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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.

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.

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)

Stephanie Budgett: University of Auckland

Relative Risk of Developing Breast Cancer(Utts, Seeing Through Statistics, p224)

Stephanie Budgett: University of Auckland

Relative Risk

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.

Or

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

Or

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

or (0.75 - 1.0)100% -25%

In words:

Thereis a reducedrisk of of developingbreastcancerforwomenwho had their first childunderthe age of 25 comparedtothosewho had their first childat 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)

Stephanie Budgett: University of Auckland

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

Stephanie Budgett: University of Auckland

Absolute Risk (Reduction) (ARR):

“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

– (0.9%)

Stephanie Budgett: University of Auckland

Number Needed to Treat (NNT)

– The number of patients that need to be treated to preventone 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

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