Paper 33: Ultrasonography and electrophysiological study of median nerve in patients with essential tremor

Author

Lee Jones - Senior Biostatistician - Statistical Review

Published

March 15, 2026

References

Lee HL, Kim J-s, Kim H, Kim Is, Kim J-w, Kim Y-e, et al. (2019) Ultrasonography and electrophysiological study of median nerve in patients with essential tremor. PLoS ONE 14(4): e0215750. https://doi.org/10.1371/journal.pone.0215750

Disclosure

This reproducibility project was conducted to the best of our ability, with careful attention to statistical methods and assumptions. The research team comprises four senior biostatisticians (three of whom are accredited), with 20 to 30 years of experience in statistical modelling and analysis of healthcare data. While statistical assumptions play a crucial role in analysis, their evaluation is inherently subjective, and contextual knowledge can influence judgements about the importance of assumption violations. Differences in interpretation may arise among statisticians and researchers, leading to reasonable disagreements about methodological choices.

Our approach aimed to reproduce published analyses as faithfully as possible, using the details provided in the original papers. We acknowledge that other statisticians may have differing success in reproducing results due to variations in data handling and implicit methodological choices not fully described in publications. However, we maintain that research articles should contain sufficient detail for any qualified statistician to reproduce the analyses independently.

Methods used in our reproducibility analyses

There were two parts to our study. First, 100 articles published in PLOS ONE were randomly selected from the health domain and sent for post-publication peer review by statisticians. Of these, 95 included linear regression analyses and were therefore assessed for reporting quality. The statisticians evaluated what was reported, including regression coefficients, 95% confidence intervals, and p-values, as well as whether model assumptions were described and how those assumptions were evaluated. This report provides a brief summary of the initial statistical review.

The second part of the study involved reproducing linear regression analyses for papers with available data to assess both computational and inferential reproducibility. All papers were initially assessed for data availability and the statistical software used. From those with accessible data, the first 20 papers (from the original random sample) were evaluated for computational reproducibility. Within each paper, individual linear regression models were identified and assigned a unique number. A maximum of three models per paper were selected for assessment. When more than three models were reported, priority was given to the final model or the primary models of interest as identified by the authors; any remaining models were selected at random.

To assess computational reproducibility, differences between the original and reproduced results were evaluated using absolute discrepancies and rounding error thresholds, tailored to the number of decimal places reported in each paper. Results for each reported statistic, e.g., regression coefficient, were categorised as Reproduced, Incorrect Rounding, or Not Reproduced, depending on how closely they matched the original values. Each paper was then classified as Reproduced, Mostly Reproduced, Partially Reproduced, or Not Reproduced. The mostly reproduced category included cases with minor rounding or typographical errors, whereas partially reproduced indicated substantial errors were observed, but some results were reproduced.

For models deemed at least partially computationally reproducible, inferential reproducibility was further assessed by examining whether statistical assumptions were met and by conducting sensitivity analyses, including bootstrapping where appropriate. We examined changes in standardized regression coefficients, which reflect the change in the outcome (in standard deviation units) for a one standard deviation increase in the predictor. Meaningful differences were defined as a relative change of 10% or more, or absolute differences of 0.1 (moderate) and 0.2 (substantial). When non-linear relationships were identified, inferential reproducibility was assessed by comparing model fit measures, including R², Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC). When the Gaussian distribution was not appropriate for the dependent variable, alternative distributions were considered, and model fit was evaluated using AIC and BIC.

Results from the reproduction of the Lee et al. (2019) paper are presented below. An overall summary of results is presented first, followed by model-specific results organised within tab panels. Within each panel, the Original results tab displays the linear regression outputs extracted from the published paper. The Reproduced results tab presents estimates derived from the authors’ shared data, along with a comprehensive assessment of linear regression assumptions. The Differences tab compares the original and reproduced models to assess computational reproducibility. Finally, the Sensitivity analysis tab evaluates inferential reproducibility by examining whether identified assumption violations meaningfully affected the results.

Summary from statistical review

This study examined the median nerve in patients with essential tremors. ANCOVA and linear regression were identified as analysis methods, the authors did not report checking the assumptions, outliers, or collinearity. Both hands of patients were used in the study without correction for repeated measures. There was no meaningful interpretation of regression coefficients other than direction. Forward modelling was used for linear regression models.

Data availability and software used

The authors have only provided half of the data with no data supplied for the control group this was not stated in their data availability statement which indicated that all relevant data are within the manuscript and its Supporting Information file, although it was mentioned in the paper itself. The data provided was a long formatted SPSS file, but the authors did not take advantage of the in-built data dictionary. All analyses were performed in SPSS.

