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Table 2 Input and estimated parameters in Approach II

From: Impact of correlation of predictors on discrimination of risk models in development and external populations

Population Input parameters for cases and controls Estimated parameters for the population
ρ Normal (μ, σ) ρ (μ, σ) Unadjusted OR * Adjusted OR ** SD of β 0+ ∑β i X i AUC
A Cases = 0.2 Ctrls = 0.2 μCases: (1, 2); Ctrls: (0, 0) \( \sigma \)Cases: (2, 2); Ctrls: (2, 2) 0.25 μ: (0.2, 0.4); \( \sigma \): (2.04, 2.15) 1.28, 1.65 1.17, 1.60 1.13 0.770
B Cases = 0.2 Ctrls = 0.4 ,, 0.40 ,, ,, 1.09, 1.60 1.09 0.765
C Cases = 0.2 Ctrls = - 0.2 ,, -0.04 ,, ,, 1.34, 1.68 1.25 0.785
D Cases = 0.1 Ctrls = 0.1 ,, 0.16 ,, ,, 1.22, 1.62 1.17 0.777
E Cases = - 0.1 Ctrls = - 0.1 ,, -0.02 ,, ,, 1.35, 1.70 1.28 0.795
F Cases = 0.2 Ctrls = 0.2 μCases: (1, 3); Ctrls: (0, 0) SD Cases: (2, 2); Ctrls: (2, 2) 0.27 μ: (0.2, 0.6); \( \sigma \): (2.04, 2.33) 1.28, 2.12 1.11, 2.07 1.77 0.858
G ,, μCases: (1, 3); Ctrls: (0, 2) SD Cases: (2, 2); Ctrls: (2, 2) 0.23 μ: (0.2, 0.2); \( \sigma \): (2.04, 2.04) 1.28, 1.28 1.23, 1.23 0.67 0.676
H ,, μCases: (1, 2); Ctrls: (0, 0) SD Cases: (2, 3); Ctrls: (2,3) 0.24 μ: (0.2, 0.4); \( \sigma \): (2.04, 3.10) 1.28, 1.25 1.21, 1.22 0.80 0.705
I ,, μCases: (1, 2); Ctrls: (0, 0) SD Cases: (2, 1); Ctrls: (2, 1) 0.27 μ: (0.2, 0.4); \( \sigma \): (2.04, 1.28) 1.28, 7.39 1.05, 7.23 2.56 0.922
  1. In each population, a disease prevalence of 20% was used
  2. Population ‘A’ is considered as reference population and all other populations are compared w.r.t. ‘A’
  3. SD: Standard Deviation; OR: Odds Ratio; Ctrls: controls
  4. ρ: Pearson correlation between two continuous predictors
  5. A risk factor X ~ Normal (μ, σ) implies ‘X’ follows a normal distribution with mean μ and variance \( \sigma \) 2
  6. *when a risk factor is normally distributed in both cases and controls and sigma is the common variance of the risk factor in both cases and controls, then unadjusted OR = exp((μ Case –μ Control )/SD2) [19]
  7. **adjusted ORs estimated by fitting logistic model