Measures of Association

Research 101 Series May 2014 Measures of Association Somjot S. Brar, MD, MPH 1,2,3 * Abstract Measures of association are used in clinical research to quantify the strength of association between variables, often an outcome and treatment or outcome and exposure. Frequently the primary results of a study are reported using measures of association, which include: relative risk, risk difference, odds ratio, and hazard ratio. These measures allow for the direction and magnitude of the association between variables to be characterized using a single statistic. The relative risk and risk difference can be calculated from simple 2x2 tables. The hazards ratio is calculated using regression methods. The odds ratio can be calculated from either a 2x2 table or using regression. This article will define measures of association used in clinical research and provide guidance on selecting which to use. Keywords Relative risk Risk difference Odds ratio Hazard ratio 1 Regional Research Committee, Dept. of Research and Evaluation, Kaiser Permanente, Pasadena, CA 2 Department of Cardiology, Kaiser Permanente, Los Angeles, CA 3 Assistant Clinical Professor of Medicine, UCLA *Corresponding author: sbrar@cvri.org Contents Introduction 1 1 Relative Risk 1 2 Risk Difference 2 3 Odds Ratio 3 4 Hazard Ratio 4 Summary 4 Introduction Measures of association are ubiquitous across all fields of clinical research. You have encountered this term countless number of times, perhaps without realizing it. The measures of association are statistics that are used to describe the relationship between two variables, often treatment and outcome or exposure and outcome. This statistic provides insight into the direction and magnitude of the association between variables. Measures of association used in clinical research include: relative risk, risk difference, odds ratio, and hazard ratio. Each of these values is typically reported with a 95% confidence interval. These measures of association are applicable to studies where the outcome or event of interest is binary. That is, events with only two possible outcomes: dead or alive, develop disease or remain disease free. Fortunately this is very common in clinical research. For studies where the outcome is a continuous variable, measures of association discussed in this review are not relevant. 1. Relative Risk Also referred to as the risk ratio, this is the simplest and most intuitive of the measures of association. The relative risk (RR) is simple to compute and interpret; it is the ratio of the event rate in two groups: Relative risk = Event rate group1 Event rate group2 = Event rate treatment Event rate control = Event rate exposed Event rate unexposed Event rates can also be interpreted as probabilities. Therefore the RR can also be defined as the ratio of probabilities for the study groups. In a randomized trial the groups of interest are treatment and control whereas in a registry study they are often labeled exposed and unexposed. Event rates by group are easily calculated from a 2 x 2 table: Dead Alive Treatment A B A+B Control C D C+D A+C B+D In this table the rows represent the study groups and the columns the outcomes. The mortality event rate or probability of death for the treatment group is A/(A + B) and for the control group is C/(C + D). Taking this one step further, the RR from the 2x2 table is defined as the ratio of the two event

Measures of Association 2/5 rates or probabilities: RR = Event rate treatment A/(A + B) = Event rate control C/(C + D) You have probably seen findings from clinical trials reported using the RR. Following is an example of how the RR may be reported: The relative risk for aspirin users vs. non-users for myocardial infarction is 0.40. This one brief statement identifies the two variables for which the association is being reported, aspirin use and myocardial infarction, and using one statistic (RR=0.40) conveys the strength and direction of this association. Since the RR is < 1, aspirin is favored. The RR in this example can be calculated from the raw data as follows: Relative Risk = Event rate aspirin user Event rate aspirin nonuser = 4% 10% = 0.04 0.10 = 0.40 This means that the risk of myocardial infarction among aspirin users is 60% lower compared to non-users. For relative risks < 1.0, subtract the RR from 1 to obtain the percent reduction: Percent Reduction = 1 RR By convention, the numerator contains the treatment or exposed group and the denominator the control or unexposed group. The later is referred to as the referent group. Shown is the RR taking non-diabetics as referent: Relative Risk = Event rate diabetic Event rate non diabetic = 15% 5% = 0.15 0.05 = 3.0 Lets assume a researcher is interested in reporting the primary outcome of a clinical trial by diabetes status. Shown is the relative risk taking diabetics as referent: Relative Risk = Event rate non diabetic Event rate diabetic = 5% 15% = 0.05 0.15 = 0.3333 It s important to identify the control or referent group (the group in the denominator) clearly. Reporting the relationship as a RR for diabetics vs. non-diabetics or non-diabetics vs. diabetics is fine. However, be clear about the order of the relationship otherwise someone reviewing your work (or even possibly you) might be confused. Most often the group of interest (treated or exposed) is placed in the numerator. Because the RR is intuitive and easy to interpret, it is recommended it be used whenever possible. Conditions that should generally be satisfied for the relative risk to be used are: 1. No loss to follow-up 2. Follow-up time is the same for all study participants 3. No need for adjusted measures of association. If an adjustment is necessary consider using odds ratios or hazard ratios. The study design that most often satisfies these conditions is the randomized controlled trial with complete follow-up in all study participants. If the randomization is properly performed the study groups will be very similar in baseline characteristics and there may not be a need to further adjust for differences or imbalances between groups. The RR is commonly used in trials with short follow-up periods where there is (very little or) no loss to follow-up. For example, in a trial exploring the role of a new medication that might reduce the need for blood transfusions during and after cardiac surgery, the outcome is likely to be ascertained in all randomized patients prior to discharge. Limitations However, there are limitations which prohibit using the RR for many studies. When imbalances between study or treatment groups exist researchers often use regression techniques to adjust for these differences. The adjustment is never perfect but can be quite good if carefully thought through. Relative risks do not lend themselves well to regression. Models do exists to do calculate adjusted RR but they can be difficult to work with and sometimes fail to converge and yield an adjusted RR. Thus, it is not common to see adjusted RR used in studies other than clinical trials. 2. Risk Difference The risk difference (RD) is quite meaningful for understanding if findings of a study are worthy of incorporating into clinical practice. If a new very expensive treatment is found to reduce the relative risk for myocardial infarction by 50% we might be very excited about this breakthrough. The RR for a 50% reduction would be 0.50. Following are two different scenarios that yield the same RR, but have significantly different implications in terms of clinical significance: Rate treatment Rate control RR RD Scenario A 5% 10% 0.50-5% Scenario B 1% 2% 0.50-1% A helpful way to understand the clinical relevance of a therapy is to calculate the number needed to treat (NNT). This is simply the inverse of the RD: NNT = 1 Risk di f f erence The NNT for scenario A is calculated as follows: NNT = 1 0.05 = 20