Regression sample

The authors used both ANCOVA and linear regression, which were all counted as linear regression. Three models were randomly selected because no primary model of interest was identified. Two models had Carpal inlet as the outcome, with each explanatory variable (Sensory NCV and Sensory amplitude) modelled separately in forward stepwise models, adjusted for age. The third model was a univariate ratio for the Carpal outlet / mid-forearm between the essential tremor and control group, as there was no control data provided, this could not be assessed for reproducibility.

Computational reproducibility results

Based on the two models, the data provided were computationally reproducible. While some confidence intervals were reported in the text, only coefficients and p-values were provided for the selected models. In addition, only a subset of the data was available, with no data provided for the control group. It was also not immediately clear on first reading that Table 3 comprised 16 separate models. Providing full model results in the supplementary materials would have improved clarity and reproducibility.

Inferential reproducibility results

Neither model was inferentially reproducible. The primary limitation was the lack of adjustment for repeated measures, as both left and right hands from each participant were included in the analyses with ICC’s >0.3. Evidence of heteroscedasticity was also observed. Consequently, linear mixed models with a wild bootstrap were fitted to account for within-participant clustering and heteroscedasticity. Comparisons of coefficients and 95% confidence intervals indicated medium (>0.10) to large (>0.20) differences in the standardized confidence-interval ranges. The direction of the coefficients and the statistical significance were consistent.

Recommended Changes

Include data for the control group.
Update the data availability statement.
Include a data dictionary.
Provide full details of models in supplementary materials, including all coefficients and 95% confidence intervals.
Consider using scatterplots to get a better understanding of the underlying relationships.
The models residuals are not independent, consider the model design and use linear mixed model or generalised estimating equations to account for the correlation due to repeated results from the same participant.
Evaluate the assumptions of the linear regression models by examining residuals, identifying influential outliers, and assessing multicollinearity among predictors. If any assumptions are violated, address them using appropriate methods.
Interpret the regression coefficients, relating results back to the outcome measures.

Model 1

Model results for Carpal inlet

Term	B	p-value
Intercept
Sensory_amplitude	−0.044	<0.001
age
SE = Standard error; Lower = lower confidence interval; Upper = upper confidence interval.

Fit statistics for Carpal inlet

R	R2	R2Adj	AIC	RMSE	F	DF1	DF2	p-value

R2 Adj = Adjusted R2; AIC = Akaike Information Criterion; RMSE = The Root Mean Squared Error; DF1 = Degrees of freedom for the model; DF2 = Degrees of freedom for the residuals.

ANOVA table for Carpal inlet

Term	SS	DF	MS	F	p-value
Sensory_amplitude
age
Residuals
SS = Sum of Squares; DF = Degrees of freedom; MS = Mean Square.

Model results for Carpal inlet

Term	B	SE	Lower	Upper	t	p-value
Intercept	19.604	1.952	15.642	23.567	10.044	<0.001
Sensory_amplitude	−0.044	0.012	−0.069	−0.020	−3.636	<0.001
age	−0.138	0.026	−0.191	−0.085	−5.291	<0.001
SE = Standard error; Lower = lower confidence interval; Upper = upper confidence interval.

Fit statistics for Carpal inlet

R	R2	R2Adj	AIC	RMSE	F	DF1	DF2	p-value
0.668	0.446	0.414	159.777	1.783	14.068	2	35	<0.001
R2 Adj = Adjusted R2; AIC = Akaike Information Criterion; RMSE = The Root Mean Squared Error; DF1 = Degrees of freedom for the model; DF2 = Degrees of freedom for the residuals.

ANOVA table for Carpal inlet

Term	SS	DF	MS	F	p-value
Sensory_amplitude	45.619	1	45.619	13.221	<0.001
age	96.602	1	96.602	27.997	<0.001
Residuals	120.763	35	3.450
SS = Sum of Squares; DF = Degrees of freedom; MS = Mean Square; Calculated using type III SS.

Visualisation of regression model

The blue line shows the best line of fit with shading representing 95% confidence intervals, while holding all other covariates constant. The dots show partial residuals, which reflect the observed data adjusted for all other predictors except the one being plotted.

Checking residuals plots for patterns

Blue line showing quadratic fit for residuals

Testing residuals for non linear relationships

Term	Statistic	p-value	Results
Sensory_amplitude	1.198	0.2393	No linearity violation
age	1.049	0.3018	No linearity violation
Tukey test	0.896	0.3704	No linearity violation
Specification test for predictors using quadratic tests, for fitted values curvature is tested through Tukey's one-degree-of-freedom test for nonadditivity.

Checking univariate relationships with the dependent variable using scatterplots

Blue line shows linear relationship, red line indicates relationship inferred by GAM modelling

Linearity results

No linearity violation was observed in either plots or tests.

Testing for homoscedasticity

Statistic	p-value	Parameter	Method
11.848	0.0027	2	studentized Breusch-Pagan test

Homoscedasticity results

The studentized Breusch-Pagan test indicates heteroscedasticity.
Some heteroscedasticity is present in plots, and a sensitivity analysis using weighted or robust regression or wild bootstrapping is recommended.

