Relative Risk, Increased Risk,
and Odds Ratios
Measures that allow us to compare two groups ...
Example:
Plaque Breaks and Heart Attacks
Plaque Burst?
Activity
Yes
Strenuous
P(Y|S) = 0.68
Normal
P(Y|N) = 0.21
Total
No
Total
146
The comparison
0.68/0.21 = 3.24
“Men who died during strenuous activity
were more than 3 times as likely to have
ruptured plaque than men who engaged
in normal activity.”
Relative risk
Disease?
Exposed to
risk?
Yes
Exposed
P(D|E)
Unexposed
P(D|EC)
No
Total
Relative Risk
= P(Diseased|Exposed) ÷ P(Diseased|Unexposed)
Example: Gender and Tattoos
Rows: gender
N
M
74
82.22
F
All
79
90.80
Columns: tattoo
Y
All
16
90
17.78
100.00
8
9.20
87
100.00
153
24
177
86.44
13.56
100.00
Cell Contents -Count
% of Row
The comparison
0.1778/0.0920 = 1.93
“Males in Fall 1998 Stat 250 classes were
almost twice (“2 times”) as likely to have
a tattoo than females in Fall 1998 Stat 250
classes.”
Interpretation of relative risk
Relative risk of 1 means that each “exposed”
group is equally likely to have the “disease.”
10% of students who take Stat 250 appreciate
statistics
10% of students who don’t take Stat 250 appreciate
statistics
RR of appreciating statistics = 0.10 /0.10 = 1
Interpretation of relative risk
• If RR = 1, the exposed and unexposed
groups are equally likely to get the disease.
• If RR < 1, the exposed group is less likely
to get the disease than the unexposed group.
• If RR > 1, the exposed group is more likely
to get the disease than the unexposed group.
Increased Risk as
an alternative to Relative Risk
• Relative risk is the “number of times more
likely” one event is to occur over another.
• Alternatively, can report the amount as a
“percent more likely”. In this case, called
“the increased risk.”
• RR of 1 means a 0 percent increased risk.
• To find increased (or decreased) risk, find
percentage above (or below) 1.
Increased Risk
What is the increased risk of male Stat
250 students having a tattoo over female
Stat 250 students?
1.93 - 1.00 = 0.93  100 = 93%
Males in Fall 1998 Stat 250 classes are 93
percent more likely to have a tattoo than
females in Fall 1998 Stat 250 classes.
Increased risk
Increased risk = (Relative Risk - 1.00)  100
Relative Risk
1.00
1.50
2.00
3.00
Equally
likely
1.5 times
more
2 times
more
3 times
more
Increased Risk
0%
50%
100%
200%
Equally
likely
50%
more
100%
more
200%
more
Increased risk
Increased risk can be negative! If so, it is
a “decreased” risk.
“Researchers found that occasional exercisers, those who did
less than the equivalent of six brisk half-hour walks a month,
were 0.70 times as likely to die than their sedentary twins.”
Relative Risk = 0.70
Increased Risk = (0.70 - 1.00) × 100 = -30%
“The researchers found that even occasional exercisers, were
30 percent less likely to die than their sedentary twins.”
Caution!
• Relative risk and increased risk by
themselves are not sufficient. Critical that
you also know the conditional probabilities
that went into calculating them.
• RR = 0.005/0.001 = 5
• RR = 0.5/0.1 = 5
• Relative risk of 5 means two very different
things in these two cases.
Odds
• If P(event) is p, then odds of an event
happening is “p/(1-p) to 1”. That is, the
probability of the event happening over the
probability of the event not happening.
• Or, if m = # with a trait and n = # without
the trait, then the odds of an event
happening is “m/n to 1”. That is, the # with
the trait over the # without the trait.
Interpreting odds
Odds = 10 to 1
For every 10 students who didn’t sleep enough
last night, I’ll find 1 who did sleep enough.
Odds = 3 to 2
For every 3 students who do their homework
daily, I’ll find 2 students who don’t do their
homework daily.
Example: Gender and Tattoos
Rows: gender
N
M
74
82.22
F
All
79
90.80
Columns: tattoo
Y
All
16
90
17.78
100.00
8
9.20
87
100.00
153
24
177
86.44
13.56
100.00
Cell Contents -Count
% of Row
Odds of Not Having a Tattoo
For males
0.8222/0.1778 to 1 = 74/16 to 1 = 4.6 to 1
For every 4.6 males I find without a tattoo, I’ll find one
male with a tattoo.
For females
0.9080/0.0920 to 1 = 79/8 to 1 = 9.9 to 1
For every 9.9 females I find without a tattoo, I’ll find
one female with a tattoo.
Odds ratio
The ratio of two odds, that is, the odds for
one group divided by the odds for another
group.
The comparison
Odds ratio = (79/8)/(74/16) = 9.9/4.6 = 2.15
The odds of finding a female without a
tattoo is 2.15 times that of the odds of
finding a male without a tattoo.
What to know?
•
•
•
•
•
Calculation of RR, IR, odds, OR
Interpretation of RR, IR, odds, OR
Relationship between RR and IR
Relationship between probability and odds
Importance of knowing probabilities and
not just RR and IR