Paper 30: Smoking and other determinants of bone turnover

Author

Lee Jones - Senior Biostatistician - Statistical Review

Published

March 15, 2026

References

Jorde R, Stunes AK, Kubiak J, Grimnes G, Thorsby PM, Syversen U (2019) Smoking and other determinants of bone turnover. PLoS ONE 14 (11): e0225539. https://doi.org/10.1371/journal.pone.0225539

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.n Criterion (BIC). When the Gaussian distribution was not appropriate, alternative distributions were considered, and model fit was evaluated using AIC and BIC.

Results from the reproduction of the Jorde 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 paper is an exploratory analysis of factors associated with bone turnover (bone resorption and formation). Stepwise linear regression was used to explore turnover biomarker markers. The authors checked for normality of continuous dependent variables. Linearity is not discussed, but scatterplots are shown. Tables were labelled with the tests used. Regression coefficients are not interpreted, conclusions are made in terms of direction and statistical significance, and there is no discussion of false positives with multiple outcomes. Collinearity was not mentioned.

Data availability and software used

Data were provided in the Supporting Information as a wide-format SPSS (.sav) file containing an embedded data dictionary. Regression analyses were performed using SPSS.

Regression sample

Seven linear regression models were fitted to evaluate predictors of bone mineral density and related biomarkers. Sex, age, BMI, and smoking status were included a priori in all models, while metabolic and biochemical markers were evaluated as candidate covariates using stepwise regression. Three outcomes were randomly selected for further evaluation: log-transformed cross-linked telopeptide of type 1 collagen (lg-CTX-1), tumour necrosis factor-α (TNF-α), and log-transformed osteoprotegerin (lg-OPG).

Computational reproducibility results

This paper were mostly computationally reproduced. The authors reported adjusted R² but when compared this corresponded to the unadjusted R². There were minor discrepancies in the second and third models attributable to typographic errors in the sex variable related to incorrect decimal point placement. The paper was straightforward to reproduce, as the data were well formatted in SPSS and required no additional data cleaning or reformatting.

Inferential reproducibility results

The models assessed for inferential reproducibility were reproducible overall. Only minor violations of model assumptions were identified, including homoscedasticity, linearity, and normality. Statistically significant quadratic terms were observed for the lg-CTX-1 and lg-OPG models; however, diagnostic plots and model fit statistics did not provide substantive support for meaningful non-linear relationships. The study design appeared independent; however, the lg-CTX-1 model showed a significant Durbin–Watson test. Further inspection suggested that this was likely due to post-randomisation sorting of control and intervention cases rather than true autocorrelation. All three models had statistically significant Shapiro–Wilk tests for residual normality; however, Q–Q plots and the Kolmogorov–Smirnov test indicated approximately normally distributed residuals. Mild heteroscedasticity was observed for the TNF-α and lg-OPG models. Bootstrapped comparisons showed that although some statistics differed by 10% or more, these changes were not meaningful, with differences in standardized regression coefficients remaining below 0.1. The direction of effects and statistical significance were consistent between the reproduced and bootstrapped models.

Recommended Changes

Correct typographic errors in the sex variable related to incorrect decimal point placement.
To facilitate assessment of independence and autocorrelation, authors are encouraged to provide datasets in the original order of randomisation, or to explicitly document any post-randomisation sorting applied to the data.
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.
Ensure complete reporting of the stepwise model selection procedure, including the direction of selection (forward, backward, or bidirectional) and the criteria for variable entry and removal.
Ensure adjusted R² is reported for multivariable models.
Place greater emphasis on the magnitude and practical relevance of regression coefficients, including their confidence intervals, rather than focusing predominantly on statistical significance.

Model 1

Model results for lg.ctx

Term	B	Lower	Upper	p-value
Intercept
sex:
Male – Female	−0.006	−0.045	0.033	0.764
age.years	0.001	0.000	0.003	0.126
BMI.base.kg.m2	−0.008	−0.011	−0.004	<0.001
smoking.status:
Smoker – Non-smoker	−0.044	−0.085	−0.004	0.032
Creatinine.umol.L.base	0.003	0.001	0.004	0.002
PTH.pmol.L.base	0.010	0.002	0.018	0.020
Calcium.mmol.L.base	0.264	0.041	0.486	0.020
SE = Standard error; Lower = lower confidence interval; Upper = upper confidence interval.

Fit statistics for lg.ctx

R	R2	R2Adj	AIC	RMSE	F	DF1	DF2	p-value
		0.134
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 lg.ctx

Term	SS	DF	MS	F	p-value
sex
age.years
BMI.base.kg.m2
smoking.status
Creatinine.umol.L.base
PTH.pmol.L.base
Calcium.mmol.L.base
Residuals
SS = Sum of Squares; DF = Degrees of freedom; MS = Mean Square.

Model results for lg.ctx

Term	B	SE	Lower	Upper	t	p-value
Intercept	−1.156	0.263	−1.674	−0.639	−4.393	<0.001
sex:
Male – Female	−0.006	0.020	−0.045	0.033	−0.300	0.7644
age.years	0.001	0.001	−0.000	0.003	1.535	0.1257
BMI.base.kg.m2	−0.008	0.002	−0.011	−0.004	−4.553	<0.001
smoking.status:
Smoker – Non-smoker	−0.044	0.021	−0.085	−0.004	−2.155	0.0318
Creatinine.umol.L.base	0.003	0.001	0.001	0.004	3.058	0.0024
PTH.pmol.L.base	0.010	0.004	0.002	0.018	2.344	0.0196
Calcium.mmol.L.base	0.264	0.113	0.041	0.486	2.330	0.0203
SE = Standard error; Lower = lower confidence interval; Upper = upper confidence interval.

Fit statistics for lg.ctx

R	R2	R2Adj	AIC	RMSE	F	DF1	DF2	p-value
0.365	0.134	0.118	−319.955	0.159	8.721	7	396	<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 lg.ctx

Term	SS	DF	MS	F	p-value
sex	0.002	1	0.002	0.090	0.7644
age.years	0.061	1	0.061	2.355	0.1257
BMI.base.kg.m2	0.536	1	0.536	20.731	<0.001
smoking.status	0.120	1	0.120	4.642	0.0318
Creatinine.umol.L.base	0.242	1	0.242	9.354	0.0024
PTH.pmol.L.base	0.142	1	0.142	5.495	0.0196
Calcium.mmol.L.base	0.140	1	0.140	5.427	0.0203
Residuals	10.247	396	0.026
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
sex
age.years	−0.536	0.5923	No linearity violation
BMI.base.kg.m2	1.893	0.0590	No linearity violation
smoking.status
Creatinine.umol.L.base	−0.445	0.6566	No linearity violation
PTH.pmol.L.base	−0.589	0.5559	No linearity violation
Calcium.mmol.L.base	−1.977	0.0488	Linearity violation
Tukey test	1.145	0.2521	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

Although the quadratic term for baseline calcium (mmol/L) was statistically significant, there was no clear evidence of curvature in the residual plots. In addition, no systematic patterns were observed in the fitted-versus-residual plots.