Model descriptives including cook’s distance and leverage to understand outliers

Term	N	Mean	SD	Median	Min	Max	Skewness	Kurtosis
Carpal inlet	38	9.513	2.426	8.650	5.800	15.300	0.969	0.073
Sensory_amplitude	38	47.368	32.264	35.500	10.000	119.000	0.912	−0.661
age	38	57.947	15.141	58.000	21.000	77.000	−0.641	−0.333
.fitted	38	9.513	1.620	9.000	7.175	12.767	0.586	−0.951
.resid	38	0.000	1.807	0.223	−3.067	3.667	0.086	−0.936
.leverage	38	0.079	0.042	0.067	0.032	0.187	1.097	0.245
.sigma	38	1.857	0.031	1.867	1.770	1.885	−1.256	0.798
.cooksd	38	0.033	0.049	0.017	0.000	0.193	1.958	2.980
.std.resid	38	0.004	1.018	0.124	−1.777	2.030	0.089	−0.929
dfb.1_	38	0.002	0.232	0.002	−0.697	0.617	−0.134	2.102
dfb.Sns_	38	−0.001	0.193	0.003	−0.625	0.498	−0.680	2.270
dfb.age	38	−0.002	0.219	0.004	−0.568	0.646	0.114	1.872
dffit	38	0.018	0.326	0.030	−0.731	0.779	0.180	−0.006
cov.r	38	1.089	0.106	1.125	0.792	1.243	−0.843	0.157
* categorical variable

Cooks threshold

Cook’s distance measures the overall change in fit, if the ith observation is removed. Potential influential observations are identified by \(\text{Cook's Distance}_i > \frac{4}{n}\), where n is the number of observations. In practice a threshold of 0.5 to 1 is often used to identify influential observations.

DFFIT threshold

DFFIT measures how many standard deviations the fitted values will change when the ith observation is removed. Potential influential observations \(\left| \text{DFFITS}_i \right| > \frac{2\sqrt{p}}{\sqrt{n}}\) where p is the number of predictors (including the intercept) and n is the number of observations. In practice this can result in a large number of points identified, a practical cut-off of 1 was used to flag observations with meaningful impact.

DFBETA threshold

DFBETAS quantify the influence of the ith observation on the jth regression coefficient as the change in that coefficient when the observation is omitted, expressed in units of the coefficient’s estimated standard error. There is a DFBETA for each parameter in the model. Potential influential observations \(|\text{DFBETA}_{ij}| > \frac{2}{\sqrt{n}}\), where n is the number of observations. In larger datasets this threshold can flag a high number of observations with only minor influence on the model. A practical cut-off of 1 was used to flag observations with meaningful impact.

Influence plot

Observations with high leverage (horizontal) and large residuals (vertical, typically at ±2 or ±3 studentized residuals) are concerning, as they may disproportionately influence the model. This combination is reflected by large bubbles with high Cook’s distance indicated by darker shadings of blue.

COVRATIO plot

COVRATIO measures the overall change in the precision (covariance) of the estimated regression coefficients if the ith observation is removed. Values close to 1 indicate little influence on the model’s precision. Values below 1 indicate inflated variances and reduced precision (wider confidence intervals), whereas values above 1 indicate deflated variances and increased precision (narrower confidence intervals). A commonly cited guideline is \(\left|\mathrm{COVRATIO}_i - 1\right| > \frac{3p}{n}\), where p is the number of parameters and n is the number of observations. A practical cut-off between 0.9 to 1.1 was used to flag observations with meaningful impact on precision, although there is no agreed universal alternative cut-off.

Observations of interest identified by the influence plot

ID	StudRes	Leverage	CookD	dfb.1_	dfb.Sns_	dfb.age	dffit	cov.r
8	2.130	0.054	0.078	0.341	−0.364	−0.230	0.509	0.792
34	1.050	0.187	0.085	0.316	−0.009	−0.368	0.504	1.220
15	2.083	0.109	0.162	0.617	−0.625	−0.487	0.730	0.854
13	−1.836	0.137	0.167	−0.697	0.498	0.646	−0.731	0.951
33	1.623	0.187	0.193	0.489	−0.014	−0.568	0.779	1.073
StudRes = studentized residuals; CookD = Cook's Distance a combined measure of leverage and influence. DFBETAS (dfb.*) measures how much a specific regression coefficient changes (in standard errors) when an observation is removed; DFFITS measures how much the fitted (predicted) value for an observation changes (in standard deviations) when that observation is removed; cov.r = coefficient covariance ratio which measures how much the overall variance (precision) of the coefficients changes when that observation is removed.

Results for outliers and influential points

Cook’s distance, DFBETAS and DFFITS were within conventional ranges. However, there were COVRATIO values that suggest multiple point may effect confidence interval widths.

