Welcome back to the next-to-last installment.
Odds ratio (OR), the relative risk (or risk ratio, RR) and the likelihood ratio (LR) are probably the most difficult concepts we will study. Even without the math, these concepts can be daunting. You may have to read this more than once and work the examples yourself on paper to see how they can make sense. It is important that you understand these concepts so when you read published data, you can understand and critique it properly. Additionally, you may find in your own work that one of them may be better than another or a better tool than an ROC curve. In any case, each of them has a place and each is imperfect and, hence, can involve the risks of both false positive and false negative results for the patient.
The OR may be used to compare whether the probability (chances) of a certain event (e.g. weight loss) is the same for two groups (for example: a group on a diet and a group which continues their regular diet but cuts back on calories). An odds ratio of 1 implies that the event (e.g. weight loss) is equally likely in both groups. An odds ratio greater than one implies that the event is more likely in the first group.
The RR is the risk of an event (e.g. developing a disease) relative to exposure. Relative risk is a ratio of the probability of the event occurring in the exposed group versus a non-exposed group.
The last of these three ratios is the likelihood ratio (LR). The likelihood ratio contains both the sensitivity and specificity of the test.
We will discuss each of these in turn with examples.
I have chosen for our first example official data from the sinking of the S. S. Titanic. I have chosen this as there are clear data from a number of groups of reasonable sizes and the outcomes of the ‘event’ are clear – there are no confounding variables such as age. We will start with two groups of males – those traveling First Class and those in Second Class. The outcomes are survival and lost at sea:
Outcome
Lost Rescued
First 118 57
Class
Second 154 14
We begin with a ratio (NOT the OR) of the First Class passengers who were lost to those who were rescued:
First Class Lost at sea/ rescued = 118/57 = 2.1.
In other words, this says that a First Class passenger is more than twice (2.1x) likely to be lost at sea than rescued. Next, we calculate the same ratio of Second Class passengers who were lost to those who were rescued or 154/14 = 11.0. A Second Class passenger is 11 times more likely to be lost than saved! The odds ratio compares the odds of being lost in each group (First vs. Second Class):
Second Class Odds/ First Class Odds = (154/14) / (118/57) = 11.0 / 2.1 = 5.3
In other words, the odds of a Second Class passenger dying are more than five-fold greater than a First Class passenger dying (11/2.1 = 5.3). Note that the OR is actually the ratio of the two relative odds. If we redo the 2 x 2 grid to make it generic we have:
Outcome
O+ O-
G1 a b
Group
G2 c d
The first ratio is a/b; the second is c/d; and the OR is a/b / c/d, which simplifies to a*d/c*d. This math is easily done on a pocket calculator or in Excel. The good news is that, more often than not, you will be reading Odds Ratios rather than calculating them. Still, it’s a good idea of know how they were obtained.
Let’s do another example from the Titanic:
Outcome
Lost Rescued
Male 709 142
Class
Female 154 308
First a/b= 709 /142 = 5.0 and then c/d = 154/ 308 =0.5 and then OR = 10.0. Men had a 10-fold greater chance of dying than women.
As mentioned, an OR of 1.0 or nearly 1.0 indicates that the odds are close to the same — similar to flipping a fair coin. A ratio greater than 1.0 indicates that the first event (in this case not surviving) is greater or more likely. An OR of less than one indicates that the event is less likely (in this case being rescued). It is important that you note which is the a, b, c and d.
Turning to the RR and using the generic box above we have RR =
a/(a+b) / c/(c+d) = a*(c+d) / c*(a+b).
In our case of males and females on the titanic, RR = 4.0. Using this formula, the relative risk or risk ratio indicates that males had a four-fold greater risk of being lost than females. This fact that the RR differs from OR is of no surprise since the formulas are different and measure different aspects of the data. RR is often used to interpret clinical trial data, where it is used to compare such data as the risk of developing a disease in people not receiving a particular treatment (or receiving a placebo) versus people who are receiving another treatment. Alternatively, it is used to compare the risk of developing a side effect in people receiving a drug as compared to the people who are not receiving the treatment (or receiving a placebo).
The same interpretation and caveat apply to the RR as they did for the OR.
Let’s look at a more detailed example from a recent publication by Mora. In this work, 26,746 healthy US women (mean age 54.6 years) were prospectively examined for Lipoprotein (a) (Lp(a)) concentrations and incidence of type 2 diabetes (n = 1670) for a follow-up period of 13 years. These data were then confirmed in 9,652 Danish men and women with diabetes (n = 419). The analyses were adjusted for risk factors: age, race, smoking, hormone use, family history, blood pressure, body mass index, hemoglobin A1c (HbA1c), C-reactive protein and lipids. That is, to avoid proposing a difference between the levels of Lp(a) was different when it was not — due to age, etc. — these factors were corrected for during the analysis.
The author concluded that “Lp(a) was inversely associated with the incident of diabetes. The ORs for nonfasting Danish men and women were 0.75, 0.64, 0.74 and 0.58 for Lp(a) quintiles 2-5 vs. quintile 1.” Quintile 1 is set at the base line or 1.0. The other 4 values are ratios of the odds for each of the quintiles to 1.0.
These data add information that may be of use to the clinician as they indicate that the lower the Lp(a), the lower the risk for Type 2 diabetes.
The last ratio is the likelihood ratio (LR). While LRs constitute an excellent way to measure and express diagnostic accuracy, LRs are rarely used, primarily because interpreting them requires a calculator to convert back and forth between probability of disease (a term familiar to all clinicians) and odds of disease (a term mysterious to most people other than statisticians and epidemiologists).
The LR of any clinical finding is the probability of that finding in patients with disease divided by the probability of the same finding in patients without disease:
LR=probability of finding in patients with disease /probability of same finding in patients without disease.
LRs may range from 0 to infinity. Findings with LRs greater than 1 argue for the diagnosis being considered — the larger the number, the more convincingly the finding suggests that disease. Findings whose LRs lie between 0 and 1 argue against the diagnosis of interest — the closer the LR is to 0, the less likely the disease. LRs that equal 1 lack diagnostic value.
In our next and final installment, we will discuss risks and how to avoid them in the post-analytical period.