Testing for homoscedasticity

Statistic	p-value	Parameter	Method
10.517	0.1611	7	studentized Breusch-Pagan test

Homoscedasticity results

The studentized Breusch-Pagan test supports homoscedasticity.
There is no distinct funnelling pattern observed, supporting homoscedasticity of residuals.

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

Term	N	Mean	SD	Median	Min	Max	Skewness	Kurtosis
lg.ctx	404	−0.463	0.171	−0.469	−0.959	0.534	0.328	2.288
sex*	404	1.525	0.500	2.000	1.000	2.000	−0.099	−1.995
age.years	404	51.817	8.666	50.000	40.000	79.000	0.741	−0.103
BMI.base.kg.m2	404	27.759	4.909	27.035	18.456	48.847	0.919	1.430
smoking.status*	404	1.213	0.410	1.000	1.000	2.000	1.398	−0.047
Creatinine.umol.L.base	404	71.252	12.387	70.000	41.000	120.000	0.509	0.399
PTH.pmol.L.base	404	6.731	2.020	6.400	3.100	17.800	1.362	3.289
Calcium.mmol.L.base	404	2.273	0.073	2.270	2.060	2.530	0.139	0.169
.fitted	404	−0.463	0.063	−0.458	−0.699	−0.265	−0.303	0.763
.resid	404	−0.000	0.159	0.006	−0.462	0.906	0.333	2.094
.leverage	404	0.020	0.012	0.017	0.006	0.107	2.490	10.176
.sigma	404	0.161	0.000	0.161	0.154	0.161	−9.912	144.001
.cooksd	404	0.003	0.006	0.001	0.000	0.085	8.797	119.003
.std.resid	404	−0.000	1.002	0.036	−2.885	5.691	0.332	2.086
dfb.1_	404	−0.000	0.050	0.000	−0.252	0.196	−0.408	4.496
dfb.sxMl	404	−0.000	0.053	0.000	−0.437	0.191	−1.164	12.223
dfb.ag.y	404	0.000	0.050	0.000	−0.276	0.242	−0.322	5.244
dfb.BMI.	404	0.000	0.054	−0.001	−0.421	0.219	−1.142	13.187
dfb.sm.S	404	−0.000	0.052	−0.000	−0.243	0.210	−0.315	5.248
dfb.Cr..L.	404	0.000	0.048	0.000	−0.209	0.281	0.263	5.217
dfb.PTH.	404	−0.000	0.054	0.000	−0.314	0.269	−0.789	10.220
dfb.Cl..L.	404	−0.000	0.051	0.000	−0.226	0.298	0.637	5.739
dffit	404	−0.001	0.146	0.004	−0.415	0.858	0.314	2.822
cov.r	404	1.021	0.038	1.028	0.526	1.115	−6.309	73.681
* 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 studentised 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.sxMl	dfb.ag.y	dfb.BMI.	dfb.sm.S	dfb.Cr..L.	dfb.PTH.	dfb.Cl..L.	dffit	cov.r
90	2.952	0.008	0.008	−0.022	−0.134	0.030	0.097	−0.053	0.014	0.054	0.001	0.259	0.864
246	1.346	0.070	0.017	0.001	0.008	−0.079	0.069	0.203	0.108	0.269	−0.054	0.369	1.058
185	−1.092	0.107	0.018	0.196	0.004	0.018	−0.018	0.013	0.080	−0.313	−0.187	−0.377	1.115
267	−1.704	0.056	0.021	0.161	−0.036	0.116	0.036	−0.213	−0.061	−0.314	−0.143	−0.415	1.019
130	5.932	0.021	0.085	−0.211	−0.437	−0.075	−0.421	−0.137	0.281	−0.249	0.298	0.858	0.526
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

Two observations had studentized residuals near or above 3. Both points had low leverage and small Cook’s distance, with DFBETAS and DFFITS within conventional ranges. However, their COVRATIO values were substantially < 1, suggesting little effect on point estimates but potential inflation of standard errors (wider confidence intervals).

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
0.030	0.8513	Asymptotic one-sample Kolmogorov-Smirnov test

Statistic	p-value	Method
0.982	<0.001	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 looks roughly normal.

Assessing collinearity with VIF

Term	VIF	Tolerance
sex	1.545	0.647
age.years	1.034	0.967
BMI.base.kg.m2	1.072	0.933
smoking.status	1.114	0.898
Creatinine.umol.L.base	1.636	0.611
PTH.pmol.L.base	1.094	0.914
Calcium.mmol.L.base	1.054	0.949
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
0.367	1.261	<0.001

Independence results

The Durbin–Watson test suggests there are auto-correlation issues.
The Durbin–Watson test was used to assess first-order autocorrelation, noting that the test assumes observations are ordered sequentially. Apparent violations may therefore arise if the dataset has been sorted after randomisation rather than reflecting true serial correlation. In this dataset, observations appear in blocks of approximately 22 control cases followed by intervention cases. This pattern may reflect post-randomisation sorting of the file, but could also indicate issues with the randomisation process itself. Consequently, results from the Durbin–Watson test were interpreted with caution.
The study design seems to be independent.

Assumption conclusions

Visual inspection of Q–Q plots and results from the Kolmogorov–Smirnov test suggested that residuals were approximately normally distributed, although the Shapiro–Wilk test indicated some departure from normality. The study design appeared independent; however, the model showed a statistically significant Durbin–Watson test. Further inspection suggested that this was likely due to block ordering of control and intervention cases resulting from post-randomisation sorting of the dataset rather than true autocorrelation.

No substantial deviations from homoscedasticity were observed. Outlier diagnostics indicated that point estimates were unlikely to be materially influenced by individual observations; however, confidence-interval widths may be affected and warrant further investigation. Minor departures from linearity were detected by formal significance tests, but no clear curvature was evident in the residual plots. Sensitivity analyses comparing linear and non-linear model specifications are therefore recommended.

Forest plot showing original and reproduced coefficients and 95% confidence intervals for lg.ctx

Change in regression coefficients

term	O_B	R_B	Change.B	reproduce.B
Intercept		−1.1563
sex:
Male – Female	−0.006	−0.0060	0.0000	Reproduced
age.years	0.001	0.0014	0.0004	Reproduced
BMI.base.kg.m2	−0.008	−0.0077	0.0003	Reproduced
smoking.status:
Smoker – Non-smoker	−0.044	−0.0445	−0.0005	Reproduced
Creatinine.umol.L.base	0.003	0.0025	−0.0005	Reproduced
PTH.pmol.L.base	0.010	0.0097	−0.0003	Reproduced
Calcium.mmol.L.base	0.264	0.2637	−0.0003	Reproduced
O_B = original B; R_B = reproduced B; Change.B = change in R_B - O_B; Reproduce.B = B reproduced.