Checking for normality of the residuals using a Q–Q plot

Normality of residuals using Shapiro-Wilk and Kolmogorov-Smirnov tests

Statistic	p-value	Method
0.089	0.8998	Exact one-sample Kolmogorov-Smirnov test

Statistic	p-value	Method
0.973	0.4893	Shapiro-Wilk normality test

Normality results

The Kolmogorov-Smirnov supports residuals being normally distributed.
The Shapiro-Wilk supports residuals being normally distributed.
QQ-plot looks roughly normal.

Assessing collinearity with VIF

Term	VIF	Tolerance
Sensory_amplitude	1.668	0.600
age	1.668	0.600
VIF = Variance Inflation Factor.

Collinearity results

All VIF values are under three, indicating no collinearity issues.
Overall, when taking into account VIF and SE, the model does not have collinearity issues.

Assessing independence with the Durbin–Watson test for autocorrelation

AutoCorrelation	Statistic	p-value
0.324	1.246	0.0160

Independence results

The Durbin–Watson test suggests there are auto-correlation issues.
The study design is not independent and should be assessed using linear mixed models or generalized estimating equations.

Assumption conclusions

The residual plots and the Breusch–Pagan test suggested some heteroscedasticity. Normality checks (Q–Q plots and tests) indicated that the residuals were approximately normally distributed, with no clear evidence of nonlinearity. Outlier diagnostics suggested that individual observations did not strongly influence point estimates, although confidence-interval widths may be affected. Evidence of autocorrelation was observed, consistent with the repeated-measures design, indicating that the independence assumption may not hold.

Forest plot showing original and reproduced coefficients and 95% confidence intervals for Carpal inlet

Change in regression coefficients

term	O_B	R_B	Change.B	reproduce.B
Intercept		19.6042
Sensory_amplitude	−0.044	−0.0444	−0.0004	Reproduced
age		−0.1378
O_B = original B; R_B = reproduced B; Change.B = change in R_B - O_B; Reproduce.B = B reproduced.

Change in p-values

Term	O_p	R_p	Change.p	Reproduce.p	SigChangeDirection
Intercept		<0.001
Sensory_amplitude	<0.001	<0.001	0.0000	Reproduced	Remains sig, B same direction
age		<0.001
O_p = original p-value; R_p = reproduced p-value; Changep = change in p-value R_p - O_p; Reproduce.p = p-values reproduced. SigChangeDirection = statistical significance and B change between original and reproduced models. Note, p-values that were <0.001 were set to 0.00099 for the purposes of comparison.

Results for p-values

The p-value was reproduced for this model.

Conclusion computational reproducibility

This model was computationally reproducible, with all reported statistics that were assessed being reproducible.

Methods

The model was successfully reproduced; however, diagnostics indicated potential heteroscedasticity and residual non-independence. To further assess the robustness of the findings, wild bootstrapping was used to obtain bootstrapped standardized regression coefficients and their 95% confidence intervals, accounting for potential heteroscedasticity. Absolute and percentage changes in coefficient estimates and confidence-interval bounds, relative to the linear model, were summarised using thresholds of a 10% change and standardized coefficient differences of <0.10 and <0.20. Concordance of coefficient direction was assessed, and the stability of statistical significance across models was evaluated.

Results

Linear vs Mixed model

The AIC and BIC were lower for mixed model compared to the linear model. The unadjusted intraclass correlation coefficient (ICC) from the null model was 0.4, suggesting a substantial degree of clustering. As ignoring clustering can lead to underestimated standard errors, inference was obtained using wild bootstrap linear mixed-effects models.

Reproduced Linear Model

Term	B	SE¹	t	95% CI		p-value
Term	B	SE¹	t	Lower	Upper	p-value
(Intercept)	0.0000	0.1242	0.0000	−0.2521	0.2521	>0.999
z_Sensory_amplitude	−0.5909	0.1625	−3.6361	−0.9209	−0.2610	<0.001
z_age	−0.8599	0.1625	−5.2913	−1.1899	−0.5300	<0.001
Sigma = 0.766; AIC = 92.4; BIC = 99.0; Residual df = 35; No. Obs. = 38
¹ SE = Standard Error

Linear Mixed Model

Term	B	SE¹	t	95% CI		p-value
Term	B	SE¹	t	Lower	Upper	p-value
(Intercept)	−0.0000	0.1527	−0.0000	−0.3106	0.3106	>0.999
z_Sensory_amplitude	−0.5727	0.1796	−3.1885	−0.9382	−0.2073	0.003
z_age	−0.8484	0.1920	−4.4195	−1.2390	−0.4578	<0.001
serial_no.SD (Intercept)	0.5879
Residual.SD (Observations)	0.4408
Sigma = 0.441; AIC = 84.4; BIC = 92.6; No. Obs. = 38
¹ SE = Standard Error

Bootstrapped results

Wild bootstrapping was performed with 10,000 iterations.

