Bayes factor testing of equality and order constraints on measures of association in social research

Figures & data

Table 1. Examples of possible tests that can be executed using the proposed methodology.

	Example hypothesis test
Precise testing	$H_0:\rho =0$ versus $H_1:\rho \ne 0$
One-sided testing	$H_0:\rho \le 0$ versus $H_1:\rho >0$
Multiple hypothesis testing	$H_0:\rho =0$ versus $H_1:\rho <0$ versus $H_2:\rho >0$
Interval testing	$H_0:\left|\rho \right|\le .1$ versus $H_1:\left|\rho \right|>.1$
Equality-constrained testing	$H_1:{\rho }_{12}={\rho }_{13}={\rho }_{14}$ versus $H_2:`\mathrm{n}\mathrm{o}\mathrm{t}{H}_1$
Order-constrained testing	$H_1:{\rho }_{12}<{\rho }_{13}<{\rho }_{14}$ versus $H_2:{\rho }_{12}>{\rho }_{13}>{\rho }_{14}$ versus $H_3:`\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}{H}_1,nor{H}_2^{\prime }$
Hypotheses with equality and order constraints	$H_1:{\rho }_{12}<{\rho }_{13}={\rho }_{14}$ versus $H_2:`\mathrm{n}\mathrm{o}\mathrm{t}{H}_1$

Table 2. Types of measures of association depending on the measurement scale of the variables.

	Scale of $Y_2$
	Dichotomous	Polychotomous-	Continuous-
Scale $Y_1$		Ordinal categories	Interval
Dichotomous	Tetrachoric	Polychoric	Biserial
Polychotomous-		Polychoric	Polyserial
Ordinal categories
Continuous-			Product-
Interval			Moment

Figure 1. Graphical model describing partial associations between life and domain satisfaction variables.

Table 3. Rough guidelines for interpreting Bayes factors [57].

$B_{12}$	Evidence
$<1/150$	Very strong evidence for $H_2$
1/150 to 1/20	Strong evidence for $H_2$
1/20 to 1/3	Positive evidence for $H_2$
1/3 to 1	Weak evidence for $H_2$
1	No preference
1 to 3	Weak evidence for $H_1$
3 to 20	Positive evidence for $H_1$
20 to 150	Strong evidence for $H_1$
$>150$	Very strong evidence for $H_1$

Figure 2. (a) Graphical representation of the subspace of $({\rho }_{21},{\rho }_{31},{\rho }_{32})$ for which the 3-dimensional correlation matrix is positive definite (taken from [59] with permission). The thick diagonal line from $(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2})$ to $(1,1,1)$ represents the correlations that satisfy ${\rho }_{21}={\rho }_{31}={\rho }_{32}$ and result in a positive diagonal correlation matrix. (b) Uniform prior ${\pi }_1^U$ for the common correlation ρ under $H_1:{\rho }_{21}={\rho }_{31}={\rho }_{32}$ in the allowed region ${\mathcal{C}}_1=\{\rho |\rho \in (-\frac{1}{2},1\left)\right\}$. (c) Uniform prior ${\pi }_2^U$ for the free parameters under $H_2:{\rho }_{31}=0$ in the allowed region ${\mathcal{C}}_2=\left\{\right({\rho }_{21},{\rho }_{32})|{\rho }_{21}^2+{\rho }_{32}^2<1\}$. (d) Uniform prior for the free correlations under $H_3:{\rho }_{31}=0,{\rho }_{21}>{\rho }_{32}$ in the allowed region ${\mathcal{C}}_3=\left\{\right({\rho }_{21},{\rho }_{32})|{\rho }_{21}^2+{\rho }_{32}^2<1,{\rho }_{21}>{\rho }_{32}\}$.

Figure 3. Left panel. The implied $beta(\frac{1}{2},\frac{1}{2})$ prior in the interval $(-1,1)$ (dotted line) in the test proposed by Wetzels and Wagenmakers' [68], and the uniform prior in $(-1,1)$ as proposed here. Right panel. Implied prior for the common correlation ρ under $H_0:{\rho }_{12}={\rho }_{13}={\rho }_{23}$ when testing against $H_1:{\rho }_{12}\ne {\rho }_{13}\ne {\rho }_{23}$ using a marginally uniform encompassing prior (dotted line), and a uniform prior for ρ on $(-\frac{1}{2},1)$ (solid line) as proposed here.

