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Chapter 9
Health and Disease Measures
9.1 Introduction
Epidemiology and clinical research always deal with strong fundamental knowledge
about probability and statistical theory. Estimation of disease indicators and health
measurement, analysis of rates and ratios (comparison and confidence interval) are
common applications of statistics in epidemiology. Because sound understanding of
statistics can provide valuable insights about the proper epidemiological and research
projects. This chapter of the book explains the conceptuality of health indicators
and the occurrence of events, rates, coefficient of agreement and how to calculate
their confidence intervals. However, since the terms ratio, rate, and proportion are
used extensively, it is preferred to first distinguish between these three terms in
epidemiology.
9.2 Difference Between Ratios, Rates, and Proportions
A ratio represents the relative magnitude or comparison of two quantities,
expressed as one measure against another. It is calculated by dividing one
phenomenon, typically with an interval or ratio scale, by another within the same
unit. The numerator and denominator do not necessarily need to have a direct
relationship. Thus, one could compare unrelated variables, such as patients with
beds or nurses with the number of physicians in a city. Ratios are used to express
relative magnitudes and are commonly employed to compare similar quantities. For
instance, in a classroom with 12 males and 18 females, the ratio of males to
females is 12 to 18 or 2 to 3. Ratios can be utilized to compare quantities of the
same type of objects as well as quantities of different types.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024
S. H. Saneii and H. Doosti, Practical Biostatistics for Medical and Health Sciences,
https://doi.org/10.1007/978-981-97-3083-4_9
177
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178 9 Health and Disease Measures
Tips
Ratio can be used to compare quantities of the same type of objects and of
different types.
A rate measures the frequency of an event in a defined population over a
specified period of time.
A proportion is a type of ratio in which the denominator includes the
numerator.
Rates are utilized to compare dissimilar quantities with distinct units, where the
numerator and denominator possess different units. Similar to ratios, rates can be
simplified to their smallest terms. Given that rates contextualize disease frequency
relative to the population size, they are valuable for comparing disease occurrence
across various locations, time periods, or demographic groups with differing
population sizes. Therefore, rates represent a specialized form of ratio that
integrates the dimension of time into the denominator. In epidemiology, a rate
quantifies the frequency of an event within a defined population over a specified
time frame. Examples include disease prevalence rates per population, mortality
rates for Covid-19 per 100,000 individuals, and measurements of speed (e.g.,
kilometers per hour) or water flow (e.g., gallons per minute).
A proportion is a type of ratio in which the denominator includes the numerator. In
other words, a proportion is a ratio that relates a part to a whole in order to compare
with the whole. A proportion might be used to describe what fraction of medical
laboratory patients tested positive for Covid-19. A proportion may be expressed as
a decimal, a fraction, or a percentage. For example, in the class with 12 males and
18 females, the total class size is 30, and the proportion of males is 12/30 or 40%.
9.3 Rates
A rate serves as a metric for the frequency of occurrence of a phenomenon relative to
the population, taking into consideration the number of events transpiring within a
specified period per unit of time (e.g., per month, per year). Given that this frequency
can fluctuate over time, the rate is expressed in terms of time, with time being
an integral component of this concept. Rates are fundamental measures of disease
occurrence in populations and provide an estimate of the speed at which disease or
mortality is occurring within a particular population. In epidemiology, the term “rate”
is commonly used in conjunction with the prevalence and incidence of diseases. The
integral components of a rate are as follows:
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9.3 Rates 179
Tip
Rate describes risk of disease among a population at risk during a given time
interval, while proportion shows the likelihood of disease among the population
at a specific point in time. The difference is in the denominator.
1. Numerator: This represents the count or frequency of the desired events observed
within a specified period of time among the population. It serves as the numerator
of the fraction.
2. Denominator: The denominator typically comprises the number of individuals
in the group at risk, whose events are included in the numerator. In some cases,
the denominator may need to be estimated as the average population size over
the duration of follow-up.
3. Specified duration (period of time): This refers to the time frame during which
the events of the study population are tallied. It commonly encompasses calendar
periods such as months, seasons, or years.
4. Multiplier: This factor, such as percent or per thousand, aids in converting the
fraction into a more convenient and easily interpretable numeric value, facilitating
better comprehension of the rate.
