Prior Predictive Checks for the Method of Covariances in Bayesian Mediation Analysis

Figures & data

Figure 1. Single mediator model depicting the indirect effect of positive interdependence (X) on learning achievement (Y) through elaboration (M). Residual variances are omitted from the figure but estimated in the model

Table 1. PrPC results for realistic ranges of M and Y are in the study by Hooper and Hannafin (1991)

	Proportion of lower percentiles below lower limit		Proportion of upper percentiles above upper limit
	M	Y	M	Y
IW	<.001	<.001	<.001	<.001
HC & LKJ	.216	.098	.058	.224
HC & TN	.217	.102	.057	.231
IG & LKJ	<.001	<.001	<.001	<.001
IG & TN	<.001	<.001	<.001	<.001
LN & LKJ	.144	.038	.008	.176
LN & TN	.139	.038	.007	.176

Italicized values indicate the lowest value in the column. IW denotes an inverse-Wishart prior, HC denotes a half-Cauchy prior, LKJ denotes the Lewandowski et al. (2009) prior, TN denotes a truncated normal prior, IG denotes an inverse-gamma prior, and LN denotes a lognormal prior.

Table 2. PrPC results for the mean difference for Hooper and Hannafin (1991)

	Proportion of data sets where generated means are significantly lower than the observed mean		Proportion of data sets where generated means are significantly higher than the observed mean		Absolute Difference between Proportion lower and higher
	M	Y	M	Y	M	Y
IW	0.149	0.144	0.143	0.143	.006	.001
HC & LKJ	0.132	0.146	0.139	0.150	.007	.004
HC & TN	0.137	0.142	0.137	0.142	<.001	<.001
IG & LKJ	0.152	0.148	0.146	0.146	.006	.002
IG & TN	0.150	0.151	0.153	0.144	.003	.007
LN & LKJ	0.143	0.144	0.141	0.133	.002	.011
LN & TN	0.144	0.140	0.141	0.134	.003	.006

Italicized values indicate the lowest value in the column. IW denotes an inverse-Wishart prior, HC denotes a half-Cauchy prior, LKJ denotes the Lewandowski et al. (2009) prior, TN denotes a truncated normal prior, IG denotes an inverse-gamma prior, and LN denotes a lognormal prior.

Table 3. PrPC results for the variance difference for Hooper and Hannafin (1991)

	Proportion of generated data sets with significantly lower variance than the observed variance			Proportion of generated data sets with significantly higher variance than the observed variance
	X	M	Y	X	M	Y
IW	.042	.061	.061	.060	.044	.042
HC & LKJ	.050	.451	.459	.051	.454	.449
HC & TN	.048	.458	.448	.045	.450	.458
IG & LKJ	.053	.054	.046	.052	.043	.049
IG & TN	.054	.055	.052	.049	.046	.048
LN & LKJ	.048	.453	.437	.050	.436	.444
LN & TN	.048	.439	.437	.050	.439	.448

Bolded values indicate high proportions. Italicized values indicate the lowest proportion in the column. IW denotes an inverse-Wishart prior, HC denotes a half-Cauchy prior, LKJ denotes the Lewandowski et al. (2009) prior, TN denotes a truncated normal prior, IG denotes an inverse-gamma prior, and LN denotes a lognormal prior.

Figure 2. 95% HPD intervals for the indirect effects when specifying the different priors based on Hooper & Hannafin (1991). The dotted line indicates the indirect effect estimate of 0. IW denotes an inverse-Wishart prior, tN denotes a truncated normal prior, lN denotes a log normal prior, IG denotes an inverse-gamma prior, and hC denotes a half-Cauchy prior

Table 4. PrPC results for a realistic range for Chang and Hwang (2017)

	Proportion of lower percentiles below lower limit		Proportion of upper percentiles above upper limit
	M	Y	M	Y
HC & LKJ	.220	.152	.041	.309
HC & TN	.222	.140	.040	.299
IG & LKJ	<.001	<.001	<.001	.002
IG & TN	<.001	<.001	<.001	<.001
LN & LKJ	.157	.076	.002	.258
LN & TN	.168	.084	.003	.257

Italicized values indicate the lowest value in the column. HC denotes a half-Cauchy prior, LKJ denotes the Lewandowski et al. (2009) prior, TN denotes a truncated normal prior, IG denotes an inverse-gamma prior, and LN denotes a lognormal prior.

Table 5. PrPC results for the mean difference for Chang and Hwang (2017)

	Proportion of data sets where generated means are significantly lower than the observed mean		Proportion of data sets where generated means are significantly higher than the observed mean		Absolute Difference between Proportion lower and higher
	M	Y	M	Y	M	Y
HC & LKJ	.050	.050	.050	.046	<.001	.004
HC & TN	.046	.047	.050	.048	.004	.001
IG & LKJ	.048	.050	.047	.049	.001	.001
IG & TN	.052	.049	.050	.049	.002	<.001
LN & LKJ	.049	.050	.051	.048	.002	.002
LN & TN	.048	.048	.050	.048	.002	<.001

Italicized values indicate the lowest value in the column. HC denotes a half-Cauchy prior, LKJ denotes the Lewandowski et al. (2009) prior, TN denotes a truncated normal prior, IG denotes an inverse-gamma prior, and LN denotes a lognormal prior.

Table 6. PrPC results for variance difference for Chang and Hwang (2017)

	Proportion of generated data sets with significantly lower variance than the observed variance			Proportion of generated data sets with significantly higher variance than the observed variance
	X	M	Y	X	M	Y
HC & LKJ	.048	.440	.436	.050	.425	.437
HC & TN	.050	.435	.440	.047	.436	.430
IG & LKJ	.047	.048	.051	.050	.049	.045
IG & TN	.051	.055	.053	.050	.046	.046
LN & LKJ	.051	.421	.419	.050	.411	.419
LN & TN	.050	.415	.428	.051	.417	.410

Bolded values indicate high proportions. Italicized values indicate the lowest proportion in the column. HC denotes a half-Cauchy prior, LKJ denotes the Lewandowski et al. (2009) prior, TN denotes a truncated normal prior, IG denotes an inverse-gamma prior, and LN denotes a lognormal prior.

Figure 3. 95% HPD intervals for the indirect effects when specifying the different priors based on Chang & Hwang (2017). The dotted line indicates the indirect effect estimate of 0. IW denotes an inverse-Wishart prior, tN denotes a truncated normal prior, lN denotes a log normal prior, IG denotes an inverse-gamma prior, and hC denotes a half-Cauchy prior

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