Recalculate 95% CIs for odds ratio for reverse reference group I am doing a meta-analysis and I need to recalculate OR and 95% CIs for the reverse reference group. For instance, I am looking OR for Q3 vs Q1 where Q1 is reference group. However, some studies reported the Q1 vs. Q3. So, I need to recalculate the OR and 95% CIs.
I found that it is possible to get the 1/OR for the odds ratio value. However, I am wondering whether is it possible to recalculate 95% CIs even when the study is not given the number of subjects for each cell.
Any input is appreciated!
For instance, I have the OR and 95% CIs as 1.14(1.02-1.30). This is for the low group compared to high group (reference). But I need to get OR and 95% CI for high group compared to low group (reference). In this case, how can I recalculate the 95% CI values. Is it possible to recalculate them? thank you again.
 A: To calculate the odds ratio for the reverse group comparison, take the reciprocal of the odds ratio. To calculate the confidence limits for the reverse group comparison, take the reciprocals of the limits and swap them. Note: This is only valid for binary categorical predictors.
That is, $\operatorname{OR}_{BA} = 1/\operatorname{OR}_{AB}$ and $L_{BA}=1/U_{AB}$ and $U_{BA}=1/L_{AB}$. Here, $L$ and $U$ denote the lower and upper confidence limits for the odds ratio, respectively.
The reason is that in a logistic regression model, the odds ratio is obtained by calculating $\operatorname{OR}=\exp(\beta)$. If we switch the reference category, the coefficients sign is reversed, i.e. $\beta_{BA} = -\beta_{AB}$. Hence, $\operatorname{OR}_{BA} = \exp(-\beta_{AB}) = 1/\exp(\beta_{AB})$.
Here is an illustration using R:
set.seed(142857)
# Sample size
n <- 200
# Simulate one continuous and one categorical binary predictor
x1 <- runif(n, 10, 50)
x2 <- rbinom(n, 1, 0.75)
# Set up the linear predictor
linpred <- -4 + log(1.2)*x1 + log(0.5)*x2
# Simulate the probabilities
probs <- plogis(linpred)
y <- rbinom(n, 1, probs)

dat <- data.frame(y = y, x1 = x1, x2 = factor(x2))

# Logistic regression with "0" being the reference category
mod1 <- glm(y~x1 + x2, data = dat, family = binomial)

# Odds ratio and confidence interval
exp(coef(mod1))[3]
0.798159
exp(confint(mod1))[3, ]
    2.5 %    97.5 % 
0.3278559 1.9074099

# Reverse the reference level to "1"
dat$x2 <- relevel(dat$x2, ref = "1")

# Re-run the logistic regression
mod2 <- glm(y~x1 + x2, data = dat, family = binomial)

# Odds ratios and confidence interval
exp(coef(mod2))[3]
1.252883
exp(confint(mod2))[3, ]
    2.5 %    97.5 % 
0.5242711 3.0501203

You see that the odds ratio where $x_2=1$ is the reference category is the reciprocal of the odds ratio where $x_2=0$ is the reference category: $1.253 = 1/0.798$. The lower confidence interval limit where $x_2 = 1$ is the reference category is the reciprocal of the upper limit where $x_2 = 0$ is the reference : $0.524 = 1/1.907$ etc.
