Table 1 Linear mixed-effects model outputs for dependent variables ‘erosion depth’ and ‘scratch depth’ versus the interaction of Group and Stage (fixed factors), i.e., Group × Stage, adjusting for repeated measurements over time.

From: Erosion-inhibiting potential of the stannous fluoride-enriched CPP-ACP complex in vitro

Parameter	Erosion depth (nm)						Scratch depth (nm)
	Mixed-effects REML regression outputs						Mixed effects-REML regression outputs
	No of observations = 160; no of groups = 40						No of observations = 160; no of groups = 40
	Log restricted-likelihood = 294.54						Log restricted-likelihood = 287.99
	Wald χ² (16) = 36,522.99; p < 0.001						Wald χ² (16) = 2220.74; p < 0.001

	Coefficient	SE	z-score	p value	95% CI		Coefficient	SE	z-score	p value	95% CI
Group
G0	Reference group: G0						Reference group: G0
G1	− 0.01	0.02	− 0.7	0.477	− 0.05	0.02	0.00	0.02	− 0.1	0.911	− 0.04	0.04
G2	− 0.01	0.02	− 0.4	0.729	− 0.04	0.03	− 0.03	0.03	− 1.1	0.298	− 0.08	0.02
G3	0.00	0.02	− 0.1	0.955	− 0.04	0.03	0.00	0.02	0.2	0.85	− 0.04	0.05
Stage
S0	Reference group: S0						Reference group: S0
S1	0.07	0.00	14.0	< 0.001	0.06	0.08	0.03	0.01	5.8	< 0.001	0.02	0.04
S2	0.25	0.01	24.5	< 0.001	0.23	0.27	0.21	0.01	19.2	< 0.001	0.19	0.23
S3	0.52	0.01	38.8	< 0.001	0.49	0.54	0.45	0.01	31.6	< 0.001	0.42	0.48
Group × Stage
G0 × S0	Reference group: G0 × S0						Reference group: G0 × S0
G1 × S1	− 0.04	0.01	− 6.1	< 0.001	− 0.06	− 0.03	− 0.02	0.01	− 2.2	0.027	− 0.030	− 0.002
G1 × S2	− 0.19	0.01	− 13.1	< 0.001	− 0.22	− 0.16	− 0.07	0.02	− 4.5	< 0.001	− 0.10	− 0.04
G1 × S3	− 0.39	0.02	− 20.5	< 0.001	− 0.42	− 0.35	− 0.14	0.02	− 6.9	< 0.001	− 0.18	− 0.10
G2 × S1	− 0.05	0.01	− 6.7	< 0.001	− 0.06	− 0.03	− 0.01	0.01	− 1.9	0.060	− 0.028	0.001
G2 × S2	− 0.20	0.01	− 13.7	< 0.001	− 0.23	− 0.17	− 0.04	0.02	− 2.5	0.013	− 0.07	− 0.01
G2 × S3	− 0.43	0.02	− 23.0	< 0.001	− 0.47	− 0.40	− 0.13	0.02	− 6.4	< 0.001	− 0.17	− 0.09
G3 × S1	− 0.06	0.01	− 8.7	< 0.001	− 0.08	− 0.05	− 0.03	0.01	− 3.5	0.001	− 0.04	− 0.01
G3 × S2	− 0.23	0.01	− 15.8	< 0.001	− 0.26	− 0.20	− 0.18	0.02	− 12.1	< 0.001	− 0.21	− 0.15
G3 × S3	− 0.48	0.02	− 25.4	< 0.001	− 0.52	− 0.44	− 0.41	0.02	− 20.3	< 0.001	− 0.45	− 0.37
Baseline value	1.01	0.01	174.6	< 0.001	1.00	1.03	1.64	0.35	4.7	< 0.001	0.95	2.33
Constant	− 0.02	0.02	− 1.3	0.199	− 0.05	0.01	− 0.05	0.03	− 1.6	0.119	− 0.10	0.01

Comparisons	p values for pairwise comparisons between different stages				p values for pairwise comparisons between different stages
Stages	Group 1	Group 2	Group 3	Control	Group 1	Group 2	Group 3	Control
S0 versus S1	< 0.001*	< 0.001*	0.105	< 0.001*	0.009*	0.002*	0.412	< 0.001*
S0 versus S2	< 0.001*	< 0.001*	0.029*	< 0.001*	< 0.001*	< 0.001*	0.034*	< 0.001*
S0 versus S3	< 0.001*	< 0.001*	0.004*	< 0.001*	< 0.001*	< 0.001*	0.004*	< 0.001*
S1 versus S2	< 0.001*	0.001*	0.122	< 0.001*	< 0.001*	< 0.001*	0.056	< 0.001*
S1 versus S3	< 0.001*	< 0.001*	0.017*	< 0.001*	< 0.001*	< 0.001*	0.006*	< 0.001*
S2 versus S3	< 0.001*	0.003*	0.117	< 0.001*	< 0.001*	< 0.001*	0.085	< 0.001*

Comparisons	p values for pairwise comparisons between different groups				p values for pairwise comparisons between different groups
Groups	Baseline	Stage 1	Stage 2	Stage 3	Baseline	Stage 1	Stage 2	Stage 3
G1 versus G0	0.477	0.003*	< 0.001*	< 0.001*	0.911	0.383	0.001*	< 0.001*
G2 versus G0	0.729	0.004*	< 0.001*	< 0.001*	0.298	0.116	0.012*	< 0.001*
G3 versus G0	0.955	0.001*	< 0.001*	< 0.001*	0.848	0.317	< 0.001*	< 0.001*
G1 versus G2	0.697	0.855	0.930	0.030*	0.328	0.378	0.824	0.552
G1 versus G3	0.508	0.765	0.159	< 0.001*	0.764	0.895	< 0.001*	< 0.001*
G2 versus G3	0.771	0.622	0.172	0.029*	0.255	0.477	< 0.001*	< 0.001*

The first level fixed factors (Group 0, Control group; and Stage 0, baseline) were used as ‘Reference group’ against which the effects of higher-level factors (Groups 1, 2 and 3; and Stages 1, 2 and 3) were compared.
Abbreviations: SE standard error of the mean, 95%CI 95% confidence interval, REML Restricted Maximum Likelihood.
Restricted maximum likelihood design was used in the linear mixed-effects models instead of the unrestricted design.
*Pairwise comparisons significant at p < 0.05.
Treatment groups: Group 1 (erosion and CPP-ACP in a saliva slurry); Group 2 (erosion and SnF₂ in a saliva slurry); and Group 3 (erosion and SnF₂/CPP-ACP in a saliva slurry). Control group was subjected to erosion and saliva only (without remineralizing agent).
Stages: Stage 0 or Baseline (prior to 1st erosion cycle); Stage 1 (after the 1st erosion cycle); Stage 2 (after the 5th erosion cycle); and Stage 3 (after the 10th erosion cycle).

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Table 1 Linear mixed-effects model outputs for dependent variables ‘erosion depth’ and ‘scratch depth’ versus the interaction of Group and Stage (fixed factors), i.e., Group × Stage, adjusting for repeated measurements over time.

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