# Relative Risk Ratio (RRR) Analysis with a Continuous Predictor

#### evopass

##### New Member
Hi, I am hoping that someone can help me with a relative risk ratio analysis. I'm trying to assess the ability of a continuous variable to predict a categorical outcome (the base group - Y) relative to other possible outcomes.

The RRR analysis gave me the RRR for the likelihood of outcome A relative to outcome Y (RRR = 0.1429, SE = 0.1279, Z = -2.17, p =.0 30, 95% CI [0.0247, 0.8261]), but I actually want to know about the likelihood of outcome Y relative to outcome A. So I took the reciprocal of the RRR and CI limits and reversed the sign of the Z score. This gave me RRR = 7.00, SE = 0.13, Z = 2.17, p = .03, 95% CI (1.21, 40.49). I kept the SE the same, although I wasn't sure if that's correct.

This was my interpretation: People who scored highly on the continuous predictor variable (+1 sd) were 7 times as likely to have outcome Y over outcome A than were people at the mean of the predictor variable. In other words, a one standard deviation increase in the predictor variable was associated with a 600% increase in the likelihood of outcome Y over outcome A.

Is this interpretation correct? Sorry for the lengthy post! I really appreciate your help.

#### victorxstc

##### Pirate
Hi and welcome

I am not a statistician, but your problem interests me and I would love to learn something here.

Your question was not lengthly. For better picturing your question to get a good solution from the community, please elaborate much more on your variables, study goal, etc. Please let us know what your variables are, or if they are confidential, give some similar examples and let us know the number and nature of each of your variables and description of your design and your aims. What analysis and software did you use which gave you RRR for a continuous variable?

This was my interpretation: People who scored highly on the continuous predictor variable (+1 sd) were 7 times as likely to have outcome Y over outcome A than were people at the mean of the predictor variable.
Shouldn't it be people scoring highly on one end versus those scored on the other high end of the range (vs people scoring very low [instead of mean])?

In other words, a one standard deviation increase in the predictor variable was associated with a 600% increase in the likelihood of outcome Y over outcome A.
Could you plz let me know how you concluded this?

#### hlsmith

##### Less is more. Stay pure. Stay poor.
I am also curious if you used a software and if so can you just flip the variables for Y and A when seting it up so that you do not need to do all of the steps you listed, plus this may confirm your original results.

#### evopass

##### New Member

You know, as silly as this sounds, I never thought to switch the base group around. Probably because it required 3 separate analyses to get the RRR's (Y vs A, Y vs B, Y vs. C). Thank you, hlsmith!!! I did this in stata and it confirmed most of the results. The only incorrect numbers were the standard errors, which were wildly different. This was the part of the analysis that I was most worried about... so it's great to have that solved.

victorxstc, the study is confidential at the moment, so I hope that you'll forgive my vagueness. However, I can give you more information if you're interested. I simplified it a lot in my previous post. The study is a psychology study where we brought people into the lab, had them fill out some questionnaires, and then gave them a choice between 4 different tasks. One of the tasks was antisocial in nature. We wanted to know which of the personality variables was the best unique predictor of antisocial choice (relative to the other options).

In my analysis, I had 6 continuous, standardized predictor variables that represent different personality constructs (e.g., agreeableness) and 4 dichotomous control variables (yes/no questions). The outcome (task choice) had 4 levels. Our hypothesis was that one of the personality variables would emerge as the sole unique predictor of antisocial task choice. This provides validity evidence for a new personality construct (and the new measure of it).

I used STATA for the analysis with the help of a colleague. See this site for an example: http://www.ats.ucla.edu/stat/stata/output/stata_mlogit.htm

As this is a logistic regression analysis and the personality variables were standardized, the coefficients index the relative risk ratios for a 1 standard deviation change in the continuous predictor. If they weren't standardized, the RRR would reference a 1 unit change in the continuous predictor in its raw scale.

