
REVIEW ARTICLE 



Year : 2007  Volume
: 10
 Issue : 4  Page : 225230 

What are relative risk, number needed to treat and odds ratio?
Kameshwar Prasad
Department of Neurology, All India Institute of Medical Sciences (AIIMS), New Delhi, India
Correspondence Address: Kameshwar Prasad Room No. 704, Department of Neurology, C. N. Centre, All India Institute of Medical Sciences, New Delhi India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09722327.37814
Abstract   
The effects of an intervention is best measured in a randomized controlled trial (RCT) and can be expressed in various ways using the measures such as risk difference, number needed to treat (NNT), relative risk or odds ratio. Risk difference (RD) is the difference in risk of the outcome event between control and experimental group. Control group is not exposed to the intervention, whereas experimental group is the one that is exposed to intervention. The risk of outcome event in the control group is also called baseline risk. The NNT is the inverse of the risk difference and indicates the number of patients required to be treated to avoid one additional outcome event. Risk difference and NNT are absolute measures of effect. Relative risk (RR) is a relative measure and is the ratio of the risk in the exposed group to that in the unexposed group. Relative risk reduction (RRR) is one minus RR and indicates the fraction (or percent) of baseline risk that reduces with exposure to the intervention. Odds ratio (OR) is ratio of odds of having the event in the exposed group to that in the unexposed group. These measures are suitable for different purposes and appeal to different constituencies. Odds ratio is the only measure suitable for use in logistic regression and case control studies.
Keywords: Number needed to treat, odds ratio, relative risk
How to cite this article: Prasad K. What are relative risk, number needed to treat and odds ratio?. Ann Indian Acad Neurol 2007;10:22530 
Introduction   
We want to know: "How good is the treatment?" We desire to have one sentence with one figure to answer this question; however, in life, one figure is not sufficient to convey the full significance of many events. For example, we ask our children: how good are your marks? If he/she replies 99%, we feel delighted. However, soon we would desire to know if it the best in the class  in other words, what is his/her position relative to the other students. How many students have marks below this? How many have marks above this? This is to know his relative position. This is described in terms of percentile. The percent of marks in absolute terms is 99, but it is possible that most of the students have it (the exam was very easy). Some testing services use percent, percentile, grade (A1, A2, B1, etc.) and GPA (grade point average) to describe the results. Each figure may give a different perspective and may appeal to different constituencies. This is why stock markets always use at least two figures to describe what happened that day: usually, one to describe the difference in actual figures (absolute difference) and the other to describe it in percent terms (relative to the opening figure).
The effects of an intervention are often expressed in more than one way. Governments often introduce economic reforms (interventions) to reduce unemployment. Usually, we read in the newspapers some facts similar to this: economic reforms decreased the jobless rate from 5% to 3%, thereby reducing it by 40%. How do they come up with these figures? I have often asked workshop participants and obtained the right answer, i.e., if the baseline jobless rate of 5% is taken as 100%, then 3% become 60% and hence a decline of 2% becomes 40%. We come across similar reports for many measures. Inflation may go down from 5 to 4% to be reported as reduction of 20%. The baseline figure of 5% in both cases has been taken as 100% and the reduction is expressed in terms of percent of the baseline value. In case of many intervention trials, the control group figure serves as the baseline figure. It is clear that the difference between the baseline and current figure is only 2% in case of jobless rates and 1% in case of inflation; however, in terms of percentage, they are 40 and 20%, respectively. Thus, we can see that changes can be expressed in terms of difference or percentage. Changes brought by therapeutic interventions can also be expressed in terms of both differences as well as percentage. For instance, a treatment reduces the risk of mortality from 5 to 3%, i.e., a difference of 2% or percent reduction of 40%. In the language of evidencebased medicine, 5% is called the baseline risk, 3% is called the relative risk, the difference of 2% is called risk difference and 40% is called the relative risk reduction. The baseline risk usually comes from the risk observed in the control groups; relative risk comes from the ratio of risk in the experimental treatment group to the risk in control group;^{ [1]} the rest can be easily calculated. The calculations are shown below with a hypothetical example.
Let us consider an example. We want to know whether treatment, for example, with rtPA (recombinant tissue plasminogen activator) saves more lives in patients with acute ischemic stroke. We conduct a randomized study, say on 400 patients  200 receive the rtPA (Group A) and the other 200 receive placebo (Group B). In Group A, 40 patients die, while in Group B, 50 patients die. Let us put the figures in a 2 × 2 table [Table  1].
