Regression models of Pearson correlation coefficient

Figures & data

Table 1. Some functions under two link functions.

	Link 1	Link 2
$h\left(x\right)$	$(1+{\mathrm{e}}^{-\mathrm{x}}{)}^{-1}$	$\tanh \left(x\right)$
$h^{\prime }\left(x\right)$	$h\left(x\right)\{1-h(x\left)\right\}$	$\{1+h(x\left)\right\}\{1-h(x\left)\right\}$
$h^{″}\left(x\right)$	$h\left(x\right)\{1-h(x\left)\right\}\{1-2h(x\left)\right\}$	$-2h\left(x\right)\{1+h(x\left)\right\}\{1-h(x\left)\right\}$
a	$-{\displaystyle \frac{{\rho }^2}{(1-\rho )(1+\rho {)}^2}}$	${\displaystyle \frac{-\rho }{(1-\rho )(1+\rho )}}$
b	${\displaystyle \frac{(1+{\rho }^2)\rho }{(1-\rho )(1+\rho {)}^2}}$	${\displaystyle \frac{(1+{\rho }^2)}{(1-\rho )(1+\rho )}}$
c	${\displaystyle \frac{{\rho }^2}{1+\rho }}$	ρ
u	${\displaystyle \frac{-{\rho }^2(2-\rho +{\rho }^2)}{(1-\rho )(1+\rho {)}^3}}$	${\displaystyle \frac{-(1+{\rho }^2)}{(1-\rho )(1+\rho )}}$
v	$-{\displaystyle \frac{\rho (-1+\rho -5{\rho }^2+{\rho }^3)}{(1-\rho )(1+\rho {)}^3}}$	${\displaystyle \frac{4\rho }{(1-\rho )(1+\rho )}}$
w	${\displaystyle \frac{{\rho }^2(1-\rho )(2+\rho )}{(1+\rho {)}^2}}$	$(1+\rho )(1-\rho )$

Table 2. Specifications of ${\text{β}}_1,\dots ,{\text{β}}_p$ under p = 2, 3 and two link functions.

	$({\text{β}}_1,{\text{β}}_2)$	$({\text{β}}_1,{\text{β}}_2,{\text{β}}_3)$
Link 1	$(1,0)$	$(1,0,0)$
Link 2	$(-1,0)$	$(-1,0,0)$

Table 3. Type-I errors (in %) of the global test (for the model significance) and individual test (for the significance of the covariates) under two types of responses, two link functions and various sample sizes.

			p = 2			p = 3
Response	Link	n	Global	${\text{β}}_1$	${\text{β}}_2$	Global	${\text{β}}_1$	${\text{β}}_2$	${\text{β}}_3$
Normal	1	100	2.5	2.6	2.3	1.6	2.6	2.1	2.4
		200	4.5	5.1	3.9	3.7	4.2	3.0	3.9
		400	5.5	4.3	5.0	4.1	3.8	3.8	5.8
		1000	6.0	4.4	5.4	5.8	5.1	5.3	5.3
	2	100	5.6	4.3	5.2	6.6	5.2	5.3	4.7
		200	5.7	3.9	6.3	7.5	6.0	6.3	5.3
		400	6.1	6.1	5.5	5.5	5.1	3.7	5.1
		1000	5.7	4.6	4.6	4.5	4.0	5.0	4.7
Binary	1	100	2.6	2.5	2.5	4.6	3.5	3.5	4.4
		200	0.1	0.8	1.0	0.1	0.5	0.6	0.4
		400	1.1	2.2	2.1	0.3	1.8	2.9	1.4
		1000	5.5	5.2	6.1	4.4	3.7	5.8	4.7
	2	100	0.6	1.1	1.2	0.5	0.9	0.8	0.7
		200	3.7	3.3	3.9	3.3	4.1	5.0	3.7
		400	4.9	5.1	4.0	4.9	4.8	5.1	6.3
		1000	5.1	4.6	4.6	5.6	5.5	4.7	6.1

Table 4. ERMSE, CR and testing power of β's of models considered in Table 1 with various sample sizes.

			p = 2					p = 3
					Power					Power
Model	Link	n	RMSE	CR(%)	Global	${\text{β}}_1$	${\text{β}}_2$	RMSE	CR(%)	Global	${\text{β}}_1$	${\text{β}}_2$	${\text{β}}_3$
Normal	1	100	1.149	100	15.0	18.7	7.8	1.296	100	14.5	18.6	13.5	13.2
		200	0.848	100	36.2	43.5	6.0	1.039	100	29.6	40.8	7.9	8.0
		400	0.619	100	69.4	75.8	6.3	0.737	100	63.2	77.1	5.4	5.8
		1000	0.373	100	98.6	99.3	4.8	0.447	100	97.4	99.0	4.3	5.1
	2	100	0.583	87.1	69.0	77.4	4.7	0.74	85.1	60.0	68.4	4.6	3.3
		200	0.401	90.5	95.5	97.5	5.4	0.493	89.1	94.0	97.3	4.6	4.8
		400	0.274	93.5	100	100	5.3	0.336	92.3	100	100	5.7	5.1
		1000	0.165	96.0	100	100	5.9	0.207	95.3	100	100	5.1	4.6
Binary	1	100	2.924	100	6.0	7.0	4.7	3.529	100	10.5	10.1	8.7	8.8
		200	1.766	100	2.9	5.7	1.8	2.319	100	2.8	6.3	2.0	1.3
		400	1.066	100	7.0	16.5	1.5	1.337	100	3.1	13.8	1.2	1.2
		1000	0.620	100	52.5	68.8	3.8	0.772	100	38.9	69.3	3.3	2.9
	2	100	0.696	86.2	20.8	40.7	1.6	0.871	83.9	4.7	26.1	0.7	0.6
		200	0.443	89.8	90.4	94.8	3.8	0.556	88.4	80.5	94.4	4.8	4.7
		400	0.308	93	99.7	99.7	4.7	0.367	91.7	99.6	100	4.4	4.6
		1000	0.196	95.5	99.9	99.9	5.1	0.235	94.8	100	100	5.0	5.0

Table 5. Rejection rates (in %) of Wilding et al. (2011)'s method and the proposed method for testing the significance of the covariate.

n	${\text{β}}_1$	Wilding et al.	Proposed
30	0	6.1	4.4
	0.5	10.6	7.2
	1	31.5	13.6
60	0	5.2	4.8
	0.5	17.2	17.1
	1	56.2	53.6
150	0	5.3	6.1
	0.5	43.2	41.8
	1	95.3	95.5

Figure 1. (a) Correlation of PCS and MCS over 20 age groups, (b) correlation of PCS and MCS over 20 age groups with different genders, (c) correlation of PCS and MCS over 20 age groups with different marital status.

Figure 2. (a) Correlation of STROKE and DIABETES over 10 age groups, (b) correlation of STROKE and DIABETES over 10 age groups with different genders, (c) correlation of STROKE and DIABETES over 10 age groups with different marital status.

Wilding, G. E., Cai, X., Hutson, A., & Yu, Z. (2011). A linear model-based test for the heterogeneity of conditional correlations. Journal of Applied Statistics, 38(10), 2355–2366. https://doi.org/10.1080/02664763.2011.559201

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Data availability statement

The data that support the findings of this study are available from the corresponding author upon request.