Bioequivalence and Bioavailability Forum

Hi ElMaestro,

wikipedia knows more than me and I think there is a good explanation of SE.
The main points to understand (about SD and SE) are:

• "the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean"
  
• "the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean"

With increasing sample size:

• SD of the sample will be more precise getting closer to the population SD
  
• SE of whatever will be closer to zero. This is the property which makes CI narrower with higher sample size.

The calculation of SE of mean (i.e. SD of mean which is different from SD of sample) is on wikipedia (as you mentioned).
I like the proofs:

(image from wiki)
Easy steps with edit the equation:
var(mean) = 1/N * var(X)
sqrt(var(mean)) = 1/sqrt(N) * sqrt(var(X))
SD(mean) = 1/sqrt(N) * SD(X)
SE of mean = SD(X)/sqrt(N)


In BA/BE 90% CI is calculated from SE of difference T-R for ln-transformed data.
(SE of difference of estimated marginal means T-R)

In the same principles as previous proof (assuming independence and using variance properties) it's possible to derive formula valid for BE in 2x2 crossover design (balanced sequences, if not: mean should be changed somehow for estimated marginal mean in the following):
var(difference T-R) = var(mean_T-mean_R) = var(mean_T) + var((-1)*mean_R) = var(mean_T) + (-1)^2 * var(mean_R) = var(mean_T) + var(mean_R) = 1/N_T * var(X_T) + 1/N_R * var(X_R)
(SE of difference is only square root of variance of difference T-R)

Moreover var(X_T) can be substituted by N_T*SE_T^2 because SE of mean=SD/sqrt(N) which implies SD=sqrt(N)*SE and then var=SD^2=N*SE^2.
So:
SE of difference T-R = SD(difference T-R) = sqrt(var(difference T-R)) = sqrt( 1/N_T * var(X_T) + 1/N_R * var(X_R) ) = sqrt( 1/N_T * N_T*SE_T^2 + 1/N_R * N_R*SE_R^2 ) = sqrt( SE_T^2 + SE_R^2 )
without between steps:
SE of difference T-R = sqrt( SE_T^2 + SE_R^2 )
where SE_T is Standard Error of the mean of ln-data of Test treatment, same for SE_R - Reference treatment.
Nevertheless it is only nice to know (not useful, I guess), GLM gives the result of difference and SE of difference without need of calculation of T and R separately.

Best regards,
zizou
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Edit: In second half of this post, for case of unbalanced sequences the mean should be somehow changed for estimated marginal mean (i.e. LSMean in SAS terminology).