It is clear that rtPA has saved some lives. If rtPA was not given, 50 patients would have died in each group. rtPA saved 160 (out of 200) patients, whereas without it, only 150 patients survived. rtPA saved 10 extra lives. For saving 10 extra lives, we treated 200 patients with rtPA. Therefore, for saving 1 extra life, the number needed to treat is 200/10 = 20 patients. Number needed to treat is also called "NNT."^{ [2]} Thus, the NNT in this study is 20. We will revisit this later.
Risk difference (absolute risk reduction)
The percentage or proportion of death is also called "risk" or the probability of death. The results of the above mentioned hypothetical trial are expressed in terms of percentage in [Table  2]. Thus, the risk of death in the treatment group (R_{ t} ) = 40/200 = 1/5 = 0.2 = 20%.
The risk in the control group (R_{ c} ) = 50/ 200 = 1/4 = 0.25 = 25%. (This risk in control group is also called "Baseline Risk" and control event rate.)
Thus, the difference in risk (Risk difference) = 20%25% = 5%.
Thus, the treatment makes a difference (reduction) of 5% in risk. Risk difference is also called Absolute Risk Reduction (ARR). The minus sign () indicates that there is a decrease in risk with the treatment. If there was no effect of the treatment, the risk in the treatment group will also be 25% and the risk difference (or ARR) will be 0 (2525%).
Thus, one measure of effect of treatment is the risk difference (also called ARR), which is simply the difference in the risk between treatment and control group (R_{ t} R_{ c} ).
Risk ratio (relative risk) and relative risk reduction
We can also consider the ratio between two risks and call it a risk ratio (RR). Conventionally, the risk in the treatment group is placed in the numerator and that in the control group in the denominator. That is,
Risk ratio = Risk in the treatment group/ Risk in the treatment group
= Rt/Rc
The risk ratio is also called "Relative risk (RR)." This tells us what proportion (or fraction) or percent of baseline risk remains in the treatment group. (Baseline risk is the risk in the control group.) You might like to remember this as Risk Remaining, which fits well with the abbreviation "RR."
If the treatment was totally ineffective, then the risk in the treatment group will be the same as that in the control group, i.e., 25%. Therefore,
In our example, the risk is reduced. It does not remain as 100%. It becomes 80% relative to the baseline risk. (Relative risk is 80%). So in relative terms reduction is 20% or the relative risk reduction (RRR) is (100  80)% = 20%.
In other words RRR = 100RR (in percentage) or 1RR (in decimals).
It can be observed that ARR (Absolute risk reduction) is an absolute measure, whereas RRR (relative risk reduction) is a relative measure. ARR tells us how much difference does the treatment make in actual (absolute) terms, whereas RRR tells us how much reduction of risk does the treatment make in relative terms (relative to the baseline risk that is taken as 100% for this).
Number needed to treat (NNT)
You can find out NNT from risk difference. Risk difference of 5% means 5 less (minus) deaths per 100 patients treated. In other words, for avoiding five deaths, we need to treat 100 patients. Therefore, for one death to be avoided, the NNT is 100/5 = 20; we can easily see the relationship. 5% in fraction terms in 5/100, whereas the NNT is 100/5. Therefore, what we should do to find out the NNT? There are three steps: First, find out the difference in risk; Second, write this in the form of a fraction (obviously, with denominator as 100); Third, invert the fraction so that you have 100 in the numerator and the difference (in %) in the denominator. Perform the cancellation and rounding. The figure you obtain is the NNT. If you prefer decimals you can perform this in two steps: First, find out the risk difference or ARR (say, 0.05). Then, take its inverse, i.e., 1/risk difference or 1/ARR.
Whenever NNT is considered, we must specify the followup period over which the difference was observed and the unfavorable outcome that was avoided. NNT usually makes sense only when the treatment has been shown to make a statistically significant difference.
How much NNT is good? Well, there is no magic figure. For an inexpensive drug with no side effects, an acceptable NNT may be 100 or even 200; whereas for risky surgery it may be only 20. Acceptable NNTs may be different for prophylactic versus "active treatment." As a general guide, for active treatments, the acceptable NNT may be 20 or 25; however, for preventive treatment, it can be even 250 or 500.
It should be noted that all the time we are assuming that the treatment decreases the risk. At times, treatments are harmful. They increase the risk. In this case, what we will get is the NNT to cause one extra death or harm. Therefore, we must understand what the treatment is doing? We need to interpret the NNT accordingly. However, it is known that even effective and helpful treatments often have adverse effects as well. We can find out NNT to prevent one adverse outcome or NNT to cause one extra harm (i.e., adverse event or NNT to cause one adverse event). The latter is at times called NNH (Number needed to harm).