Change in lower 95% confidence intervals for coefficients

term	O_lower	R_lower	Change.lci	Reproduce.lower
Intercept		−1.6737
sex:
Male – Female	−0.045	−0.0451	−0.0001	Reproduced
age.years	0.000	−0.0004	−0.0004	Reproduced
BMI.base.kg.m2	−0.011	−0.0110	0.0000	Reproduced
smoking.status:
Smoker – Non-smoker	−0.085	−0.0850	0.0000	Reproduced
Creatinine.umol.L.base	0.001	0.0009	−0.0001	Reproduced
PTH.pmol.L.base	0.002	0.0016	−0.0004	Reproduced
Calcium.mmol.L.base	0.041	0.0412	0.0002	Reproduced
O_lower = original lower confidence interval; R_lower = reproduced lower confidence interval; change.lci = change in R_lower - O_lower; Reproduce.lower = lower confidence interval reproduced.

Change in upper 95% confidence intervals for coefficients

term	O_upper	R_upper	Change.uci	Reproduce.upper
Intercept		−0.6389
sex:
Male – Female	0.033	0.0332	0.0002	Reproduced
age.years	0.003	0.0033	0.0003	Reproduced
BMI.base.kg.m2	−0.004	−0.0044	−0.0004	Reproduced
smoking.status:
Smoker – Non-smoker	−0.004	−0.0039	0.0001	Reproduced
Creatinine.umol.L.base	0.004	0.0042	0.0002	Reproduced
PTH.pmol.L.base	0.018	0.0179	−0.0001	Reproduced
Calcium.mmol.L.base	0.486	0.4863	0.0003	Reproduced
O_upper = original upper confidence interval; R_upper = reproduced upper confidence interval; change.uci = change in R_upper - O_upper; Reproduce.upper = upper confidence interval reproduced.

Change in adjusted R²

O_R2Adj	R_R2Adj	Change.R2Adj	Reproduce.R2Adj
0.134	0.1182	−0.0158	Not Reproduced
O_R2Adj = original R2 Adjusted; R_R2Adj = reproduced R2 Adjusted; Change.R2Adj = change in R2Adj (R2Adj - O_R2Adj); Reproduce.R2Adj = R2 Adjusted reproduced.

Change in p-values

Term	O_p	R_p	Change.p	Reproduce.p	SigChangeDirection
Intercept		<0.001
sex:
Male – Female	0.764	0.7644	0.0004	Reproduced	Remains non-sig, B same direction
age.years	0.126	0.1257	−0.0003	Reproduced	Remains non-sig, B same direction
BMI.base.kg.m2	<0.001	<0.001	0.0000	Reproduced	Remains sig, B same direction
smoking.status:
Smoker – Non-smoker	0.032	0.0318	−0.0002	Reproduced	Remains sig, B same direction
Creatinine.umol.L.base	0.002	0.0024	0.0004	Reproduced	Remains sig, B same direction
PTH.pmol.L.base	0.020	0.0196	−0.0004	Reproduced	Remains sig, B same direction
Calcium.mmol.L.base	0.020	0.0203	0.0003	Reproduced	Remains sig, B same direction
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.

Bland Altman plot showing differences between original and reproduced p-values for lg.ctx

Results for p-values

The mean difference for p-values in this model was close to zero as they were all reproduced.

Conclusion computational reproducibility

This model was mostly computationally reproduced. Regression coefficients and confidence intervals were reproduced. However, the authors reported adjusted R² but when compared this corresponded to the unadjusted R². P-values were reproduced and had the same interpretation, and regression coefficients did not change direction.

Methods

The model was successfully reproduced; however, there were indications that the residuals may not be normally distributed. Diagnostic checks suggested a minor deviation from linearity for one variable. A quadratic (squared) term was fitted, and AIC and BIC were used to compare model fit with the linear specification. If the linear model was retained, to further verify the findings, non-parametric BCa bootstrapping (10,000 resamples) was used to obtain standardised coefficients and 95% confidence intervals (CIs), and CI widths were examined. Percentage and absolute changes in estimates and confidence-interval bounds relative to the linear model were summarised using thresholds of 10% change and standardised coefficient differences of <0.10 and <0.20. Coefficient directions were assessed for concordance, and the stability of statistical significance was evaluated.

Results

Comparison of linear and squared model

Characteristic	Linear			Squared Calcium
Characteristic	Beta	95% CI¹	p-value	Beta	95% CI¹	p-value
(Intercept)	0.0706	-0.0822, 0.2233	0.364	0.1396	-0.0273, 0.3066	0.101
sex
Female	—	—		—	—
Male	-0.0348	-0.2630, 0.1934	0.764	-0.0510	-0.2790, 0.1769	0.660
z_age.years	0.0728	-0.0205, 0.1661	0.126	0.0688	-0.0242, 0.1618	0.147
z_BMI.base.kg.m2	-0.2203	-0.3154, -0.1252	<0.001	-0.2206	-0.3153, -0.1258	<0.001
smoking.status
Non-smoker	—	—		—	—
Smoker	-0.2591	-0.4955, -0.0227	0.032	-0.2443	-0.4803, -0.0083	0.042
z_Creatinine.umol.L.base	0.1825	0.0652, 0.2997	0.002	0.1768	0.0598, 0.2938	0.003
z_PTH.pmol.L.base	0.1151	0.0186, 0.2117	0.020	0.1271	0.0302, 0.2240	0.010
z_Calcium.mmol.L.base	0.1121	0.0175, 0.2066	0.020	0.1230	0.0282, 0.2179	0.011
z_Calcium.mmol.L.base2				-0.0642	-0.1281, -0.0004	0.049
¹ CI = Confidence Interval

Comparison of linear and squared model fit

Model_type	R2adj	sigma	df	logLik	AIC	BIC	deviance	df.residual	nobs
Linear	0.118	0.93747	7	−543.13	1,104.3	1,140.3	348.03	396	404
Squared Calcium	0.125	0.93405	8	−541.14	1,102.3	1,142.3	344.62	395	404

Linearity conclusion

After re-running the models and re-evaluating assumptions, statistically significant deviations from linearity were identified. However, these did not improve model fit, as indicated by similar AIC and Adjusted R² values. Visual inspection suggested that the apparent non-linearity arose from sparse data at the higher end of the predictor ranges rather than a meaningful improvement in model performance.

Bootstrapped results

As the inclusion of non-linear terms did not improve model fit, the original model was retained. Further model assessment was conducted using Non-parametric BCa bootstrapping with 10,000 resamples.

