文档介绍：Generalized Least Squares for trend estimation
of summarized dose-response data
1st Nordic and Baltic Stata Users Group meeting
September 26, 2005
Nicola Orsini
Institute Environmental Medicine, Karolinska Institutet
Rino o
Dept. Medical Epidemiology and Biostatistics, Karolinska Institutet
Sander Greenland
Dept. Epidemiology, UCLA School of Public Health
1
Outline
• Motivating example - Incidence Rate Data
• The statistical model
• The estimation method
• How to fit the variance-covariance matrix
• Confidence bounds for the covariance estimates
2
Summarized Incidence Rate Data
Larsson ., Bergkvist L., Wolk A., Magnesium Intake in Rela-
tion to Risk of Colorectal Cancer in Women, JAMA, 2005.
. use http://nicolaorsini./1SSM/magnsd, clear
. list , clean
class dose adjirr lb ub cases ptimed
1. <209 198 180 65886168
2. 209-224 218 171 69336072
3. 225-237 232 147 63346672
4. 238-254 246 154 67493232
5. >=255 268 153 66696056
3
log(Adjusted IRR)
−1 −.8 −.6 −.4 −.2 0
198
Data source: Larsson et al 2005, JAMA
Magnesium intake and risk of colorectal cancer
218
Dose (mg/day)
232
246
268
4
Linear regression model (single study)
y = Xβ+ ǫ
where
y is a n × 1 vector of beta coefficients (log odds ratios, log rate
ratios, log risk ratios)
X is a n × p fixed-effects design matrix (no intercept). xi1 is as-
sumed to be the exposure variable, where i = 1, 2, ..., n identifies
non-reference exposure levels
β is a p × 1 vector of unknown coefficients
ǫ is a n × 1 vector of random errors, such that ǫ ∼ N(0, Σ)
5
Generalized Least Squares
Suppose for now that the variance-covariance matrix of the error
Σ is known.
This method involves minimizing (y − Xβ)′Σ−1(y − Xβ) with re-
spect to β.
The resulting estimator β of the regression coefficients β is
b
β= (X′Σ−1X)−1X′Σ−1y
b
and the estimated covariance matrix V of β is
b
V = Cov(β) = (