Normally, you would interpret an OR as follows: for a one-unit increase in your continuous predictor variable, the odds of the dependent variable being positive (=1) increase by factor x (in your case 2.6). However, since the odds ratio in your analysis is non-significant, you should not conclude a substantial relationship of the two variables in your data. 

http://www.ats.ucla.edu/stat/mult_pkg/faq/general/odds_ratio.htm

Statistical software fits your data to a specified model to obtain this odds ratio. 

As explained above, an odds ratio of 2.6 means that a one unit increase in continuous variable leads to a 2.6 fold increase in the odds of the event.

Dear Suleyman,

It also depends how 'insignificant' your results are. The fact of the matter is, and an unpopular opinion to boot, that the p-value is an arbitrary value. P = .05 just means that  there's a 1 in 20 chance this level of change was by accident. With that said, is there really a big difference between p = .051 and p = .049? 

At this point it's really a judgment call to be frank. You could: a) stick to the guns of science and anything p > .05 is not worth discussing; or b) document the result as 'trending.' But where the upper limit of a trending p-value lies is entirely, as I've mentioned, a judgment call. I've seen published work using .08 as an upper limit on trending associations.

To put a long story short, if you feel like this association is worth mentioning, you could dictate it as such, "Predictor variable X was associated with increased odds of outcome Y (OR 2.6; 95% CI ----, p = 0.51), however this association was trending."

If you were conducting an epidemiological study with disease prevalence, your OR can be used as an estimate of effect size. This also influences whether you think your result is worth mentioning. According to Chen, Cohen, and Chen, "[their] calculations indicate that OR = 1.68, 3.47, and 6.71 are equivalent to Cohen's d = 0.2 (small), 0.5 (medium), and 0.8 (large), respectively, when disease rate is 1% in the nonexposed group; Cohen's d < 0.2 when OR 0.8 when OR > 5." [1]

Hope this helps.

Reference:

[1] Chen H, Cohen P, Chen S. How big is a big odds ratio? Interpreting the magnitudes of odds ratios in epidemiological studies. Communications in Statistics—Simulation and Computation®. 2010 Mar 31;39(4):860-4.

Dear Alexander Pabst, Mehmet Sinan Iyisoy, Shaun Hiu

Does this interpretation also applies to descriptive cross-section study or its limited to case control and longitudinal studies only?

Thanks for your response.

Yes, the odds ratio can be used in both settings, i.e. case-control/longitudinal and cross-sectional to express the association between an exposure and an outcome of interest. However, the meaning is somewhat different: In case-control studies, an odds ratio indicates how much higher the odds of exposure are among cases than among controls to develop the outcome over a specified period of time. In cross-sectional studies, the odds ratio is usually interpreted as prevalence ratio, which indicates how much higher the prevalence of an outcome at a fixed point in time is in the exposed group relative to the unexposed group. Also very well explained here (post by Eduardo J Simoes):

https://www.researchgate.net/post/Prevalence_Ratio_Odds_Ratio_and_Relative_Risk

Any thoughts from the group on interpreting a significant odds ratio less than 1 (0.77 to be exact) with a continuous predictor from a logistic regression model?

My initial interpretation would be: "for every 1 unit increase in X, there was .77 times the odds of success." However, is there a more intuitive interpretation based on percentage? Ie, can I say, for every 1 unit increase in X, the odds of success decreased by _____"?

Thank you!