Figure 4. Posterior probabilities of $H_1:{\rho }_{21}={\rho }_{31}={\rho }_{32}$ (solid line), $H_1:{\rho }_{21}>{\rho }_{31}>{\rho }_{32}$ (dotted line), and $H_2:$ not $H_1,H_2$ (dashed line) for different effects ρ and different sample sizes n.

Table 4. Posterior probabilities for the competing hypotheses from Examples 1 and 2.

Application 1: Life, leisure,and relationship satisfaction
Hypothesis test 1	Posterior probabilities
$H_{1a}$: ${\rho }_{g1y2y1}>{\rho }_{g1y3y1}>{\rho }_{g1y3y2}$	0.0020
$H_{1b}$:${\rho }_{g1y3y1}>{\rho }_{g1y2y1}>{\rho }_{g1y3y2}$	0.0085
$H_{1c}$:${\rho }_{g1y3y1}={\rho }_{g1y2y1}={\rho }_{g1y3y2}$	0.9879
$H_{1d}$: neither $H_{1a}$, $H_{1b}$, nor $H_{1c}$	0.0016
Hypothesis test 2
$H_{2a}$: ${\rho }_{g2y2y1}>{\rho }_{g2y3y1}>{\rho }_{g2y3y2}$	0.0001
$H_{2b}$: ${\rho }_{g2y3y1}>{\rho }_{g2y2y1}>{\rho }_{g2y3y2}$	0.2955
$H_{2c}$: ${\rho }_{g2y3y1}={\rho }_{g2y2y1}={\rho }_{g2y3y2}$	0.2104
$H_{2d}$: neither $H_{2a}$, $H_{2b}$, nor $H_{2c}$	0.4940
Hypothesis test 3
$H_{3a}$: ${\rho }_{g1y2y1}>{\rho }_{g2y2y1}$	0.1588
$H_{3b}$: ${\rho }_{g1y2y1}={\rho }_{g2y2y1}$	0.8278
$H_{3c}$: ${\rho }_{g1y2y1}<{\rho }_{g2y2y1}$	0.0135
Hypothesis test 4
$H_{4a}$: ${\rho }_{g1y3y1}>{\rho }_{g2y3y1}$	0.0809
$H_{4b}$: ${\rho }_{g1y3y1}={\rho }_{g2y3y1}$	0.9017
$H_{4c}$: ${\rho }_{g1y3y1}<{\rho }_{g2y3y1}$	0.0174
Application 2: Support for justice principles
Hypothesis test 5
$H_{5a}$: ${\rho }_{g1y2y1}>{\rho }_{g2y2y1}>{\rho }_{g3y2y1}>{\rho }_{g4y2y1}$	0.0044
$H_{5b}$: ${\rho }_{g1y2y1}={\rho }_{g2y2y1}={\rho }_{g3y2y1}={\rho }_{g4y2y1}$	0.0000
$H_{5c}$: ${\rho }_{g1y2y1}={\rho }_{g2y2y1}>{\rho }_{g3y2y1}={\rho }_{g4y2y1}$	0.9899
$H_{5d}$: neither $H_{5a}$, $H_{5b}$, nor $H_{5c}$	0.0057

Figure A1. Left panel: Trace plot of latent $z_{31}$ observed in category 1 of the 3th (ordinal) outcome variable (blue line), $z_{32}$ observed in category 2 (orange line), $z_{33}$ observed in category 3 (green line), and $z_{34}$ observed in category 4 (purple line), as well as the corresponding threshold parameters ${\gamma }_{31}$, ${\gamma }_{32}$, and ${\gamma }_{34}$ for the 3th outcome variable. Right panel: Scatter plot of posterior draws of $({\eta }_{21},{\eta }_{31})$ (red dots) with additional contour plot and univariate density plots (solid lines) and normal approximations (dashed lines).

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