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What’s the Risk: Differentiating Risk Ratios, Odds Ratios, and Hazard Ratios?

Andrew george.

1 Emergency Medicine, Brown University, Providence, USA

Thor S Stead

2 Emergency Medicine, Warren Alpert Medical School, Providence, USA

Latha Ganti

3 Emergency Medicine, University of Central Florida College of Medicine, Orlando, USA

4 Emergency Medicine, Envision Physician Services, Nashville, USA

5 Emergency Medical Services, Polk County Fire Rescue, Bartow, USA

Risk ratios, odds ratios, and hazard ratios are three common, but often misused, statistical measures in clinical research. In this paper, the authors dissect what each of these terms define, and provide examples from the medical literature to illustrate each of these statistical measures. Finally, the correct and incorrect methods to use these measures are summarized.

Introduction and background

Risk ratios, odds ratios, and hazard ratios are three ubiquitous statistical measures in clinical research, yet are often misused or misunderstood in their interpretation of a study’s results [ 1 ]. A 2001 paper looking at the use of odds ratios in obstetrics and gynecology research reported 26% of studies (N = 151) misinterpreted odds ratios as risk ratios [ 2 ], while a 2012 paper found similar results within published literature on obesity, with 23.2% of studies (N = 62) published across a one-year period in two leading journals misrepresenting odds ratios [ 3 ].

Understanding the three measures’ applicability and usage allows for a more accurate interpretation of study results and a better understanding of what each value demonstrates and, equally importantly, what each does not.

Confidence intervals and p-values

In order to entertain any discussion of statistical analysis, it is important to first understand the concept of population statistics. Plainly, population statistics are the values of any measure within the population of interest, and estimating them is the goal of most studies [ 4 ]. For instance, in a study looking at obesity rates for patients on a certain medication, the population statistic could be the average obesity rate for all patients on the medication.

However, identifying this value would require having data for every single individual that falls into this category, which is impractical. Instead, a randomized sample can be gathered, from which sample statistics can be obtained. These sample statistics serve as estimates of the corresponding population statistics and allow a researcher to make conclusions about a population of interest.

A significant limitation exists in that these constructed samples must be representative of the larger population of interest. While there are many steps that can be taken to reduce this limitation, sometimes its effects (so-called sampling bias [ 5 ]) go beyond the control of the researcher. Additionally, even in a theoretical situation with no sampling bias, randomization could result in a misrepresentative sample. In the previous example, suppose that the population rate of obesity among all adults eligible for the medication was 25%. In a simple random sample of 30 patients from this population, there is a 19.7% chance that at least 10 patients will be obese, resulting in a sample obesity rate of 33.3% or even higher. Even if there is no relationship between the medication and obesity rates, it is still possible to encounter a rate that appears to be different from the overall obesity rate, which occurred through randomness in sampling alone. This effect is the reason for reporting confidence intervals and p-values in clinical research.

Confidence intervals are intervals in which the population statistic could lie. They are constructed based on the sample statistic and certain features of the sample that gauge how likely it is to be representative and are reported to a certain threshold [ 6 ]. A 95% confidence interval is an interval constructed such that, on average, 95% of random samples would contain the true population statistic within their 95% confidence interval. Thus, a threshold for significant results is often taken as 95%, with the understanding that all values within the reported range are equally valid as the possible population statistic.

The p-value reports similar information in a different way. Rather than constructing an interval around a sample statistic, a p-value reports the probability that the sample statistic was produced from random sampling of a population, given a set of assumptions about the population, referred to as the “null hypothesis” [ 7 ]. Taking the example study on obesity rates again, the obesity rate among the sample (a sample of patients on the medication) could be reported alongside a p-value determining the chance that such a rate could be produced from randomly sampling the overall population of patients eligible for the medication. In the case of the study, the null hypothesis is that the population rate of obesity among patients on the medication is equal to the overall rate of obesity among all patients eligible for the medication, that is, 25%. A one-tailed p-value can be used if there is reason to believe that an effect would occur in only one direction (for instance, there may be reason to believe the medication would increase weight gain but not decrease it), whereas a two-tailed p-value should be used in all other cases. When using a symmetric distribution, such as the normal distribution, two-tailed p-values are simply twice the one-tailed p-value.

Suppose again that a sample of 30 patients on the medication contains 12 obese individuals. With a one-tailed test, our p-value is 0.0216 (using the binomial distribution). Thus, we can say that our observed rate of 40% is significantly different from the hypothesized rate of 25% at a significance level of 0.05. In another sense, the 95% confidence interval for the observed proportion is 25.6% to 61.07%. Confidence intervals correspond to two-tailed tests, where a two-tailed test is rejected if and only if the confidence interval does not contain the value associated with the null hypothesis (in this case, 25%).

If a calculated p-value is small, it is likely the population is not structured as originally stated in the null hypothesis. If we obtain a low p-value, we have evidence that there was some effect or reason for the observed difference - the medication, in this case. A threshold of 0.05 (or 5%) is typically used, with a p-value having to be below this threshold for its corresponding attribute to be statistically significant.

Risk ratios

Risk, another term for probability, is another fundamental principle of statistical analysis. Probability is a comparison of observing a specific event occurring as a result to the total unique results. A coin flip is a trivial example: the risk of observing a heads is ½ or 50%, as of all possible unique trials (a flip resulting in heads or a flip resulting in tails), only one is the event of interest (heads).

Using only risk allows predictions about a single population. For instance, looking at obesity rates within the U.S. population, the CDC reported that 42.4% of adults were obese in 2017-2018. So, the risk of an individual in the U.S. being obese is around 42.4% [ 8 ]. However, most studies look at the effect of a specific intervention or other item (such as mortality) on another. Earlier, we supposed that the obesity rate of eligible patients was 25%, but here we will use the 42.4% associated with the U.S. adult population. Suppose we observe a risk of 25% in a random sample of patients on the medication as well. To conceptualize the effect of the medication on obesity, a logical next step would be dividing the risk of obesity in the U.S. population on the medication with the risk of obesity in the U.S. population, which results in a risk ratio of 0.590.

This calculation - a ratio of two risks - is what is meant by the eponymous risk ratio (RR) statistic, also known as relative risk. It allows a specific number to be given for how much more risk an individual in one category bears compared to an individual in another category. In the example, an individual taking the medication bears 0.59 times as much risk as an adult from the general U.S. population. However, we have assumed that the population eligible for the medication had an obesity rate of 25% - perhaps only a group of young adults, who may be healthier on average, are eligible to take the medication. When investigating the effect of the medication on obesity, this is the proportion that should be used as the null hypothesis. If we observe an obesity rate on the medication of 40%, with a p-value less than the significance level of 0.05, this is evidence that the medication increases the risk of obesity (with an RR, in this scenario, of 1.6). As such, it is important to carefully choose the null hypothesis to make relevant statistical predictions.

With RR, a result of 1 signifies that both groups have the same amount of risk, while results not equal to 1 indicate that one group bore more risk than another, a risk that is assumed to be due to the intervention examined by the study (formally, the assumption of causal direction).

To illustrate, we look at the results of a 2009 study published in the Journal of Stroke and Cerebrovascular Diseases. The study reports that patients with a prolonged electrocardiographic QTc interval were more likely to die within 90 days compared with patients without a prolonged interval (relative risk [RR]=2.5; 95% confidence interval [CI] 1.5-4.1) [ 9 ]. Having a confidence interval between 1.5 and 4.1 for the risk ratio indicates that patients with a prolonged QTc interval were 1.5-4.1 times more likely to die in 90 days than those without a prolonged QTc interval.

A second example- in a landmark paper demonstrating that the blood pressure curve in acute ischemic stroke is U-shaped rather than J-shaped [ 10 ], the investigators found that the RR increased nearly two-fold in patients with mean arterial blood pressure (MAP) >140 mmHg or <100 mmHg (RR=1.8, 95% CI 1.1-2.9, p=0.027). Having a CI of 1.1-2.9 for the RR means that patients with a MAP outside the range of 100-140 mmHg were 1.1-2.9 times more likely to die than those who had initial MAP within this range.

For another example, a 2018 study on Australian naval recruits found that those with prefabricated orthoses (a type of foot support) had a 20.3% risk of suffering at least one adverse effect, while those without had a risk of 12.4% [ 11 ]. A risk ratio here is given by 0.203/0.124, or 1.63, suggesting that recruits with foot orthoses bore 1.63 times the risk of having some adverse consequence (e.g. foot blister, pain, etc.) than those without. However, the same study reports a 95% confidence interval for the risk ratio of 0.96 to 2.76, with a p-value of 0.068. Looking at the confidence interval, the 95% reported range (the commonly accepted standard) includes values under 1, 1, and values above 1. Remembering that all values are equally likely to be the population statistic, at 95% confidence, there is no way to exclude the possibility that foot orthoses have no effect, have a significant benefit, or have a significant detriment. Additionally, the p-value is greater than the standard of 0.05, therefore this data does not provide significant evidence of foot orthoses having any consistent effect on adverse events such as blisters and pain. As stated previously, this is no coincidence - if they are calculated using the same or similar methods and the p-value is two-tailed, confidence intervals and p-values will report the same results.

When utilized correctly, risk ratios are a powerful statistic that allow an estimation in a population of the change in risk one population bears over another. They are quite easy to understand (the value is how many times the risk one group bears over another), and with the assumption of causal direction, quickly show whether an intervention (or other tested variable) has an effect on outcomes.

However, there are limitations. Firstly, RRs cannot be applied in all cases. Because risk in a sample is an estimate of risk in a population, the sample must be reasonably representative of the population. As such, case-control studies, by simple virtue of the fact that ratios of outcomes are controlled, cannot have a risk ratio reported. Secondly, as with all the statistics discussed herein, RR is a relative measure, providing information about the risk in one group relative to another. The problem here is that a study where two groups had a risk of 0.2% and 0.1% bears the same RR, 2, as one where two groups had a risk of 90% and 45%. Though in both cases it is true that those with the intervention were at twice the risk, this equates to only 0.1% more risk in one case while 45% more risk in another case. Thus, reporting only the RR exaggerates the effect in the first instance, while potentially even minimizing the effect (or at least decontextualizing it) in the second instance.

Odds ratios

While risk reports the number of events of interest in relation to the total number of trials, odds report the number of events of interest in relation to the number of events not of interest. Stated differently, it reports the number of events to nonevents. While the risk, as determined previously, of flipping a coin to be heads is 1:2 or 50%, the odds of flipping a coin to be heads is 1:1, as there is one desired outcome (event), and one undesired outcome (nonevent) (Figure  1 ). 

Just as with RR, where the ratio of two risks was taken for two separate groups, a ratio of two odds can be taken for two separate groups to produce an odds ratio (OR). Instead of reporting how many times the risk one group bears relative to the other, it reports how many times the odds one group bears to the other.

For most, this is a more difficult statistic to understand. Risk is often a more intuitive concept than odds, and thus understanding relative risks is often preferred to understanding relative odds. However, OR does not suffer from the same causal assumption limitations as RR, making it more widely applicable.

For instance, odds are a symmetric measure, meaning that while risk only examines outcomes given interventions, odds can also examine interventions given outcomes. Thus, a study can be constructed where, rather than choosing trial groups and measuring outcomes, outcomes can be chosen, and other factors can be analyzed. The following is an example of a case-control study, a situation where RR cannot be used but OR can.

A 2019 case-control study proves a good example. Seeking to find potential correlation between a hepatitis A virus (HAV) infection prominent in Canada and some causing factor, a study was constructed based on the outcome (in other words, individuals were categorized based on their HAV status, as the “intervention”, or causal event, was unknown). The study looked at those with HAV and those without and what foods they had eaten prior to HAV infection [ 12 ]. From this, multiple odds ratios were constructed comparing a specific food item to HAV status. For example, the data found that among those subjects who had exposure to shrimp/prawns, eight were positive for HAV while seven were not, while for those without exposure two were positive for HAV while 29 were not. An odds ratio is taken by (8:7)/(2:29) which equals approximately 16.6. The study data reported an OR of 15.75, with the small discrepancy likely originating from any pre-calculation adjustments for confounding variables that was not discussed in the paper. A p-value of 0.01 was reported, thus providing statistical evidence for this OR being significant.

This can be interpreted in two equal ways. Firstly, the odds of shrimp/prawn exposure for those with HAV are 15.75 times higher than for those without. Equivalently, the odds for HAV-posiitve versus HAV-negative is 15.75 times higher for those exposed to shrimps/prawns than for those not exposed.

Overall, OR provides a measure of the strength of association between two variables on a scale of 1 being no association, above 1 being a positive association, and below 1 being a negative association. While the previous two interpretations are correct, they are not as directly understandable as an RR would have been, had it been possible to determine one. An alternative interpretation is that there is a strong positive correlation between shrimp/prawn exposure and HAV.

Because of this, in some specific cases, it is appropriate to approximate RR with OR. In such cases, the rare disease assumption must hold. That is, a disease must be exceedingly rare within a population. Under this case, the risk of the disease within the population (p/(p+q)) approaches the odds of the disease within the population (p/q) as p becomes insignificantly small relative to q. Thus, the RR and OR converge as the population gets larger. However, if this assumption fails, the difference becomes increasingly exaggerated. Mathematically, in p+q trials, decreasing p increases q to maintain the same total trials. With risk, only the numerator changes, whereas with odds both the numerator and denominator change in opposite directions. As a result, for cases where the RR and OR are both below 1, the OR will underestimate the RR, while for cases where both are above 1, the OR will overestimate the RR.

Misreporting of the OR as the RR, then, can often exaggerate data. It is important to remember that OR is a relative measure just as RR, and thus sometimes a large OR can correspond with a small difference between odds.

For the most faithful reporting, then, OR should not be presented as an RR, and should only be presented as an approximation of RR if the rare disease assumption can reasonably hold. If possible, a RR should always be reported.

Hazard ratios

Both RR and OR concern interventions and outcomes, thus reporting across an entire study period. However, a similar but distinct measure, the hazard ratio (HR), concerns rates of change (Table  1 ).

HRs are in tandem with survivorship curves, which show the temporal progression of some event within a group, whether that event is death, or contracting a disease. In a survivorship curve, the vertical axis corresponds to the event of interest and the horizontal axis corresponds to time. The hazard of the event is then equivalent to the slope of the graph, or the events per time.

A hazard ratio is simply a comparison of two hazards. It can show how quickly two survivorship curves diverge through comparison of the slopes of the curves. An HR of 1 indicates no divergence - within both curves, the likelihood of the event was equally likely at any given time. An HR not equal to 1 indicates that two events are not occurring at an equal rate, and the risk of an individual in one group is different than the risk of an individual in another at any given time interval.

An important assumption that HRs make is the proportional rates assumption. To report a singular hazard ratio, it must be assumed that the two hazard rates are constant. If the slope of the graph is to change, the ratio will likewise change over time, and thus will not apply as a comparison of likelihood at any given time.

Consider the trial of a novel chemotherapeutic agent seeking to extend life expectancy of patients with a specific cancer. In both the intervention and the control group, 25% had died by week 40. Since both groups decreased from 100% survival to 75% survival over the 40-week period, the hazard rates would be equal and thus the hazard rate equal to 1. This suggests that an individual receiving the drug is just as likely to die as one not receiving the drug at any time.

However, it is possible that in the intervention group, all 25% died between weeks six to 10, while for the control group, all 25% died within weeks one to six. In this case, comparing medians would display a higher life expectancy for those on the drug despite the HR not showing any difference. In this case, the proportional hazards assumption fails, as the hazard rates change (quite dramatically) over time. In cases such as this, HR is not applicable.

Because it is sometimes difficult to determine whether the proportional hazards assumption reasonably applies, and because taking an HR strips the original measurement (hazard rates) of the time unit, it is common practice to report HR in conjunction with median times.

