Three-Step Latent Class Analysis with Inverse Propensity Weighting in the Presence of Differential Item Functioning

Figures & data

Figure 1. Latent class model with confounded treatment effect.

Figure 2. Latent class model with confounded treatment effect where both treatment and the confounders have direct effects on some observed indicators.

Table 1. Bias of the estimate of the ATE.

DIF variable	Z				C1
${\gamma }_Z$	1		2		1		2
${\beta }_1$	1	2	1	2	1	2	1	2
Regression adjusted	0.069	0.068	0.049	0.050	–0.002	–0.003	–0.003	–0.004
One-step IPW	0.073	0.073	0.049	0.049	–0.001	–0.001	–0.002	–0.002
Three-step IPW	0.066	0.065	0.054	0.054	–0.001	–0.001	–0.003	–0.002
Regression adjusted with DIF	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
One-step IPW with DIF	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Three-step IPW with DIF	0.000	0.000	0.000	0.000	0.000	0.000	–0.001	–0.001

Note. These results are based on the analysis of synthetic data sets that were generated to be exactly in agreement with the population values. The data sets resemble a sample size of 1,000.

Figure 3. Overlap of the propensity scores.

Table 2. Profiles of the two identified classes.

	Lanza et al.		Three-step
	Class 1	Class 2	Class 1	Class 2
	0.70	0.30	0.68	0.32
Alcohol
No use	0.57	0.18	0.55	0.05
Light use	0.29	0.35	0.38	0.33
Binge	0.14	0.47	0.07	0.62
Cigarette
No use	0.91	0.47	0.82	0.53
Use	0.09	0.53	0.18	0.47
Marijuana
No use	0.99	0.83	0.99	0.79
Use	0.01	0.17	0.01	0.21
Crack/Cocaine
No use	1.00	0.94	1.00	0.95
Use	0.00	0.06	0.00	0.05

Note. These parameter estimates result from estimating the first step, i.e., the measurement model without the inclusion of covariates. Note that the first step of the three-step analysis includes direct effects between college-alcohol use and college-cigarette use, while the analysis by Lanza et al. (2013) does not.

Table 3. Balance of the college enrolled vs. not enrolled group before and after IPW.

	No weights				Inverse propensity weights
	Not enrolled	Enrolled	p-value	SMD	Not enrolled	Enrolled	p-value	SMD
N	651	441			1086.3	1119.9
Age (mean, SD)	17.7 (0.8)	17.4 (0.6)	<0.001	0.415	17.6 (0.7)	17.6 (0.8)	0.578	0.078
Gender (female)	312 (47.9%)	247 (56.0%)	0.009	0.162	545.8 (50.2%)	543.8 (48.6%)	0.746	0.034
Race			0.048	0.154			0.623	0.101
White	413 (63.4%)	310 (70.3%)			677.3 (62.4%)	642.7 (57.4%)
Black	210 (32.3%)	119 (27.0%)			365.8 (33.7%)	427.8 (38.2%)
Other	28 (4.3%)	12 (2.7%)			43.1 (4.0%)	49.4 (4.4%)
Household income (mean, SD)	16055 (11323)	22855 (15214)	<0.001	0.507	17661 (11868)	18116 (13185)	0.695	0.036
Nr. siblings			<0.001	0.416			0.948	0.222
0	19 (2.9%)	12 (2.7%)			30.0 (2.8%)	24.3 (2.2%)
1	62 (9.5%)	87 (19.7%)			136.6 (12.6%)	165.4 (14.8%)
2	121 (8.6%)	109 (24.7%)			235.6 (21.7%)	230.7 (20.6%)
3	155 (23.8%)	78 (17.7%)			248.9 (22.9%)	253.5 (22.6%)
4	88 (13.5%)	56 (12.7%)			134.0 (12.3%)	96.6 (8.6%)
5	68 (10.4%)	37 (8.4%)			95.5 (8.8%)	115.7 (10.3%)
6 or more	138 (21.2%)	62 (14.0%)			205.9 (18.9%)	233.6 (21.0%)
Language			0.911	0.027			0.186	0.216
English	538 (82.6%)	360 (81.6%)			865.0 (79.6%)	828.2 (74.0%)
Spanish	80 (12.3%)	57 (12.9%)			150.6 (13.9%)	147.9 (13.2%)
Other	33 (5.1%)	24 (5.4%)			70.7 (6.5%)	143.7 (12.8%)
Maternal education (mean, SD)	10.5 (2.8)	11.9 (3.1)	<0.001	0.491	11.1 (2.9)	11.07 (3.34)	0.970	0.004
Education aspiration years (mean, SD)	13.2 (1.7)	15.7 (1.6)	<0.001	1.578	14.1 (2.1)	14.2 (2.0)	0.795	0.029
Education aspiration (yes)	138 (21.2%)	205 (46.5%)	<0.001	0.555	365.0 (33.6%)	440.1 (39.3%)	0.284	0.119
Parent figure (both)	512 (78.6%)	387 (87.8%)	<0.001	0.245	868.9 (80.0%)	841.9 (75.2%)	0.364	0.115
Metropolitan status (urban)	478 (73.4%)	367 (83.2%)	<0.001	0.239	852.1 (78.4%)	924.3 (82.5%)	0.260	0.103
College preparation program			<0.001	0.817			0.924	0.065
Vocational	161 (24.7%)	44 (10.0%)			202.8 (18.7%)	204.3 (18.2%)
Commercial	21 (3.2%)	15 (3.4%)			30.1 (2.8%)	34.6 (3.1%)
College preparatory	122 (18.7%)	239 (54.2%)			358.6 (33.0%)	400.3 (35.7%)
General program	347 (53.3%)	143 (32.4%)			494.7 (45.5%)	480.7 (42.9%)