Change in regression coefficients

Term	B	boot.B	B_diff	%_Diff	Diff_10%	Diff_0.1	Diff_0.2
Intercept	0.0000	0.0003	−0.0003	−1,000.0000	Yes	No	No
z_Sensory_amplitude	−0.5909	−0.5721	−0.0189	−3.1900	No	No	No
z_age	−0.8599	−0.8474	−0.0126	−1.4600	No	No	No
B = standardized regression coefficient reproduced B; boot.B = boostrapped standardized reproduced B; B_diff = change in B - boot.B; %_Diff = percentage difference, percentage changes were truncated at ±1000%; Diff_10% = difference ≥10% ; Diff_0.1 and Diff_0.2 = absolute difference ≥0.1 and ≥0.2, respectively.

Change in lower 95% confidence interval

Term	Lower	boot.Lower	Lower_diff	%_Diff	Diff_10%	Diff_0.1	Diff_0.2
Intercept	−0.2521	−0.3218	0.0697	27.6300	Yes	No	No
z_Sensory_amplitude	−0.9209	−0.9854	0.0645	7.0000	No	No	No
z_age	−1.1899	−1.3684	0.1785	15.0000	Yes	Yes	No
Lower = standardized reproduced lower CI; boot.Lower = boostrapped standardized reproduced lower CI; Lower_diff = change in Lower - boot.Lower; %_change = percentage difference, percentage changes were truncated at ±1000%; Diff_10% = difference ≥10% ; Diff_0.1 and Diff_0.2 = absolute difference ≥0.1 and ≥0.2, respectively.

Change in upper 95% confidence interval

Term	Upper	boot.Upper	Upper_diff	%_Diff	Diff_10%	Diff_0.1	Diff_0.2
Intercept	0.2521	0.3292	−0.0771	−30.5600	Yes	No	No
z_Sensory_amplitude	−0.2610	−0.1684	−0.0926	−35.5000	Yes	No	No
z_age	−0.5300	−0.3317	−0.1983	−37.4100	Yes	Yes	No
Upper = standardized reproduced upper CI; boot.Upper = boostrapped standardized reproduced upper CI; Upper_diff = change in Upper - boot.Upper; %_change = percentage difference, percentage changes were truncated at ±1000%; Diff_10% = difference ≥10% ; Diff_0.1 and Diff_0.2 = absolute difference ≥0.1 and ≥0.2, respectively.

Change in Range of 95% confidence interval

Term	Range	boot.Range	Range_Diff	%_Diff	Diff_10%	Diff_0.1	Diff_0.2
Intercept	0.5042	0.6509	0.1467	29.1000	Yes	Yes	No
z_Sensory_amplitude	0.6599	0.8170	0.1571	23.8100	Yes	Yes	No
z_age	0.6599	1.0367	0.3768	57.1000	Yes	Yes	Yes
Range = standardized reproduced CI range; boot.B = boostrapped standardized reproduced CI range; Range_diff = change in CI Range ; %_change = percentage difference, Diff_10% = difference ≥10% ; Diff_0.1 and Diff_0.2 = absolute difference ≥0.1 and ≥0.2, respectively.

Change in p-value significance and regression coefficient direction

Term	p-value	boot.p-value	changep	SigChangeDirection
Intercept	1.0000	0.9987	0.0013	Remains non-sig, B same direction
z_Sensory_amplitude	<0.001	0.0018	−0.0009	Remains sig, B same direction
z_age	<0.001	<0.001	−0.0002	Remains sig, B same direction
p-value = standardized reproduced p-value; boot.p-value = boostrapped standardized reproduced p-value; changep = change in p-value - boot.p-value; SigChangeDirection = statistical significance and B change between reproduced and bootstrapped model.

Check the distribution of bootstrap estimates

The bootstrap distribution of each coefficient appeared approximately normal and centered near the original estimate (red dashed line), suggesting that the estimates are relatively stable. No strong skewness or multimodality was observed.

Conclusions based on bootstrapped model

Substantial correlation of error residuals was observed as the study design had repeated measures with the unadjusted ICC being 0.4 and the AIC was for the linear mixed model was 8 units lower than the linear model. The linear mixed model was bootstrapped, and the CI range standardized differences for sensory amplitude was 0.16, and age 0.38 when compared to the linear model and was not inferentially reproducible.

Model 2

Model results for Carpal inlet

Term	B	p-value
Intercept
Sensory_NCV	−0.269	<0.001
age
SE = Standard error; Lower = lower confidence interval; Upper = upper confidence interval.

