Solved – Interpretation of odds ratio (OR) below 1

logisticodds-ratioregression

I am running two logistic regression analyses, and I am very confused about the interpretation of the odds ratio, specifically in the case of an OR below 1. I have looked into all kind of different related posts on this forum, but nowhere I can seem to find a similar situation with an explanation. So I hope you will be able to help me out! This is the situation:

Model 1:
I have a dichotomous outcome: yes/no diagnosis
I have a continuous predictor: score A

Model 2:
I have a dichotomous outcome: yes/no diagnosis
I have a continuous predictor: score B

For model 1 I have an OR above 1 (Exp(B), thus the OR, =1.17), I interpreted this as:
"the odds of having a diagnosis was 1.17 times greater with one point increase in score A"
I think this is correct isn't it?

For model 2 I have an OR below 1 (Exp(B), thus OR, =0.91). Based on some other posts on stats forums I figured I have to take the reciprocal of 0.91 (being 1.10), so I interpreted this as:
"the odds of having a diagnosis was 1.10 times smaller with one point increase score B"
Is this correct?
I started doubting because somewhere else I read that I should not take the reciprocal but instead do 1-0.91=0.09 and use this number for my interpretation. Although I am not sure how exactly I would use this number in my interpretation…

I hope someone can shed some light on the different numbers and ways of interpreting my OR's!

Thanks a lot!
Sabine

Best Answer

The odds of having a diagnosis in model 2 decreases by a factor 0.91 for a unit increase in score B. Some people don't like these numbers less than 1, so they take the invers. This is now interpreted that the odds of a diagnosis increases by a factor 1,10 for a unit decrease in factor B.

If you multiply something by 1.10, you increase it by 10%. In general the relationship between a factor increase and the percentage change is (f - 1) * 100%. So an odds ratio of 0.91 corresponds to a (0.91 - 1)*100% = -9% change in odds for a unit increase in factor B, or a 9% decrease in the odds for a unit change in factor B.

Related Solutions

Solved – Calculating risk ratio using odds ratio from logistic regression coefficient

Zhang 1998 originally presented a method for calculating CIs for risk ratios suggesting you could use the lower and upper bounds of the CI for the odds ratio.

This method does not work, it is biased and generally produces anticonservative (too tight) estimates of the risk ratio 95% CI. This is because of the correlation between the intercept term and the slope term as you correctly allude to. If the odds ratio tends towards its lower value in the CI, the intercept term increases to account for a higher overall prevalence in those with a 0 exposure level and conversely for a higher value in the CI. Each of these respectively lead to lower and higher bounds for the CI.

To answer your question outright, you need a knowledge of the baseline prevalence of the outcome to obtain correct confidence intervals. Data from case-control studies would rely on other data to inform this.

Alternately, you can use the delta method if you have the full covariance structure for the parameter estimates. An equivalent parametrization for the OR to RR transformation (having binary exposure and a single predictor) is:

$$RR = \frac{1 + \exp(-\beta_0)}{1+\exp(-\beta_0-\beta_1)}$$

And using multivariate delta method, and the central limit theorem which states that $\sqrt{n} \left( [\hat{\beta}_0, \hat{\beta}_1] - [\beta_0, \beta_1]\right) \rightarrow_D \mathcal{N} \left(0, \mathcal{I}^{-1}(\beta)\right)$, you can obtain the variance of the approximate normal distribution of the $RR$.

Note, notationally this only works for binary exposure and univariate logistic regression. There are some simple R tricks that make use of the delta method and marginal standardization for continuous covariates and other adjustment variables. But for brevity I'll not discuss that here.

However, there are several ways to compute relative risks and its standard error directly from models in R. Two examples of this below:

x <- sample(0:1, 100, replace=T)
y <- rbinom(100, 1, x*.2+.2)
glm(y ~ x, family=binomial(link=log))
library(survival)
coxph(Surv(time=rep(1,100), event=y) ~ x)

http://research.labiomed.org/Biostat/Education/Case%20Studies%202005/Session4/ZhangYu.pdf

Solved – Computation and Interpretation of Odds Ratio with continuous variables with interaction, in a binary logistic regression model

As others have noted, it is probably easier to interpret this graphically. I will make certain assumptions to demonstrate the thought process for interpreting interactions like this:

$A$ is my predictor of interest, so I will interpret the odds ratio of $A$ at varying levels of $B$, and
$B$ has a range $[0, 100]$ with mean of 50 and most values falling in $[25, 75]$ such that this is the range of interest.

In the scenario I have set up, the actual log odds of $A$, 0.756, is probably not of interest since it is the log odds of $A$ when $B=0$ and $B=0$ applies to so few people in the data that we do not care for it.

I will calculate the log-odds of $A$ when $B=\{25,50,75\}$. This results in:

\begin{align} \beta_1 + \beta_3 \times \{25, 50, 75\}&{}=\\ 0.756 -0.00303 \times \{25, 50, 75\}&{}=\\ \{0.756 -0.07575, 0.756 -0.15150, 0.756 -0.22725\}&{}=\\ \{0.68025, 0.60450, 0.52875\} \end{align}

The odds ratio of $A$ will then be $1.97\ (e^{0.68025})$, $1.83\ (e^{0.60450})$, $1.70\ (e^{0.52875})$ when $B=\{25, 50, 75\}$ respectively.

So you find that the odds ratio of $A$ drops as the value of $B$ increases. We can also graph the set-up at varying values of B:

To create the graph, I used all integer values of $B$ in $[25, 75]$.

Best Answer

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Solved – Calculating risk ratio using odds ratio from logistic regression coefficient

Solved – Computation and Interpretation of Odds Ratio with continuous variables with interaction, in a binary logistic regression model

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