The denominator of a rate encompasses entities such as population, patients, or
subjects, which can fluctuate both temporally and geographically. The primary
function of computing a rate is to standardize the denominator, enabling valid
comparisons between different groups, times, or locations. However, it’s crucial to
note that a rate is a reliable tool only under stable conditions throughout the period
for which it is calculated. In cases where the population undergoes variation within
this period, such as at the beginning and end of a year, it is customary to utilize the
average population (typically the middle of the year) to ensure consistency and
accuracy in the calculation.
Rates can only be averaged if the denominator of the fractions is the same. For
example, if the prevalence rate of a disease in a county is 3 per 1000 per year and in
another county is 5 per 1000 per year, the rate for both counties together will not be
4 per 1000 per year unless both counties have equal population. In these cases, the
weighted mean is recommended, which takes into account the relative frequencies.
Rates describe risk of disease among a population at risk during a given time
interval, while proportions describe the likelihood of disease among the population
at a specific point in time.
9.3.1 Incidence
In epidemiology, incidence refers to the occurrence of new cases of a disease,
disorder, or specific health-related event within a population during a defined time
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180 9 Health and Disease Measures
period, such as a month or a year. I n the calculation of incidence, the numerator
represents the number of new events that arise within the specified time frame,
while the denominator signifies the population at risk of experiencing the event
during this period. Achieving an accurate incidence, whether as a proportion or
rate, necessitates a precise definition of the denominator, which comprises the
population at risk. Additionally, two other critical elements include a well-defined
numerator, representing the number or frequency of new events occurring without
any prior history of the disease in question, and a clearly defined duration or period
of time over which the incidence is calculated.
Tip
The incidence of disease indicates how quickly subjects are developing the
disease, disorder or incident.
Incidence can typically be computed as either a proportion or a rate, each serving
distinct purposes. The proportion indicates the risk of an event occurring within a
given time frame, while the rate quantifies the number of new cases in a population
over time.
Incidence proportion (IP), also known as proportional incidence, measures the
likelihood that an individual will develop a disease or disorder during a specified
period. This proportion is calculated by dividing the number of new events (disease
or disorder) occurring during the period by the total population at risk. Incidence
proportion is commonly utilized in studies with a short follow-up duration, as it
requires complete follow-up and observation of all subjects at risk for the outcome
throughout the study period, or at least until they experience the outcome. It assumes
that individuals were initially free of the disease. Incidence proportion is also referred
to as cumulative incidence.
Incidence rate (IR) represents the number of new cases divided by the total person-
time at risk. The incidence rate is typically more precise than incidence proportion
for long-term studies, as it accommodates individuals entering and leaving study
populations.
Incidence rate
=
Number of new cases in a period of time
Number of person − years when people were at risk of getting the disease
Incidence proportion
=
Number of new cases in a period of time
Number of people at risk of getting the disease at the start of the period
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9.3 Rates 181
Year (1 Jan)
7
5
2
3.5
4
2
7
1.5
5
7
Person-Time (Years)
Individuals
2014
2015 2016
2017
2018
2019
1
2
3
4
5
6
7
8
9
10
2020
2021
Disease free
Disease period
Lost on follow-up
Death
On June 30, 2020, there were 3 people who were sick out of the remaining 5 people
in the group. This yields a prevalence of 3 out of 5, which is equivalent to 60%. The
total number of person-years followed up in this study is obtained by summing the
numbers to the right of the graph, which equals 44.0 person-years.
7 + 5 + 2 + 3.5 + 4 + 2 + 7 + 1.5 + 5 + 7 = 44
Incidence rate
=
number of new cases in a period of time
Number of person − years when people were at risk of getting the disease
Incidence proportion
=
number of new cases in a period of time
Number of people at risk of getting the disease at the start of the period
Example 1 In a sample population of a city with 50,000 people, we want to determine
the incidence rate of developing Covid-19 over a period of 6 months. At the beginning
of the study (t = 0), 30 cases of Covid-19 were identified. Assuming these people
cannot get Covid-19 for the second time, they will not be counted in the population.
In the follow-up 2 months later ( t = 2), 1000 new cases of Covid-19, and in the
second follow-up at the end of the study (t = 6), 2000 new cases were identified.
Find the incidence and prevalence rate of Covid-19.
Solution: To measure the prevalence rate, we simply divide the total number of cases
(30 + 1000 + 2000 = 3030) by the total number of sample population (50,000). So,
the prevalence is equal to:
PR =
x
n
=
3030
50000
= 0.0606.
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182 9 Health and Disease Measures
This measure tells the widespread of Covid-19 for any person in our sample
population but provides little information about the actual risk of developing of
Covid-19 in coming month.
