Distribution of Odds Ratio (OR) – Statistical Insights

distributionsodds-ratio

I have a bunch of articles presenting "OR" with a- 95% CI (confidence intervals).

I want to estimate from the articles the P value for the observed OR. For that, I need an assumption regarding the OR distribution. What distribution can I safely assume/use?

Best Answer

The log odds ratio has a Normal asymptotic distribution :

$\log(\hat{OR}) \sim N(\log(OR), \sigma_{\log(OR)}^2)$

with $\sigma$ estimated from the contingency table. See, for example, page 6 of the notes:

Asymptotic Theory for Parametric Models

Related Solutions

Solved – Converting odds ratios to Cohen’s d for meta analysis

Hi Erica and welcome to the site. Have a look at this (page 3) document and this paper. The basic formula for the conversion is $$ d=\mathrm{LogOR}\times \frac{\sqrt{3}}{\pi} $$ Applying the delta-method, we get the following expression for the the variance of $d$ (the standard error of $d$ is just the square root of its variance): $$ \mathrm{Var}_{d}=\mathrm{Var}_{\mathrm{LogOR}}\times \frac{3}{\pi^{2}} $$

Where $\mathrm{LogOR}$ denotes the log of the odds ratio and $\mathrm{Var}_{\mathrm{LogOR}}$ denotes the variance of the log odds ratio.

To get the variance of the log odds ratio, you can use the information given by the confidence interval. To get the standard error of the log odds ratio, use the following formula: $$ \mathrm{SE}_{\mathrm{LogOR}}=\frac{\log(\mathrm{CI}_{upper}) - \log(\mathrm{CI}_{lower})}{2\times z_{1-\alpha/2}} $$

Where $\mathrm{CI}_{upper}$ denotes the upper and $\mathrm{CI}_{lower}$ the lower bound of the confidence interval for the odds ratio (as given in the papers) and $z_{1-\alpha/2}$ is the $1-\alpha/2$ quantile of the standard normal distribution. For a 95%-CI, $\alpha = 0.05$ and $z_{1-\alpha/2}\approx 1.96$. To get the variance of the log odds ratio, just square the standard error. In your example, the standard error of the log odds ratio is about $0.247$. Hence, the variance of the log odds ratio is $0.247^{2}\approx0.061$. To calculate the confidence interval of $d$, you need the standard error of $d$, which is simply $$ \mathrm{SE}_{d}=\mathrm{SE}_{\mathrm{LogOR}}\times\frac{\sqrt{3}}{\pi} $$

Solved – Calculate z-score from odds-ratio

Odds ratios standard errors, CIs, etc are calculated on the log of the odds.

Here's some R code. First we'll take the log of the odds:

OR <- 0.997804
ORL95 <- 0.970573

logor <- log(OR)
logorl95 <- log(ORL95)

Then calculate the difference between the odds ration and the lower (or upper, it doesn't matter, they're the same) confidence limit.

diff <- logor - logorl95

The confidence limit is 1.96 * se, so:

seor <- diff / 1.96

z is estimate / se.

z <- logor / seor
z

Let's calculate p, just to make sure we didn't make a foolish mistake along the way:

(1 - pnorm(abs(z))) * 2

Here are the important bits of the output:

> z 
[1] -0.155723

> (1 - pnorm(abs(z))) * 2
[1] 0.8762514

Best Answer

Related Solutions

Solved – Converting odds ratios to Cohen’s d for meta analysis

Solved – Calculate z-score from odds-ratio

Related Question