Change in regression coefficients

Term	B	boot.B	B_diff	%_Diff	Diff_10%	Diff_0.1	Diff_0.2
Intercept	0.0706	0.0718	−0.0013	−1.8200	No	No	No
sexMale	−0.0348	−0.0360	0.0012	3.4600	No	No	No
z_age.years	0.0728	0.0729	−0.0001	−0.1400	No	No	No
z_BMI.base.kg.m2	−0.2203	−0.2216	0.0013	0.6000	No	No	No
smoking.statusSmoker	−0.2591	−0.2611	0.0020	0.7800	No	No	No
z_Creatinine.umol.L.base	0.1825	0.1820	0.0004	0.2400	No	No	No
z_PTH.pmol.L.base	0.1151	0.1152	−0.0001	−0.0700	No	No	No
z_Calcium.mmol.L.base	0.1121	0.1133	−0.0012	−1.0900	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	−0.0822	−0.0994	0.0173	21.0300	Yes	No	No
sexMale	−0.2630	−0.2823	0.0193	7.3400	No	No	No
z_age.years	−0.0205	−0.0172	−0.0033	−15.8900	Yes	No	No
z_BMI.base.kg.m2	−0.3154	−0.3219	0.0065	2.0700	No	No	No
smoking.statusSmoker	−0.4955	−0.5006	0.0052	1.0400	No	No	No
z_Creatinine.umol.L.base	0.0652	0.0720	−0.0069	−10.5200	Yes	No	No
z_PTH.pmol.L.base	0.0186	0.0161	0.0025	13.3300	Yes	No	No
z_Calcium.mmol.L.base	0.0175	0.0194	−0.0019	−11.0300	Yes	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	0.2233	0.2532	−0.0299	−13.3900	Yes	No	No
sexMale	0.1934	0.1961	−0.0027	−1.4000	No	No	No
z_age.years	0.1661	0.1636	0.0025	1.4900	No	No	No
z_BMI.base.kg.m2	−0.1252	−0.1250	−0.0002	−0.1200	No	No	No
smoking.statusSmoker	−0.0227	−0.0193	−0.0034	−14.8700	Yes	No	No
z_Creatinine.umol.L.base	0.2997	0.2948	0.0050	1.6600	No	No	No
z_PTH.pmol.L.base	0.2117	0.2180	−0.0064	−3.0200	No	No	No
z_Calcium.mmol.L.base	0.2066	0.2114	−0.0048	−2.3200	No	No	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	0.3054	0.3526	0.0472	15.4500	Yes	No	No
sexMale	0.4564	0.4785	0.0220	4.8300	No	No	No
z_age.years	0.1865	0.1808	−0.0057	−3.0700	No	No	No
z_BMI.base.kg.m2	0.1902	0.1969	0.0067	3.5200	No	No	No
smoking.statusSmoker	0.4728	0.4813	0.0085	1.8000	No	No	No
z_Creatinine.umol.L.base	0.2346	0.2227	−0.0118	−5.0400	No	No	No
z_PTH.pmol.L.base	0.1931	0.2019	0.0089	4.5900	No	No	No
z_Calcium.mmol.L.base	0.1891	0.1920	0.0029	1.5100	No	No	No
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	0.3643	0.4254	−0.0611	Remains non-sig, B same direction
sexMale	0.7644	0.7690	−0.0045	Remains non-sig, B same direction
z_age.years	0.1257	0.1128	0.0129	Remains non-sig, B same direction
z_BMI.base.kg.m2	<0.001	<0.001	−0.0000	Remains sig, B same direction
smoking.statusSmoker	0.0318	0.0349	−0.0031	Remains sig, B same direction
z_Creatinine.umol.L.base	0.0024	0.0013	0.0011	Remains sig, B same direction
z_PTH.pmol.L.base	0.0196	0.0257	−0.0061	Remains sig, B same direction
z_Calcium.mmol.L.base	0.0203	0.0184	0.0019	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 the bootstrapped model

This model was inferentially reproducible. While some statistics changed by 10% or more, these differences were not meaningful with a change in standardized regression coefficients of less than 0.1. The direction of effects and statistical significance remained consistent between the reproduced and bootstrapped models.

Model 2

Model results for TNF.pg.ml.base

Term	B	Lower	Upper	p-value
Intercept
sex:
Male – Female	0.035	0.144	0.465	<0.001
age.years	0.001	−0.008	0.010	0.837
BMI.base.kg.m2	0.014	−0.006	0.034	0.176
smoking.status:
Smoker – Non-smoker	0.085	−0.102	0.272	0.370
HOMA.IR.base	0.043	0.005	0.081	0.027
SE = Standard error; Lower = lower confidence interval; Upper = upper confidence interval.

Fit statistics for TNF.pg.ml.base

R	R2	R2Adj	AIC	RMSE	F	DF1	DF2	p-value
		0.095
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 TNF.pg.ml.base

Term	SS	DF	MS	F	p-value
sex
age.years
BMI.base.kg.m2
smoking.status
HOMA.IR.base
Residuals
SS = Sum of Squares; DF = Degrees of freedom; MS = Mean Square.

Model results TNF.pg.ml.base

Term	B	SE	Lower	Upper	t	p-value
Intercept	1.640	0.371	0.911	2.369	4.422	<0.001
sex:
Male – Female	0.305	0.082	0.144	0.465	3.736	<0.001
age.years	0.001	0.005	−0.008	0.010	0.206	0.8366
BMI.base.kg.m2	0.014	0.010	−0.006	0.034	1.354	0.1764
smoking.status:
Smoker – Non-smoker	0.085	0.095	−0.102	0.272	0.897	0.3704
HOMA.IR.base	0.043	0.019	0.005	0.081	2.225	0.0266
SE = Standard error; Lower = lower confidence interval; Upper = upper confidence interval.

Model fit for TNF.pg.ml.base

R	R2	R2Adj	AIC	RMSE	F	DF1	DF2	p-value
0.308	0.095	0.084	954.798	0.773	8.388	5	399	<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 TNF.pg.ml.base

Term	SS	DF	MS	F	p-value
sex	8.465	1	8.465	13.956	<0.001
age.years	0.026	1	0.026	0.043	0.8366
BMI.base.kg.m2	1.112	1	1.112	1.834	0.1764
smoking.status	0.488	1	0.488	0.804	0.3704
HOMA.IR.base	3.003	1	3.003	4.952	0.0266
Residuals	242.008	399	0.607
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
sex
age.years	−1.897	0.0586	No linearity violation
BMI.base.kg.m2	−1.459	0.1454	No linearity violation
smoking.status
HOMA.IR.base	−1.512	0.1314	No linearity violation
Tukey test	−1.333	0.1826	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
9.455	0.0922	5	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
TNF.pg.ml.base	405	2.400	0.814	2.360	0.030	5.320	0.320	0.484
sex*	405	1.523	0.500	2.000	1.000	2.000	−0.094	−1.996
age.years	405	51.840	8.667	50.000	40.000	79.000	0.734	−0.116
BMI.base.kg.m2	405	27.774	4.912	27.053	18.456	48.847	0.910	1.403
smoking.status*	405	1.212	0.409	1.000	1.000	2.000	1.402	−0.036
HOMA.IR.base	405	3.476	2.701	2.738	0.444	21.207	2.469	8.915
.fitted	405	2.400	0.251	2.401	1.997	3.446	0.743	1.156
.resid	405	−0.000	0.774	−0.037	−2.410	2.528	0.223	0.435
.leverage	405	0.015	0.011	0.012	0.006	0.128	4.613	33.271
.sigma	405	0.779	0.002	0.779	0.769	0.780	−3.066	11.679
.cooksd	405	0.003	0.006	0.001	0.000	0.080	7.017	78.842
.std.resid	405	−0.000	1.002	−0.048	−3.116	3.271	0.224	0.431
dfb.1_	405	0.000	0.047	0.001	−0.228	0.178	−0.348	3.974
dfb.sxMl	405	−0.000	0.050	0.001	−0.193	0.197	−0.142	1.581
dfb.ag.y	405	−0.000	0.054	0.001	−0.276	0.291	−0.243	6.193
dfb.BMI.	405	0.000	0.048	−0.001	−0.216	0.284	0.519	6.029
dfb.sm.S	405	0.000	0.051	0.000	−0.197	0.278	0.867	6.692
dfb.HOMA	405	−0.000	0.055	−0.000	−0.259	0.552	2.120	29.521
dffit	405	−0.003	0.128	−0.005	−0.431	0.696	0.433	2.588
cov.r	405	1.015	0.026	1.021	0.876	1.155	−1.774	9.711
* categorical variable