To calculate the incidence rate, we must take into account how many months
each person participated in the study and when they developed Covid-19. When it
is not clear exactly when a person develops the disease in question, epidemiologists
frequently use actuarial method interpolating, and it is assumed that the person was
developed half-way point between follow-ups. In this calculation:
In the first 2 months, 1000 new cases were identified, so we assume that they
developed Covid-19 at half-way (one month), thus 1000 (
1000 × 1) person-months
of disease-free life.
Two thousand new cases identified in the sixth month were free of Covid-19
disease for the first two months. Therefore, we assume that they were infected half-
way between two and 6 months and had disease-free life for 4 (2 + 2) months. Thus
contributing 8000 (
4 × 2000) person-months of disease-free life. That is a total of
(8000 + 1000) = 9000 person-months so far.
On the other hand, 46,970 people (50,000 − 2000 − 1000 − 30) who have never
developed Covid-19 over the 6-month period should be accounted as 281,820 (46,970
× 6) contributing person-months of disease-free life.
That is a total of 290,820 (281,820 + 9000) person-months of life. Now, by
dividing 3000 new cases of Covid-19 by 290,820, the rate of 0.0103 or 103 cases
of Covid-19 per 10,000 populations per month can be obtained. In other words, if
you were to follow 10,000 people for one month, you would see 103 new cases of
Covid-19. This is a much more accurate measure of risk than prevalence rate.
IR =
3000
29, 820
= 0.0103.
Confidence interval for incidence rate
First, we obtain the standard error of the incidence rate using the normal
approximation for the situation with x new cases existing in n person-time:
Se
(
Ir
)
=
1 − Ir
x
.
Then the 95% confidence interval of the incidence rate is calculated by the
following formula.
95%CI =
e
ln Ir−1.96×Se
(
Ir
)
to e
ln Ir+1.96×Se
(
Ir
)
.
However, due to the ease of calculation, the confidence interval for “the number
of new cases” can be obtained based on the Poisson distribution from the simpler
relation below.
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9.3 Rates 183
CI =
z
1−
α
2
2
−
√
x
2
to
z
1−
α
2
2
+
√
x + 1
2
.
Example 2 In example 1, the incidence of Covid-19 in a 6-month period was 0.0103
with 3000 new cases. Calculate a 95% confidence interval for the incidence rate.
Solution: In this example x = 3000 and z
1−
α
2
= 1.96. The confidence interval for
incidence rate is:
Se
(
Ir
)
=
1 − Ir
x
=
1 − 0.0103
3000
= 0.01816
95%CI =
e
ln Ir−1.96×Se
(
Ir
)
to e
ln Ir+1.96×Se
(
Ir
)
=
e
−4.576−1.96×0.01816
to e
−4.576+1.96×0.01816
=
e
−4.6116
to e
−4.5404
=
(
0.0099 to 0.01067
)
.
The confidence interval for the number of new cases of the disease is:
CI =
z
1−
α
2
2
−
√
x
2
to
z
1−
α
2
2
+
√
x + 1
2
=
1.96
2
−
√
3000
2
to
1.96
2
+
√
3001
2
= 2893 to 3109).
9.3.2 Prevalence
In epidemiology, prevalence, also known as point prevalence, is determined by
dividing the total number of individuals in a population who have a disease or
disorder at a specific point in time by the total population, typically expressed as a
percentage. Prevalence encompasses both new and existing cases of the disease. It
provides a snapshot of the situation at a single moment, hence its alternative name,
“point” prevalence. In contrast, periodic prevalence refers to the proportion of the
population affected by the disease at any given time period.
The prevalence can be defined as follows:
prevalence rate =
Total number of exixting cases at a given time
Total number of people i n the population
.
Confidence interval for prevalence rate.
First, the standard error of the prevalence rate must be calculated.
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184 9 Health and Disease Measures
Se
(
Pr
)
=
x
(
n − x
)
n
3
.
Then obtain the confidence interval for prevalence rate as the confidence interval
of the rates from the following formula:
95% CI = Pr ± z
1−
α
2
.Se(Pr).
Example 3 In the population of 50,000 people, for example 1, obtain a 95%
confidence interval for the prevalence rate.
Solution: In this example x = 3030 and z
1−
α
2
= 1.96. The confidence interval for
prevalence rate is:
Se
(
Pr
)
=
3030
(
50, 000 − 3030
)
50, 000
3
= 0.00107
95%CI = Pr ± z
1−
α
2
.Se(Pr)
= 0.0606 ± 1.96 × 0.00107
=
(
0.0585 to 0.0627
)
.
