The bases of summary measures at it easiest.

1. NJILLA STEEV-BECCA
SC17P276

2. 1. Ratio
2. Proportion
3. Rate
Indices Used to Summarize Information

3. • A ratio can be written as one number divided by another (a
fraction) of the form a/b.
• Both a and b refer to the frequency of some event or
occurrence.
Ratio

4. • R = a/b „
• Often a ratio R is rescaled by multiplying by a constant k,
Where k is a number such as 10, 100, 1,000, or 10,000 „
• R is always > 0
• R may or may not have units
Properties of Ratios

5. Example 1:
R= 3,151 ⁄ 189 × 1 = 16.7:1
Examples of Ratios:

6. Example 2:
 Bangu, a village in the west of Cameroon has a population
size of 20,000 people with approximately 4 hospitals.
a) How many hospitals are the per person?
b) What is the ratio of hospitals per 10,000 people?
Solution;
• R = number of hospitals / (population size).

7. Cont,,,,
a) R = 4 hospitals/20,000 people
= 0.0002 hospitals per person
b) R*k = 0.0002 * 10,000
= 2 hospitals per 10,000 people
• Units = hospitals per 10,000 people

8. Example 3: Odds
• p = proportion of people with disease „
• 1–p = proportion of people without disease „
• O = p / (1–p) = “odds” of disease
• „No units

9. Example 3: Odds Ratio
• OR = odds ratio.
• OR =
odds of disease in Population1
odds of disease in Population 2
• OR =
𝑂1
𝑂2
• No units

10. Cont,,,
 Suppose country X has a population of 10,000,000 with
4100,000 of the population infected with malaria. While
country Y has a population of 12,000,000 with 3,600,000 of the
population infected with malaria.
a) what is the odds of having the disease in Country X and Y.
b) What is the “odd ratio” of the disease of country x to that of
country y.

11. Cont,,,,
a) O= p/( 1-p)
= (4100000/10000000)

12. Example 4: Standardized Mortality Ratio.
• SMR = standardized mortality ratio „
• SMR = the ratio of the number of events observed in the study
population to the number that would be expected if the study
population were exposed to the same specific rates as the
standard population.
• SMR = O/E

13. Cont,,,
 Example
Suppose a study of AIDS in a Country A involving two samples X and Y in
which sample x where exposed to some condtions, C and and sample y where
not revealed 40 cases of AIDS in town x and 20 in town y. What is the SMR.
 SMR=
observed number of AIDS cases in sample x
expected number of AIDS cases in 𝑠𝑎𝑚𝑝𝑙𝑒 𝑥
: 40 cases / 20 cases = 2
No units.

14. • A proportion is a ratio in which the numerator is a subset (or
part) of the denominator and can be written as a/(a+b)
Proportion

15. • n = the number of individuals in a population.
• x = the number of individuals in the same population which
possess characteristic C.„
• p = proportion in the population with characteristic C is equal
to x/n.
Properties of proportions

16. Cont….
• p takes on values between 0 and 1 (p is a fraction).
• p has no units.
• „p may be multiplied by a constant k
− Where k is a number such as 100, 1,000, or 100,000

17. Cont,,,
Example1;
 In a sample of four pets - a bird, a fish, a dog, and a cat. One
might ask what proportion has four legs. Only two pets (the
dog and the cat) have four legs.
-- Therefore, the proportion of pets with four legs is 2/4
or 0.50

18. Cont,,,
Example 2:
 Calculate the proportion of men in the NHANES follow-up
study who were diabetics.
 Numerator = 189 diabetic men
 Denominator = Total number of men = 189 + 3,151 = 3,340
 Proportion = (189 ⁄ 3,340) × 100 = 5.66%
Example 3:
Calculate the proportion of deaths among men.
Numerator = deaths in men
= 100 deaths in diabetic men + 811 deaths in nondiabetic men
= 911 deaths in men

