Journal of Physiotherapy 62 (2016) 172–174

Journal of
PHYSIOTHERAPY
journal homepage: www.elsevier.com/locate/jphys

Appraisal Research Note

Interpretation of dichotomous outcomes: risk, odds, risk ratios, odds ratios and
number needed to treat
Introduction having occurred during a specific period of observation, as in this
example, this measure of risk is called an incidence (Strictly
Clinical research often investigates whether a patient’s speaking, this is an incidence proportion, ie, the proportion of
diagnosis, prognostic characteristics or treatment are related to people in whom the outcome occurs during a specified period of
his or her clinical outcomes. The outcomes may be continuous time. An incidence rate measures the occurrence of the outcome
measures, which can occupy any point on a scale, for example: pain per unit of person-time). Although the outcome may be positive
intensity on a Visual Analogue Scale (0 to 100)1 or body (eg, recovery) or negative (eg, falling), we still talk about the ‘risk’
temperature in degrees.2 Other outcomes are measured on ordinal of that outcome. If the outcome of interest was recorded at a single
scales, where the differences between adjacent categories may time point, such as the number of people in a community sample
vary along the scale, for example: manual muscle testing grades of who have neck pain, this measure is called a prevalence.
none, trace, poor, fair, good and normal.3 Some continuous
outcomes may not be strictly linear (because each step on the Odds
scale may not have exactly the same magnitude), but are often
treated as such, for example: the Berg Balance Scale (0 to Odds are calculated differently from risk and are defined as the
56 points).4 In contrast, it is generally clear whether an outcome number of people in whom the outcome (eg, a fall) occurred
is dichotomous because it only has two states (such as yes/no, divided by the number of people in whom the outcome did not
better/not better, or dead/alive). Dichotomous outcomes can also occur. The odds of falling (using the same data from Maki et al6
be derived from continuous scales (eg, categorising temperatures above) would be 59/37 or 1.59. Note that the odds and the risk for
as febrile/not febrile) or from ordinal scales (eg, categorising the same data are not the same (Box 1), because their
muscle contraction as present/absent). Such dichotomisation is denominators are different.
generally not recommended, primarily due to loss of information.5
However, if the threshold used is nominated a priori and based on a Comparing risk or odds between groups
clinically relevant point on the scale, then dichotomisation of a
continuous outcome may be appropriate.5 The example above from Maki et al6 describes the risk or odds of
The reporting and interpretation of studies with dichotomous an event (a fall) in one population (independent ambulatory older
outcomes can be challenging. There are several ways to report the people). However, many clinical studies investigate the difference
findings about dichotomous outcomes, including: proportions, in risk or odds for a dichotomous outcome based on certain patient
percentages, risk, odds, risk ratios, odds ratios, number needed to characteristics or exposures. For example, we may want to know if
treat, likelihood ratios, sensitivity, specificity, and pre-test and older people fall more often than younger people, or if trial
post-test probability – each of which has a different meaning. The participants who were randomised to receive a Tai Chi interven-
purpose of this two-part series is to describe the correct tion fell less often than those who were randomised to no
interpretation of commonly used methods of reporting dichoto- treatment. This is commonly performed by calculating the ratio of
mous outcomes. This first paper focuses on risk, odds, risk ratio, either the risk (risk ratio) or the odds (odds ratio) between the
odds ratio and number needed to treat. The second paper will focus groups of interest.
on sensitivity, specificity and likelihood ratios. A risk ratio (also known as relative risk) is calculated by dividing
the risk of the outcome (eg, a fall) in people with a characteristic or
Risk and odds exposure (eg, received a Tai Chi intervention) by the risk of the
same outcome in the people who do not have that characteristic or
The chance of a dichotomous outcome occurring can be exposure (eg, control group). In a similar way, odds ratios are
described in terms of risk or odds. While these terms may sound calculated by dividing the odds of the outcome in people who have
similar, they have different meanings and methods of calculation. a particular characteristic by the odds of the same outcome in
people who do not have that characteristic. A worked example for
Risk calculating both risk ratios and odds ratios can be seen in Box 1. It is
critical to note that the value of the relative risk and the odds ratio
Risk is defined as the probability of an event during a specified are different, despite being based on the same data, and therefore
period of time. In clinical studies, risk is often calculated as the the interpretation is also different.
number of people in whom an outcome occurs divided by the total It is useful to recognise that risk ratios and odds ratios can be
number of people assessed. There are various ways to express risk. greater than or less than 1. When greater than 1, this indicates that
The raw count of a dichotomous outcome can be summarised as a people with the clinical characteristic (positive test result,
proportion or a percentage. For example, in an observational prognostic characteristic or treatment exposure) are more likely
study6 of 96 independent ambulatory older people, 59 had a fall to have the outcome of interest (diagnosis, prognostic outcome or
during a 1-year period. This could be reported as a proportion (59/ treatment outcome) than people without that clinical characteris-
96 = 0.61) or as a percentage (61%). If the outcome was recorded as tic. When risk ratios or odds ratios are less than 1, this indicates