Note. SMD <0.2 and p-values > 0.05 indicate good balance between the groups.

Table 4. Class membership probabilities estimated in the last step of our newly proposed analysis strategy.

College	Class 1	Class 2
No	0.65	0.35
Yes	0.75	0.25

Note. The ATE can be depicted in an odds ratio of 0.64 compared to an odds ratio of 0.16 reported by Lanza et al. (2013).

Table 5. Direct effects between college and alcohol use.

		Alcohol
College		No use	Light use	Binge
No	Class 1	0.60	0.33	0.08
No	Class 2	0.05	0.28	0.66
Yes	Class 1	0.50	0.44	0.06
Yes	Class 2	0.05	0.40	0.55

Note. These direct effects are estimated without weighting in our proposed analysis strategy.

Table 6. Conditional response probabilities for cigarette use per treatment group.

		Cigarette
College		No use	Use
No	Class 1	0.77	0.23
No	Class 2	0.45	0.55
Yes	Class 1	0.90	0.10
Yes	Class 2	0.68	0.32

Note. These direct effects are estimated without weighting in our proposed analysis strategy.

Table 7. Class membership probabilities estimated with the modified analysis strategy based on Lanza et al. (2013).

College	Class 1	Class 2
No	0.58	0.42
Yes	0.76	0.24

Note. The ATE can be depicted in an odds ratio of 0.44.

Table 8. Conditional response probabilities for cigarette use, marijuana use, and crack/cocaine use estimated using the modified analysis strategy based on Lanza et al. (2013).

		Cigarette
		No use				Use
		Marijuana
		No use		Use		No use		Use
		Crack/Cocaine
College		No use	Use	No use	Use	No use	Use	No use	Use
No	Class 1	0.74	0.00	0.00	0.00	0.25	0.00	0.00	0.00
No	Class 2	0.45	0.01	0.08	0.01	0.36	0.02	0.06	0.02
Yes	Class 1	0.90	0.00	0.00	0.00	0.09	0.00	0.00	0.00
Yes	Class 2	0.66	0.01	0.12	0.01	0.16	0.01	0.03	0.01

Lanza, S. T., Coffman, D. L., & Xu, S. (2013). Causal inference in Latent Class analysis. Structural Equation Modeling, 20, 361–383. https://doi.org/10.1080/10705511.2013.797816

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