How to calculate confidence intervals for pooled odd ratios in meta-analysis?

Question

I have two datasets from genome-wide association studies. The only information available are the odd ratios and their confidence intervals (95%) for each genotyped SNP. My want to generate a forest plot comparing these two odds ratios, but I can't find the way to calculate the combined confidence intervals to visualize the summary effects. I used the program PLINK to perform the meta-analysis using fixed effects, but the program did not show these confidence intervals.

• How can I calculate such confidence intervals?

The data available is:

• Odd ratios for each study,
• 95% confidence intervals and
• Standard errors.

Answer 1

In most meta-analysis of odds ratios, the standard errors $se_i$ are based on the log odds ratios $log(OR_i)$. So, do you happen to know how your $se_i$ have been estimated (and what metric they reflect? $OR$ or $log(OR)$)? Given that the $se_i$ are based on $log(OR_i)$, then the pooled standard error (under a fixed effect model) can be easily computed. First, let's compute the weights for each effect size: $w_i = \frac{1}{se_i^2}$. Second, the pooled standard error is $se_{FEM} = \sqrt{\frac{1}{\sum w}}$. Furthermore, let $log(OR_{FEM})$ be the common effect (fixed effect model). Then, the ("pooled") 95% confidence interval is $log(OR_{FEM}) \pm 1.96 \cdot se_{FEM}$.

Update

Since BIBB kindly provided the data, I am able to run the 'full' meta-analysis in R.

library(meta)
or <- c(0.75, 0.85)
se <- c(0.0937, 0.1029)
logor <- log(or)
(or.fem <- metagen(logor, se, sm = "OR"))

> (or.fem <- metagen(logor, se, sm = "OR"))
    OR            95%-CI %W(fixed) %W(random)
1 0.75  [0.6242; 0.9012]     54.67      54.67
2 0.85  [0.6948; 1.0399]     45.33      45.33

Number of trials combined: 2 

                         OR           95%-CI       z  p.value
Fixed effect model   0.7938  [0.693; 0.9092] -3.3335   0.0009
Random effects model 0.7938  [0.693; 0.9092] -3.3335   0.0009

Quantifying heterogeneity:
tau^2 < 0.0001; H = 1; I^2 = 0%

Test of heterogeneity:
    Q d.f.  p.value
 0.81    1   0.3685

Method: Inverse variance method

References

See, e.g., Lipsey/Wilson (2001: 114)

Answer 2

Actually, you could use software like METAL which is specifically designed for meta-analyses in GWA context.

It's awkward that plink doesn't give the confidence interval. However, you can get the CI because you have the final OR (take $\log(\text{OR})$) and the $p$-value (hence the $z$) for the fixed effect.

Bernd's method is even more precise.

Beware that I would be more worried about the effect direction as it looks like you only have summary stats for each study but nothing to be sure which is the OR allele. Unless you know it is done on the same allele.

Christian

Answer 3

Here is code to get CIs for meta-analysis as in PLINK:

getCI = function(mn1, se1, method){
    remov = c(0, NA)
    mn    = mn1[! mn1 %in% remov]
    se    = se1[! mn1 %in% remov]
    vars  <- se^2
    vwts  <- 1/vars

    fixedsumm <- sum(vwts * mn)/sum(vwts)
    Q         <- sum(((mn - fixedsumm)^2)/vars)
    df        <- length(mn) - 1
    tau2      <- max(0, (Q - df)/(sum(vwts) - sum(vwts^2)/sum(vwts)) )

    if (method == "fixed"){ wt <- 1/vars } else { wt <- 1/(vars + tau2) }

    summ <- sum(wt * mn)/sum(wt)
    if (method == "fixed") 
         varsum <- sum(wt * wt * vars)/(sum(wt)^2)
    else varsum <- sum(wt * wt * (vars + tau2))/(sum(wt)^2)

    summtest   <- summ/sqrt(varsum)
    df         <- length(vars) - 1
    se.summary <- sqrt(varsum)
    pval       = 1 - pchisq(summtest^2,1)
    pvalhet    = 1 - pchisq(Q, df)
    L95        = summ - 1.96*se.summary
    U95        = summ + 1.96*se.summary
    # out = c(round(c(summ,L95,U95),2), format(pval,scientific=TRUE), pvalhet)   
    # c("OR","L95","U95","p","ph")
    # return(out)

    out = c(paste(round(summ,3), ' [', round(L95,3), ', ', round(U95,3), ']', sep=""),
            format(pval, scientific=TRUE), round(pvalhet,3))
    # c("OR","L95","U95","p","ph")
    return(out)
}

Calling R function:

getCI(log(plinkORs), plinkSEs)

Answer 4

This is a comment (don't have enough rep. points). If you know the sample size (#cases and #controls) in each study, and the odds ratio for a SNP, you can reconstruct the 2x2 table of case/control by a/b (where a and b are the two alleles) for each of the two studies. Then you can just add those counts to get a table for the meta-study, and use this to compute the combined odds-ratio and confidence intervals.