In a study evaluating the prognostic performance of The Rapid Emergency Medicine Score (REMS) and the Worthing Physiological Scoring system (WPSS), the investigators found that the risk of 30-day mortality was increased by 30% for each additional REMS unit (HR: 1.28; 95% confidence interval (CI): 1.23-1.34) and by 60% for each additional WPSS unit (HR: 1.6; 95% CI: 1.5-1.7). In this case, the death rate did not change, but rather the scoring system to predict it did, so the HR can be used. Having a confidence interval between 1.5 and 1.7 for the WPSS hazards ratio indicates that the mortality curve for those with a higher WPS declines at a faster rate (about 1.5-1.7 times). Since the low end of the interval is still above 1, we are confident that the true hazard of death within 30 days is higher for the group with higher WPS [ 13 ].

In a 2018 study on binge drinking amongst individuals with certain risk factors, a survival curve was constructed plotting the rate of achieving binge drinking for controls, those with a family history, male sex, those with high impulsivity, and those with a higher response to alcohol. For men and those with a family history, statistically significant evidence for a higher rate of achieving binge drinking was reported (an HR of 1.74 for men and 1.04 for those with a family history) [ 14 ]. However, for those with high impulsivity, though the HR was 1.17, the 95% confidence interval ranged from 1.00 to 1.37. Thus, to a 95% confidence level, it is impossible to rule out that the HR was 1.00.

Because of the exaggeration present, it is important to avoid representing ORs as RRs, and similarly, it is important to recognize that a reported OR rarely provides a good approximation of relative risks but rather simply provides a measure of correlation.

Because of its ability to make firm conclusions and understandability, RR should be reported if possible, however in the cases where its causality assumption is violated (such as case-control studies and logistic regression), OR can be used.

HRs are used with survival curves and assume that hazard rates are equal over time. While useful to compare two rates, they should be reported with median times to justify the proportional hazards assumption.

Finally, regardless of the value of the HR/RR/OR statistic, an interpretation should only be made after determining whether the result provides statistically significant evidence towards a conclusion (as determined by the p-value or confidence interval). Remembering these principles and the framework of HR/RR/OR minimizes misrepresentation and prevents one from drawing incorrect conclusions from the results of a published study concerning various samples. Figure  2  summarizes correct and incorrect usage of these various risk ratios.

Conclusions

Medical literature is full of statistical measures designed to help clinicians make inferences about a particular intervention or association between variables or the effect of an intervention over time. Thus understanding the meaning of each of these measures is paramount for making patient care decisions.

Acknowledgments

This research was supported (in whole or in part) by HCA Healthcare and/or an HCA Healthcare affiliated entity. The views expressed in this publication represent those of the author(s) and do not necessarily represent the official views of HCA Healthcare or any of its affiliated entities.

The content published in Cureus is the result of clinical experience and/or research by independent individuals or organizations. Cureus is not responsible for the scientific accuracy or reliability of data or conclusions published herein. All content published within Cureus is intended only for educational, research and reference purposes. Additionally, articles published within Cureus should not be deemed a suitable substitute for the advice of a qualified health care professional. Do not disregard or avoid professional medical advice due to content published within Cureus.

The authors have declared that no competing interests exist.

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• Published: 18 May 2017

Method to estimate relative risk using exposed proportion and case group data

• Yoichi Yada 1  

Scientific Reports volume  7 , Article number:  2131 ( 2017 ) Cite this article

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• Ecological epidemiology
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A change in risk of an event occurring, which is affected with a factor, is a common issue in many research fields, and relative risk is widely used because of intuitive interpretation. Estimating relative risk has required data from two follow-up groups and can thus be cost and time consuming. Subjects for whom an event occurred (case group) are often observed but generally analyzed in comparison to those for whom an event did not (control group); however, estimating relative risk using case group data without approximation is hindered. In this study, an obstacle to estimate relative risk using case control data is clarified as a mathematical expression and a new equation to estimate relative risk using the exposed proportion and case group data is proposed. The proposed equation is derived without using the Bayesian methods. A method to estimate the confidence interval for the proposed estimator is also provided. The usefulness of the proposed equation, which requires neither control nor follow-up groups, is demonstrated for both theoretical and real-life examples.

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Introduction

A change in risk of an event occurring associated with exposure to a factor is generally studied in many fields, such as medicine and social science 1 , 2 . Relative risk ( RR ), also known as “rate ratio”, is widely used as a measure of association and can be interpreted intuitively 3 , 4 because of its simple definition:

where π 1 and π 0 are the probabilities of an event occurring (i.e., risks) for subjects exposed and unexposed to a factor. Estimating RR requires the estimators of both π 1 and π 0 , such as the prevalence or cumulative incidence rate.

The probability estimators can be calculated using existing data of large-scale epidemiological studies or should be obtained from a smaller study designed for the estimation. Let N be the total number of subjects to be studied, such as population, and N 1 and N 0 be the exposed and unexposed parts of N . The N 1 is written as

where E is the exposed proportion. The probabilities of an event occurring can be written as

where N 11 and N 01 are the numbers of subjects for whom an event occurred among N 1 and N 0 . When p 1 and p 0 are the estimators of π 1 and π 0 , they should be defined as

where n 1 and n 0 are the observed numbers of exposed and unexposed subjects and n 11 and n 01 are the numbers of subjects for whom the event occurred among n 1 and n 0 . Thus, eRR , which is defined as

is used as the estimator of relative risk. The groups of n 11 and n 01 can be found in groups of exposed and unexposed subjects, who were followed to the event occurring (called “cohort”). However, appropriate cohorts may be occasionally found in epidemiological survey results or should be obtained from a fresh study designed for the purpose (i.e., cohort study).

