. "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
In particular, if then
The difficulty arises when we wish to also estimate the variance of for example to then obtain a confidence interval around the point estimate . To do so, we typically need to resort to linearization techniques that allow us to compute an approximation to the variance of a non-linear function of the parameters. A method that can be used for this purpose is called the Delta method and is described below.
Least-squares means of the response, also known as adjusted means or marginal means can be computed for each classification or qualitative effect in the model. Examples of qualitative effects in our models include type of institution (two levels: public or private) discipline (with six categories in our study), gender of chair of search committee, and others. Least-squares means are predicted population margins or within-effect level means adjusted for the other effects in the model. If the design is balanced, the least-squares means (LSM) equal the observed marginal means. Our study design is highly unbalanced and thus the LSM of the response variable for any effect level will not coincide with the simple within-effect level mean response.
Each least-squares mean is computed as for a given vector L. For example, in a model with two factors A and B, where A has three levels and B has two levels, the least squares mean response for the first level of factor A is given by:
where the first coefficient 1 in L corresponds to the intercept, the next three coefficients correspond to the three levels of factor A and the last two coefficients correspond to the two levels of factor B. If the model also includes an interaction between A and B, then L and has an additional 3 × 2 elements. The corresponding values of the additional six elements in L would be ½ for the two interaction levels involving the first level of factor A (A1B1, A1B2 ) and 0 for the four interaction levels that do not involve the first level of factor A (A2B1, AsB2, A3B1, A3B2). The coefficient vector L is constructed in a similar way to compute the LSM of y (or a transformation of y) for the remaining two levels of A, two levels of B, and even for the six levels of the interaction between A and B if it is present in the model.
When the response variable has been transformed prior to fitting the model, the LSM is computed in the transformed scale and must be then transformed back into the original scale. If we have MLEs of the regression coefficients, we can easily compute the LSMs in the original scale simply by applying the inverse