http://dx.doi.org/10.1016/j.jphys.2016.02.016
1836-9553/ß 2016 Australian Physiotherapy Association. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://
creativecommons.org/licenses/by-nc-nd/4.0/).
Appraisal Research Note 173

Box 1. Calculation of risk, odds, risk ratios, risk reduction and odds ratios using hypothetical data.

In a randomised controlled trial involving 500 female athletes followed for 1 year, 125 report an injury by 1 year.
Risk 125/500 = 0.25 or 25%

Odds 125/375 = 0.33 or 33%

In the trial, 250 received preventative treatment and 250 did not; 75 of the untreated athletes were injured and 50 of the
treated athletes were injured.

Injured Uninjured Row total

Untreated 75 175 250
Treated 50 200 250
Column total 125 375 500

Risk ratio (also known as (75/250) / (50/250) = 1.5

relative risk) or 30% / 20% = 1.5
The risk of injury among the untreated athletes is 1.5 times the risk among the treated athletes.

Alternatively, the inverse calculation can be made.

(50/250) / (75/250) = 2/3
or 20% / 30% = 2/3
The risk of injury among the treated athletes is two-thirds of the risk among the untreated
athletes.

Absolute risk reduction Risk in untreated athletes is 75 /250 = 30%

Risk in treated athletes is 50/250 = 20%
Absolute risk reduction is 30% – 20% = 10%
The absolute difference between the risk of an injury in a treated athlete and an untreated
athlete over a 1 -year period is 10%.

Relative risk reduction Relative risk is (risk in treated) / (risk in untreated) = 20% / 30% = 67%
Relative risk reduction = 100% – relative risk = 100% – 67% = 33%
The relative reduction in risk associated with being treated is 33%.

Number needed to treat 100% / absolute risk reduction = 100% / 10% = 10

For every 10 athletes that undergo the preventative treatment, one is likely to remain
uninjured when she would otherwise have had an injury.

Odds ratio (75/175) / (50/200) = 1.71

or 0.43 / 0.25 = 1.71
The odds of injury among the untreated athletes is 1.71 times higher than the odds among the
treated athletes.

Alternatively, the inverse calculation can be made.

(50/200) / (75/175) = 0.58
or 0.25 / 0.43 = 0.58
The odds of injury among the treated athletes is 0.58 times the odds among the untreated
athletes.

Italic text presents the interpretation of the statistic in sentence format.

that these people with the clinical characteristic are less likely to that the people with generalised joint laxity had 2.8 times the risk
have the outcome of interest. of having an anterior cruciate injury compared with those without
The possible range of odds ratios is from 0 to infinity. In generalised joint laxity. As mentioned previously, risk ratios of less
contrast, the range of risk ratios starts at 0 but has an upper limit than 1 mean the risk of the outcome is less in those with the
that depends on the risk in the reference group (the people who do characteristic than in those without. For example, Kiely et al8
not have the clinical characteristic). For example, if the risk in the investigated the risk of stroke among people with moderate-to-
reference group is 50%, the maximum value that a relative risk can high physical activity and compared it with the risk among people
have is 2.0, because this would indicate that 100% of people with with low physical activity. They reported an risk ratio of 0.8. This
the clinical characteristic have the outcome of interest. was correctly interpreted as: moderate-to-high physical activity
was associated with a relative reduction in the risk of stroke by 20%
compared with low physical activity.
Interpretation of risk ratios