The 600% increase is determined by the RRR =7. It may be easier to think about some lower numbers first. If people at 1 standard deviation above the mean were twice as likely to choose the antisocial choice over task 1 than were people at the mean (ie., RRR =2) , that would mean that there's a 100% increase in the probability of antisocial choice going from 0 (mean) to 1 sd. If RRR = 3, that would mean there was a 200% increase. These websites give some good explanations:
http://stats.org/in_depth/faq/absolute_v_relative.htm
http://sph.bu.edu/otlt/MPH-Modules/EP/EP713_Association-Brooks/EP713_Association-Brooks_print.html

Here I found that RRR= 7, so a 1 sd increase is associated with a 600% increase in the likelihood of antisocial choice relative to the other task. You can also get this by calculating (RRR - 1 ) x 100%

In any case, thank you again for your help. Hopefully this post will help someone else in the future too!

#### victorxstc

##### Pirate
victorxstc, the study is confidential at the moment, so I hope that you'll forgive my vagueness. However, I can give you more information if you're interested.
Of course I am, but I asked so because it allows other community members to help you better if they want to.

This provides validity evidence for a new personality construct (and the new measure of it).
Now that interests me personally! I look forward to see it in DSM VI!

As this is a logistic regression analysis and the personality variables were standardized, the coefficients index the relative risk ratios for a 1 standard deviation change in the continuous predictor. If they weren't standardized, the RRR would reference a 1 unit change in the continuous predictor in its raw scale.

The 600% increase is determined by the RRR =7. It may be easier to think about some lower numbers first. If people at 1 standard deviation above the mean were twice as likely to choose the antisocial choice over task 1 than were people at the mean (ie., RRR =2) , that would mean that there's a 100% increase in the probability of antisocial choice going from 0 (mean) to 1 sd. If RRR = 3, that would mean there was a 200% increase. These websites give some good explanations:
http://stats.org/in_depth/faq/absolute_v_relative.htm
http://sph.bu.edu/otlt/MPH-Modules/E...oks_print.html

Here I found that RRR= 7, so a 1 sd increase is associated with a 600% increase in the likelihood of antisocial choice relative to the other task. You can also get this by calculating (RRR - 1 ) x 100%
Thanks a lot for your detailed response. Indeed it will help many future readers. It seems that everything is already clear for you. Then I think we now know that your fiirst interpretation was correct (which was: )

This was my interpretation: People who scored highly on the continuous predictor variable (+1 sd) were 7 times as likely to have outcome Y over outcome A than were people at the mean of the predictor variable. In other words, a one standard deviation increase in the predictor variable was associated with a 600% increase in the likelihood of outcome Y over outcome A.
Then remains only standard errors for the logistic regression coefficients? I had a similar study once. I reported the original RRR, ARR, OR, etc. Since they were all less than 1, it was difficult for the reader to easily see the impact. It was also difficult to me to discuss the results. But I did not inverse all the values (all, including the RRRs, CIs, etc.). I just reported their original format with the confidence intervals and all other necessary items for the original format. Then in the first paragraph of the discussion, clarified the reciprocal of those RRRs, ARRs, ORs, etc. and discussed them. I think not all those values (CI, Z, standard errors, etc.) should be necessarily reversed as they do not interest the reader. They are only very important to future studies, the authors of which would themselves deal with inversing the original values reported by you. But that was only my opinion.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Vic or anyone else,

I only skimmed this thread, so I don't know the content - but by inverse are you referring to flipping the two groups in the RR, etc., so the values are now positive (e.g., so instead of protection now you have risk, for a lack of better words). If so, is this a typical use for this word in regards to contingency style data?

#### victorxstc

##### Pirate
but by inverse are you referring to flipping the two groups in the RR, etc., so the values are now positive (e.g., so instead of protection now you have risk, for a lack of better words). If so, is this a typical use for this word in regards to contingency style data?
Not necessarily flipping the two groups, but when b/a is reported instead of a/b (but flipping the two groups can happen in a "b-a" instead of a "a-b" form too). Actually I used the term "reciprocal" in my manuscript and I am not sure whether the word "inverse" is the correct or typical term for this purpose. By inverse here, I meant writing that for example "this treatment was 4-fold more likely to be successful than the other one (OR = 0.25)" [so I guessed 4 is inverse of 0.25 but I am not sure since my English sucks!].

#### hlsmith

##### Less is more. Stay pure. Stay poor.
I was reading a stats book last night and its author also used the phrase recipricol of the Odds Ratio and also pointed out that you can do the same thing with the confidence intervals - to bring them along for the ride.