Measures of Association 3/5 Note that the RD needs to be in decimal format. This reveals that 20 patients would need to be treated with this new very expensive therapy to prevent one myocardial infarction. For comparison, the NNT for scenario B is 100; thus 100 patients would need to be treated to prevent one event. We would of course want to know more about the treatment: how costly? is it invasive? risks? Integrating these details, along with the RD and NNT, will help us make a more informed decision as to whether the new treatment should be adopted. When the effect of a treatment is assessed in terms of risk difference, the magnitude of the treatment effect often looks less impressive. Thus, many publications do not to report this measure of association. However, the RD provides more insight into the importance of the findings than the RR. Publications often emphasize the RR, but don t forget to calculate the RD and NNT as this is likely more clinically relevant. 3. Odds Ratio The ratio of odds or odds ratio (OR) can be a confusing measure of association. It is easily calculated from a 2x2 table but is more often calculated using logistic regression. If the OR can be calculated using a 2x2 table, then it s probably best to report the RR instead. Before defining the OR, let s define the term odds. Many people are not comfortable thinking in terms of odds. The odds of an event occurring is simply: Odds = p 1 p where p is the probability of the event occurring and 1 p is the probability of the event not occurring. Odds are often used in gambling. For example, the odds of a horse winning may be reported as 20 to 1. This means that for 21 races the horse is expected to lose 20 and win 1. In clinical research, success (1 win) is often stated first and the failure second (20 losses). The probability of a win is thus 1/21 or 4.8% and the odds of winning are 1/20 or 5%. In this case the probability and odds of the outcome are quite similar, but this is not always the case (as is discussed below). The ratio of the odds for the outcome in the study groups (treatment vs. control or exposed vs. unexposed) yields the odds ratio. The OR can also be calculated from a 2x2 table. Following is the familiar 2x2 table: Dead Alive Treatment A B A+B Control C D C+D A+C B+D The odds of dying with treatment from the above table is the probability of the event occurring (A/A + B) divided by the probability of the event not occurring (B/A + B) or simply A/B. The calculation for the odds of dying with control therapy is similar (C/D). The OR is calculated as the ratio of two odds: OR = p/(1 p) treatment p/(1 p) control = A/B C/D = AD BC Odds ratio is often used when RR is not appropriate for the reasons previously outlined. Also, the OR is the appropriate measure of association for case-control studies. When there is suspected imbalance in baseline characteristics between study groups, as is routinely the case in registry studies, OR prove quite useful. In a registry study the treatment or exposure is not randomly assigned. A patient is likely to receive specific treatments because it is in the patient s best interest as determined by the treating physician. This selection bias creates important imbalances between the groups being compared, and when not accounted for yields misleading or biased measures of association. The problem with imbalances or differences between groups is that this can have a profound impact on the study outcome. For example, a study reports that aspirin is ineffective at decreasing the rate of myocardial infarction in patients with known coronary artery disease. However, aspirin users were on average 15 years older than non-users. Knowing this, you are probably not surprised with the result given the marked difference in age between aspirin users and non-users. When a variable is different between groups and influences the outcome it is referred to as a confounder. Not all variables that are different between groups are confounders. For example, if the non-users were more likely to have green eyes, it is irrelevant. Having green eyes does not increase or decrease the risk of being prescribed aspirin or increase the risk for myocardial infarction. Imbalances in important variables between groups is common in registry studies, retrospective or prospective. There are various methods for dealing with the imbalance. The most common are mathematical models that allow for adjusting for these important differences. The modeling method that yields odds ratios is referred to as logistic regression. Another strategy is to limit or restrict the study population. If the effect of aspirin is being assessed in a registry study of patients with coronary artery disease, restricting the study population to subjects over 65 may create groups with comparable ages. Some authors report and interpret the OR as a RR, which should be discouraged. The OR and RR differ by the term in the denominator. For RR, the denominator is comprised of all subjects whereas for the OR the denominator is comprised of subjects without the outcome of interest. The OR can approximate the RR when the probability of the outcome is rare ( < 10%). However the OR can be considerably larger than the RR when the probability of the outcome, in the referent group, is common. The following plot illustrates this relationship.

Measures of Association 4/5 Table 1. Measures of Association Measure Study type Source Notes RR Clinical trial 2x2 table Simplest to understand Ratio of probabilities in study groups Mathematically intuitive Need complete follow-up for all subjects Is meaningful when study groups are balanced or similar RD Clinical trial 2x2 table Useful for understanding clinical significance Use to calculate NNT OR Clinical trial 2x2 table Less intuitive than RR Registry study Logistic regression Ratio of odds in study groups Useful when there is imbalance between groups Use when outcome is binary (yes/no; dead/alive) Use when time-to-event is not relevant or important To identify predictors of outcome Must use for case-control studies HR Clinical trial Cox proportional Use when time to event is relevant Registry study hazards model Use when follow-up times are variable Used when there is loss to follow-up Interpretation is similar to RR Mathematically complex, let the computer do the work! RR if OR = 2 by base probability referent group. As the probability in the treatment or exposed group reaches 0, the OR approaches 0, and as the probability reaches 1, the OR approaches infinity. Relative Risk 1.0 1.4 1.8 4. Hazard Ratio When time-to-event is important to account for, the hazard ratio (HR) is the preferred measure of association. The HR is generated from a Cox proportional hazards model. This is a mathematically complex regression model that takes into consideration time to event, differences in follow-up time, and loss to follow-up. The HR can be interpreted similar to the RR. 0.0 0.2 0.4 0.6 0.8 1.0 Probability in reference group For a fixed OR of 2, the corresponding RR ranges between approximately 1.8-2.0 when the probability of the outcome in the reference group is < 10% or 0.10. For comparison, when the probability of the outcome is high (e.g., 80% or 0.80), the RR is close to 1.2 for a corresponding OR of 2.0. This discrepancy becomes more pronounced for higher OR. In summary, OR should not be interpreted as RR. Also, the RR is constrained by the probability in the referent group (e.g., control or unexposed). For example, if the probability of the event in the control group is 60% or 0.60 then the RR can be no greater than 1/0.60 or 1.667. In contrast, the OR is not constrained by the probability in the Summary Now that you know the measures of association, you can better understand why some are chosen over others. Sometimes the investigator has considerable latitude in deciding which to use; often the study design dictates which should be used. As you design your own research proposals you will need to decide how to report results. When reading published literature try to understand why certain measures of association were used and if it was the best choice. A summary is provided in table 1. All measures of association should be reported with 95% confidence intervals. Remember, this discussion is relevant to studies with binary outcomes (e.g., dead/alive). When the outcome is a continuous variable the measures of association discussed are not applicable. For retrospective or prospective registry studies, regression analyses should be be used as there will likely be impor-

Measures of Association 5/5 tant imbalances between study groups. Regression methods can take into account measured imbalances between groups and yield adjusted measures of association. If you are not interested in the time to event and have not collected data such that specific timing of the endpoints is ascertained then logistic regression is appropriate and the measure of association will be the OR. If weeks, months, or years pass during your study period, there is loss of follow-up, or variable follow-up times then strongly consider reporting hazard ratios. For clinical trials, there is considerable latitude as more than one measure of association may be appropriate, especially when there is no loss to follow-up (RR, OR, HR, RD). However, for the reasons previously discussed it is preferable to use the RR or HR over OR. Use RR if the follow-up is short and complete and the HR if follow-up is longer, incomplete, or their is interest in understanding differences in time-toevent between groups. Use the RD to quantify the clinical significance of study results. The RD and NNT provide added insight into the clinical relevance of differences in outcomes between groups. Disclaimer The views expressed are those of the author and not supported or endorsed by Kaiser Permanente. The articles in this series were written on nights and weekends; while each manuscript has been revised multiple times, it is possible some errors were missed. If you do identify any please email these to me. Also email any comments or suggestions for future topics. I will try to reply to these the best I can.