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);

## 16 comments:

HI,

I really appreciated this question! I am currently trying to write an odds ratio sentence for a continuous variable that is less than one. How do I go about writing this when there are not categories?

Many thanks!

Danielle

You are dealing with a continuous predictor variable. The interpretation for odds ratio is straight forward.

If you use a unit of 1 for the continuous variable, you would just say that the odds for xxx is xx% higher per unit of xxx.

if you use a unit of 10 for the continuous predictor variable, you would just say that the odds for xxx is xx% higher per 10 unit of xxx.

If you use SAS software for your calculation, Proc Logistic has an option UNITS to allow you to specify it is 1 unit or 10 unit or any other unit you specify.

http://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#statug_logistic_sect026.htm

If the odds ratio for 1 unit change in a continuous predictor variable is small, it is preferable to use a larger unit (such as 5 or 10).

how should we interpret an OR with value 0?

Odds ratio is around 1. You can not have a odds ratio = 0.

There may be a situation with odds ratio of A/B close to 0 (very very small). In this case, if you flip round, it means the odds ratio of B/A is very large.

Thank you! This was so helpful and clear.

Many thanks for the above clarification. how do you report an OR of 1.24 when comparing two groups for a clinical outcome?

you would say "the odds for group A are 24% higher than the odds for group B for a clinical outcome".

for details, you may refer to http://www.ats.ucla.edu/stat/mult_pkg/faq/general/odds_ratio.htm

I have these results:

OR (CI) P- value

1.3(0.3-5.4) 0.70

2.5(1.2-5.3) 0.01

1.2(0.04-0.7) 0.01

1.1(0.6-2.3) 0.67

3.4(2.0-5.7) 0.01

How would I interpret them?

For the first one and the fourth one, the odds ratio is not significant (p-value >0.05 and the 95% confidence interval include 1. you don't have to interpret the odds and just simply conclude that there is no difference between two groups.

For the second one, you may say "the odds of achieving the success in one group is 2.5 times higher than that in control group" or something like that.

for the third one, you may have an error because the confidence interval does not cover the point estimate of odds ratio.

For the first one, you may say "the odds or achieving the success in one group is 3.4 times higher than that in control group" or something like that.

How would I interpret the following odds for a survey item response. Thanks in advance.

Strongly Agree(reference group)Disagree: 0.24 (0.12-0.47), p<0.001Agree: 0.27 (0.16-0.47), p<0.001Strongly Agree: 0.17 (0.10-0.31), p<0.001Odds ratio? Can it be negative? Or it is always positive?

Odds ration can not be negative. It is always positive. Whether or not there is a protective effect or negative effect will be judged by the odds ratio greater than 1 or less than one.

The same applies to other ratios such as hazard ratio, risk ratio

There seems to be an issue with the results you presented below. You have 'Strongly Agree' category presented twice. Not sure if you are talking about the odds ratio or odds. Odds ratio is supposed to be the comparison of two groups or two categories.

How would I interpret the following odds for a survey item response. Thanks in advance.

Strongly Agree (reference group)

Disagree: 0.24 (0.12-0.47), p<0.001

Agree: 0.27 (0.16-0.47), p<0.001

Strongly Agree: 0.17 (0.10-0.31), p<0.001

Sorry about the typo as I was typing from my mobile device. The reference group is Strongly Disagree.

Strongly Disagree (reference group)

Disagree: 0.24 (0.12-0.47), p<0.001

Agree: 0.27 (0.16-0.47), p<0.001

Strongly Agree: 0.17 (0.10-0.31), p<0.001

The output is from the multivariate binary logistic regression showing odds ratio, 95% CI and p value.

BTW, the Strongly Disagree, Disagree, Agree, and Strongly Agree responses were each dummy coded as 0 and 1 (and then compared to the regular variable with the original 4 Likert categorical responses and the output were the same).

What makes the output hard for me to interpret is that the odd ratios are very similar regardless of the level of agreement. Thanks again for sharing your thoughts on this.

Thank you, dr, for this post.

This probably is a silly question, but I need to ask it and I hope you have the time to answer it:

The transformation of an OR lower than 1 by using 1/OR - can this be used on the CI as well? I shouldn't be a problem since it's "Only" a transformation that does not change anything else - right?

yes, if you transform the point estimate of the OR, the corresponding confidence interval will also be transformed.

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