VARIANCE OF A NONLINEAR FUNCTION OF 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 L в ■ Pro­grams including SAS will also output an estimate of the variance of L в ■ We can compute the estimate of the mean in the original scale by applying the inverse transformation g-1 to Lв as described above. In order to obtain an estimate of the variance of g-1 (L в), 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 в, И(в) and given s2, the variance of в, we can obtain an expression for the variance of И(в) 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 L в, with estimated variance <OL■ The estimate of the mean in the original scale is obtained by apply­ing the inverse transformation to the LSM:

m = LSM.. = exp (L в)

original

The variance of m is given by:

Suppose now that the response variable was binary and that we used a logit transformation so that

Given an MLE в and an estimate of L в the least squares mean in the trans­formed scale, we compute m and &m as follows:

exp ( l в)

1 + exp (L в)

Г 2 = exP (L ‘P) Г

m |^1 + exp( L’P) ів

Given a point estimate of the least squares mean in the original scale and an approximation to its variance, we can compute an approximate 100(1-a)% con­fidence interval for the true mean in the original scale in the usual manner:

100(1- a)% for m = m ± tdfa,2Г,

where df is the appropriate degrees of freedom. In our case, and due to relatively large sample sizes everywhere, the t critical value can be replaced by the corre­sponding upper al2 tail of the standard normal distribution.

Main Considerations for Taking a Position by Number of Respondents Saying

“Yes”