9.3.3 Mortality Rate
Mortality rate quantifies and depicts the trend and tendency of a particular population
to die (or due to a specific cause) during a given time period. Mortality rate is typically
expressed in units of deaths per 1000 individuals and always refers to a time period,
usually 1 year. However, sometimes mortality rates are calculated for shorter time
periods such as monthly or even daily. The mortality rate is major (inverse) health
indicators, because death is the ultimate situation of “disease”, or “unhealthiness”.
An important specific mortality measure is the crude death rate, which looks at
mortality from all causes in a given time interval for a given population. The annual
crude death rate (m) can be calculated as a proportion:
m =
d
P
× 10
k
,
where d is the count of deaths during the year and p is the total population or
“population at risk”, which is generally approximated by the population at midyear.
If k = 3, the result is per thousand or when k = 4 the result is per ten thousand.
We can compute different types of mortality rates such as: age-specific,
sex-specific, or age-and-sex-specific mortality rates to gauge health conditions in
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9.4 Standardization 185
Table 9.1 Formulas of frequently used mortality measures
Measure Numerator Denominator Multiplier
(
10
k
)
Crude death rate Total number of deaths during a
given time interval
Mid-interval population 1000 or
100,000
Cause-specific
death rate
Number of deaths assigned to a
specific cause during a given time
interval
Mid-interval population 100,000
Proportionate
mortality
Number of deaths assigned to a
specific cause during a given time
interval
Total number of deaths
from all causes during the
same time interval
100 or
1000
Death-to-case
ratio
Number of deaths assigned to a
specific cause during a given time
interval
Number of new cases of
same disease reported
during the same time
interval
100
Neonatal
mortality rate
Number of deaths among children
< 28 days of age during a given
time interval
Number of live births
during the same time
interval
1000
Postneonatal
mortality rate
Number of deaths among children
28–364 days of age during a given
time interval
Number of live births
during the same time
interval
1000
Infant mortality
rate
Number of deaths among children
<1 year of age during a given time
interval
Number of live births
during the same time
interval
1000
Maternal
mortality rate
Number of deaths assigned to
pregnancy-related causes during a
given time interval
Number of live births
during the same time
interval
100,000
specific demographic groups. Table 9.1 summarizes the most commonly used
mortality measures formulas.
9.4 Standardization
Comparison of health indicators is common in epidemiological studies, but this
comparison is misleading regardless of differences in population’s characteristics.
Because the populations are different in terms of demographic, age and gender
structure. For example, age is one of the most common and important confounding
factors in health studies. When the age distribution of the groups being compared is
different and the age is related to the outcomes of interest (such as death or
prevalence of disease), comparisons can be disrupted. Therefore, a method should
be employed to eliminate the confounding effect of age so that we can make a
sound and logical comparison. Standardization is a classic epidemiological method
for comparing populations disease or death experiences of rates. There are two
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186 9 Health and Disease Measures
Fig. 9.1 Direct standardazition
main methods for standardization to characterize whether the population
distribution used is standard (direct method) or uses a set of specific measures
(indirect method). These two methods are presented below.
9.4.1 Procedure for Direct Standardization
Compute the mortality rates specific to each age group within both populations.
Select one population to serve as the standard (reference) population. Typically, this
is the national population when comparing mortality rates of a specific community.
For the other population being studied, determine the expected number of deaths
by multiplying its age-specific mortality rates by the number of individuals in each
age group of the standard population.
Sum up the expected deaths across all age groups.
Subsequently, calculate the age-adjusted mortality rates by dividing the total
expected deaths by the standard population size.
Finally, compare the age-standardized mortality rates of the two populations to
draw conclusions (Fig. 9.1).
9.4.2 Procedure for Indirect Standardization
We propose the following method for computing indirect standardization:
•
Select a reference or standard population.
•
Determine the observed number of deaths in the population(s) under investigation.
•
Utilize the age-specific mortality rates from the chosen reference population and
apply them to the population(s) of interest.
•
Multiply the population count in each age group of the population(s) of interest
by the age-specific mortality rate corresponding to the equivalent age group in the
reference population.
•
Aggregate the total expected deaths for each population of interest.
•
Calculate the standardized mortality ratio (SMR) by dividing the total observed
deaths in the population(s) of interest by the expected deaths (Fig.
9.2).
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