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Risk
Extension
Article by Dr Tim Kenny
http://www.patient.co.uk/print/4875
• A bat and a ball cost $1.10. If the bat costs $1
more than the ball, how much is the ball?
• A bat and a ball cost $1.10. If the bat costs $1
more than the ball, how much is the ball?
• 5c
• A bat and a ball cost $1.10. If the bat costs $1
more than the ball, how much is the ball?
• 5c
• Intuition suggests 10c
• We have the same problem with risk.
• Many reports in the media about the benefits
of treatments present risk results as relative
risk reductions rather than absolute risk
reductions.
• This often makes the treatments seem better
than they actually are.
• This is why we study ‘RISK’
Absolute risk
• Absolute risk of a disease is your risk of
developing the disease over a time period.
Relative risk
• Relative risk is used to compare the risk in two
different groups of people.
Example
• Say the absolute risk of developing a disease is
4 in 100 in non-smokers.
• Say the relative risk of the disease is increased
by 50% in smokers.
• The 50% relates to the 4 - so the absolute
increase in the risk is 50% of 4, which is 2. So,
the absolute risk of smokers developing this
disease is 6 in 100.
An example when talking about
treatments
• Say men have a 2 in 20 risk of developing a
certain disease by the time they reach the age of
60.
• Then, say research shows that a new treatment
reduces the relative risk of getting this disease by
50%.
• The 50% is the relative risk reduction, and is
referring to the effect on the 2. 50% of 2 is 1. So
this means that the absolute risk is reduced from
from 2 in 20, to 1 in 20
Number needed to treat
• A figure which is often quoted in medical
research is the number needed to treat (NNT).
This is the number of people who need to take
the treatment for one person to benefit from
the treatment.
• For example, say a pharmaceutical company
reported that medicine X reduced the relative
risk of developing a certain disease by 25%.
• If the absolute risk of developing the disease
was 4 in 100 then this 25% reduction in
relative risk would reduce the absolute risk to
3 in 100.
• But this can be looked at another way. If 100
people do not take the medicine, then 4 in
those 100 people will get the disease.
• If 100 people do take the medicine, then only
3 in those 100 people will get the disease.
• Therefore, 100 people need to take the
treatment for one person to benefit and not
get the disease. So, in this example, the NNT is
100.
• A quick way of obtaining the NNT for a
treatment is to divide 100 by the absolute
reduction in percentage points in risk when
taking the medicine.
• Or
• divide 1 by the absolute reduction in
proportion in risk when taking the medicine
Being vaccinated).
• Say the absolute risk of developing complications from
a certain disease is 4 in 20.
• Say a medicine reduces the relative risk of getting
these complications by 50%.
• This reduces the absolute risk from 4 in 20, to 2 in 20.
• In percentage terms, 4 in 20 is 20%, and, 2 in 20 is 10%.
Therefore, the reduction in absolute risk in taking this
medicine is from 20% to 10% - a reduction of 10
percentage points. The NNT would be 100 divided by
10. That is, 10 people would need to take the medicine
for one to benefit.
Helping to decide about taking a
treatment
• The decision on whether to take a treatment needs to
balance various things, such as:
• What is the absolute risk of getting the disease to start
with?
• How serious is the disease anyway?
• How much is the absolute risk reduced with
treatment?
• The risks or side-effects in taking the treatment.
• How much does the treatment cost? Is it worth it to an
individual if the individual is paying, or is it worth it to
the country if the government is paying?
Example:
• Say your absolute risk of developing a certain
disease is 4 in 1,000.
• If a treatment reduces the relative risk by 50%,
it means the 4 is reduced by 50%.
• Therefore, the treatment reduces the absolute
risk from 4 in 1,000 to 2 in 1,000. Not really
much in absolute terms.
• If it were a minor disease, one which you are
likely to recover from, then you are not likely
to bother to take the treatment.
• If it is a fatal disease, you might consider
taking the treatment - any reduction in risk
may be better than none. However:
• Say there was a 1 in 100 risk of developing
serious side-effects from treatment. You are
then not likely to want the treatment, as the
risk from serious side-effects is higher than
the risk from the disease.
• If there were no risk from the treatment, you
might consider the treatment worthwhile.
• If the treatment were very expensive: then
you may not be able to afford it and decide to
take the risk without treatment;
• if the government is paying, it might decide
not to fund this treatment, as the reduction in
absolute risk is not great and many people
would need treatment to benefit one person.
Example:
• On the other hand, say your absolute risk of
developing a disease is 4 in 10 and a
treatment reduces the relative risk by 50%.
Your absolute risk goes down to 2 in 10 - a big
reduction.