Risk ratios are relatively easy and intuitive to interpret. For Absolute risk reduction and relative risk reduction
example, a study investigating risk factors for non-contact anterior
cruciate injury reported a risk ratio of 2.8 for the presence of To correctly interpret the clinical importance of risk ratios, it is
generalised joint laxity.7 A correct interpretation would be to say critical to understand that the absolute risk reduction (also called
174 Appraisal Research Note

absolute risk difference, because the change in risk is not always a lower than risk ratios. They only become similar when the
reduction) and relative risk reduction mean different things. In the incidence or prevalence is very low.9 Holcomb et al9 found that in
stroke example above, the relative risk was 0.8, indicating a 20% nearly half of the studies they reviewed, the odds ratio was more
relative reduction in risk associated with moderate-to-high than 20% larger than the risk ratio when using the same data. They
physical activity. How much this reduces the absolute risk of also found that 26% of the studies they reviewed incorrectly
stroke is dependent on the baseline risk. If the risk of stroke in the interpreted odds ratios as risk ratios. It could be asked why odds
low physical activity participants were 10%, an relative risk of ratios are used if they are often misinterpreted. One explanation is
0.8 would mean that the risk of a stroke in those who performed that odds ratios have mathematical properties that make them
moderate-to-high physical activity would be 8% (0.10 x 0.8). This more easily managed within some statistical procedures. More
means the absolute risk reduction was 2% (ie, from 10% down to 8%). thorough explanations are available for interested readers.10
If, however, the risk of stroke in the low physical activity
participants were 1%, then the same relative risk would only Confidence intervals
result in an absolute risk reduction of 0.2%. This demonstrates the
importance of distinguishing the relative risk reduction and the All of the statistics mentioned above, from basic risk and odds
absolute risk reduction. The baseline risk is very important to the through to NNT, can be calculated with 95% CIs. Briefly, with each
absolute risk reduction but not considered in relative risk statistic, the 95% CI indicates the range of uncertainty around the
reduction. The same risk ratio (eg, 0.8) will have a greater impact estimate. An excellent and more detailed explanation of the
on absolute risk reduction, the more common the outcome is. interpretation of 95% CI for dichotomous measures has previously
been presented in this journal.11 With risk ratios or odds ratios, a
95% CI that crosses 1 indicates that the result is not statistically
Number needed to treat
significant, whereas a 95% CI that does not include 1 indicates that
the difference between groups can be attributed to the distin-
Another statistic that is used to help in the interpretation of risk
guishing treatment, exposure or characteristic between groups.
ratios is the number needed to treat (NNT), which is a simple way of
The 95% CI around the NNT is easy to interpret when the result is
understanding how many patients need to be treated for one
statistically significant, but it is particularly unintuitive when the
patient, on average, to benefit. For example, let us imagine that we
result is not statistically significant, so readers are referred to
conducted a clinical trial to determine if participants who were
further explanation elsewhere.12
randomised to receive a Tai Chi intervention fell less often than
those who were randomised to no treatment. If the incidence of
Summary
falling was 30% in the Tai Chi group and 60% in the no-treatment
group, the absolute risk reduction would be 30% and the risk ratio
Statistics that summarise dichotomous outcome measures,
would be 0.5. The formula for the NNT is 100%/absolute risk
including risk ratios, odds ratios, absolute risk reduction and
reduction. So, in our example, this would be 100%/30% = 3.3. This
relative risk reduction, are commonly used, but have different
means that 3.3 patients would need to be treated with the Tai Chi
meanings. A good understanding of these terms will enable readers
intervention to prevent one fall. Because it is not possible to treat a
of clinical studies to ensure that they correctly interpret the clinical
fraction of a patient, the NNT may be reported with the decimal
importance of the findings reported.
places rounded off (in this example, to 3) or conservatively
Acknowledgements: Nil.
rounded up to the next highest whole number of participants (in
Competing interests:Nil.
this example, to 4).
Provenance: Not commissioned. Peer reviewed.
It is worth noting that, while the treatment was very effective (it
halved the rate of falls), the NNT was affected by both the
effectiveness of the treatment and the risk in the reference group
Mark Hancocka and Peter Kentb,c
(the no-treatment group). So, if the risk in the no-treatment group a
Faculty of Medicine and Health Sciences, Macquarie University,
had been only 10%, a treatment with the same relative risk would
Sydney, Australia
have an NNT of 20 (100%/absolute risk reduction = 100/5 = 20). The b
School of Physiotherapy and Exercise Science, Curtin University,
reason for this is that most people (nine out of 10) had no fall
Perth, Australia
during the follow-up period, so even a very effective treatment c
Department of Sports Science and Clinical Biomechanics, University of
would need to be given to many people in that population before it
Southern Denmark, Denmark
prevented a fall.

Interpretation of odds ratios References

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