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Table 3 Fit statistics for the formative indicator models

From: Response shift in health-related quality of life measures in the presence of formative indicators

Model χ2 df p-value RMSEA CFI Δχ2 Δdf p-value
Step 1: invariance of reflective indicators (OH, QoL, FC) used for identification Purpose
Baseline 2.59 3 0.458 0 1
Constrained modela 16.06 11 0.139 0.04 0.99 13.13 8 0.108
Step 2: invariance of items used to create the two domain-level formative indicators (PA, FA)
Baseline 30.30 24 0.175 0.03 0.99
Constrained modelb 74.14 35  < 0.001 0.07 0.97 44.56 11  < 0.001
Constrained modelc 43.26 33 0.109 0.03 0.99 13.01 9 0.162
Step 3: RS detection—invariance in the formative indicator model
Baseline 51.59 30 0.01 0.05 0.98
Constrained modeld 54.80 33 0.01 0.05 0.98 3.59 3 0.310
  1. Step 1–step 3 are illustrated in Fig. 3
  2. OH, overall health; QoL, overall quality of life; FC, functioning scales; PA, pain; FA, fatigue; RMSEA, root mean square approximation; CFI, comparative fit index
  3. aLoadings, intercepts and residual variances of the three indicators, OH, QoL and FC, are constrained to be equal across the two time points; the mean of the latent variable is set to zero at t = 1 and freely estimated at t = 2
  4. bLoadings, intercepts and residual variances of the five items measuring the latent variables PA and FA are constrained to be equal across the two time points; the means of the latent variables are set to zero at t = 1 and freely estimated at t = 2
  5. cAfter removing two error variance equality constraints
  6. dPA and FA regression coefficients and the unexplained variance of the latent variable is constrained to be equal across the two time points and the model intercept is set to zero at t = 1 and freely estimated at t = 2