• If it were a minor disease that you are likely to
recover from, then you may still take the
treatment if there were no risk of side-effects,
so as not to be troubled with the disease.
• If it is a fatal disease, you are likely to
definitely want treatment provided the risk of
side-effects was much lower than the risk of
getting the disease.
• Treatments for medical conditions are often
quoted in the press along the lines ... "New
treatment reduces your risk of X disease by 25%".
However, although this sounds good, it usually
refers to the relative risk. But, the benefit really
depends on how common or rare the disease is.
A large reduction of relative risk for a rare disease
might not mean much reduction in the absolute
risk. For example, a 75% reduction in relative risk
for something that has a 4 in a million absolute
risk of happening brings the absolute risk down
to 1 in a million.
Stephanie Budgett: University of Auckland
Stephanie Budgett: University of Auckland
Stephanie Budgett: University of Auckland
Stephanie Budgett: University of Auckland
Conveying Risk
• Absolute risk
• Baseline risk
• Relative risk
• Risk difference
• Increased risk/Reduced risk
• Odds Ratio
• Number needed to treat
Stephanie Budgett: University of Auckland
Absolute risk
The incidence of an event in a particular group
– For example, women in New Zealand have a
0.1 risk of developing breast cancer over their
lifetime
– (This risk will vary according to a woman’s age,
family history, lifestyle,…)
Stephanie Budgett: University of Auckland
Baseline risk
– This is the risk without a specified treatment or
behaviour.
If we want to find out if taking an aspirin helps
prevent heart attacks, the baseline risk is…
• the risk of having a heart attack without taking
aspirin.
If we want to investigate the risk of smoking and
getting lung cancer, the baseline risk is…
• the risk of getting lung cancer without smoking
Stephanie Budgett: University of Auckland
Relative risk
– The ratio of the risks for two groups
e.g. Relative risk of cancer due to smoking
= Risk (prob) of Cancer for a smoker
Risk (prob) Cancer for a nonsmoker
Stephanie Budgett: University of Auckland
Relative Risk
– It is useful to compare the risk of disease (e.g.
heart attacks) for those with a certain
characteristic (e.g. taking aspirin) to the baseline
risk of that disease (e.g. heart attacks in those
not taking aspirin).
– It doesn’t usually matter which way round we
calculate the ratio, but relative risks of greater
than 1 are easier to interpret than those
between 0 and 1.
Stephanie Budgett: University of Auckland
Relative Risk of Developing Breast Cancer
(Utts, Seeing Through Statistics, p224)
First Child
at age 25
Yes
No
Total
Breast
Cancer
31
65
96
No Beast
Cancer
1597
4475
6072
Total
1628
4540
6168
Stephanie Budgett: University of Auckland
Relative Risk of Developing Breast Cancer
(Utts, Seeing Through Statistics, p224)
First Child
at age 25
Yes
No
Total
Breast
Cancer
31
65
96
No Beast
Cancer
1597
4475
6072
Total
1628
4540
6168
Stephanie Budgett: University of Auckland
Relative Risk
31
having first child ³ 25
= 1628 = 1.33
65
having first child < 25
4540
Stephanie Budgett: University of Auckland
In words
The risk of developing breast cancer for women
who had their first child at age 25 or older is
1.33 times the risk of developing breast cancer
for women who had their first child under the
age of 25
Stephanie Budgett: University of Auckland
OR the other way up
i.e. comparing “under 25” to “over 25”
Relative risk = 0.0143 / 0.0190 = 0.75
The risk of developing breast cancer for women
who had their first child under the age of 25 is
0.75 times the risk of developing breast cancer
for women who had their first child at the age of
25 or older.
Stephanie Budgett: University of Auckland
Risk difference
– The difference in risk, of lung cancer say, associated
with smoking, is simply
Risk for those exposed (smokers) – Baseline risk
(nonsmokers)
Risk for the exposed – Risk for the unexposed
(simple difference between the 2 probabilities)
Seldom used and quoted
– because for small probabilities ratios tend to be much
more stable measures of effect (from population to
population) than differences
Stephanie Budgett: University of Auckland
Risk difference (Attributable Risk)
– The difference in risk of breast cancer
associated with a woman having her first child at
the age of 25 or older (compared with under the
age of 25) is:
Risk for those ‘exposed’ (first child > 25)
– Baseline Risk ‘unexposed’ (first child ≤ 25)
= 0.0190 – 0.0143 = 0.0047
Stephanie Budgett: University of Auckland
Increased/Decreased risk
– Sometimes the change in risk is expressed as a
percentage increase (or decrease) instead of a
multiple.
