25) = 12.03. The p-value is determined for a Χ2
statistic with 2 degrees of freedom. The odds ratio in this instance is b/c = 5/25 = .2,
The chi square, Fishers exact and McNemar test are used to determine the statistical significance
of proportions such as odds ratios or relative risks. The chi square test is used where all of the
cells in a 2X2 table have at least 5 observations while a Fishers exact test is used when one or
more of the cells have fewer than 5 observations. The McNemar test is used when cases and
controls are matched by one of more variables and accounts for the lack of independence
between observations.
Cochran Mantel-Haenszel summary odds ratio (often called the Mantel-Haenszel test)
is a hypothesis test for association between two variables while controlling for one or more
nuisance or control variables. Mantel and Haenszel proposed stratification techniques to
account for confounding.
How to Perform a Mantel-Haenszel test of a series of fourfold (2x2) tables.
Given two variables where each variable has exactly two possible outcomes (typically
defined as success and failure), we define the odds ratio as:
o = (N11/N12)/ (N21/N22)
= (N11N22)/ (N12N21)
where
N11 = number of successes in sample 1
N21 = number of failures in sample 1
N12 = number of successes in sample 2
N22 = number of failures in sample 2
The first definition shows the meaning of the odds ratio clearly, although it is more
commonly given in the literature with the second definition.
The log odds ratio is the logarithm of the odds ratio:
l(o) = LOG{(N11/N12)/ (N21/N22)}
= LOG{(N11N22)/ (N12N21)}
Alternatively, the log odds ratio can be given in terms of the proportions
l(o) = LOG{(p11/p12)/ (p21/p22)}
= LOG{(p11p22)/ (p12p21)}
where
p11 = N11/ (N11 + N21)
= proportion of successes in sample 1
p21 = N21/ (N11 + N21)
= proportion of failures in sample 1
p12 = N12/ (N12 + N22)
= proportion of successes in sample 2
p22 = N22/ (N12 + N22)
= proportion of failures in sample 2
Success and failure can denote any binary response. Dataplot expects "success" to be
coded as "1" and "failure" to be coded as "0".
The bias corrected version of the statistic is:
l'(o) = LOG[{(N11+0.5) (N22+0.5)}/ {(N12+0.5) (N21+0.5)}]
In addition to reducing bias, this statistic also has the advantage that the odds ratio is still
defined even when N12 or N21 is zero (the uncorrected statistic will be undefined for these
cases).
Note that N11, N21, N12, and N22 defines a 2x2 contingency table. These types of
contingency tables are also referred to as fourfold tables.
Fleiss, Levin, and Paik also use the following formulation for the ith 2x2 table:
Outcome Variable
Sample Present
Absent Total
1
Xi
ni1 - Xi
ni1
2
mi - Xi
Xi - li
ni2
Total
mi
ni. - mi
ni.
where li = mi + ni2 - ni..
The Mantel-Haenszel test can be used to estimate the common odds ratio and to test
whether the overall degree of association is significant. It is a consistent estimator in the
following two cases:
1.
When the number of tables is fixed, and possibly small, but each table has large marginal
frequencies.
2.
The number of tables is large. The marginal frequencies can be small in the individual
tables.
Define the following quantities
o
o
o
o
o
o
The Mantel-Haenszel estimate of the common odds ratio is
where g denotes the number of groups.
An estimate of the variance of
is
A confidence interval for the log(odds ratio) is then
where
is the normal percent point function and SE is the standard error of the
estimate (= square root of the variance).
The Mantel-Haenszel chi-square statistic for the significance of the overall degree of
association is
where
Pi1 = n11/ ni1
Pi2 = n12/ ni2
= (ni1 Pi1 + ni2 Pi2)/ni.
i=1- i
i
The test statistic is compared to a chi-square distribution with one degree of
freedom.