Comparing Populations

Comparing Groups in a Population > Explanation

In addition to comparing rates between two population groups we often want to compare the probabilities of groups in a population having a particular attribute.

[EXPAND]

We can do this by looking at their relative risks or odds. For example, to compare the likelihood of an event A occurring for individuals in one group to that for individuals in another group (e.g. men and women, experimental group and a control group) we could use either the relative risk or the odds ratio.

You can view a video about this below.

Probability can be viewed as the fraction of times that event A occurs over the long run. It can be considered a long-term rate of occurrence.

EXAMPLE

If the rate of a disease is 5 per 10,000 in a certain population then the probability of developing that disease is:

p =
5 / 10,000
= 0.0005 = 0.05%

Rate =
Number of people with disease / All people with and without disease

Since each individual either does or does not develop the disease, the probability of not developing the disease is: 1-p = 0.9995.

END COMPARING GROUPS EXPLANATION

[COLLAPSE]

Comparing Groups in a Population > Odds Ratios

Odds are another way of looking at uncertainty, and are related to probability in the following way.

[EXPAND]

The odds of event A are:

Odds A =
pr(A happens) / pr(A does not happen)

=
prA / 1-prA

=
pr(A) / pr(A)

The odds of an event A occurring are the ratio of:

A occurring : A not occurring.

EXAMPLE

Probability Odds Calculation Odds
1/4 (1/4) / (3/4) = 1/3 1/3 or 1 : 3
1/3 (1/3) / (2/3) = 1/2 1/2 or 1 : 2

EXERCISE

Using the table above as a guide calculate the odds for the following probabilities of A occurring: 1/5, 2/3, 0.7.

Probability Odds Calculation Odds
1/5 (1/5) / (4/5) = or
2/3 (2/3) / = or
/ = or
Check
The completed table should look like the one below.

Probability Odds Calculation Odds
1/5 (1/5) / (4/5) = 1/4 1/4 or 1:4
2/3 (2/3) / (1/3) = 2/1 2 or 2:1
7/10 (7/10) / (3/10) = 7/3 7/3 or 7:3

Relationship between odds and probability

  • Odds = 1, means the event is just as likely to happen as not to happen.
    That is, the probability is 0.5 or 50%
  • Odds < 1, means the event is less likely to happen than not to happen.
    That is, the probability is less than 0.5 (less than 50%)
  • Odds > 1, means the event is more likely to happen than not to happen. That is, the probability is more than 0.5 (greater than 50%)
  • If the odds of A = 5, the event is 5 times more likely to happen than not to happen.
  • If the odds of A = 0.1, the event is only a tenth as likely to happen as not to happen.

Odds vs Probability


If the probabilities of the event in each of two groups are p1 (first group) and p2 (second group), then the odds ratio is:

p1 / 1 - p1
p2 / 1 - p2

This simplifies to:

p1 / q1
p2 / q2

This simplifies to:

p1q2
p2q1

where qx = 1 - px

  • An odds ratio of 1 indicates that the event is equally likely to occur in both groups.
  • An odds ratio greater than 1 indicates that the event is more likely to occur in the first group than the second. For example, an odds ratio = 6, means that the odds of A happening is 6 times higher for the first than the second group.
  • An odds ratio less than 1 indicates that the event is less likely to occur in the first group than the second. For example, an odds ratio = 0.1, means that the odds of A happening for group 1 is one tenth of the odds of A happening for group 2.

EXAMPLE 1

Of 49 swimmers with enamel erosion (the cases) 32 reported swimming 6 or more hours per week compared with 118 out of 245 swimmers without enamel erosion (the controls). Do swimmers who swim 6 or more hours per week have a greater risk of enamel erosion.

Swim time per week Erosion of enamel Total
Yes (Cases) No (Controls)
> 6 hours 32 118 150
< 6 hours 17 127 144
Total 49 245 294
Odds ratio =
32 x 127 / 17 x 118
= 2.0 approx

i.e. People swimming 6 or more hours per week have double the risk of enamel erosion.

EXERCISE 1

Chances of Getting the Death Penalty
(Michael Radelet), University of Florida

In a study Radelet classified 326 murderers by race of the victim and type of sentence given to the murderer. 36 of the convicted murderers received the death sentence. Of this group, 30 had murdered a white person whereas 184 of the group that did not receive the death sentence had murdered a white person. Radelet's view was, that if you killed a white person in Florida, the chances of getting the death penalty were three times greater than if you had killed a black person. (Source: American Sociological Review 1981) He came to this conclusion by using an odds ratio.

  • a. Construct a table for this data.
  • b. Estimate the probability of a murderer receiving the death sentence.
  • c. Estimate the probability of a murderer receiving the death penalty given that the victim was white.
  • d. What is the probability of a murderer receiving the death penalty given that the victim was black?
  • e. What is the odds ratio of a murderer getting the death penalty if they killed a white person compared to if they killed a black person? (answer to two decimal places.)

a.

Victim Death Sentence Total
Yes No
White 30
Black
Total 326
Check

The completed table should like the one below.

Victim Death Sentence Total
Yes No
White 30 183 214
Black 6 106 112
Total 36 290 326

b. The final answer is to 4 decimal places.

=
/
=
=
36 / 326
= 0.1104
Check

c. The final answer is to 4 decimal places.

=
/
=
=
30 / 214
= 0.1402
Check

d. The final answer is to 4 decimal places.

=
/
=
=
6 / 112
= 0.0536
Check

e.

Odds ratio =
Odds if victim white / Odds if victim black
=
Check

Calculate the odds victim is white:

30 / 214
184 / 214

Calculate the odds victim is black:

6 / 112
106 / 112

Use these to calculate the odds ratio:

=
30 / 184
x
106 / 6
= 2.88

This is close to the claimed value of 3.


EXERCISE 2

What is the odds ratio for attending lectures for a student who gained an 'A' compared to a student who gained a 'C'. Note that this table is headed 'Lecture Attendance". The final answer is to two decimal places.

Lecture Attendance
Grade Yes No Total
A 31 5 36
B 33 2 35
C 37 15 52
D 20 40 60
TOTAL 121 62 183

The answer is:

31 x 15 / 37 x 5
= 2.51
Check

EXAMPLE 2

The table below is used here as an example of comparing relative risk and odds ratios.

Survival of passengers on the Titanic
Gender Alive Dead Total
Female 308 154 462
Male 142 709 851
Total 450 863 1,313

Relative risk compares the probability of death in each group.

Probability of death for females =
154 / 462
= 0.3333
Probability of death for males =
709 / 851
= 0.8331

Therefore, the relative risk of death for males than for females is:

Probability of death for males / Probability of death for females
=
0.8331 / 0.3333
= 2.5

That is, there was a 2.5 greater probability of death for males than for females.

The Odds ratio compares the odds of death in each group.

Odds of death for females =
154 / 308
= 0.5

That is, exactly 2 to 1 against death.

Odds of death for males =
709 / 142
= 4.993

That is, almost 5 to 1 for death.

Therefore the odds ratio of death for males than for females is:

Odds of death for males / Odds of death for females
4.993 / 0.5
= 9.986

That is, the odds of death for males are almost ten times those for females.

Although relative risk might be easier to interpret, some research designs (e.g. case control trials) don't always allow it to be computed. For example, if we change the number of controls in the trial we can change the relative risk. However, the odds ratio always remains the same so this is what is used. When an event is rare the relative risk and the odds ratio will be very close.

END ODDS RATIOS

[COLLAPSE]