Counterfactual Mediation Analysis with a Latent Class Exposure

Figures & data

Figure 1. Simulated mediation model; U_p = binary latent class indicators; X = latent class exposure; M = binary manifest mediator; Y = binary manifest outcome.

Table 1. Data used for simulation (sample = 5,000).

Conduct trajectory (X)	Peer deviance (M)	Hazardous alcohol use (Y)			Total
		0: No	1: Yes	% Y = 1
1: EOP	0: No	150	120	44.4%	270
2: AO		200	125	38.5%	325
3: CL		295	180	37.9%	475
4: Low		1950	980	33.4%	2930
1: EOP	1: Yes	50	80	61.5%	130
2: AO		75	100	57.1%	175
3: CL		65	60	48.0%	125
4: Low		325	245	43.0%	570

EOP: Early-Onset Persistent; AO: Adolescent Onset; CL: Childhood Limited.

Table 2. Values for direct and indirect effects based on simulated data in Table 1.

	Estimate (log RR)	RR
EOP vs Low
Total Effect (TE)	0.3567	1.4286
Total Natural Indirect Effect (TNIE)	0.0570	1.0587
Pure Natural Direct effect (PNDE)	0.2996	1.3494
AO vs Low
Total Effect (TE)	0.2513	1.2857
Total Natural Indirect Effect (TNIE)	0.0809	1.0842
Pure Natural Direct effect (PNDE)	0.1704	1.1858
CL vs Low
Total Effect (TE)	0.1335	1.1429
Total Natural Indirect Effect (TNIE)	0.0116	1.0116
Pure Natural Direct effect (PNDE)	0.1220	1.1297

EOP: Early-Onset Persistent; AO: Adolescent Onset; CL: Childhood Limited; RR: Risk Ratio.

Table 3. Performance measures (Monte Carlo standard errors) for each method and data generating mechanism for the total effect (TE) of Early-Onset Persistent versus Low conduct problems (true value = 0.357); N = 457 simulated datasets with good entropy (0.9), N = 434 simulated datasets with medium entropy (0.8), N = 424 simulated datasets with poor entropy (0.7).

Performance measure (for θ)	Data-generating mechanism	One-step	Bias-adjusted three-step (BCH)	Modal class assignment	Non-inclusive pseudo class draws	Inclusive pseudo class draws	Updated pseudo class draws
Bias; % bias	Good entropy	0.002 (0.003); 0.6%	−0.002 (0.003); 0.6%	−0.025 (0.003); 7.1%	−0.029 (0.002); 8.0%	0.002 (0.003); 0.7%	−0.002 (0.003); 0.6%
	Medium entropy	0.004 (0.003); 1.1%	−0.005 (0.003); 1.5%	−0.048 (0.003); 13.6%	−0.063 (0.002); 17.5%	0.005 (0.003); 1.3%	−0.003 (0.003); 0.8%
	Poor entropy	0.002 (0.004); 0.4%	−0.009 (0.004); 2.5%	−0.075 (0.003); 21.0%	−0.102 (0.003); 28.5%	0.003 (0.004); 0.8%	−0.009 (0.004); 2.4%
Coverage	Good entropy	95.8% (0.009)	96.1% (0.009)	93.7% (0.011)	96.7% (0.008)	95.4% (0.010)	96.1% (0.009)
	Medium entropy	95.9% (0.010)	95.9% (0.010)	87.1% (0.016)	92.6% (0.013)	94.7% (0.011)	95.9% (0.010)
	Poor entropy	95.3% (0.010)	95.0% (0.011)	73.1% (0.022)	73.3% (0.021)	89.6% (0.015)	94.6% (0.011)
Bias-eliminated coverage	Good entropy	95.8% (0.009)	95.6% (0.010)	94.5% (0.011)	97.4% (0.007)	95.4% (0.010)	96.1% (0.009)
	Medium entropy	95.6% (0.010)	95.6% (0.010)	95.2% (0.010)	98.4% (0.006)	94.2% (0.011)	95.6% (0.010)
	Poor entropy	95.5% (0.010)	95.3% (0.010)	94.6% (0.011)	99.3% (0.004)	89.9% (0.015)	94.8% (0.011)
Empirical standard error	Good entropy	0.058 (0.002)	0.060 (0.002)	0.055 (0.002)	0.051 (0.002)	0.058 (0.002)	0.057 (0.002)
	Medium entropy	0.063 (0.002)	0.066 (0.002)	0.056 (0.002)	0.049 (0.002)	0.063 (0.002)	0.062 (0.002)
	Poor entropy	0.078 (0.003)	0.080 (0.003)	0.062 (0.002)	0.052 (0.002)	0.080 (0.003)	0.073 (0.003)
Average model standard error	Good entropy	0.060 (0.0002)	0.060 (0.0002)	0.055 (0.0002)	0.059 (0.0001)	0.058 (0.0002)	0.059 (0.0002)
	Medium entropy	0.067 (0.0003)	0.067 (0.0003)	0.057 (0.0002)	0.064 (0.0002)	0.062 (0.0002)	0.065 (0.0002)
	Poor entropy	0.085 (0.001)	0.077 (0.0005)	0.059 (0.0003)	0.070 (0.0004)	0.066 (0.0004)	0.081 (0.001)

