If the primary scientific objective was...

If the primary scientific objective was to describe changes in CSP over the 12-min follow-up period and determine whether the pattern of change differed between groups, then we could fit an LMM including treatment effect and time as a continuous covariate with an interaction term to capture non-parallel growth trends. Despite Fig. 2B indicating some non-linearity towards the end of the study follow-up, we note that we have made a strong assumption of linearity in this example. Fitting this model (Table 2) indicates that there is a significant increase in CSP during follow-up in the control group [i.e. a significant effect for time; 0.08 (95% confidence interval (CI) 0.05–0.12)], and no discernible difference from this trend in group ECD (0 weeks) [i.e. non-significant interaction term with time; −0.02 (95% CI −0.08 to 0.03)]. The ECD (3-week) group interaction term is significant (P < 0.001), and despite not reaching significance, there was a tendency for CSP to be reduced over time in the sympathectomy group (−0.05; 95% CI −0.10 to 0.00). Moreover, both terms are negative, which is consistent with Fig. 2B, where the time course for these 2 groups is relatively flat. We could formally test this using appropriate contrasts. One could also perform post hoc tests to establish treatment effect differences at each measurement time (Fig. 2A), but one would need to correct for multiple comparisons (not implemented here). None of the groups admitted a significant main treatment effect relative to the control group. Code to fit this model using the R statistical software package is shown in Supplementary Material, Appendix.

Table 1:

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Methodologies for analysing repeated-measures data, their advantages and disadvantages and some software options

Method	Advantages	Disadvantages	Software
Two-stage methods	Analysis is based on familiar univariate analysis methods Data summary methods may facilitate interpretation, e.g. AUC and rate of change are well-understood concepts in biomedicine research Multiple summary methods can be used	Can be difficult to specify the correct summary statistic in advance Reduced data summary statistics are relatively less efficient Reduced data summary statistics can lose information or fail to capture features of the time course Summary methods not readily implemented in statistical software, but the summary measures are generally rudimentary to calculate Missing data can result in sample bias	Standard tests for independent groups (e.g. t-test, ANOVA, Mann–Whitney U-test, Kruskal–Wallis test) are standard in all statistics software packages Summary statistics can be calculated ‘by hand’ or using a simple programme written in a spreadsheet or statistics package

RM-ANOVA	Includes the data at all time points Simple to implement and conceptually an extension of the ubiquitous ANOVA	Requires complete data on each subject Depends on restrictive sphericity assumption, which is highly questionable for longitudinal data Cannot handle mis-timed/unbalanced measurements Results provide limited information on how the groups differ, often requiring post hoc analyses	SPSS: ‘general linear model: repeated measures’ SAS: PROC GLM R: aov, ANOVA (in the car [20] package), ezANOVA (in the ez [21] package) Stata: ANOVA

LMMs	Includes data at all time points Missing data can be straightforwardly handled if missing (completely) at random Allows flexible modelling of the time effect Permits unbalanced data with greatly different numbers of measurements per subject Allows for time-varying covariates Permits estimation of individual trends Can be augmented with more complex covariance structures that captures more features of the correlation patterns and hierarchically	Implementation and complexity of fitting is relatively more difficult Assumptions can be harder to assess	SPSS: ‘mixed models’ SAS: PROC MIXED R: lme (nlme [22] package) or lmer (lme4 [23] package) Stata: xtmixed

Method	Advantages	Disadvantages	Software
Two-stage methods	Analysis is based on familiar univariate analysis methods Data summary methods may facilitate interpretation, e.g. AUC and rate of change are well-understood concepts in biomedicine research Multiple summary methods can be used	Can be difficult to specify the correct summary statistic in advance Reduced data summary statistics are relatively less efficient Reduced data summary statistics can lose information or fail to capture features of the time course Summary methods not readily implemented in statistical software, but the summary measures are generally rudimentary to calculate Missing data can result in sample bias	Standard tests for independent groups (e.g. t-test, ANOVA, Mann–Whitney U-test, Kruskal–Wallis test) are standard in all statistics software packages Summary statistics can be calculated ‘by hand’ or using a simple programme written in a spreadsheet or statistics package

RM-ANOVA	Includes the data at all time points Simple to implement and conceptually an extension of the ubiquitous ANOVA	Requires complete data on each subject Depends on restrictive sphericity assumption, which is highly questionable for longitudinal data Cannot handle mis-timed/unbalanced measurements Results provide limited information on how the groups differ, often requiring post hoc analyses	SPSS: ‘general linear model: repeated measures’ SAS: PROC GLM R: aov, ANOVA (in the car [20] package), ezANOVA (in the ez [21] package) Stata: ANOVA