Cooks threshold

DFFIT threshold

DFBETA threshold

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

Observations of interest identified by the influence plot

ID	StudRes	Leverage	CookD	dfb.1_	dfb.sxMl	dfb.ag.y	dfb.BMI.	dfb.sm.S	dfb.HOMA	dffit	cov.r
363	−0.709	0.128	0.012	−0.021	0.048	−0.021	0.090	0.006	−0.248	−0.271	1.155
318	3.154	0.009	0.015	0.121	0.169	−0.027	−0.124	−0.099	−0.039	0.305	0.884
217	3.312	0.016	0.028	0.150	0.071	−0.207	−0.068	−0.061	0.255	0.419	0.876
213	−1.491	0.077	0.031	0.025	0.026	0.108	−0.076	0.004	−0.259	−0.431	1.064
236	2.304	0.084	0.080	0.127	−0.081	−0.191	−0.129	0.278	0.552	0.696	1.023
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

Two observations had studentized residuals above 3. Both points had low leverage and small Cook’s distance, with DFBETAS and DFFITS within conventional ranges. However, their COVRATIO values were substantially < 1, suggesting little effect on point estimates but potential inflation of standard errors (wider confidence intervals).

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
0.044	0.4085	Asymptotic one-sample Kolmogorov-Smirnov test

Statistic	p-value	Method
0.993	0.0405	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 looks roughly normal.

Assessing collinearity with VIF

Term	VIF	Tolerance
sex	1.109	0.901
age.years	1.027	0.974
BMI.base.kg.m2	1.680	0.595
smoking.status	1.010	0.990
HOMA.IR.base	1.805	0.554
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
0.007	1.983	0.7860

Independence results

The Durbin–Watson test suggests there are no auto-correlation issues.
The study design seems to be independent.

Assumption conclusions

The model was found to meet the assumptions of linearity, normality, and independence with no substantial outliers identified. While the model was not statistically heteroscedastic, it visually displayed signs of mild heteroscedasticity, and could be examined further.

Forest plot showing Original and Reproduced coefficients and 95% confidence intervals for TNF.pg.ml.base

Change in regression coefficients

term	O_B	R_B	Change.B	reproduce.B
Intercept		1.6402
sex:
Male – Female	0.035	0.3049	0.2699	Not Reproduced
age.years	0.001	0.0009	−0.0001	Reproduced
BMI.base.kg.m2	0.014	0.0138	−0.0002	Reproduced
smoking.status:
Smoker – Non-smoker	0.085	0.0853	0.0003	Reproduced
HOMA.IR.base	0.043	0.0429	−0.0001	Reproduced
O_B = original B; R_B = reproduced B; Change.B = change in R_B - O_B; Reproduce.B = B reproduced.

Change in lower 95% confidence intervals for coefficients

term	O_lower	R_lower	Change.lci	Reproduce.lower
Intercept		0.9111
sex:
Male – Female	0.144	0.1444	0.0004	Reproduced
age.years	−0.008	−0.0080	0.0000	Reproduced
BMI.base.kg.m2	−0.006	−0.0063	−0.0003	Reproduced
smoking.status:
Smoker – Non-smoker	−0.102	−0.1017	0.0003	Reproduced
HOMA.IR.base	0.005	0.0050	0.0000	Reproduced
O_lower = original lower confidence interval; R_lower = reproduced lower confidence interval; change.lci = change in R_lower - O_lower; Reproduce.lower = lower confidence interval reproduced.

Change in upper 95% confidence intervals for coefficients

term	O_upper	R_upper	Change.uci	Reproduce.upper
Intercept		2.3693
sex:
Male – Female	0.465	0.4653	0.0003	Reproduced
age.years	0.010	0.0098	−0.0002	Reproduced
BMI.base.kg.m2	0.034	0.0339	−0.0001	Reproduced
smoking.status:
Smoker – Non-smoker	0.272	0.2722	0.0002	Reproduced
HOMA.IR.base	0.081	0.0808	−0.0002	Reproduced
O_upper = original upper confidence interval; R_upper = reproduced upper confidence interval; change.uci = change in R_upper - O_upper; Reproduce.upper = upper confidence interval reproduced.

Change in adjusted R²

O_R2Adj	R_R2Adj	Change.R2Adj	Reproduce.R2Adj
0.095	0.0838	−0.0112	Not Reproduced
O_R2Adj = original R2 Adjusted; R_R2Adj = reproduced R2 Adjusted; Change.R2Adj = change in R2Adj (R2Adj - O_R2Adj); Reproduce.R2Adj = R2 Adjusted reproduced.

Change in p-values

Term	O_p	R_p	Change.p	Reproduce.p	SigChangeDirection
Intercept		<0.001
sex:
Male – Female	<0.001	<0.001	0.0000	Reproduced	Remains sig, B same direction
age.years	0.837	0.8366	−0.0004	Reproduced	Remains non-sig, B same direction
BMI.base.kg.m2	0.176	0.1764	0.0004	Reproduced	Remains non-sig, B same direction
smoking.status:
Smoker – Non-smoker	0.370	0.3704	0.0004	Reproduced	Remains non-sig, B same direction
HOMA.IR.base	0.027	0.0266	−0.0004	Reproduced	Remains sig, B same direction
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.

Bland Altman plot showing differences between original and reproduced p-values for TNF.pg.ml.base

Results for p-values

The mean difference for p-values in this model was close to zero as they were all reproduced.

Conclusion computational reproducibility

This model was mostly computationally reproduced with a typographic error in the sex coefficient. There was also a reporting error for adjusted R² which corresponded to the unadjusted R². P-values were reproduced and had the same interpretation, and regression coefficients did not change direction.