To recapitulate, we have covered four measures of treatment effect in our example: (1) Risk difference or ARR = 5% or 0.05; (2) NNT = 20; (3) RR (Risk ratio or relative risk) = 80% or 0.8; and (4) Relative risk reduction (RRR) = 20% or 0.2.
The measures RD (ARR), NNT, RR and RRR are generally adequate for communication among clinicians and are sufficient to deal with most situations. However, at times, as observed in casecontrol studies, none of the measures can be applied. The measure, which applies to this type of studies as well as others, is based on odds (unlike all the above mentioned ones that are based on probabilities or risks).
Odds ratio (OR)^{ [3]}
We usually say that the odds of England team winning the cricket match is 1:4. What does it mean? It implies that if there is one chance of winning, there are four chances of losing. In other words, there are one out of five (20%) chances of winning and four out of five (80%) chances of losing. Chance is the probability. Odds of 1:4 implies 20% probability of winning and 80% probability of losing. Thus, the odds consider both the sides of the coin  winning vs. losing, death vs. survival, improvement vs. deterioration. The odds of 1:4 is equal to ¼, i.e., 0.25 or 25%. You can see that 25% odds of winning means 20% probability of winning. You may ignore this interrelationship. It should be kept in mind that the odds expression requires probability of one side of the coin (winning, for example) in the numerator and probability of the other side of coin (losing, in our example) in the denominator.
Consider an example, say, 20% of the patients in the treatment group died, which means 80% survived in the treatment group. Therefore, what are the odds of death in the treatment group? Remember, for odds, we require the chance (probability) of death in the numerator, i.e., 20% and chance of survival in the denominator, i.e., 80%. Therefore, the odds will be 20%/80% (in decimals, 0.2/0.8); this is equal to ¼.
Now, say, 25% of the patients in the control group died, which implies that 75% survived. Therefore, the odds of death in the control group are 25%/75%. (or 0.25/0.75) = 1/3.
Therefore, the odds ratio that usually has the odds of death (or any adverse event) in the treatment group as the numerator and odds of death in the placebo group in the denominator will be equal to ¼/⅓ = ¼ χ ⅓ = ¼ × ^{3}/_{1} = ¾ = 0.75 (or 75%) Thus, one way of expressing the treatment effect is odds ratio = 0.75 (=75%). Again OR can be interpreted as "Odds Remaining." Therefore, the OR is 75%. Therefore, odds reduction is (100  75)% = 25%.
There are two other common situations in which the odds ratios are used:
Case control studies : The estimation of risk requires followup (cohort) studies. Case control studies do not involve followup. Investigators conducting such studies select subjects on the basis of presence (cases) or absence (controls) of outcome and then determine their exposure status. The only measure of effect that can be estimated in case control studies is the "odds ratio." Case control studies are the design of choice when the frequency of disease is low; and in such situations, odds ratios approximate the risk ratios.
Logistic regression : When the outcome is binary (yes/no, death/survival, remission/no remission), then independent role of multiple determinants (variables) of the outcome can be efficiently studied using a type of multivariable analyses, called logistic regression. This regression estimates the odds ratios associated with each determinant in relation to the outcome. The interpretation of these odds ratio remains the same as explained above (OR equal to one indicates no independent association between the variable and outcome and values of OR significantly less or more than one indicate independent association between the variable and the outcome.
Treatment effect in our example can be expressed in several ways:
a. Risk of death is decreased from 25 to 20%
b. Risk difference or Absolute risk reduction is 5% (5/100)
c. NNT is 100/5, i.e., 20
d. Risk ratio or Relative risk = 20/25% = 0.8 or 80%
e. Relative risk reduction = 20%
f. Odds ratio = 0.75 or 75%
g. Odds reduction is 25%
If you are the drug manufacturer, what will be your choicest measure? Obviously, the odds reduction of 25%. What will be your second choice? The relative risk reduction of 20%. Both are relative to the odds or risks in the control group (taken as 100%), which will be reduced by 25 or 20%, respectively. In actual terms, how much difference does the treatment make? Only 5%. This is well captured by NNT, i.e., 20 patients must be treated to prevent one extra death. All are technically correct measures, each giving one perspective on the effect of the treatment.
Why are there so many effect measures (relative merits and demerits)?
Suppose you are satisfied that there is no threat to the validity of the study. You are ready to look at the results. But, let us digress for a moment.