Unfortunately, few existing results provide appropriate cohorts and long-term observations of cohorts, for example, over several years or decades, are likely to be costly and time consuming, and thus, estimating relative risk can be burdensome for researchers. Meanwhile, because case groups are commonly observed, studies comparing them to a control group (case control study) and estimating the change in risk tend to be less costly and time consuming. Although a case control study is often conducted, estimating relative risk using case control data is hindered. To demonstrate, let m 1 and m 0 be the numbers of observed subjects in a case group and control group and m 11 and m 01 be the numbers of exposed subjects in the case and control groups (see Table  1 ). When meRR is defined similarly to the estimator of relative risk as

meRR may be misused as an estimator of relative risk but will largely vary with observing conditions that researchers can designate, such as the size of m 1 . Moreover, researchers cannot perceive the effects of those observing conditions. Thus, meRR is not appropriate for the estimation. Although this obstacle for estimating relative risk caused by observation is well known to epidemiologists 1 , few studies have clarified the effects of observing conditions as a mathematical expression.

According to Cornfield (1951), relative risk can be approximated using an odds ratio ( OR ) 5 , which is defined as

when π 0 is small (so-called “rare disease assumption”). Thus, the estimator of OR ( eOR ), which is defined as

is often computed instead of estimating relative risk. However, OR always overstates the association and the divergence of overstatement depends on RR or π 0 6 , 7 and thus, using eOR may be misleading.

In addition, some study designs that reduce costs and estimate relative risk were proposed 8 , 9 , 10 , although they still require cohorts or the likes. Few studies have focused on deriving the above equations. Zhang and Yu (1998) proposed an equation that can compute relative risk from the odds ratio 11 as follows:

This equation served as a new method to estimate relative risk using case control data; however, the estimator of π 0 or π 1 is still required to perform the calculation.

Other than above, the Bayesian methods also provide an equation of relative risk. When P o and P e are the probabilities of finding subjects for whom an event occurred and who were exposed to a factor, the Bayes’ theorem 12 can be written as

where P eo is the probability of finding subjects who were exposed to a factor among subjects for whom an event occurred. Because π 0 can be written as

However, because P eo and P e will vary depending on methods of observation, precise estimation with using this equation should require follow-up data of all subjects or a carefully collected random sample of that. Moreover, because of difference in probability definitions, such as using “the probability of finding exposed subjects” rather than “the exposed proportion”, there is resistance toward the Bayesian methods among some researchers, such as traditional statisticians.

This study illustrates an obstacle, which prevent relative risk from being estimated using case control data, as a mathematical expression of inconsistency in the observations and proposes a new equation to estimate relative risk, which requires case group data and the exposed proportion. The proposed equation is derived without the Bayesian methods, and do not require the probability estimators; that is, neither control groups nor cohorts are needed. Theoretical and real-life examples that demonstrate validity and wide applicability of the proposed equation are also provided.

To clarify an obstacle in estimating relative risk using case control data and derive an equation to estimate relative risk, let us introduce a proportion of observed subjects among all subjects of interest (hereinafter, “observed proportion”). For example, the number of observed individuals exposed to a factor divided by the exposed population constitutes the observed proportion of exposed individuals. As a expression, the observed proportion is the same as “the sampling proportion”, which is the proportion of a sample among all subjects of interest. However, the observed proportion cannot be estimated while the sampling proportion can be even assigned by researchers.

In cohort studies, the observed proportions can be defined as follows:

where OP exp and OP unexp are the observed proportions of exposed and unexposed subjects and d exp and d unexp are constants. Cohort studies must be designed as follows:

such that d exp and d unexp are sufficiently small to be ignored. Inserting equations ( 13 ) and ( 14 ) into equation ( 5 ), we obtain

When d exp  = 0 and d unexp  = 0,

Therefore, eRR can be used to estimate the relative risk in cohort studies.

In case control studies, the observed proportions may be defined as follows:

where OP case and OP cont are the observed proportions of case group and control group and d case and d cont are constants. Case control studies must be designed as

such that d case and d cont should be sufficiently small to be ignored. Substituting equations ( 19 ) and ( 20 ) in equation ( 8 ), we obtain

When d case  = 0 and d cont  = 0,

Therefore, eOR can be used to estimate the odds ratio.

However, inserting equations ( 19 ) and ( 20 ) into equation ( 6 ), we must obtain

when d case  = 0 and d cont  = 0. Thus assuming OP case is equivalent to OP cont , meRR can estimate the relative risk. Unfortunately, the equivalence of OP case and OP cont cannot be estimated but must be tested.

Equation ( 25 ) is a mathematical expression that illustrates an obstacle to estimate relative risk using case control data. Thus, excluding both OP case and OP cont would clearly remove this obstacle in estimating relative risk.