Fit statistics for Carpal inlet

R	R2	R2Adj	AIC	RMSE	F	DF1	DF2	p-value

R2 Adj = Adjusted R2; AIC = Akaike Information Criterion; RMSE = The Root Mean Squared Error; DF1 = Degrees of freedom for the model; DF2 = Degrees of freedom for the residuals.

ANOVA table for Carpal inlet

Term	SS	DF	MS	F	p-value
Sensory_NCV
age
Residuals
SS = Sum of Squares; DF = Degrees of freedom; MS = Mean Square.

Model results Carpal inlet

Term	B	SE	Lower	Upper	t	p-value
Intercept	28.104	3.066	21.881	34.327	9.168	<0.001
Sensory_NCV	−0.269	0.055	−0.380	−0.158	−4.913	<0.001
age	−0.115	0.020	−0.155	−0.075	−5.833	<0.001
SE = Standard error; Lower = lower confidence interval; Upper = upper confidence interval.

Model fit for Carpal inlet

R	R2	R2Adj	AIC	RMSE	F	DF1	DF2	p-value
0.740	0.548	0.522	152.022	1.610	21.215	2	35	<0.001
R2 Adj = Adjusted R2; AIC = Akaike Information Criterion; RMSE = The Root Mean Squared Error; DF1 = Degrees of freedom for the model; DF2 = Degrees of freedom for the residuals.

ANOVA table for Carpal inlet

Term	SS	DF	MS	F	p-value
Sensory_NCV	67.911	1	67.911	24.138	<0.001
age	95.728	1	95.728	34.025	<0.001
Residuals	98.471	35	2.813
SS = Sum of Squares; DF = Degrees of freedom; MS = Mean Square; Calculated using type III SS.

Visualisation of regression model

Checking residuals plots for patterns

Blue line showing quadratic fit for residuals

Testing residuals for non linear relationships

Term	Statistic	p-value	Results
Sensory_NCV	−1.264	0.2149	No linearity violation
age	0.528	0.6012	No linearity violation
Tukey test	1.693	0.0904	No linearity violation
Specification test for predictors using quadratic tests, for fitted values curvature is tested through Tukey's one-degree-of-freedom test for nonadditivity.

Checking univariate relationships with the dependent variable using scatterplots

Blue line shows linear relationship, red line indicates relationship inferred by GAM modelling

Linearity results

No linearity violation was observed in either plots or tests.

Testing for homoscedasticity

Statistic	p-value	Parameter	Method
1.310	0.5195	2	studentized Breusch-Pagan test

Homoscedasticity results

The studentized Breusch-Pagan test supports homoscedasticity.
Some heteroscedasticity is present in plots, and a sensitivity analysis using weighted or robust regression or wild bootstrapping is recommended.

Model descriptives including cook’s distance and leverage to understand outliers

Term	N	Mean	SD	Median	Min	Max	Skewness	Kurtosis
Carpal inlet	38	9.513	2.426	8.650	5.800	15.300	0.969	0.073
Sensory_NCV	38	44.395	5.460	45.000	32.000	55.000	−0.293	−0.452
age	38	57.947	15.141	58.000	21.000	77.000	−0.641	−0.333
.fitted	38	9.513	1.796	9.077	6.353	13.869	0.475	−0.528
.resid	38	0.000	1.631	−0.505	−2.751	4.134	0.697	−0.368
.leverage	38	0.079	0.045	0.061	0.031	0.219	1.385	1.296
.sigma	38	1.677	0.033	1.690	1.542	1.702	−2.185	5.255
.cooksd	38	0.031	0.044	0.011	0.000	0.182	1.876	2.992
.std.resid	38	0.001	1.015	−0.311	−1.730	2.504	0.677	−0.440
dfb.1_	38	0.001	0.201	−0.000	−0.463	0.652	0.471	2.115
dfb.S_NC	38	−0.000	0.184	0.002	−0.630	0.402	−0.778	2.517
dfb.age	38	−0.002	0.198	0.000	−0.674	0.497	−0.341	2.780
dffit	38	0.004	0.314	−0.068	−0.599	0.754	0.644	−0.127
cov.r	38	1.092	0.126	1.127	0.623	1.392	−1.363	4.042
* categorical variable

Cooks threshold

DFFIT threshold

DFBETA threshold

Influence plot

COVRATIO plot

Observations of interest identified by the influence plot

ID	StudRes	Leverage	CookD	dfb.1_	dfb.S_NC	dfb.age	dffit	cov.r
34	0.220	0.219	0.005	0.080	−0.045	−0.110	0.117	1.392
1	2.724	0.031	0.067	0.035	0.071	−0.138	0.488	0.623
8	2.158	0.057	0.085	0.396	−0.390	−0.148	0.531	0.787
15	1.782	0.137	0.158	0.652	−0.630	−0.337	0.710	0.967
33	1.558	0.190	0.182	0.343	−0.085	−0.674	0.754	1.095
StudRes = studentized residuals; CookD = Cook's Distance a combined measure of leverage and influence. DFBETAS (dfb.*) measures how much a specific regression coefficient changes (in standard errors) when an observation is removed; DFFITS measures how much the fitted (predicted) value for an observation changes (in standard deviations) when that observation is removed; cov.r = coefficient covariance ratio which measures how much the overall variance (precision) of the coefficients changes when that observation is removed.