Different people may look at the same result for different purposes and in different ways.^{ [4],[5]} A typical example for this is how the result of a given investigation of a patient is viewed by different care providers. Consider the brain CT of a patient suspected to have hemorrhagic stroke. Emergency physician examines the CT to see if this confirms his suspicion. If yes, he refers the case to a neurologist. The neurologist examines it to determine whether the site is typical of a hypertensive bleed and calculates the volume of the hematoma to inform the patient or relatives regarding the patient's prognosis. If he thinks that the prognosis is not favorable with medical treatment alone, then he refers the case to a neurosurgeon who attempts to determine whether hematoma is operable or what is the benefit vs. risk of surgery in this hematoma (there are many other issues, but let us use only the above one for illustration). Similarly, the results of a study have many clients. Each client has a different purpose in mind when he looks at the results.
Let us consider a study that compared the treatment of stroke in "stroke unit" versus "general medical ward." The results have many clients with different objectives in mind.
 Hospital administration wants to know the cost effectiveness of stroke unit as compared with the current treatment policy of stroke patients in general medical ward.
 The physician wants to know how much benefit it would offer to his patients. He sees different types of patients. Some patients are young with few risk factors and mild stroke; say with 2% risk of institutionalization. Some patients are old with many risk factors and severe stroke with (say) 90% risk of death or dependence. The physician wants to know the benefit in each type of patients.
Accordingly, we want effect measures that:
 are easy to understand,
 provide an idea of the cost effectiveness,
 are meaningfully applicable to all types of patients, and
 convey the same idea even when you measure unfavorable (e.g., Death) or favorable (e.g., survival) outcomes.
Number Needed to Treat (NNT)
Let us consider a hypothetical study showing that 50% of stroke patients treated in the general medical ward (hereafter called "ward") were institutionalized, whereas only 25% were so in the group treated in a stroke unit. Thus, the stroke unit made a difference of 25% (2550%), yielding a NNT of 4. This implies that four patients need to be treated in the stroke unit to avoid institutionalization of one patient with stroke.
The hospital administrator calculates cost of the stroke unit. On an average, if it is $5000, he can easily see that $20000 needs to be spent to prevent one institutionalization. He can easily compare the cost of institutionalization vs. stroke unit treatment and take a decision. Thus, one advantage of NNT is that it gives a quick (even if very rough) idea regarding costeffectiveness.
The NNT has another advantage. This becomes apparent when we are dealing with a very low frequency outcome. For example, mammography reduces the incidence of death rate from breast cancer from 0.08 to 0.02%, i.e., a difference of 0.06%. The NNT turns out to be 1666. This implies that 1666 ladies must have regular mammography for 7 years to prevent one death from breast cancer. You may find this representation more easier and understandable than a difference of 0.06%. Thus, NNT helps to convert a small decimal into a round figure that is more comprehensible. Thus, the two advantages of NNT are as follows:
(i) It provides an easy way to obtain an idea of the costbenefit.
(ii) It is more easily understood by policy makers and physicians.
However, NNT has disadvantages too:
(i) For example, if you are communicating with the patient, and say, four patients must be treated to save one additional patient, the patient might ask: Am I likely to be the saved one or one among the other three who may possibly die? In other words, this is not easily interpretable for an individual patient.
Risk difference (RD) or absolute risk reduction (ARR)
Risk difference has three merits:
(i) Easy to calculate and interpret: You have to perform only a subtraction and RD tells you how much difference the intervention could make.
(ii) It is symmetrical, i.e., conveys the same effect whether you measure the favorable or unfavorable outcome. In the stroke example, if you measured the favorable outcome such as "going home," still the difference will be the same. Fifty percent went home in the "ward" group and 75% in the stroke unit group  a difference of 25%, which is the same in magnitude as mentioned earlier.
(iii) It helps in calculation of NNT.
(iv) The fourth merit of RD is that its confidence interval can be calculated even when no patient had the outcome of interest in any group. For example, no patient died in any group.
However, it has some demerits:
(i) At times, it is too small to be pronounced and interpreted easily (for example, see the mammography example mentioned above)
(ii) It cannot apply equally to all types of patients. Consider the two patients with acute stroke  one mild and one severe. You might assume (although it is not correct) that the risk of death/institutionalization of the severe patient would be down to 65% (9025); however, what about the mild patient  as such his risk is 2%? How can stroke unit make a difference of 25%, when the total risk is 2%? This illustrates the difficulty in using the RD (or ARR) from the study data. (However, RR is equally applicable in both cases  see next paragraph)
Risk ratio or relative risk (RR)
It has the merit of applicability to all types of patients. For example, in the stroke example RR would be 25/50% = 0.5 (=50%). That implies that the risk with treatment in the stroke unit is 50% of that with treatment in general medical ward. Thus, it would be 45% (half of 90) with the stroke unit treatment in the severe case, whereas it would be 1% (half of two percent) in the mild case; risk ratio easily applies to both.
