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.

Подпись: Var(H (в)) = Подпись: dH (в) дв VARIANCE OF A NONLINEAR FUNCTION OF PARAMETERS

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

Подпись: O2 = Подпись: д exp (L в) dL 'в Подпись: oL, в =[exp(Lв)] О,в■

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:

Подпись: m =

VARIANCE OF A NONLINEAR FUNCTION OF PARAMETERS

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”

Consideration

Gender of Respondent Male

Female

Pay

90

88

Benefits

65

62

Promotion opportunities

101

91

Start-up package

131

117

Funding opportunities

96

100

Family-related reasons

120

168

Job location

156

176

Collegiality

170

209

Reputation of department or university

184

224

Quality of research facilities

152

155

Access to research facilities

130

134

Opportunities for research collaboration

179

216

Desire to build or lead a new program or area of research

165

152

This was the only offer I received

52

48

NOTE: There were a total of 612 males and 666 females that responded in each category.