Methods

The model was successfully reproduced; however, it may exhibit slight heteroscedasticity. To further verify the findings, bootstrapped standardized regression coefficients and their 95% confidence intervals were examined. Percentage and absolute changes in estimates and confidence-interval bounds relative to the linear model were summarised using thresholds of 10% change and standardised coefficient differences of <0.10 and <0.20. Coefficient directions were assessed for concordance, and the stability of statistical significance was evaluated. Wild bootstrapping was used to account for potential heteroscedasticity.

Results

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	−0.2198	−0.2220	0.0022	0.9900	No	No	No
sexMale	0.3751	0.3780	−0.0029	−0.7600	No	No	No
z_age.years	0.0100	0.0108	−0.0008	−8.0200	No	No	No
z_BMI.base.kg.m2	0.0837	0.0841	−0.0004	−0.4700	No	No	No
smoking.statusSmoker	0.1049	0.1051	−0.0002	−0.2300	No	No	No
z_HOMA.IR.base	0.1424	0.1407	0.0017	1.1800	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	−0.3661	−0.3637	−0.0025	−0.6800	No	No	No
sexMale	0.1777	0.1815	−0.0038	−2.1400	No	No	No
z_age.years	−0.0849	−0.0894	0.0045	5.3200	No	No	No
z_BMI.base.kg.m2	−0.0378	−0.0322	−0.0055	−14.6800	Yes	No	No
smoking.statusSmoker	−0.1251	−0.1209	−0.0042	−3.3400	No	No	No
z_HOMA.IR.base	0.0166	0.0119	0.0047	28.4800	Yes	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	−0.0735	−0.0790	0.0055	7.4600	No	No	No
sexMale	0.5725	0.5715	0.0011	0.1800	No	No	No
z_age.years	0.1048	0.1084	−0.0036	−3.4600	No	No	No
z_BMI.base.kg.m2	0.2052	0.1986	0.0065	3.1800	No	No	No
smoking.statusSmoker	0.3349	0.3313	0.0035	1.0600	No	No	No
z_HOMA.IR.base	0.2683	0.2699	−0.0017	−0.6200	No	No	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	0.2926	0.2847	−0.0080	−2.7200	No	No	No
sexMale	0.3948	0.3899	−0.0049	−1.2300	No	No	No
z_age.years	0.1897	0.1979	0.0081	4.2900	No	No	No
z_BMI.base.kg.m2	0.2430	0.2309	−0.0121	−4.9700	No	No	No
smoking.statusSmoker	0.4600	0.4523	−0.0077	−1.6800	No	No	No
z_HOMA.IR.base	0.2517	0.2580	0.0064	2.5300	No	No	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	0.0033	0.0022	0.0011	Remains sig, B same direction
sexMale	<0.001	<0.001	0.0001	Remains sig, B same direction
z_age.years	0.8366	0.8307	0.0059	Remains non-sig, B same direction
z_BMI.base.kg.m2	0.1764	0.1523	0.0241	Remains non-sig, B same direction
smoking.statusSmoker	0.3704	0.3628	0.0076	Remains non-sig, B same direction
z_HOMA.IR.base	0.0266	0.0348	−0.0082	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

Conclusions based on bootstrapped model

Model 3

Model results for lg.opg

Term	B	Lower	Upper	p-value
Intercept
sex:
Male – Female	−0.012	−0.035	0.001	0.286
age.years	0.006	0.005	0.007	<0.001
BMI.base.kg.m2	−0.002	−0.005	0.001	0.237
smoking.status:
Smoker – Non-smoker	0.032	0.005	0.058	0.018
HOMA.IR.base	0.007	0.002	0.013	0.006
SE = Standard error; Lower = lower confidence interval; Upper = upper confidence interval.

Fit Statistics lg.opg

R	R2	R2Adj	AIC	RMSE	F	DF1	DF2	p-value
		0.216
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 lg.opg

Term	SS	DF	MS	F	p-value
sex
age.years
BMI.base.kg.m2
smoking.status
HOMA.IR.base
Residuals
SS = Sum of Squares; DF = Degrees of freedom; MS = Mean Square.

Model results for lg.opg

Term	B	SE	Lower	Upper	t	p-value
Intercept	2.201	0.052	2.099	2.304	42.204	<0.001
sex:
Male – Female	−0.012	0.011	−0.035	0.010	−1.069	0.2858
age.years	0.006	0.001	0.005	0.007	9.330	<0.001
BMI.base.kg.m2	−0.002	0.001	−0.005	0.001	−1.183	0.2375
smoking.status:
Smoker – Non-smoker	0.032	0.013	0.005	0.058	2.376	0.0180
HOMA.IR.base	0.007	0.003	0.002	0.013	2.740	0.0064
SE = Standard error; Lower = lower confidence interval; Upper = upper confidence interval.

Fit statistics for lg.opg

R	R2	R2Adj	AIC	RMSE	F	DF1	DF2	p-value
0.464	0.216	0.206	−634.130	0.109	21.923	5	399	<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 lg.opg

Term	SS	DF	MS	F	p-value
sex	0.014	1	0.014	1.142	0.2858
age.years	1.044	1	1.044	87.040	<0.001
BMI.base.kg.m2	0.017	1	0.017	1.400	0.2375
smoking.status	0.068	1	0.068	5.647	0.0180
HOMA.IR.base	0.090	1	0.090	7.506	0.0064
Residuals	4.786	399	0.012
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
sex
age.years	0.833	0.4055	No linearity violation
BMI.base.kg.m2	1.464	0.1439	No linearity violation
smoking.status
HOMA.IR.base	2.020	0.0441	Linearity violation
Tukey test	0.590	0.5549	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

Although the quadratic term for baseline Homa was statistically significant, there was no clear evidence of curvature in the residual plots, with spares data at the higher end of the scale. No systematic patterns were observed in the fitted-versus-residual plots.