However, the demerit of RR is that it is not symmetrical. As discussed above, the stroke unit halves the risk of unfavorable outcome. If you measure the favorable outcome here (such as "going home"), then it should double its rate; however, calculations using RR do not yield such results. With 75% going home in the stroke unit group and 50% in the ward treatment group, the RR of "going home" is 75/50% = 1.5, rather than two. The other demerit which you might have noticed is that it does not sound right to say the risk of "going home." Going home is a favorable outcome and risk is a rather loaded concept that sounds awkward in association with favorable outcome.
In summary, the merits of RR are:
(i) Its applicability to all types of patients
(ii) Easier to interpret than odds ratio
Its demerits are as follows:
(i) Asymmetry: If there is 10% risk of death in experimental group and 40% risk in the control group; RR = 0.25, i.e., RRR = 1  0.25 = 0.75 or 75% risk reduction. If we counted the number of survivals.
(ii) Lack of neutrality: The risk of survival in experimental and control groups will be 90 and 60%, respectively; (RR = 1.5) It implies that the relative risk of increase of survival is 50%. In one way, it is 75%, and the other way it is 50%; this is the asymmetry. The risk of survival sounds awkward. Risk sounds alright only for unfavorable outcomes, not the favorable ones. Therefore, this is not a neutral concept.
(iii) There is no way to calculate the confidence interval (C.I.) of RR when there is zero event in both the treatment groups, for example, no death in any of the two groups in a study.
Odds ratio
The merits of OR are as follows:
(i) Similar to RR, it is applicable to all kinds of patients, irrespective of their levels of risk without the treatment.
(ii) It is not a loaded concept. It is neutral. The odds of going home sound as appropriate as the odds of institutionalization or death. Just as the odds of winning or losing both sound acceptable.
(iii) It is symmetrical. In the stroke example, the odds of institutionalization in the "stroke unit" group is 50:50 = 1, whereas in the "general ward" group, it is 75:25 = 3. The odds ratio for institutionalization with stroke unit vs. general ward is 1/3. Now, let us see what happens if we measured the odds of going home. This is 50:50 (=1) with stroke unit group and 25:75 with the general ward group is 1/3. Therefore, the odds ratio of going home is 1 χ (1/3) = 1 × (3/1) = 3. Thus, the odds of institutionalization with stroke unit care is 1/3 of that with general ward. Similarly, the odds of going home with stroke unit is three times that with general ward. The symmetry is clear and no matter what you measure  the favorable or unfavorable outcome, it gives the same impression.
(iv) The fourth merit of OR is that it can be used in one of the commonest form of adjusted analysis (using logistic regression), whereas RD or RR cannot be
(v) It has certain mathematical properties that make it a favored measure for some statistical calculation including metaanalysis.
The demerits of OR are as follows:
(i) It is a difficult concept to understand and interpret for health professionals
(ii) If interpreted like RR, it overestimates the treatment effects. OR and RR are similar only when events in the control and experimental group are 10% or less or when they are close to one.
(iii) Similar to that in RR, there is no way to calculate the confidence interval around OR, when there is zero event in both the treatment arms. Only RD lends itself to calculation of C.I. in this situation.
References   
1.  Guyatt G, Rennie, editors. User's guides to the medical literature: A manual for evidencebased clinical practice. AMA Press: Chicago; 2002. Available from: http://www.amaassn.org. 
2.  Laupacis A, Sackett DL, Roberts RS. An assessment of clinically useful measures of the consequences of treatment. N Engl J Med 1988;318:172833. [PUBMED] 
3.  Peto R, Pike MC, Armitage P, Breslow NE, Cox DR, Howard SV, et al. Design and analysis of randomized clinical trials requiring prolonged observation of each patient II: Analysis and examples. Br J Cancer 1977;35:139. [PUBMED] 
4.  Malenka DJ, Baron JA, Johansen S, Wahrenberger JW, Ross JM. The framing effect of relative and absolute risk. J Gen Intern Med 1993;8:5438. [PUBMED] 
5.  Naylor CD, Chen E, Strauss B. Measured enthusiasm: Does the method of reporting trial results alter perceptions of therapeutic effectiveness? Ann Intern Med 1992;117:91621. [PUBMED] 
[Table  1], [Table  2]
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