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. Author manuscript; available in PMC: 2023 Nov 1.
Published in final edited form as: Epidemiology. 2022 Oct 5;33(6):843–853. doi: 10.1097/EDE.0000000000001534

Table 2.

Average predicted values of Y at X percentiles for each combination of X generation method (XG) and Y data generation method (YG), estimated with four linear regression models (n=500; iteration=1000)

XG Analytic model X percentile YG
YGAdd
 YGMult
E(Y) Mean %bias Coverage Average ll E(Y) Mean %bias Coverage Average ll
XG LogN GLM with untransformed X 25th 0.27 −0.24 0.95 −708 −0.04 −97.63 0.89 −712
50th 0.46 −0.23 0.94 −708 0.13 −48.16 0.73 −712
75th 0.78 −0.22 0.93 −708 0.29 −37.88 0.36 −712
RLM with untransformed X 25th 0.27 −0.10 0.94 −708 −0.04 −95.50 0.88 −712
50th 0.46 −0.12 0.93 −708 0.13 −48.03 0.72 −712
75th 0.78 −0.13 0.93 −708 0.29 −37.27 0.38 −712
GLM with ln(X) 25th 0.27 8.07 0.92 −732 −0.04 2.97 0.95 −708
50th 0.46 37.34 0.05 −732 0.13 −1.11 0.94 −708
75th 0.78 23.41 0.13 −732 0.29 −0.60 0.94 −708
RLM with ln(X) 25th 0.27 11.62 0.89 −732 −0.04 1.73 0.95 −708
50th 0.46 35.11 0.08 −732 0.13 −0.64 0.93 −708
75th 0.78 19.59 0.25 −732 0.29 −0.34 0.93 −708
XG TrncN GLM with untransformed X 25th 0.27 0.19 0.95 −709 −0.04 180.48 0.80 −715
50th 0.46 0.03 0.94 −709 0.13 −92.30 0.33 −715
75th 0.78 −0.06 0.94 −709 0.29 −30.89 0.53 −715
RLM with untransformed X 25th 0.27 0.37 0.94 −709 −0.04 174.25 0.80 −715
50th 0.46 0.14 0.93 −709 0.13 −90.83 0.34 −715
75th 0.78 0.00 0.93 −709 0.29 −30.48 0.53 −715
GLM with ln(X) 25th 0.27 67.82 0.06 −723 −0.04 −2.23 0.94 −709
50th 0.46 41.79 0.02 −723 0.13 −0.24 0.94 −709
75th 0.78 8.28 0.76 −723 0.29 −0.49 0.94 −709
RLM with ln(X) 25th 0.27 66.90 0.08 −723 −0.04 −3.54 0.94 −709
50th 0.46 41.37 0.03 −723 0.13 0.13 0.93 −709
75th 0.78 8.10 0.75 −723 0.29 −0.33 0.94 −709
XG Mixed GLM with untransformed X 25th 0.27 0.08 0.93 −708 −0.04 −62.78 0.92 −719
50th 0.46 0.02 0.93 −708 0.13 −63.97 0.64 −719
75th 0.78 −0.01 0.94 −708 0.29 −49.23 0.14 −719
RLM with untransformed X 25th 0.27 0.23 0.93 −709 −0.04 −69.30 0.91 −719
50th 0.46 0.12 0.92 −709 0.13 −62.15 0.66 −719
75th 0.78 0.05 0.93 −709 0.29 −48.45 0.15 −719
GLM with ln(X) 25th 0.27 68.84 0.15 −772 −0.04 −3.04 0.94 −709
50th 0.46 68.98 0 −772 0.13 0.09 0.94 −709
75th 0.78 39.65 0 −772 0.29 −0.30 0.94 −709
RLM with ln(X) 25th 0.27 66.25 0.17 −773 −0.04 −4.34 0.93 −709
50th 0.46 62.94 0 −773 0.13 0.46 0.93 −709
75th 0.78 33.48 0 −773 0.29 −0.14 0.94 −709
XG MErr GLM with untransformed X 25th 0.27 30.28 0.67 −720 −0.04 49.34 0.90 −718
50th 0.46 6.07 0.89 −720 0.13 −79.00 0.47 −718
75th 0.78 −8.05 0.71 −720 0.29 −43.12 0.27 −718
RLM with untransformed X 25th 0.27 19.38 0.79 −722 −0.04 59.58 0.89 −719
50th 0.46 3.54 0.92 −722 0.13 −79.33 0.46 −719
75th 0.78 −5.70 0.80 −722 0.29 −41.33 0.30 −719
GLM with ln(X) 25th 0.27 69.27 0.06 −724 −0.04 −11.66 0.93 −709
50th 0.46 40.71 0.03 −724 0.13 −3.04 0.94 −709
75th 0.78 6.53 0.83 −724 0.29 −4.11 0.94 −709
RLM with ln(X) 25th 0.27 68.27 0.07 −724 −0.04 −11.67 0.93 −709
50th 0.46 40.27 0.03 −724 0.13 −2.84 0.94 −709
75th 0.78 6.36 0.83 −724 0.29 −3.94 0.94 −709

Abbreviations: Coverage, 95% confidence interval coverage; ll, log-likelihood.

a

XGLogN: ln(X) ∼ N(μ=0.42, σ=0.8)

b

XGTruncN: X ∼ TN (a=0.01, μ=0.42, σ=2.4)

c

XGMixed: 80% of X ∼ TN (a=0.01, μ=0.42, σ=2.4), 20% of X ∼ TN (a=0.01, μ=0.63, σ=7.2)

d

XGMErr; 95% of X ∼ XTruth, 5% of X ∼ XTruth*MErr, where XTruth ∼TN(a=0.01, μ=0.42, σ=2.4) and MErr ∼ TN(a=0.01, μ=1, σ=2).

e

YGAdd; Y = 0.3X + N(0,1).

f

YGMult: Y = 0.3ln(X) + N(0,1).