Results for outliers and influential points

Cook’s distance, DFBETAS and DFFITS were within conventional ranges. However, there were COVRATIO values that suggest multiple point may effect confidence interval widths. ##### Checking for normality of the residuals using a Q–Q plot

Normality of residuals using Shapiro-Wilk and Kolmogorov-Smirnov tests

Statistic	p-value	Method
0.136	0.4474	Exact one-sample Kolmogorov-Smirnov test

Statistic	p-value	Method
0.941	0.0435	Shapiro-Wilk normality test

Normality results

The Kolmogorov-Smirnov supports residuals being normally distributed.
The Shapiro-Wilk normality test indicates residuals may not be normally distributed.
QQ-plot indicates the residuals are not normally distributed.

Assessing collinearity with VIF

Term	VIF	Tolerance
Sensory_NCV	1.172	0.853
age	1.172	0.853
VIF = Variance Inflation Factor.

Collinearity results

All VIF values are under three, indicating no collinearity issues.
Overall, when taking into account VIF and SE, the model does not have collinearity issues.

Assessing independence with the Durbin–Watson test for autocorrelation

AutoCorrelation	Statistic	p-value
0.318	1.148	0.0060

Independence results

The Durbin–Watson test suggests there are auto-correlation issues.
The study design is not independent and should be assessed using linear mixed models or generalized estimating equations.

Assumption conclusions

Residual diagnostics suggested mild heteroscedasticity, although the Breusch–Pagan test was not statistically significant. The Q–Q plot indicated minor deviations from normality, while there was no clear evidence of nonlinearity. Outlier and influence diagnostics showed that individual observations did not materially affect point estimates, although confidence-interval widths may be sensitive to influential cases. Evidence of autocorrelation was observed, consistent with the repeated-measures design, indicating that the assumption of independent observations may not be fully satisfied.

Forest plot showing Original and Reproduced coefficients and 95% confidence intervals for Carpal inlet

Change in regression coefficients

term	O_B	R_B	Change.B	reproduce.B
Intercept		28.1040
Sensory_NCV	−0.269	−0.2686	0.0004	Reproduced
age		−0.1150
O_B = original B; R_B = reproduced B; Change.B = change in R_B - O_B; Reproduce.B = B reproduced.

Change in p-values

Term	O_p	R_p	Change.p	Reproduce.p	SigChangeDirection
Intercept		<0.001
Sensory_NCV	<0.001	<0.001	0.0000	Reproduced	Remains sig, B same direction
age		<0.001
O_p = original p-value; R_p = reproduced p-value; Changep = change in p-value R_p - O_p; Reproduce.p = p-values reproduced. SigChangeDirection = statistical significance and B change between original and reproduced models. Note, p-values that were <0.001 were set to 0.00099 for the purposes of comparison.

Results for p-values

The p-value for this model was reproduced.

Conclusion computational reproducibility

This model was computationally reproducible, with all reported statistics that were assessed being reproducible.

Methods

The model was successfully reproduced; however, diagnostics suggested potential heteroscedasticity, non-normality and non-independence of residuals. To further assess the robustness of the findings, wild bootstrapping was used to obtain bootstrapped standardized regression coefficients and their 95% confidence intervals, accounting for potential heteroscedasticity. Absolute and percentage changes in coefficient estimates and confidence-interval bounds, relative to the linear model, were summarised using thresholds of a 10% change and standardized coefficient differences of <0.10 and <0.20. Concordance of coefficient direction was assessed, and the stability of statistical significance was evaluated across models.

Results

Linear vs Mixed model

The AIC and BIC were lower for the mixed model compared to the linear model. The unadjusted intraclass correlation coefficient (ICC) from the null model was 0.309, suggesting a substantial degree of clustering.

Reproduced Linear Model

Term	B	SE¹	t	95% CI		p-value
Term	B	SE¹	t	Lower	Upper	p-value
(Intercept)	0.0000	0.1121	0.0000	−0.2277	0.2277	>0.999
z_Sensory_NCV	−0.6045	0.1230	−4.9130	−0.8543	−0.3547	<0.001
z_age	−0.7177	0.1230	−5.8331	−0.9675	−0.4679	<0.001
Sigma = 0.691; AIC = 84.7; BIC = 91.2; Residual df = 35; No. Obs. = 38
¹ SE = Standard Error

Linear Mixed Model

Term	B	SE¹	t	95% CI		p-value
Term	B	SE¹	t	Lower	Upper	p-value
(Intercept)	0.0000	0.1351	0.0000	−0.2748	0.2748	>0.999
z_Sensory_NCV	−0.5556	0.1305	−4.2574	−0.8212	−0.2901	<0.001
z_age	−0.6990	0.1457	−4.7957	−0.9955	−0.4024	<0.001
serial_no.SD (Intercept)	0.5013
Residual.SD (Observations)	0.4368
Sigma = 0.437; AIC = 79.4; BIC = 87.6; No. Obs. = 38
¹ SE = Standard Error

Bootstrapped results

As ignoring clustering can lead to underestimated standard errors, inference was obtained using wild bootstrap linear mixed-effects models with 10000 resamples.

