Dear all,
I am hoping someone could help me with this problem where different
parameterization is used. I am meta-analyzing coefficients and standard
errors from multiple studies where the raw data is not shared.
Here is an illustrative example in R for one analyst:
set.seed(1066)
status <- rbinom( 1000, 1, 0.2 )
males <- rbinom( 1000, 1, 0.6 )
smoke <- rbinom( 1000, 1, 0.3 )
age <- runif(1000, min=20, max=80)
f1 <- glm( status ~ males*smoke + age, family="binomial" )
Now, unfortunately some analysts coded sex as females instead of males.
Using the same dataset, I get the following output with females:
females <- 1 - males
f2 <- glm( status ~ females*smoke + age, family="binomial" )
My question is how to 'convert' from one set of coefficients to another.
coef( summary( f1 ) )
# Estimate Std. Error z value Pr(>|z|)
# (Intercept) -1.520399871 0.284464584 -5.3447774 9.052825e-08
# males 0.213851446 0.201717381 1.0601538 2.890746e-01
# smoke -0.123103049 0.292346483 -0.4210861 6.736922e-01
# age -0.001056007 0.004612947 -0.2289223 8.189293e-01
# males:smoke 0.283775173 0.362821438 0.7821345 4.341355e-01
coef(summary(f2))
# Estimate Std. Error z value Pr(>|z|)
# (Intercept) -1.306548425 0.262573162 -4.9759405 6.493160e-07
# females -0.213851446 0.201717381 -1.0601538 2.890746e-01
# smoke 0.160672124 0.214923130 0.7475795 4.547138e-01
# age -0.001056007 0.004612947 -0.2289223 8.189293e-01
# females:smoke -0.283775173 0.362821438 -0.7821345 4.341355e-01
I have worked out algebrically (and numerically) the following:
Beta(females) = -Beta(males)
Var(females) = Var(males)
Beta(females:smoke) = -Beta(males:smoke)
Var(females:smoke) = Var(males:smoke)
Beta(smoke | fit1) = Beta(smoke | fit2) + Beta(females:smoke)
= 0.160672124 -0.283775173
= -0.1231030
But I am really struggling on defining the relationship between
Var(smoke | fit1) and Var(smoke | fit2) ? I tried but ended up with a
covariance term which I could not solve.
Any suggestions is much appreciated. Many thanks in advance.
Regards, Adai
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