How to interpret odds ratios that are smaller than 1?

Here is a question posted on the web about the interpretation of odds ratios that are less than 1.

"I know that OR estimates= 1 mean that both groups/categories have the same odds. I also know that if OR estimates are greater than 1, e.g, 1.24 for Young vs. Old persons, then I can say: Young people have 24% increase in the odds of living in an apartment than older people. Or, I also know I can say, for example, for an OR of 0.322 Non-White vs. White, that the odds of Whites are 1/.322 = about 3 times higher than those of Non-Whites, to live in a house they own. Now, how would I say the odds are of a NON-White person in the example above, to live in a house they own? Is is 1-.322=.678 less likely, with respect to odds, to live in a house they own? Or, similarly, they have 67.8% lower odds to live in a house they own? "

If we have to say the odds for a Non-White person, we may say "Non Whites have odds .322 times as great as those of Whites".

In an article "When can odds ratio misled?", Davies et al stated:

"the odds of an event is the number of those who experience the event divided by the number of those who do not. It is expressed as a number from zero (event will never happen) to infinity (event is certain to happen). Odds are fairly easy to visualise when they are greater than one, but are less easily grasped when the value is less than one. Thus odds of six (that is, six to one) mean that six people will experience the event for every one that does not (a risk of six out of seven or 86%). An odds of 0.2 however seems less intuitive: 0.2 people will experience the event for every one that does not. This translates to one event for every five non-events (a risk of one in six or 17%). "

Another webblog described the issue in interpreting the odds ratio that is less than one.

"When you are interpreting an odds ratio (or any ratio for that matter), it is often helpful to look at how much it deviates from 1. So, for example, an odds ratio of 0.75 means that in one group the outcome is 25% less likely. An odds ratio of 1.33 means that in one group the outcome is 33% more likely."

In an article "The odds ratio: calculation, usage, and interpretation" in Biochemia Medica, the author clear suggest converting the odds ratio to be greater than 1 by arranging the higher odds of the evnet to avoid the difficulties in interpreting the odds ratio that is less than 1.

“An OR of less than 1 means that the first group was less likely to experience the event. However, an OR value below 1.00 is not directly interpretable. The degree to which the first group is less likely to experience the event is not the OR result. It is important to put the group expected to have higher odds of the event in the first column. It is not valid to try to determine how much less the first group’s odds of the event was than the second group’s. When the odds of the first group experiencing the event is less than the odds of the second group, one must reverse the two columns so that the second group becomes the first and the first group becomes the second. Then it will be possible to interpret the difference because that reversal will calculate how many more times the second group experienced the event than the first. If we reverse the columns in the example above, the odds ratio is: (5/22)/(45/28) = (0.2273/1.607) = 0.14 and as can be seen, that does not tell us that the new drug group died 0.14 times less than the standard treatment group. In fact, this arrangement produces a result that can only be interpreted as “the odds of the first group experiencing the event is less than the odds of the second group experiencing the event”. The degree to which the first group’s odds are lower than that of the second group is not known.”

In practice, when dealing with the odds ratio less than 1, when possible, I almost always try to reverse the column or recode the response variable to get the odds ratio larger than 1 before I do an interpretation. It is easier for people (especially non-statisticians) to understand the odds ratio with the value greater than 1.

In an example below, the treatment group is actually less effective in terms of the response.

Treatment	Failure (0)	Success (1)
No (0)	21	30
Yes (1)	32	17

The following SAS code can be easily used to calculate the odds ratio:
Data test; input Trt resp count; datalines;
1 1 17
1 0 32
0 1 30
0 0 21
;
proc logistic data=test descending; weight count; model resp=trt;
run;

From the SAS outputs, we get the odds ratio of 0.372, which indicates that the treatment group has odds 0.372 times lower compared to the non-treatmetn group in terms of the success. The interpretation is somewhat difficult to understand.

The program can be easily revised to calculate the odds ratio of failure rate, which gives an odds ratio of 1/0.372 = 2.689. The odds ratio can be intepretated as "the odds of achieve the success in non-treatment group is 2.689 times higher than that in treatment group".

proc logistic data=test; weight count; model resp=trt;
run;

In SAS PROC Logistic, with descending option, probability modeled is response=1 (success); without descending option, probability modeled is response=0 (failure);

Design and Analysis of Bioequivalence Studies for Highly Variable Drugs (HVD) or Highly Variable Drug Products (HVDP)

For bioequivalence studies, it is often for us to show the average bioequivalence by declaring the bioequivalence if the 90% confidence interval of the geometric least squares mean ratio is within 80-125%. The associated study design is typically 2x2x2 cross over design with reasonable sample size (for example, 12 subjects, 24 subjects,…) if the within subject variable is not so big. This approach has been outlined in several FDA’s guidelines:

-         Statistical Approaches to Establishing Bioequivalence

-         Bioavailability and Bioequivalence Studies for Orally Administered Drug Products — General Considerations

-         Food-Effect Bioavailability and Fed Bioequivalence Studies

Recently, there are a lot of discussions about the bioequivalence studies for a product with high variability (high variable drugs). Highly Variable Drugs refer to the type of drugs with higher within subject variability and  is Defined as one for which the root mean square error (RMSE) from the ANOVA bioequivalence analysis > 0.3 for either AUC or Cmax.

For highly variable drugs, if we employ the common study design, the required sample size will be very large, which will cause the ethic concerns to implement such studies.

FDA had several advisory committee meeting in discussing this issue. The most recent meetings were in 2004 and 2009. In 2009 meeting, the slide presentation  by Dr Conner from FDA summarized the development in dealing with this issue and FDA’s position (see slide presentation “Bioequivalence Methods for Highly Variable Drugs and Drug Products”).

Among various approaches to address the bioequivalence issue for highly variable drugs, reference-scaled average BE approach has been suggested. This approach requires less subjects in the study, but with replicated treatment design such as three-period, reference- replicated, crossover design with sequences of TRR, RTR, & RRT or four-period design with sequences of TRTR and RTRT. The replicated crossover designs were also discussed in FDA guidance “Statistical Approaches to Establish Bioequivalance”, but was for dealing with the carryover effects. Here, the replicated crossover designs are for dealing with highly variable drugs.

The implementation of the reference-scaled average BE approaches have been detailed and discussed in FDA guidance (draft) many publications. The most relevant ones are:

-         FDA Guidance on Progesterone (2011)

-         Sample Sizes for Designing Bioequivalence Studies for Highly Variable Drugs by Endrenyi and Tothfalusi (2012)

-         Bioequivalence of Highly Variable Drugs: A Comparison of the Newly Proposed Regulatory Approaches by FDA and EMA by Karalis et al (2011)

The European Medicines Agency also recognizes certain drugs as highly variable drug products (HVDP) and  is willing to accept a wider difference (i.e., a wider 90% confidence interval) in Cmax for bioequivalence evaluation. In its guidance "Guideline on the Investigation of Bioequivalence", section 4.1.10 specifically discussed the HVDP:

Other Readings:

-         Bioequivalence Approaches for Highly Variable Drugs and Drug Products by Haidar et al

-         Generic Drug Bioequivalence by Dr Aaron Sigler

-         An Example of How to Write the Statistical Section of a Bioequivalence Study Protocol for FDA Review

-         Global Bioequivalence Guidelines

-         Using SAS Proc Power to Perform Model-based Power Analysis for Clinical Pharmacology Studies