Here, let us focus on the exposure odds, which is the ratio of exposed subjects to unexposed ones. Let EOC be the exposure odds in a case group and defined as

Inserting equation ( 19 ) into equation ( 26 ) leads

When d case  = 0, substituting equations ( 2 ) and ( 3 ) into equation ( 27 ) leads

Assume that a random sample is selected from all subjects and eE is the proportion of exposed subjects among the sample. Thus, eE can be written as

where l is the size of a random sample and l 1 is the number of exposed subjects among the sample. The observed proportion of a random sample (that is, the sampling proportion) may be defined as

where d sample is a constant. Inserting equation ( 30 ) into equation ( 29 ),

Because the random sampling should provide

then d sample is sufficiently small to be ignored. When d sample  = 0, inserting equation ( 2 ) into equation ( 31 ) leads

Thus, let PRR be defined as

Substituting equations ( 26 ) and ( 29 ) into equation ( 34 ) leads

Both d case and d sample should be sufficiently small to be ignored when a random sample is selected from all subjects of whom a case group represents an event-occurring part. When d case  = 0 and d sample  = 0, combining equations ( 28 ), ( 33 ), and ( 35 ), we must obtain

Therefore, PRR must be an estimator of relative risk when subjects among whom a case group is observed and subjects from whom a random sample is selected are the same.

This estimator is computed from the exposure odds in a case group and those in all subjects to be studied, and thus, no control group is required. In addition, the estimation is performed without a cohort.

Equation ( 34 ) is quite similar to equation ( 12 ), but note that PRR was derived without using the Bayesian methods and can be applicable to more general data: data of a case group and a random sample.

Therefore, by considering the observed proportions, an observational inconsistency preventing relative risk from being estimated in the case control studies was clarified as a mathematical expression, and a new equation to estimate relative risk using the exposed proportion and a case group was proposed; the proposed equation requires neither control groups nor cohorts.

Application to Model Data

Suppose the probabilities of disease Y developing among people exposed and unexposed to chemical compound X are 0.03 and 0.01 (i.e., relative risk is 3).

When the proportion of exposed people in a city, which has a population of 100000, is 30%, researchers should observe the following data: 30 patients are found among 1000 exposed participants and 10 patients among 1000 unexposed participants during a follow-up period; 180 exposed patients are observed in a case group of 320 and 97 exposed participants are observed in a control group of 328; and 300 exposed people are found in a random sample of 1000 participants (see Table  2 ). The observed proportions of the case and control groups, which are unavailable for the researchers, are then 1/5 and 1/300.

Thus, estimating relative risk from cohort data must be

Estimating odds ratio from case-control data is

and meRR should be

Finally, the proposed estimator PRR can be computed as

Note that the proposed equation will estimate the relative risk as precisely as the estimation in a cohort study but does not require follow-up group data, such as cohort data.

Confidence Interval

The proposed estimator PRR is the ratio of two odds.

On estimating the odds ratio as \(eOR={m}_{11}\cdot ({m}_{0}-{m}_{10})\cdot {m}_{10}^{-1}\cdot {({m}_{1}-{m}_{m11})}^{-1}\) , the following eSE ( ln eOR ) is known as the maximum likelihood estimator for the standard deviation of ln  eOR 13 :

Let us apply this formula to PRR for estimating confidence interval (CI).

When these two odds are nonzero, the estimator of the standard deviation of the logarithm of PRR will be

Thus, the following formulas would provide the 100(1 −  α )% confidence limits for PRR .

where LCL and UCL are the lower and upper limits of CI and Z α /2 represents the α /2 point of the normal distribution, such as 1.96 for 95% interval.

To prove this estimators for CI, computer simulation was conducted. It is assumed that 30% of the population 100000 was exposed. The total number of exposed and unexposed people for whom an event occurred was determined by using two sets of risks, in which the relative risk is 3: π 1  = 0.03 and π 0  = 0.01 or π 1  = 0.3 and π 0  = 0.1. Samples, exposed case-groups, and unexposed case-groups were picked from the corresponding people based on each six sets of the observed proportions, and the CI was computed each time. Each set of six proportions was chosen so that each group should be close to the size used generally in research.

Table  3 demonstrates the number of times the true relative risk was included in the 95% CI in each one million trials. It is shown that the true value (relative risk: 3) is included at a rate of approximately 95%; this method will well estimate CI.

Application to Real-Life Data

The suicide rate among the youth of Japan is considerably high and suicide accounts for nearly half of the causes of death among those in their twenties 14 . Meanwhile, unemployment is suggested to increase suicide risk 2 , 15 .

The proposed equation was applied to the latest suicide and employment data in Japan as real-life data, and confidence intervals at 95% were also estimated. The prevalence of suicide and employment among individuals in their twenties in 2015 was obtained from a statistics report published by the Ministry of Health, Labour and Welfare 16 and the Labour Force Survey 17 . The data used are presented in Table  4 . Suicide victims who were unemployed are treated as “No occupation”. Although the Labour Force Survey was conducted in a specific month in 2015 using random sampling, the indicators should represent the characteristics of the Japanese population in that year.

The estimation of relative risk for unemployed women is

and the 95% confidence interval for this relative risk can be estimated as follows:

The estimation for men can be done in the same way. Thus, the estimated relative risk is 0.82 (95% CI: 0.52–1.30) for women and 0.78 (95% CI: 0.60–1.00) for men. Unemployment did not increase the risk of suicide.