Table 4. Performance measures (Monte Carlo standard errors) for each method and data generating mechanism for the total natural indirect effect (TNIE) of Early-Onset Persistent versus Low conduct problems (true value = 0.057); N = 457 simulated datasets with good entropy (0.9), N = 434 simulated datasets with medium entropy (0.8), N = 424 simulated datasets with poor entropy (0.7).

Performance measure (for θ)	Data-generating mechanism	One-step	Bias-adjusted three-step (BCH)	Modal class assignment	Non-inclusive pseudo class draws	Inclusive pseudo class draws	Updated pseudo class draws
Bias; % bias	Good entropy	0.002 (0.001); 2.8%	0.001 (0.001); 2.5%	−0.001 (0.001); 2.0%	−0.002 (0.001); 3.1%	0.0001 (0.001); 0.2%	0.0003 (0.001); 0.6%
	Medium entropy	0.002 (0.001); 3.0%	0.001 (0.001); 2.1%	−0.004 (0.001); 7.3%	−0.006 (0.001); 11.0%	−0.001 (0.001); 2.4%	−0.0003 (0.001); 0.5%
	Poor entropy	0.002 (0.001); 3.1%	0.002 (0.001); 2.9%	−0.007 (0.001); 13.1%	−0.012 (0.001); 21.0%	−0.003 (0.001); 5.3%	−0.001 (0.001); 0.9%
Coverage	Good entropy	95.2% (0.010)	95.0% (0.010)	92.8% (0.012)	95.6% (0.010)	96.1% (0.009)	94.7% (0.010)
	Medium entropy	95.4% (0.010)	95.6% (0.010)	92.9% (0.012)	95.9% (0.010)	96.3% (0.009)	96.3% (0.009)
	Poor entropy	95.5% (0.010)	92.9% (0.012)	88.0% (0.016)	92.0% (0.013)	95.5% (0.010)	95.5% (0.010)
Bias-eliminated coverage	Good entropy	95.2% (0.010)	94.5% (0.011)	94.3% (0.011)	96.4% (0.009)	96.1% (0.009)	94.7% (0.010)
	Medium entropy	95.2% (0.010)	95.2% (0.010)	95.4% (0.010)	97.7% (0.007)	96.5% (0.009)	96.3% (0.009)
	Poor entropy	94.3% (0.011)	92.2% (0.013)	90.8% (0.014)	98.1% (0.007)	97.4% (0.008)	95.8% (0.010)
Empirical standard error	Good entropy	0.022 (0.001)	0.023 (0.001)	0.020 (0.001)	0.019 (0.001)	0.020 (0.001)	0.022 (0.001)
	Medium entropy	0.023 (0.001)	0.024 (0.001)	0.018 (0.001)	0.016 (0.001)	0.018 (0.001)	0.023 (0.001)
	Poor entropy	0.030 (0.001)	0.031 (0.001)	0.020 (0.001)	0.015 (0.001)	0.020 (0.001)	0.027 (0.001)
Average model standard error	Good entropy	0.022 (0.0002)	0.022 (0.0002)	0.020 (0.0002)	0.021 (0.0001)	0.022 (0.0002)	0.022 (0.0002)
	Medium entropy	0.026 (0.0002)	0.026 (0.0002)	0.019 (0.0002)	0.021 (0.0002)	0.023 (0.0002)	0.025 (0.0002)
	Poor entropy	0.033 (0.001)	0.031 (0.0004)	0.020 (0.0003)	0.021 (0.0002)	0.025 (0.0003)	0.032 (0.001)

Table 5. Performance measures (Monte Carlo standard errors) for each method and data generating mechanism for pure natural direct effect (PNDE) of Early-Onset Persistent versus Low conduct problems (true value = 0.300); N = 457 simulated datasets with good entropy (0.9), N = 434 simulated datasets with medium entropy (0.8), N = 424 simulated datasets with poor entropy (0.7).