19. Cont,,,
 Notice that the numerator (911 deaths in men) is a subset of the
denominator.
 Denominator = all deaths
= 911 deaths in men + 72 deaths in diabetic women + 511
deaths in nondiabetic women
= 1,494 deaths
 Proportion = 911 ⁄ 1,494 = 60.98% = 61%

20. Cont,,,,
 Note!!!
 Proportions can easily be converted to ratios.
 If the numerator is the number of women (179) who attended a clinic
and the denominator is all the clinic attendees (341), the proportion of
clinic attendees who are women is 179 ⁄ 341, or 52% (a little more
than half). To convert to a ratio, subtract the numerator from the
denominator to get the number of clinic patients who are not women,
i.e., the number of men (341 − 179 = 162 men.)Thus, ratio of women
to men could be calculated from the proportion as:
Ratio = 179 ⁄ (341 − 179) × 1
= 179 ⁄ 162
= 1.1 to 1 female-to-male ratio

21. • A rate is a ratio of the form a*/ (a+b)
• a* = the frequency of events during a certain time period.
• a+b = the number at risk of the event during that time period. „
• A rate may or may not be a proportion
Rate

22. • The calendar time period is the same in both the numerator and
denominator of a rate. „
• A rate expresses the relative frequency of an event per unit
time (“risk”).
Properties of rate

23. Exercise

24. Infant mortality rate (IMR) = number of infant deaths per
1,000 live births during a calendar year
• The IMR is a ratio
Examples of Rates in Vital Statistics

25.  Fertility rate = number of live births per 1,000 women aged
15–44 during a calendar year.
• The fertility rate is both a ratio and a proportion.
 Annual crude death rate
=
total # deaths in a calendar year
totalmidyear population

26.  Annual age-specific death rate for ages 1–4
=
total # deaths aged1−4 in a calendar year
midyear population aged1−4
 Percent of all deaths which are ages 1–4
=
total # deaths aged1 −4 in calendar year
total # deaths in calendar year
* 100
 Percent of all deaths ages 1–4 due to malignancy
=
# cancer −related deaths aged1 −4 in calendar year
total # deaths in calendar year
* 100

27. 1. Incidence:
 It is the number of new cases of a disease occurring in an at-
risk population during a defined period time interval.
Examples of Rates: Incidence and
Prevalence

28. Cont….
Incidence proportion;
- It is a measure of risk
- Incidence proportion(per 1000)
=
# new cases of a disease during a specific period of time
# persons at risk of developing the disease during that period of time
- Example; number of sick people who ate egg salad divided by
the total number of people who ate salad at a luncheon.

29. Cont..
 Incidence rate;
=
# new cases of specific disease in calendar year
total midyear population
 Example; the incidence rate of tuberculosis = 25 per 10,000
person-years.
 Incidence density is same as incidence rate.

30. 2. Prevalence:
 It is the proportion of the population with the disease.
 Prevalence per 1000:
=
# of cases of disease present in the pop at a spec𝑖𝑓𝑖𝑐 𝑡𝑖𝑚𝑒
# 𝑜𝑓 𝑝𝑒𝑟𝑠𝑜𝑛𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑜𝑝 𝑎𝑡 𝑡ℎ𝑎𝑡 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑡𝑖𝑚𝑒
* 1,000
 Example; Global prevalence of Mycobacterium TB infection was
32% (1.86 billion people ).
 Two types of prevalence;
- Point prevalence
- Period prevalence

31. Cont…
 Incidence can be high while prevalence is low and vice versa.
 Consider this two cases;
 Case 1;
A chronic, incurable disease such as diabetes, can have a low incidence
and high prevalence because it is not very fatal but it cannot be completely cured
either.
 Case 2;
A short- duration, curable disease such as common cold, can have a high
incidence but low prevalence because many people get a cold each year but
it last for a short time.

