. "Appendix 3-7: Marginal Mean and Variance of Transformed Response Variables." Gender Differences at Critical Transitions in the Careers of Science, Engineering, and Mathematics Faculty. Washington, DC: The National Academies Press, 2010.
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Gender Differences at Critical Transitions in the Careers of Science, Engineering, and Mathematics Faculty
transformation to . For example, if g(μ) = log(μ) = xβ and is the least squares mean in the transformed scale, we can compute the LSM in the original scale as
If the transformation was the logit transformation, the LSM in the original scale is computed as
VARIANCE OF A NONLINEAR FNCTION PARAMETERS
Suppose that we fit a model to a response variable that has been transformed using some function g as above, and obtain an estimate of a mean . Programs including SAS will also output an estimate of the variance of . We can compute the estimate of the mean in the original scale by applying the inverse transformation g–1 to as described above. In order to obtain an estimate of the variance of , however, we need to make use of, for example, the Delta method, which we now explain.
Given any non-linear function H of some scalar-valued random variable θ, H(θ) and given σ2, the variance of θ, we can obtain an expression for the variance of H(θ) as follows:
For example, suppose that we used a log transformation on a response variable and obtained an LSM in the transformed scale that we denote , with estimated variance . The estimate of the mean in the original scale is obtained by applying the inverse transformation to the LSM:
The variance of is given by:
Suppose now that the response variable was binary and that we used a logit transformation so that