# The effect of a lack of dose-response results on odds ratio

This is part of a quantitative reasoning assignment I was working on.

The study hypothesized that exercise may reduce the risk of disability for activities of daily living(ADL). However after identifying confounders and adjusting for their effects, walking and calisthenics happen to have an OR of >1, in other words both exercises are instead positively associated instead of the hypothesised negatively association. I need to come up with a reason why this is the case, and this is my reason.

The OR for walking and calisthenics is more than 1 after adjustment. This could be due to the limitation that was identified by the researchers that exercise types were captured as Categorical Nominal variables, but intensity, duration and frequency was not collected. Dose-response relationships cannot be analysed to examine the association. Assuming that numerical data could be used instead to derive an association, it might be the case that all exercises are negatively associated to ADL disability. Now suppose that the scatter plot of walking and calisthenics are rather gently negatively sloped, such that using categorical nominal instead will cause a positive association between the exercise and ADL disability. A possible reason for this is that age is positively associated to disability from Table 1 and given a Y/N question, they are more likely to reply yes to participation.

I don’t know if my analysis is right and I’m using the right jargons, do inform me if more information is required about the research or pertains to this post. Can someone help me to verify this? Thanks.

• You may want to consider the uncertainty around the estimate. – Björn Apr 2 '19 at 3:53
• Are you referring to the size of the CI? If so, it's because the exercises are arranged in descending sample size down the row, so random error generally increases down the row and increase the size of OR(I didn't put my data here). I'm actually interested to know why OR got to >1 for the 2 exercises – Prashin Jeevaganth Apr 2 '19 at 3:56
• Yes, but do you think you really have much to go I that the "true" ORs are really >1 or is this quite possibly just noise? – Björn Apr 2 '19 at 4:02
• I'm actually not too sure if I should believe that OR that is outside the bracket as well, I acknowledge that it's a log scale but the log of middle value of the range doesn't quite fit the range of the log of the 2 end points in 3 sf. Is that also important? – Prashin Jeevaganth Apr 2 '19 at 4:05

## 1 Answer

@Bjorn is trying to get you to see that having odds ratios above 1 in the sample doesn't tell you whether those odds ratios would be above 1 in the population. It's entirely consistent for there to be an odds ratio less than 1 in the population (i.e., that walking and calisthenics do indeed reduce the odds of ADL disability) and for you to have observed the odds ratios you found in your sample, simply as a result of random chance.

Let's make it simpler. Consider you're trying to estimate the relationship between $$x$$ and $$y$$. Let's say there is a population of 200 individuals, and in that population, the regression slope on the line $$y=\beta_0+\beta_1X+\epsilon$$ is 1, as plotted below.

Let's say we draw a random sample of 30 from this population and estimate the slope of the regression line in the sample.

The slope is negative. Does that mean there is confounding or something causing that observed negative relationship? Or did we just happen to draw a sample with a negative slope by chance? It would be pointless and misleading to try to rationalize and explain why the slope in the sample is negative when it was simply due to chance.

The confidence interval provides a range of plausible values for the odds ratio in the population. The range includes values less than 1, so there is no reason to put so much stock into the odds ratios for walking and calisthenics being slightly greater than 1. Based on your data, it's totally plausible for walking and calisthenics to have odds ratios less than 1 in the population. Having the point estimate be greater than 1 could just be due to chance. Based on the analysis presented, we simply don't have enough information to claim whether these activities increase the risk, decrease the risk, or have no effect on the risk of ADL disability. That's the only claim you can make.