Testing for homoscedasticity

Statistic	p-value	Parameter	Method
6.634	0.2493	5	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
lg.opg	405	2.488	0.123	2.485	2.134	2.925	0.123	0.468
sex*	405	1.523	0.500	2.000	1.000	2.000	−0.094	−1.996
age.years	405	51.840	8.667	50.000	40.000	79.000	0.734	−0.116
BMI.base.kg.m2	405	27.774	4.912	27.053	18.456	48.847	0.910	1.403
smoking.status*	405	1.212	0.409	1.000	1.000	2.000	1.402	−0.036
HOMA.IR.base	405	3.476	2.701	2.738	0.444	21.207	2.469	8.915
.fitted	405	2.488	0.057	2.477	2.389	2.668	0.727	0.034
.resid	405	−0.000	0.109	0.001	−0.312	0.409	0.213	0.632
.leverage	405	0.015	0.011	0.012	0.006	0.128	4.613	33.271
.sigma	405	0.110	0.000	0.110	0.108	0.110	−3.528	16.688
.cooksd	405	0.003	0.005	0.001	0.000	0.051	4.699	28.229
.std.resid	405	0.001	1.001	0.012	−2.876	3.754	0.218	0.634
dfb.1_	405	0.000	0.054	−0.000	−0.342	0.226	−0.573	7.539
dfb.sxMl	405	−0.000	0.049	−0.001	−0.174	0.178	−0.114	1.126
dfb.ag.y	405	−0.000	0.056	0.001	−0.202	0.398	1.112	10.890
dfb.BMI.	405	−0.000	0.047	−0.000	−0.220	0.237	0.472	5.393
dfb.sm.S	405	0.000	0.054	−0.001	−0.263	0.319	0.957	8.677
dfb.HOMA	405	0.000	0.050	−0.000	−0.215	0.440	2.149	22.198
dffit	405	0.003	0.124	0.001	−0.384	0.555	0.718	2.177
cov.r	405	1.015	0.027	1.021	0.826	1.160	−1.786	11.191
* 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.sxMl	dfb.ag.y	dfb.BMI.	dfb.sm.S	dfb.HOMA	dffit	cov.r
363	0.473	0.128	0.005	0.014	−0.032	0.014	−0.060	−0.004	0.165	0.181	1.160
19	3.817	0.009	0.022	0.163	−0.146	−0.176	0.013	−0.094	−0.107	0.369	0.826
13	3.215	0.019	0.032	0.196	−0.166	−0.202	−0.070	0.319	0.023	0.444	0.887
355	3.060	0.023	0.036	−0.342	0.175	0.398	0.155	−0.103	−0.153	0.473	0.904
236	1.838	0.084	0.051	0.101	−0.064	−0.152	−0.103	0.222	0.440	0.555	1.053
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

Three observations had studentized residuals > 3. these observations had low leverage and small Cook’s distance, with DFBETAS and DFFITS within conventional ranges. However, their COVRATIO values were substantially < 1, suggesting little effect on point estimates but potential inflation of standard errors (wider confidence intervals).

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
0.027	0.9333	Asymptotic one-sample Kolmogorov-Smirnov test

Statistic	p-value	Method
0.992	0.0279	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 looks roughly normal.

Assessing collinearity with VIF

Term	VIF	Tolerance
sex	1.109	0.901
age.years	1.027	0.974
BMI.base.kg.m2	1.680	0.595
smoking.status	1.010	0.990
HOMA.IR.base	1.805	0.554
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
−0.022	2.037	0.7780

Independence results

The Durbin–Watson test suggests there are no auto-correlation issues.
The study design seems to be independent.

Assumption conclusions

The model met the assumptions of normality and independence. However, residual plots and diagnostic tests indicated potential issues with linearity, suggesting that a non-linear relationship may not have been adequately captured, and residuals may be mildly heteroscedastic.

Forest plot showing original and reproduced coefficients and 95% confidence intervals for lg.opg

Change in regression coefficients

term	O_B	R_B	Change.B	reproduce.B
Intercept		2.2012
sex:
Male – Female	−0.012	−0.0123	−0.0003	Reproduced
age.years	0.006	0.0059	−0.0001	Reproduced
BMI.base.kg.m2	−0.002	−0.0017	0.0003	Reproduced
smoking.status:
Smoker – Non-smoker	0.032	0.0318	−0.0002	Reproduced
HOMA.IR.base	0.007	0.0074	0.0004	Reproduced
O_B = original B; R_B = reproduced B; Change.B = change in R_B - O_B; Reproduce.B = B reproduced.

Change in lower 95% confidence intervals for coefficients

term	O_lower	R_lower	Change.lci	Reproduce.lower
Intercept		2.0987
sex:
Male – Female	−0.035	−0.0348	0.0002	Reproduced
age.years	0.005	0.0047	−0.0003	Reproduced
BMI.base.kg.m2	−0.005	−0.0045	0.0005	Reproduced
smoking.status:
Smoker – Non-smoker	0.005	0.0055	0.0005	Reproduced
HOMA.IR.base	0.002	0.0021	0.0001	Reproduced
O_lower = original lower confidence interval; R_lower = reproduced lower confidence interval; change.lci = change in R_lower - O_lower; Reproduce.lower = lower confidence interval reproduced.

Change in upper 95% confidence intervals for coefficients

term	O_upper	R_upper	Change.uci	Reproduce.upper
Intercept		2.3038
sex:
Male – Female	0.001	0.0103	0.0093	Not Reproduced
age.years	0.007	0.0072	0.0002	Reproduced
BMI.base.kg.m2	0.001	0.0011	0.0001	Reproduced
smoking.status:
Smoker – Non-smoker	0.058	0.0581	0.0001	Reproduced
HOMA.IR.base	0.013	0.0128	−0.0002	Reproduced
O_upper = original upper confidence interval; R_upper = reproduced upper confidence interval; change.uci = change in R_upper - O_upper; Reproduce.upper = upper confidence interval reproduced.

Change in adjusted R²

O_R2Adj	R_R2Adj	Change.R2Adj	Reproduce.R2Adj
0.216	0.2057	−0.0103	Not Reproduced
O_R2Adj = original R2 Adjusted; R_R2Adj = reproduced R2 Adjusted; Change.R2Adj = change in R2Adj (R2Adj - O_R2Adj); Reproduce.R2Adj = R2 Adjusted reproduced.

Change in p-values

Term	O_p	R_p	Change.p	Reproduce.p	SigChangeDirection
Intercept		<0.001
sex:
Male – Female	0.286	0.2858	−0.0002	Reproduced	Remains non-sig, B same direction
age.years	<0.001	<0.001	0.0000	Reproduced	Remains sig, B same direction
BMI.base.kg.m2	0.237	0.2375	0.0005	Reproduced	Remains non-sig, B same direction
smoking.status:
Smoker – Non-smoker	0.018	0.0180	0.0000	Reproduced	Remains sig, B same direction
HOMA.IR.base	0.006	0.0064	0.0004	Reproduced	Remains sig, B same direction
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.

Bland Altman Plot showing differences between original and reproduced p-values for lg.opg

Results for p-values

The mean difference for p-values in this model was close to zero as they were all reproduced.

Conclusion computational reproducibility

This model was mostly computationally reproducible, with a typographic error in the sex upper 95% confidence interval. P-values were reproduced and had the same interpretation, and regression coefficients did not change direction. There was also a reporting error for adjusted R² which corresponded to the unadjusted R².

Methods

The model was successfully reproduced; however, there were indications that the residuals may not be normally distributed and may exhibit slight heteroscedasticity. Diagnostic checks suggested a minor deviation from linearity for one variable. A quadratic (squared) term was fitted, and AIC and BIC were used to compare model fit with the linear specification. If the linear model was retained, to further verify the findings, bootstrapped standardized regression coefficients and their 95% confidence intervals were examined. Percentage and absolute changes in estimates and confidence-interval bounds relative to the linear model were summarised using thresholds of 10% change and standardised coefficient differences of <0.10 and <0.20. Coefficient directions were assessed for concordance, and the stability of statistical significance was evaluated. Wild bootstrapping was used to account for potential heteroscedasticity.