Performance measure (for θ)	Data-generating mechanism	One-step	Bias-adjusted three-step (BCH)	Modal class assignment	Non-inclusive pseudo class draws	Inclusive pseudo class draws	Updated pseudo class draws
Bias; % bias	Good entropy	0.001 (0.003); 0.2%	−0.004 (0.003); 1.2%	−0.024 (0.003); 8.0%	−0.027 (0.003); 8.9%	0.002 (0.003); 0.8%	−0.002 (0.003); 0.8%
	Medium entropy	0.002 (0.003); 0.7%	−0.006 (0.003); 2.2%	−0.044 (0.003); 14.8%	−0.056 (0.002); 18.8%	0.006 (0.003); 2.0%	−0.002 (0.003); 0.8%
	Poor entropy	−0.0002 (0.004); 0.1%	−0.011 (0.004); 3.5%	−0.067 (0.003); 22.5%	−0.090 (0.003); 29.9%	0.006 (0.004); 2.0%	−0.008 (0.004); 2.7%
Coverage	Good entropy	95.4% (0.010)	93.4% (0.012)	94.% (0.011)	96.3% (0.009)	94.3% (0.011)	96.3% (0.009)
	Medium entropy	96.1% (0.009)	95.9% (0.010)	91.2% (0.014)	94.5% (0.011)	94.9% (0.011)	96.3% (0.009)
	Poor entropy	95.8% (0.010)	95.3% (0.010)	80.9% (0.019)	84.9% (0.017)	90.8% (0.014)	95.9% (0.010)
Bias-eliminated coverage	Good entropy	95.4% (0.010)	93.4% (0.012)	93.7% (0.011)	97.4% (0.007)	94.7% (0.010)	96.1% (0.009)
	Medium entropy	96.3% (0.009)	95.9% (0.010)	96.1% (0.009)	98.6% (0.006)	95.6% (0.010)	96.8% (0.008)
	Poor entropy	95.8% (0.010)	95.3% (0.010)	94.3% (0.011)	99.1% (0.005)	91.5% (0.014)	95.9% (0.010)
Empirical standard error	Good entropy	0.063 (0.002)	0.066 (0.002)	0.059 (0.002)	0.054 (0.002)	0.062 (0.002)	0.062 (0.002)
	Medium entropy	0.069 (0.002)	0.071 (0.002)	0.059 (0.002)	0.051 (0.002)	0.067 (0.003)	0.067 (0.002)
	Poor entropy	0.084 (0.003)	0.087 (0.003)	0.064 (0.002)	0.053 (0.002)	0.082 (0.003)	0.078 (0.003)
Average model standard error	Good entropy	0.066 (0.0002)	0.066 (0.0002)	0.060 (0.0002)	0.064 (0.0002)	0.064 (0.0002)	0.065 (0.0002)
	Medium entropy	0.074 (0.0003)	0.074 (0.0003)	0.061 (0.0003)	0.069 (0.0002)	0.068 (0.0003)	0.072 (0.0003)
	Poor entropy	0.093 (0.001)	0.086 (0.001)	0.063 (0.0004)	0.074 (0.0004)	0.073 (0.0005)	0.089 (0.001)

Figure 3. (a) Total effects (TE) for the applied example by method and class comparison; (b) Total natural indirect effects (TNIE) for the applied example by method and class comparison; (c) Pure natural direct effects (PNDE) for the applied example by method and class comparison; Effect estimates shown are log-risk ratios and 95% confidence intervals for each latent class (Early-Onset Persistent, Adolescent Onset, and Childhood Limited) versus the Low class, N = 3,039; Methods = one-step, bias-adjusted threestep (bch), modal class assignment (modal), non-inclusive PCDs (npcd), inclusive PCDs (incpcd), and updated PCDs (upcd); latent class comparisons = Early-Onset Persistent (eop) versus Low, Adolescent Onset (ao) versus Low; Childhood Limited (cl) versus Low.

Data availability statement

The Mplus input files to create the simulated data that support the findings of this study are openly available in Github at https://github.com/gemmahammerton/latentclass-mediation. Access to ALSPAC data (used in the applied example) is through a system of managed open access (http://www.bristol.ac.uk/alspac/researchers/access/).