Incidentally, the proportions of victims who were classified under “No occupation” are comparatively large for both women and men, and thus, the situation of no occupation might increase risk. Let us, on trial, assume that a person who is neither employed nor attending school is the same as an individual with no occupation. The number of women in no occupation is then 1.04 million (6.21 − 4.40 − 0.77 = 1.04); the estimates of the relative risk and confidence limits for women in no occupation can be computed as follows:

For men, the number is 0.53 million (6.56 − 5.02 − 1.01 = 0.53); the estimation can be done in the same way. Thus, the relative risk would be estimated to be 4.36 (95% CI: 3.72–5.10) for women and 4.20 (95% CI: 3.78–4.67) for men.

Although the calculations were not adjusted and the definition of no occupation is tentative, these results suggest that being neither employed nor educated may substantially increase the risk of suicide among the young Japanese population. It might be also suggested that the Japanese governments should consider the indicator of unemployment.

Note that relative risks were estimated without a fresh cohort study, which is generally difficult to conduct.

Evaluating a change in risk of an event occurring caused by exposure to (or the presence of/occupation as) a factor is generally attempted in many research fields, such as epidemiology, medicine, social science, politics, and product development. Relative risk, which is the ratio of the risks, can be easily interpreted and widely used, but has been believed to require large-scale epidemiological research or a smaller cohort study designed for the estimation. A case control study, which compares the case and control group, is more convenient than the cohort study, but relative risk cannot be estimated using case control data. The estimator of the odds ratio, which can be calculated using case control data, is often used instead of relative risk, because the former can sometimes approximate the latter. A method to calculate relative risk using the odds ratio was also proposed. Unfortunately, the odds ratio may be misleading to interpret the change in risk and calculating relative risk using the ratio still requires either estimator of risks. Furthermore, control group data are still required, burdening researchers in terms of cost and effort.

In this study, introducing the observed proportion, an observational inconsistency preventing relative risk from being estimated in case control studies was clarified as a mathematical expression; by excluding this inconsistency, a new equation that estimates relative risk using case data was proposed. The proposed equation, which serves as an estimator of relative risk itself without approximation, requires only the exposure odds of a case group and that of all subjects to be studied; no control group is then needed. The calculation is done without using risk estimators, and thus, cohorts are also not needed. Therefore, evaluating a change in risk can be easily conducted without additional costs, efforts, and time generally needed in a fresh study. Moreover, the proposed equation was derived without using the Bayesian probabilities nor the Bayes’ theorem and is free from researcher’s resistance toward the Bayesian methods.

A method of estimating confidence limits of the proposed estimator was also presented and proved to estimate that successfully. Although there may be a more appropriate estimation method of confidence interval, pursuing the best method is beyond the scope of this paper.

Once the exposed proportions by various characteristics are investigated, changes in every risk associated with the exposure will able to be estimated by applying the proposed equation to appropriate case group data. Even the estimation of a change in risk, which has been believed to be impossible, can be done, such as the adverse effect of a social situation on the suicide rate, the effect of a policy on birthrate, or the impact of a new drug for a pandemic on survival rate. There are two caveats: the case group must comprise subjects from whom the exposed proportion was computed and the exposure to the factor must precede the occurring event. Existing statistical methods, such as adjusting confounding factors, should be also applicable for the proposed estimator.

Although the proposed equation is quite simple, its advantages will not only reduce the costs of epidemiological studies but may also make itself a powerful tool in almost all research fields that treat risks.

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Yada, Y. Method to estimate relative risk using exposed proportion and case group data. Sci Rep 7 , 2131 (2017). https://doi.org/10.1038/s41598-017-02302-1

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Overview of Analytic Studies

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Epi_Tools.XLSX

All Modules

Case-Control Studies

Cohort studies have an intuitive logic to them, but they can be very problematic when:

• The outcomes being investigated are rare;
• There is a long time period between the exposure of interest and the development of the disease; or
• It is expensive or very difficult to obtain exposure information from a cohort.

In the first case, the rarity of the disease requires enrollment of very large numbers of people. In the second case, the long period of follow-up requires efforts to keep contact with and collect outcome information from individuals. In all three situations, cost and feasibility become an important concern.

A case-control design offers an alternative that is much more efficient. The goal of a case-control study is the same as that of cohort studies, i.e. to estimate the magnitude of association between an exposure and an outcome. However, case-control studies employ a different sampling strategy that gives them greater efficiency.   As with a cohort study, a case-control study attempts to identify all people who have developed the disease of interest in the defined population. This is not because they are inherently more important to estimating an association, but because they are almost always rarer than non-diseased individuals, and one of the requirements of accurate estimation of the association is that there are reasonable numbers of people in both the numerators (cases) and denominators (people or person-time) in the measures of disease frequency for both exposed and reference groups. However, because most of the denominator is made up of people who do not develop disease, the case-control design avoids the need to collect information on the entire population by selecting a sample of the underlying population.

To illustrate this consider the following hypothetical scenario in which the source population is Plymouth County in Massachusetts, which has a total population of 6,647 (hypothetical). Thirteen people in the county have been diagnosed with an unusual disease and seven of them have a particular exposure that is suspected of being an important contributing factor. The chief problem here is that the disease is quite rare.

If I somehow had exposure and outcome information on all of the subjects in the source population and looked at the association using a cohort design, it might look like this:

Therefore, the incidence in the exposed individuals would be 7/1,007 = 0.70%, and the incidence in the non-exposed individuals would be 6/5,640 = 0.11%. Consequently, the risk ratio would be 0.70/0.11=6.52, suggesting that those who had the risk factor (exposure) had 6.5 times the risk of getting the disease compared to those without the risk factor. This is a strong association.