LMMs	Includes data at all time points Missing data can be straightforwardly handled if missing (completely) at random Allows flexible modelling of the time effect Permits unbalanced data with greatly different numbers of measurements per subject Allows for time-varying covariates Permits estimation of individual trends Can be augmented with more complex covariance structures that captures more features of the correlation patterns and hierarchically	Implementation and complexity of fitting is relatively more difficult Assumptions can be harder to assess	SPSS: ‘mixed models’ SAS: PROC MIXED R: lme (nlme [22] package) or lmer (lme4 [23] package) Stata: xtmixed

AUC: area under the curve; GLM: generalized linear model; LMMs: linear mixed models; RM-ANOVA: repeated-measures analysis of variance.

Table 1:

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Methodologies for analysing repeated-measures data, their advantages and disadvantages and some software options

Method	Advantages	Disadvantages	Software
Two-stage methods	Analysis is based on familiar univariate analysis methods Data summary methods may facilitate interpretation, e.g. AUC and rate of change are well-understood concepts in biomedicine research Multiple summary methods can be used	Can be difficult to specify the correct summary statistic in advance Reduced data summary statistics are relatively less efficient Reduced data summary statistics can lose information or fail to capture features of the time course Summary methods not readily implemented in statistical software, but the summary measures are generally rudimentary to calculate Missing data can result in sample bias	Standard tests for independent groups (e.g. t-test, ANOVA, Mann–Whitney U-test, Kruskal–Wallis test) are standard in all statistics software packages Summary statistics can be calculated ‘by hand’ or using a simple programme written in a spreadsheet or statistics package

RM-ANOVA	Includes the data at all time points Simple to implement and conceptually an extension of the ubiquitous ANOVA	Requires complete data on each subject Depends on restrictive sphericity assumption, which is highly questionable for longitudinal data Cannot handle mis-timed/unbalanced measurements Results provide limited information on how the groups differ, often requiring post hoc analyses	SPSS: ‘general linear model: repeated measures’ SAS: PROC GLM R: aov, ANOVA (in the car [20] package), ezANOVA (in the ez [21] package) Stata: ANOVA

LMMs	Includes data at all time points Missing data can be straightforwardly handled if missing (completely) at random Allows flexible modelling of the time effect Permits unbalanced data with greatly different numbers of measurements per subject Allows for time-varying covariates Permits estimation of individual trends Can be augmented with more complex covariance structures that captures more features of the correlation patterns and hierarchically	Implementation and complexity of fitting is relatively more difficult Assumptions can be harder to assess	SPSS: ‘mixed models’ SAS: PROC MIXED R: lme (nlme [22] package) or lmer (lme4 [23] package) Stata: xtmixed

Method	Advantages	Disadvantages	Software
Two-stage methods	Analysis is based on familiar univariate analysis methods Data summary methods may facilitate interpretation, e.g. AUC and rate of change are well-understood concepts in biomedicine research Multiple summary methods can be used	Can be difficult to specify the correct summary statistic in advance Reduced data summary statistics are relatively less efficient Reduced data summary statistics can lose information or fail to capture features of the time course Summary methods not readily implemented in statistical software, but the summary measures are generally rudimentary to calculate Missing data can result in sample bias	Standard tests for independent groups (e.g. t-test, ANOVA, Mann–Whitney U-test, Kruskal–Wallis test) are standard in all statistics software packages Summary statistics can be calculated ‘by hand’ or using a simple programme written in a spreadsheet or statistics package

RM-ANOVA	Includes the data at all time points Simple to implement and conceptually an extension of the ubiquitous ANOVA	Requires complete data on each subject Depends on restrictive sphericity assumption, which is highly questionable for longitudinal data Cannot handle mis-timed/unbalanced measurements Results provide limited information on how the groups differ, often requiring post hoc analyses	SPSS: ‘general linear model: repeated measures’ SAS: PROC GLM R: aov, ANOVA (in the car [20] package), ezANOVA (in the ez [21] package) Stata: ANOVA

LMMs	Includes data at all time points Missing data can be straightforwardly handled if missing (completely) at random Allows flexible modelling of the time effect Permits unbalanced data with greatly different numbers of measurements per subject Allows for time-varying covariates Permits estimation of individual trends Can be augmented with more complex covariance structures that captures more features of the correlation patterns and hierarchically	Implementation and complexity of fitting is relatively more difficult Assumptions can be harder to assess	SPSS: ‘mixed models’ SAS: PROC MIXED R: lme (nlme [22] package) or lmer (lme4 [23] package) Stata: xtmixed

AUC: area under the curve; GLM: generalized linear model; LMMs: linear mixed models; RM-ANOVA: repeated-measures analysis of variance.

