. Author manuscript; available in PMC: 2017 Jun 15.

Published in final edited form as: Stat Med. 2016 Jan 5;35(13):2133–2148. doi: 10.1002/sim.6856

Table 5.

Log odds ratios for the ATBC Colorectal Cancer Study with Institute of Medicine cut-points and seasonal adjustment to week 39 and to week 1.

Log odds ratio estimates (with standard errors)

Adjusted for season and calibration

No adjustment (Raw)

Season only

Calibration only

Reference week 39

CLR using (10) with x̂₀₁

CLR using (10) with x̂₀₃

Normal model

Q₂

−10.9(78.2)

−1.81*(.684)

−4.53*(1.16)

.138(.237)

−.749*(.276)

−.653*(.302)

Q₃

−10.8(78.2)

−1.54*(.674)

−3.89*(1.05)

−.510(.331)

−.950*(.331)

−.572(.330)

Q₄

−11.1(78.2)

−1.97*(.712)

−4.46*(1.11)

−.603(.671)

−1.20*(.572)

−1.36*(.431)

CAT

−.219(.142)

−.175(.137)

−.308*(.148)

−.177(.133)

−.418*(.144)

−.334*(.126)

{\hat{α}}_{1}^{*} / {\hat{b}}_{1} {\hat{κ}}_{W}

{\hat{α}}_{1}^{*} / {\hat{b}}_{3} {\hat{κ}}_{W}

CONT

−.143* (.061)

−.168* (.070)

−.171* (.070)

−.140* (.061)

Reference week 1

CLR using (10) with x̂₀₁

CLR using (10) with x̂₀₃

Normal model

Q₂

−.583*(.278)

−.300(.249)

−.415(.330)

.138(.237)

.166(.233)

−.653*(.302)

Q₃

−.586(.305)

−.457(.304)

−.399(.342)

−.510(.331)

−.303(.334)

−.573(.330)

Q₄

−1.53*(.505)

−1.25*(.526)

−1.29*(.570)

−.603(.671)

−.223(.846)

−1.36*(.431)

CAT

−.366*(.127)

−.311*(.125)

−.300*(.133)

−.117(.133)

−.074(.139)

−.334*(.126)

{\hat{α}}_{1}^{*} / {\hat{b}}_{1} {\hat{κ}}_{W}

{\hat{α}}_{1}^{*} / {\hat{b}}_{3} {\hat{κ}}_{W}

CONT

−.143* (.061)

−.168* (.070)

−.171* (.070)

−.140* (.061)

Standard errors are in parentheses. Asterisk * indicates P < 0.05. “NORMAL” indicates the normal model procedure in equations (11) and (12). “CLR using (10) with x̂₀₁” and “CLR using (10) with x̂₀₃” denote procedures that substitute those calibrated and seasonally adjusted estimates of x₀ into the conditional logistic likelihood (10). The quantities Q₂, Q₃ and Q₄ denote log odds ratios for categories 2, 3, and 4 compared to category 1. “CAT” indicates the log odds per category increase in a trend model with scores 0,1,2 and 3. “CONT” refers to the continuous trend model with the trend parameter being the log odds per 10 nmol/L increase in vitamin D. Estimates of (τ, a, b, γ₀, γ₁, γ₂, γ₃, γ₄) from the ATBC data were (15.52, 3.86, 1.20, 39.43, −5.84, −4.49, 1.44, −0.444).