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

Kundu, Suman; Mazumdar, Madhu; Ferket, Bart

doi:10.1186/s12874-017-0345-1

BMC Medical Research Methodology

Table 1 Input and estimated parameters in Approach I

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

Population	Input parameters			Estimated parameters
Population	ρ	Normal (μ, σ)	Adjusted OR	Cases		Controls		SD of β ₀ + ∑β _i X _i	AUC
				ρ	(μ, σ)	ρ	(μ, σ)
A	0.2	μ: (0, 0); \( \sigma \): (1, 1)	(1.5, 1.5)	0.17	μ: (0.37, 0.35); \( \sigma \): (0.97, 0.97)	0.17	μ: (-0.09, -0.09); \( \sigma \): (0.98, 0.98)	0.61	0.663
B	-0.1	,,	,,	-0.12	μ: (0.27, 0.28); \( \sigma \): (0.98, 0.98)	-0.12	μ: (-0.07, -0.07); \( \sigma \): (0.99, 0.99)	0.54	0.645
C	- 0.2	,,	,,	-0.22	μ: (0.26, 0.24); \( \sigma \): (0.99, 0.99)	-0.22	μ: (-0.06, -0.06) \( \sigma \): (0.99, 0.99)	0.51	0.639
D	0.1	,,	,,	0.07	μ: (0.33, 0.33) \( \sigma \): (0.99, 0.99)	0.07	μ: (-0.08, -0.08) \( \sigma \): (0.99, 0.99)	0.59	0.660
E	0.4	,,	,,	0.37	μ: (0.42, 0.42) \( \sigma \): (0.97, 0.97)	0.37	μ: (-0.10, -0.10) \( \sigma \): (0.98, 0.98)	0.67	0.676
F	0.2	,,	(1.5, 1.2)	0.18	μ: (0.34, 0.20) \( \sigma \): (0.98, 0.98)	0.19	μ: (-0.08, -0.05) \( \sigma \): (0.99, 0.99)	0.47	0.629
G	,,	,,	(1.2, 1.2)	0.19	μ: (0.16, 0.17) \( \sigma \): (1, 1)	0.19	μ: (-0.04, -0.04) \( \sigma \): (1, 1)	0.27	0.575
H	,,	,,	(1.5, 3)	0.10	μ: (0.41, 0.76) \( \sigma \): (0.97, 0.97)	0.14	μ: (-0.10, -0.19) \( \sigma \): (0.98, 0.98)	1.25	0.789
I	,,	,,	(0.8, 0.8)	0.20	μ: (-0.20, -0.20) \( \sigma \): (1, 1)	0.19	μ: (0.05, 0.05) \( \sigma \): (0.99, 0.99)	0.33	0.593
J	-0.1	,,	(1.5, 0.8)	-0.09	μ: (0.37, -0.20); \( \sigma \): (0.99, 0.99)	-0.08	μ: (-0.08, 0.05); \( \sigma \): (0.98, 0.98)	0.49	0.632
K	0.2	,,	,,	0.21	μ: (0.28, -0.11) \( \sigma \): (0.99, 0.99)	0.21	μ: (-0.07, 0.03) \( \sigma \): (0.99, 0.99)	0.42	0.616
L	0.4	,,	,,	0.40	μ: (0.24, -0.05); \( \sigma \): (0.99, 0.99)	0.41	μ: (-0.06, 0.01); \( \sigma \): (0.99, 0.99)	0.37	0.603
M	,,	Mean: (0, 0); SD: (1, 3)	(1.5, 1.5)	0.10	μ: (0.41, 2.47) \( \sigma \): (0.97, 0.97)	0.14	μ: (-0.10, -0.62) \( \sigma \): (0.98, 0.98)	1.37	0.804
N	- 0.2	,,	,,	-0.25	μ: (0.11, 2.25) \( \sigma \): (1, 1)	-0.24	μ: (-0.03, -0.56) \( \sigma \): (1, 1)	1.21	0.781
O	0.1	,,	,,	0.01	μ: (0.34, 2.38) \( \sigma \): (0.98, 0.98)	0.04	μ: (-0.08, -0.59) \( \sigma \): (0.99, 0.99)	1.31	0.795
P	0.4	,,	,,	0.30	μ: (0.56, 2.56) \( \sigma \): (0.94, 0.94)	0.33	μ: (-0.14, -0.63) \( \sigma \): (0.96, 0.96)	1.42	0.810

In each population, a disease prevalence of 20% was used
Population ‘A’ is considered as reference population; all other populations are compared w.r.t ‘A’
SD standard deviation, OR odds ratio
ρ: Pearson correlation between two continuous predictors
A risk factor X ~ Normal (μ, σ) implies ‘X’ follows a normal distribution with mean μ and variance σ ²
In Approach I, the adjusted ORs were pre-specified and thus considered as input parameters
Numbers are rounded to two decimals except for AUC estimates

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