Table 2:

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Results from analysis of laboratory experiment longitudinal data

Linear mixed-effects model^a
	Estimate	SE	95% CI	P-value
Intercept	4.05	0.17	(3.72 to 4.37)	<0.001
Group
ECD (3 weeks)	−0.44	0.23	(−0.90 to 0.03)	0.064
ECD (0 weeks)	−0.33	0.24	(−0.82 to 0.17)	0.19
Sympathectomy	−0.32	0.23	(−0.80 to 0.15)	0.18
Time (min)	0.08	0.02	(0.05 to 0.12)	<0.001
Time × ECD (3 weeks)	−0.09	0.03	(−0.14 to −0.04)	<0.001
Time × ECD (0 weeks)	−0.02	0.03	(−0.08 to 0.03)	0.43
Time × sympathectomy	−0.05	0.03	(−0.10 to 0.00)	0.054

Summary statistic (Kruskal–Wallis rank-sum tests)

	df		χ² statistic	P-value

Slope	3		8.53	0.036
Final value	3		11.14	0.011

Linear mixed-effects model^a
	Estimate	SE	95% CI	P-value
Intercept	4.05	0.17	(3.72 to 4.37)	<0.001
Group
ECD (3 weeks)	−0.44	0.23	(−0.90 to 0.03)	0.064
ECD (0 weeks)	−0.33	0.24	(−0.82 to 0.17)	0.19
Sympathectomy	−0.32	0.23	(−0.80 to 0.15)	0.18
Time (min)	0.08	0.02	(0.05 to 0.12)	<0.001
Time × ECD (3 weeks)	−0.09	0.03	(−0.14 to −0.04)	<0.001
Time × ECD (0 weeks)	−0.02	0.03	(−0.08 to 0.03)	0.43
Time × sympathectomy	−0.05	0.03	(−0.10 to 0.00)	0.054

Summary statistic (Kruskal–Wallis rank-sum tests)

	df		χ² statistic	P-value

Slope	3		8.53	0.036
Final value	3		11.14	0.011

Fitted by restricted maximum likelihood.

CI: confidence interval; df: degrees of freedom; ECD: extrinsic cardiac denervation; SE: standard error.

Table 2:

Open in new tab

Results from analysis of laboratory experiment longitudinal data

Linear mixed-effects model^a
	Estimate	SE	95% CI	P-value
Intercept	4.05	0.17	(3.72 to 4.37)	<0.001
Group
ECD (3 weeks)	−0.44	0.23	(−0.90 to 0.03)	0.064
ECD (0 weeks)	−0.33	0.24	(−0.82 to 0.17)	0.19
Sympathectomy	−0.32	0.23	(−0.80 to 0.15)	0.18
Time (min)	0.08	0.02	(0.05 to 0.12)	<0.001
Time × ECD (3 weeks)	−0.09	0.03	(−0.14 to −0.04)	<0.001
Time × ECD (0 weeks)	−0.02	0.03	(−0.08 to 0.03)	0.43
Time × sympathectomy	−0.05	0.03	(−0.10 to 0.00)	0.054

Summary statistic (Kruskal–Wallis rank-sum tests)

	df		χ² statistic	P-value

Slope	3		8.53	0.036
Final value	3		11.14	0.011

Linear mixed-effects model^a
	Estimate	SE	95% CI	P-value
Intercept	4.05	0.17	(3.72 to 4.37)	<0.001
Group
ECD (3 weeks)	−0.44	0.23	(−0.90 to 0.03)	0.064
ECD (0 weeks)	−0.33	0.24	(−0.82 to 0.17)	0.19
Sympathectomy	−0.32	0.23	(−0.80 to 0.15)	0.18
Time (min)	0.08	0.02	(0.05 to 0.12)	<0.001
Time × ECD (3 weeks)	−0.09	0.03	(−0.14 to −0.04)	<0.001
Time × ECD (0 weeks)	−0.02	0.03	(−0.08 to 0.03)	0.43
Time × sympathectomy	−0.05	0.03	(−0.10 to 0.00)	0.054

Summary statistic (Kruskal–Wallis rank-sum tests)

	df		χ² statistic	P-value

Slope	3		8.53	0.036
Final value	3		11.14	0.011

Fitted by restricted maximum likelihood.

CI: confidence interval; df: degrees of freedom; ECD: extrinsic cardiac denervation; SE: standard error.

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