Results

Comparison of linear and squared model

Characteristic	Linear			Squared HOMA
Characteristic	Beta	95% CI¹	p-value	Beta	95% CI¹	p-value
(Intercept)	-0.0034	-0.1397, 0.1328	0.960	-0.0551	-0.1998, 0.0896	0.455
sex
Female	—	—		—	—
Male	-0.0999	-0.2837, 0.0839	0.286	-0.0770	-0.2614, 0.1075	0.412
z_age.years	0.4191	0.3308, 0.5075	<0.001	0.4262	0.3379, 0.5144	<0.001
z_BMI.base.kg.m2	-0.0681	-0.1812, 0.0450	0.237	-0.0421	-0.1576, 0.0733	0.474
smoking.status
Non-smoker	—	—		—	—
Smoker	0.2589	0.0447, 0.4730	0.018	0.2496	0.0361, 0.4631	0.022
z_HOMA.IR.base	0.1633	0.0461, 0.2804	0.006	0.0397	-0.1279, 0.2073	0.642
z_HOMA.IR.base2				0.0419	0.0011, 0.0827	0.044
¹ CI = Confidence Interval

Comparison of linear and squared model fit

Model_type	R2adj	sigma	df	logLik	AIC	BIC	deviance	df.residual	nobs
Linear	0.206	0.89221	5	−525.45	1,064.9	1,092.9	317.62	399	405
Squared HOMA	0.212	0.88878	6	−523.39	1,062.8	1,094.8	314.39	398	405

Linearity conclusion

Bootstrapped results

As the inclusion of non-linear terms did not improve model fit, the original model was retained. Further model assessment was conducted using Wild bootstrapping with 10,000 iterations.

Change in regression coefficients

Term	B	boot.B	B_diff	%_Diff	Diff_10%	Diff_0.1	Diff_0.2
Intercept	−0.0034	−0.0036	0.0001	3.9700	No	No	No
sexMale	−0.0999	−0.1004	0.0005	0.4800	No	No	No
z_age.years	0.4191	0.4188	0.0003	0.0700	No	No	No
z_BMI.base.kg.m2	−0.0681	−0.0683	0.0002	0.3300	No	No	No
smoking.statusSmoker	0.2589	0.2603	−0.0014	−0.5600	No	No	No
z_HOMA.IR.base	0.1633	0.1634	−0.0001	−0.0900	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	−0.1397	−0.1300	−0.0097	−6.9500	No	No	No
sexMale	−0.2837	−0.2742	−0.0096	−3.3700	No	No	No
z_age.years	0.3308	0.3239	0.0069	2.0800	No	No	No
z_BMI.base.kg.m2	−0.1812	−0.1729	−0.0082	−4.5500	No	No	No
smoking.statusSmoker	0.0447	0.0353	0.0094	21.0300	Yes	No	No
z_HOMA.IR.base	0.0461	0.0520	−0.0059	−12.7200	Yes	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	0.1328	0.1184	0.0144	10.8300	Yes	No	No
sexMale	0.0839	0.0731	0.0107	12.8100	Yes	No	No
z_age.years	0.5075	0.5166	−0.0091	−1.8000	No	No	No
z_BMI.base.kg.m2	0.0450	0.0372	0.0078	17.3200	Yes	No	No
smoking.statusSmoker	0.4730	0.4841	−0.0111	−2.3500	No	No	No
z_HOMA.IR.base	0.2804	0.2752	0.0053	1.8800	No	No	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	0.2725	0.2484	−0.0241	−8.8400	No	No	No
sexMale	0.3676	0.3473	−0.0203	−5.5300	No	No	No
z_age.years	0.1766	0.1927	0.0160	9.0700	No	No	No
z_BMI.base.kg.m2	0.2262	0.2102	−0.0161	−7.1000	No	No	No
smoking.statusSmoker	0.4283	0.4488	0.0205	4.7900	No	No	No
z_HOMA.IR.base	0.2343	0.2232	−0.0111	−4.7500	No	No	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	0.9604	0.9553	0.0051	Remains non-sig, B same direction
sexMale	0.2858	0.2581	0.0278	Remains non-sig, B same direction
z_age.years	<0.001	<0.001	0.0000	Remains sig, B same direction
z_BMI.base.kg.m2	0.2375	0.2047	0.0328	Remains non-sig, B same direction
smoking.statusSmoker	0.0180	0.0238	−0.0058	Remains sig, B same direction
z_HOMA.IR.base	0.0064	0.0040	0.0024	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 bootstrapped estimates

References

Disclosure

Methods used in our reproducibility analyses

Summary from statistical review

Data availability and software used

Regression sample

Computational reproducibility results

Inferential reproducibility results

Recommended Changes

Model 1

Model results for lg.ctx

Fit statistics for lg.ctx

ANOVA table for lg.ctx

Model results for lg.ctx

Fit statistics for lg.ctx

ANOVA table for lg.ctx

Visualisation of regression model

Checking residuals plots for patterns

Testing residuals for non linear relationships

Checking univariate relationships with the dependent variable using scatterplots

Linearity results

Testing for homoscedasticity

Homoscedasticity results

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

Cooks threshold

DFFIT threshold

DFBETA threshold

Influence plot

COVRATIO plot

Observations of interest identified by the influence plot

Results for outliers and influential points

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

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

Normality results

Assessing collinearity with VIF

Collinearity results

Assessing independence with the Durbin–Watson test for autocorrelation

Independence results

Assumption conclusions

Forest plot showing original and reproduced coefficients and 95% confidence intervals for lg.ctx

Change in regression coefficients

Change in lower 95% confidence intervals for coefficients

Change in upper 95% confidence intervals for coefficients

Change in adjusted R2

Change in p-values

Bland Altman plot showing differences between original and reproduced p-values for lg.ctx

Results for p-values

Conclusion computational reproducibility

Methods

Results

Comparison of linear and squared model

Comparison of linear and squared model fit

Linearity conclusion

Bootstrapped results

Change in regression coefficients

Change in lower 95% confidence interval

Change in upper 95% confidence interval

Change in Range of 95% confidence interval

Change in p-value significance and regression coefficient direction

Check the distribution of bootstrap estimates

Conclusions based on the bootstrapped model

Model 2

Model results for TNF.pg.ml.base

Fit statistics for TNF.pg.ml.base

ANOVA table for TNF.pg.ml.base

Model results TNF.pg.ml.base

Model fit for TNF.pg.ml.base

ANOVA table for TNF.pg.ml.base

Visualisation of regression model

Checking residuals plots for patterns

Testing residuals for non linear relationships

Checking univariate relationships with the dependent variable using scatterplots

Linearity results

Testing for homoscedasticity

Homoscedasticity results

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

Cooks threshold

DFFIT threshold

DFBETA threshold

Influence plot

Change in adjusted R²

Change in adjusted R²

Change in adjusted R²