In this hypothetical example, I had data on all 6,647 people in the source population, and I could compute the probability of disease (i.e., the risk or incidence) in both the exposed group and the non-exposed group, because I had the denominators for both the exposed and non-exposed groups.

The problem , of course, is that I usually don't have the resources to get the data on all subjects in the population. If I took a random sample of even 5-10% of the population, I might not have any diseased people in my sample.

An alternative approach would be to use surveillance databases or administrative databases to find most or all 13 of the cases in the source population and determine their exposure status. However, instead of enrolling all of the other 5,634 residents, suppose I were to just take a sample of the non-diseased population. In fact, suppose I only took a sample of 1% of the non-diseased people and I then determined their exposure status. The data might look something like this:

With this sampling approach I can no longer compute the probability of disease in each exposure group, because I no longer have the denominators in the last column. In other words, I don't know the exposure distribution for the entire source population. However, the small control sample of non-diseased subjects gives me a way to estimate the exposure distribution in the source population. So, I can't compute the probability of disease in each exposure group, but I can compute the odds of disease in the case-control sample.

The Odds Ratio

The odds of disease among the exposed sample are 7/10, and the odds of disease in the non-exposed sample are 6/56. If I compute the odds ratio, I get (7/10) / (5/56) = 6.56, very close to the risk ratio that I computed from data for the entire population. We will consider odds ratios and case-control studies in much greater depth in a later module. However, for the time being the key things to remember are that:

• The sampling strategy for a case-control study is very different from that of cohort studies, despite the fact that both have the goal of estimating the magnitude of association between the exposure and the outcome.
• In a case-control study there is no "follow-up" period. One starts by identifying diseased subjects and determines their exposure distribution; one then takes a sample of the source population that produced those cases in order to estimate the exposure distribution in the overall source population that produced the cases. [In cohort studies none of the subjects have the outcome at the beginning of the follow-up period.]
• In a case-control study, you cannot measure incidence, because you start with diseased people and non-diseased people, so you cannot calculate relative risk.
• The case-control design is very efficient. In the example above the case-control study of only 79 subjects produced an odds ratio (6.56) that was a very close approximation to the risk ratio (6.52) that was obtained from the data in the entire population.
• Case-control studies are particularly useful when the outcome is rare is uncommon in both exposed and non-exposed people.

The Difference Between "Probability" and "Odds"?

• The odds are defined as the probability that the event will occur divided by the probability that the event will not occur.

If the probability of an event occurring is Y, then the probability of the event not occurring is 1-Y. (Example: If the probability of an event is 0.80 (80%), then the probability that the event will not occur is 1-0.80 = 0.20, or 20%.

The odds of an event represent the ratio of the (probability that the event will occur) / (probability that the event will not occur). This could be expressed as follows:

Odds of event = Y / (1-Y)

So, in this example, if the probability of the event occurring = 0.80, then the odds are 0.80 / (1-0.80) = 0.80/0.20 = 4 (i.e., 4 to 1).

• If a race horse runs 100 races and wins 25 times and loses the other 75 times, the probability of winning is 25/100 = 0.25 or 25%, but the odds of the horse winning are 25/75 = 0.333 or 1 win to 3 loses.
• If the horse runs 100 races and wins 5 and loses the other 95 times, the probability of winning is 0.05 or 5%, and the odds of the horse winning are 5/95 = 0.0526.
• If the horse runs 100 races and wins 50, the probability of winning is 50/100 = 0.50 or 50%, and the odds of winning are 50/50 = 1 (even odds).
• If the horse runs 100 races and wins 80, the probability of winning is 80/100 = 0.80 or 80%, and the odds of winning are 80/20 = 4 to 1.

NOTE that when the probability is low, the odds and the probability are very similar.

On Sept. 8, 2011 the New York Times ran an article on the economy in which the writer began by saying "If history is a guide, the odds that the American economy is falling into a double-dip recession have risen sharply in recent weeks and may even have reached 50 percent." Further down in the article the author quoted the economist who had been interviewed for the story. What the economist had actually said was, "Whether we reach the technical definition [of a double-dip recession] I think is probably close to 50-50."

Question: Was the author correct in saying that the "odds" of a double-dip recession may have reached 50 percent?

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Odds ratios and risk ratios: what's the difference and why does it matter?

Affiliation.

• 1 Department of Family Medicine, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA. [email protected]
• PMID: 18580722
• DOI: 10.1097/SMJ.0b013e31817a7ee4

Odds ratios (OR) are commonly reported in the medical literature as the measure of association between exposure and outcome. However, it is relative risk that people more intuitively understand as a measure of association. Relative risk can be directly determined in a cohort study by calculating a risk ratio (RR). In case-control studies, and in cohort studies in which the outcome occurs in less than 10% of the unexposed population, the OR provides a reasonable approximation of the RR. However, when an outcome is common (iY 10% in the unexposed group), the OR will exaggerate the RR. One method readers can use to estimate the RR from an OR involves using a simple formula. Readers should also look to see that a confidence interval is provided with any report of an OR or RR. A greater understanding of ORs and RRs allows readers to draw more accurate interpretations of research findings.

• Case-Control Studies
• Cohort Studies
• Odds Ratio*