You can calculate/approximate the standard errors via the p-values. First, convert the two-sided p-values into one-sided p-values by dividing them by 2. So you get $p = .0115$ and $p = .007$. Then convert these p-values to the corresponding z-values. For $p = .0115$, this is $z = -2.273$ and for $p = .007$, this is $z = -2.457$ (they are negative, since the odds ratios are below 1). These z-values are actually the test statistics calculated by taking the log of the odds ratios divided by the corresponding standard errors (i.e., $z = log(OR) / SE$). So, it follows that $SE = log(OR) / z$, which yields $SE = 0.071$ for the first and $SE = .038$ for the second study.
Now you have everything to do a meta-analysis. I'll illustrate how you can do the computations with R, using the metafor package:
library(metafor)
yi <- log(c(.85, .91)) ### the log odds ratios
sei <- c(0.071, .038) ### the corresponding standard errors
res <- rma(yi=yi, sei=sei) ### fit a random-effects model to these data
res
Random-Effects Model (k = 2; tau^2 estimator: REML)
tau^2 (estimate of total amount of heterogeneity): 0 (SE = 0.0046)
tau (sqrt of the estimate of total heterogeneity): 0
I^2 (% of total variability due to heterogeneity): 0.00%
H^2 (total variability / within-study variance): 1.00
Test for Heterogeneity:
Q(df = 1) = 0.7174, p-val = 0.3970
Model Results:
estimate se zval pval ci.lb ci.ub
-0.1095 0.0335 -3.2683 0.0011 -0.1752 -0.0438 **
Note that the meta-analysis is done using the log odds ratios. So, $-0.1095$ is the estimated pooled log odds ratio based on these two studies. Let's convert this back to an odds ratio:
predict(res, transf=exp, digits=2)
pred se ci.lb ci.ub cr.lb cr.ub
0.90 NA 0.84 0.96 0.84 0.96
So, the pooled odds ratio is .90 with 95% CI: .84 to .96.
Systematic Reviews in Health Care: Meta-analysis In Context is an excellent resource if you're looking to do a meta-analysis (in healthcare or otherwise - if you're not in health, ignore their overt fondness for clinical trials).
They also include extensive documentation for conducting a meta-analysis in Stata beyond just a pooled estimate of effect. That is where I would head to first.
Best Answer
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:
Then calculate the difference between the odds ration and the lower (or upper, it doesn't matter, they're the same) confidence limit.
The confidence limit is 1.96 * se, so:
z is estimate / se.
Let's calculate p, just to make sure we didn't make a foolish mistake along the way:
Here are the important bits of the output: