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. 2012 Oct 19;14(2):366–381. doi: 10.1093/biostatistics/kxs037

Table 1.

Mean values of PCF and PNF estimated using observed risks R in a population, assuming that the model is well calibrated; risk estimates in a case–control sampling when the disease prevalence μ is known, and based on observations of (R,Y) in the population

Inline graphic
 Inline graphic
 ARE
p PCF true R CC (R,Y) R CC (R,Y) CC/R (R,Y)/R (R,Y)/CC
α=6.55,β=124.45†
0.10 0.18 0.18 0.18 0.18 0.01 2.67 2.67 386.71 387.23 1.00
0.20 0.32 0.32 0.31 0.31 0.01 4.03 4.03 367.45 368.10 1.00
0.30 0.44 0.44 0.44 0.44 0.01 4.63 4.65 362.90 363.84 1.00
0.40 0.55 0.55 0.55 0.55 0.01 4.70 4.68 369.90 368.69 1.00
α=1,β=19‡
0.10 0.32 0.32 0.32 0.32 0.06 3.80 3.81 59.41 59.64 1.00
0.20 0.51 0.51 0.51 0.51 0.08 4.60 4.61 59.27 59.39 1.00
0.30 0.65 0.65 0.65 0.65 0.07 4.30 4.31 62.52 62.70 1.00
0.40 0.76 0.76 0.76 0.76 0.05 3.53 3.53 68.18 68.19 1.00
α=0.3,β=5.7§
0.10 0.51 0.51 0.51 0.51 0.22 4.23 4.32 19.07 19.48 1.02
0.20 0.73 0.73 0.73 0.73 0.16 3.65 3.71 22.48 22.86 1.02
0.30 0.86 0.86 0.86 0.86 0.08 2.35 2.38 28.20 28.46 1.01
0.40 0.93 0.93 0.93 0.93 0.03 1.28 1.29 38.74 38.98 1.01
Inline graphic Inline graphic ARE
q PNF true R CC (R,Y) R CC (R,Y) CC/R (R,Y)/R (R,Y)/CC
α=6.55,β=124.45†
0.90 0.60 0.60 0.60 0.60 0.07 6.11 6.15 92.04 92.64 1.01
0.80 0.45 0.45 0.45 0.45 0.06 4.52 4.51 73.59 73.40 1.00
0.70 0.34 0.34 0.34 0.34 0.05 3.34 3.34 66.08 66.02 1.00
0.60 0.26 0.26 0.26 0.26 0.04 2.42 2.42 60.15 60.15 1.00
α=1,β=19.0‡
0.90 0.60 0.60 0.60 0.60 0.06 6.11 6.07 95.58 94.83 0.99
0.80 0.45 0.45 0.45 0.45 0.06 4.52 4.54 74.24 74.62 1.01
0.70 0.34 0.34 0.35 0.35 0.05 3.34 3.36 62.44 62.80 1.01
0.60 0.26 0.26 0.26 0.26 0.04 2.42 2.43 59.11 59.46 1.01
α=0.3,β=5.7§
0.90 0.35 0.35 0.38 0.38 0.11 3.58 3.65 32.19 32.81 1.02
0.80 0.25 0.25 0.26 0.26 0.08 1.89 1.93 24.23 24.75 1.02
0.70 0.18 0.18 0.19 0.19 0.05 1.14 1.14 21.79 21.79 1.00
0.60 0.13 0.13 0.14 0.14 0.04 0.71 0.72 20.15 20.33 1.01

Results are based on 500 simulations for each set of parameters (α,β) for the beta distribution and values of q and p. Each simulation has N=10 000 samples with μ=0.05.

AREs are computed as the ratio of the influence function-based variances.ARE=asymptotic relative efficiency, the ratio of the influence functions-based variances CC/R=var(TCC)/var(TR), (R,Y)/R=var(T(R,Y))/var(TR), (R,Y)/CC=var(T(R,Y))/var(TCC), where T=PCF or T=PNF, respectively.

†AUC=0.63.

‡AUC=0.79.

§AUC=0.92.