Change in regression coefficients

Term	B	boot.B	B_diff	%_Diff	Diff_10%	Diff_0.1	Diff_0.2
Intercept	0.0000	0.0007	−0.0007	−1,000.0000	Yes	No	No
z_Sensory_NCV	−0.6045	−0.5552	−0.0493	−8.1600	No	No	No
z_age	−0.7177	−0.6983	−0.0194	−2.7000	No	No	No
B = standardized regression coefficient reproduced B; boot.B = boostrapped standardized reproduced B; B_diff = change in B - boot.B; %_Diff = percentage difference, percentage changes were truncated at ±1000%; Diff_10% = difference ≥10% ; Diff_0.1 and Diff_0.2 = absolute difference ≥0.1 and ≥0.2, respectively.

Change in lower 95% confidence interval

Term	Lower	boot.Lower	Lower_diff	%_Diff	Diff_10%	Diff_0.1	Diff_0.2
Intercept	−0.2277	−0.2832	0.0555	24.3900	Yes	No	No
z_Sensory_NCV	−0.8543	−0.8853	0.0310	3.6300	No	No	No
z_age	−0.9675	−1.0429	0.0755	7.8000	No	No	No
Lower = standardized reproduced lower CI; boot.Lower = boostrapped standardized reproduced lower CI; Lower_diff = change in Lower - boot.Lower; %_change = percentage difference, percentage changes were truncated at ±1000%; Diff_10% = difference ≥10% ; Diff_0.1 and Diff_0.2 = absolute difference ≥0.1 and ≥0.2, respectively.

Change in upper 95% confidence interval

Term	Upper	boot.Upper	Upper_diff	%_Diff	Diff_10%	Diff_0.1	Diff_0.2
Intercept	0.2277	0.2893	−0.0616	−27.0800	Yes	No	No
z_Sensory_NCV	−0.3547	−0.2314	−0.1233	−34.7700	Yes	Yes	No
z_age	−0.4679	−0.3493	−0.1186	−25.3400	Yes	Yes	No
Upper = standardized reproduced upper CI; boot.Upper = boostrapped standardized reproduced upper CI; Upper_diff = change in Upper - boot.Upper; %_change = percentage difference, percentage changes were truncated at ±1000%; Diff_10% = difference ≥10% ; Diff_0.1 and Diff_0.2 = absolute difference ≥0.1 and ≥0.2, respectively.

Change in Range of 95% confidence interval

Term	Range	boot.Range	Range_Diff	%_Diff	Diff_10%	Diff_0.1	Diff_0.2
Intercept	0.4553	0.5725	0.1172	25.7300	Yes	Yes	No
z_Sensory_NCV	0.4996	0.6539	0.1543	30.9000	Yes	Yes	No
z_age	0.4996	0.6936	0.1941	38.8400	Yes	Yes	No
Range = standardized reproduced CI range; boot.B = boostrapped standardized reproduced CI range; Range_diff = change in CI Range ; %_change = percentage difference, percentage changes were truncated at ±1000%, Diff_10% = difference ≥10% ; Diff_0.1 and Diff_0.2 = absolute difference ≥0.1 and ≥0.2, respectively.

Change in p-value significance and regression coefficient direction

Term	p-value	boot.p-value	changep	SigChangeDirection
Intercept	1.0000	0.9951	0.0049	Remains non-sig, B same direction
z_Sensory_NCV	<0.001	<0.001	−0.0002	Remains sig, B same direction
z_age	<0.001	<0.001	−0.0002	Remains sig, B same direction
p-value = standardized reproduced p-value; boot.p-value = boostrapped standardized reproduced p-value; changep = change in p-value - boot.p-value; SigChangeDirection = statistical significance and B change between reproduced and bootstrapped model.

Check distribution of bootstrap estimates

Conclusions based on the bootstrapped model

Substantial correlation of error residuals was observed as the study design had repeated measures with the unadjusted ICC being 0.309 and the AIC was for the linear mixed model was 5 units lower than the linear model. The linear mixed model was bootstrapped, and the CI range standardized differences for Sensory_NCV was 0.1543, for age (0.1941) when compared to the linear model and was not inferentially reproducible.