Change in Risk
Increased risk=
´100%
Baseline Risk
Or
Increased Risk= ( Relative Risk - 1) ´100%
Stephanie Budgett: University of Auckland
Increased risk
The risk of developing breast cancer for women who had
their first child at age 25 or older is 1.33 times that for
those who had their first child under the age of 25.
Thus
0.0190 - 0.0143
Increased risk=
´100 = 33%
0.0143
Or
(1.33-1) ´100% = 33%
In words: There is an increased risk of 33% of developing
breast cancer for women who had their first child at age
25 or older compared
Stephanie Budgett: University of Auckland
Decreased risk
For women who have their first child under the age of 25, the
risk of developing breast cancer is 0.75 times that for women
who had their first child at age 25 or older.
Thus
0.0143 - 0.0190
Decreased risk=
´100% = -25%
0.0190
or
(0.75 - 1.0) 100% -25%
In words:
There is a reduced risk of of developing breast cancer for
women who had their first child under the age of 25
compared to those who had their first child at age 25 or older.
Stephanie Budgett: University of Auckland
Odds Ratio
– Very common in technical reporting of risk
– Idea is more complicated than that of “relative
risk”
– BUT when we are comparing small
probabilities the Relative risk and odds ratio
are numerically almost identical
Stephanie Budgett: University of Auckland
Lots of important forms of statistical analysis
naturally produce odds ratios (e.g. logistic
regression)
Relative riskof cancer due to smoking=
Prob of cancer for a smoker
Prob of cancer for a nonsmoker
Odds of cancer for a smoker
Odds ratio =
Odds of cancer for a nonsmoker
Stephanie Budgett: University of Auckland
Prob of cancer for a smoker
Odds of cancer =
Prob of cancer for a nonsmoker
First Child
at age 25
Yes
No
Total
Breast
Cancer
31
65
96
No Beast
Cancer
1597
4475
6072
Total
1628
4540
6168
Odds of cancer risk for a woman having first child ≥ 25
31
31
1628
=
=
1597
1597
1628
Prob of cancer for a smoker
Odds of cancer =
Prob of cancer for a nonsmoker
First Child at
Breast
age 25
Cancer
Yes
31
No
65
Total
96
Breast cancer odds
No Beast
Cancer
1597
4475
6072
Total
1628
4540
6168
for a woman having first child ≥ 25= 0.0194
for a woman having first child < 25= 65 / 4475 = 0.0145
Odds ratio = 0.0194 / 0.0145
= 1.34
Relative risk = 1.33
First Child at
Breast
age 25
Cancer
Yes
31
No
65
Total
96
Breast cancer odds
No Beast
Cancer
1597
4475
6072
Total
1628
4540
6168
for a woman having first child ≥ 25= 0.0194
for a woman having first child < 25= 65 / 4475 = 0.0145
Odds ratio = 0.0194 / 0.0145
= 1.34
Odds
– If the risk of disease is small, the odds ratio
and the relative risk will be approximately equal.
– Relative risk is more intuitive, but the odds
ratio is easy to deal with statistically.
Stephanie Budgett: University of Auckland
Conveying Risk
What is the benefit of a cholesterol‐lowering
drug on the risk of coronary heart disease?
“People with high cholesterol can rapidly
reduce…their risk of death by 22% by taking a
widely prescribed drug.”
What does
this mean?
Stephanie Budgett: University of Auckland
“22% risk reduction”
Does it mean that out of 100 with high
cholesterol, 22 can be prevented from
becoming heart attack victims?
NO!
Stephanie Budgett: University of Auckland
Here are some of the results from the trial
Deaths
(per 1,000 with high cholesterol)
Active Drug
Placebo
32
41
RR= 32
= 0.78
41
( 22% reduction)
Stephanie Budgett: University of Auckland
Absolute Risk (Reduction) (ARR):
Deaths
(per 1,000 with high cholesterol)
Active Drug
32
Placebo
41
“What is the effect of treatment?”
– If we treat 1000 people …. (on average and
taking everything at face value) instead 41 dying
(as would if untreated) we’d have 32 die a saving
of 41‐32 = 9 lives per 1000 people treated
Stephanie Budgett: University of Auckland
– (0.9%)
Stephanie Budgett: University of Auckland
Number Needed to Treat (NNT)
– The number of patients that need to be
treated to prevent one bad outcome.
Stephanie Budgett: University of Auckland
Number needed to treat (NNT):
– “How many people do we need to treat to
prevent one death?” (on average and taking
everything at face value)
9 deaths per 1,000 treated are prevented by the
Drug so on average etc, we need to treat
1000/9 =111 people to prevent one death
Stephanie Budgett: University of Auckland