32.  Prevalence rate (point prevalence):
=
# cases [old or new ] of specific disease at time t
total population at time t
 Example; Do you currently have asthma?
 Prevalence rate (period prevalence):
=
# cases diagnosed with a specific disease in a time period
total population in the time period
 Example; Have you had asthma during the last n years?

33.  Prevalence = Incidence * duration.
- Higher incidence results in higher prevalence
- Longer duration results in higher prevalence
Relationship between incidence and prevalence

34.  Individuals may be exposed to the risk of an event for varying
amounts of time during a total time period of a certain length
due to:
− Entering the time period later
− Leaving the time period earlier
− Experiencing the event of interest
Person-Time and Rates

35. „Person-time;
 Is a calculation combining persons and time.
 Is the sum of the individual units of time that people have
been exposed to the risk of an event.
 Is used in the denominator of person-time rates.
 Is often used in epidemiology and vital statistics

36. • T = length of the time period of interest „N(T) = number of people exposed
to risk of the event during T.
• „E(T) = sum of the time units that each person is exposed to risk of the
event (total person-time)
• „D(T) = number of people with the event during T
R=
D(T)
E(T)
=
number of events
total person−time of exposure
Definitions Useful in Person-Time Analysis

37.  Suppose during a two-year period of time, 10 episodes of diarrhea at
a day-care center were reported. Thirty-five children attend the day-
care center, for varying fractions of the two-year period, for a total
of 50 child-years.
R= 10 diarrhea episodes/ 50 child −years of observation
= 0.20 episodes per child-year
„
Example 1: Person-Years

38. Note!!!
 In vital statistics, the exact exposure times rarely are
known.„
 E(T), the denominator, may be approximated by
multiplying the mid-period population, N, by the length of
the time period, T.
 Then, R =
D(T)
N ∗ T

39. • Suppose during a two-year period of time, 10 episodes of
diarrhea at a day-care center were reported. Suppose 30
children were enrolled in the day-care center at the mid-period
of one year.
• R =
(10 diarrhea episodes)
30 children attending the daycare center for 2 years
• = 10/60
• = 0.17 episodes per child-year.
Example 2: Person-Years

40.  Absolute (arithmetic) change = rate2 – rate1 „
 Relative change = rate2/rate1 „
 Proportional (percent) change = (rate2–rate1)/rate1
Assessing Change in Two Rates

41.  Annual mortality rate from all causes (per 1000 pop):
=
Total no of deaths from all causes in one year
No of persons in the pop at midyeaar
* 1000
Mortality rate

42.  Annual mortality rate from all causes for children under the
age of 10 (per 1,000 population):
= Total no of deaths from all causes in one year in childre𝑛 under age 10
No of children in the pop under age 10 at midyeaar
* 1000
Age-specific mortality rate

43.  Annual mortality rate from lung cancer (per 1,000 population):
=
Total no of deaths from lung cancer in one year
No of persons in the pop at midyeaar
* 1000
Cause-specific mortality rate

44.  Annual mortality rate from leukemia for children under the age
of 10 (per 1,000 population):
= Total no of deaths fom leukemia in one year in childre𝑛 under age 10
No of children in the pop under age 10 at midyeaar
* 1000
Age-and-cause specific mortality rate

45.  Case fatality rate (%):
=
No of individuals dying during a specified onset period of time
after disease onset or diagnosis
No of individuals with the
specifed disease
* 100
Case fatality rate

46.  Assume a population of 100,000 people
- 20 are sick with disease “ x”
- In one year, 18 die fro disease “x”
 The mortality rate in that year from disease “x” = 18/ 100,000.
= 0.00018
 The case fatality rate from “x” = 18/20
= 0.9 or 90%
Comparison of mortality rate and case fatality rate

47.  Proportionate mortality from a specific disease in a particular
year:
=
No of deaths from cardiovascular disease in the U.S in 2000
Total deaths in the U.S in 2000
Proportionate mortality