Short version

Material and Methods

To conduct the dose-response meta-analysis (i.e. all illustrations and results presented here are based on non-linear input data from corresponding papers), the dosresmeta package in R was employed, focusing on the association between exposure and response. Two estimation methods, Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML), were used to evaluate variance components and model fit. The analysis utilized a restricted cubic spline model with three knots to capture potential non-linear associations. Model selection was performed using ML based on Akaike Information Criterion (AIC), with subsequent re-estimation using REML to obtain unbiased variance estimates. The Wald test was also applied to assess the deviation from linearity of the spline model, allowing comparison between linear and non-linear model fits (NB! linear models are fitted with non-linear input data from corresponding papers, which differs from our main results where linear input data are used for linear meta-analisis).

Results

The restricted cubic spline ML model with three knots was identified as the best-fitting model according to AIC. However, the Wald test suggested that a linear model provided an equally good fit, indicating that linearity could be assumed for simplicity. The final model was thus chosen as the linear REML model, providing robust variance estimates while maintaining practical interpretability. Comparisons also revealed that ML estimates exhibited downward bias in variance components, whereas REML provided more reliable estimates for reporting purposes. Confidence intervals of the fitted lines were consistent across models, further supporting the decision to use the simpler linear approach.

Generate and plot results for IHD, MI, and Stroke outcomes

Linear Model

Restricted Cubic Spline Model

Linear Model

Restricted Cubic Spline Model

Linear Model

Restricted Cubic Spline Model

Long version

Maximum Likelihood (ML) Estimation:

ML estimates all parameters of the model, including fixed effects and variance components (random effects).
ML maximizes the likelihood function for the entire model, which includes both fixed and random components. However, because the fixed effects contribute to the likelihood calculation, the variance components are often underestimated. This is because the estimation of fixed effects does not account for the uncertainty in estimating these random effects.
In practice, this means the variance of random effects estimated by ML tends to be biased downward.
ML is commonly used when comparing different models, such as using AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), or Likelihood Ratio Tests (LRT) to determine the best fitting model.

2. Restricted Maximum Likelihood (REML) Estimation:

REML is designed to provide unbiased estimates of variance components by taking into account the loss of degrees of freedom from estimating fixed effects. This correction ensures that the estimation of variance components is less biased compared to ML.
REML focuses only on estimating the random effects part of the model. The fixed effects are estimated separately, which makes REML a better choice for variance component estimation.
However, REML estimates cannot be used to compare models with different fixed effects structures. This is because the REML likelihood only pertains to the random effects part, and different fixed effects will yield different likelihood functions, making comparisons based on criteria like AIC or BIC meaningless.

In Practical Application:

Model Selection: ML is preferred for model selection purposes since it provides a full likelihood that encompasses both fixed and random components. For comparing different models (e.g., deciding whether a linear or cubic spline model fits better), ML estimation provides meaningful values for model comparison statistics like AIC, BIC, and LRT.
Variance Estimation: Once the best model structure is identified using ML, the final model may be re-estimated using REML to produce less biased estimates of variance components. This is particularly useful when reporting the estimated variance components and providing confidence intervals for the model parameters.

IHD results

MI results

Stroke results

Sensitivity analysis

For sensitivity analysis I used RCS with 3 knots (as the best model for all outcomes).

Restrict curves to noise levels 75 dB or below (only few above this exposure – Mette checked in Dk population). Not performed yet but Andrei expects that it will not affect results.
Exclude studies with very high “lower level”, reference level

IHD sensitivity

Only for Pyko et all original data was affected, for the rest of the studies over 75 estimates are linearised from the original data.

Studies with risk estimates >= 75 dB
Study	subtype	Tranformed exposure, Lden	Categorical RR (95% CI) adjusted for SES, lifestyle
Cai et al., 2018 // EPIC-OXFORD	IHD	76.5	NA
Cai et al., 2018 // EPIC-OXFORD	IHD	81.5	NA
Hao et al., 2022	IHD	75.3	NA
Hoffmann et al. 2015	IHD	79.1	NA
Pyko et al., 2023 // Recalculated	IHD	75.0	Model 2: 40 dB: 1.00 (0.95-1.05), 45 dB: 1.00 (0.98-1.03); 50 dB: 0.99 (0.97-1.01); 55 dB: 1.00 (0.98-1.02); 60 dB: 1.02 (1.00-1.04); 65 dB: 1.04 (1.01-1.06); 70 dB: 1.08 (1.04-1.13); 75 dB: 1.16 (1.06-1.27) Size of studybase and number of cases in analysis provided in column AH.

For IHD NO studies with reference exposure levels below 60 dB were excluded - i.e. ORIGINAL graph is presented.

MI Sensitivity

Studies with risk estimates >= 75 dB
Study	subtype	Tranformed exposure, Lden	Categorical RR (95% CI) adjusted for SES, lifestyle
Babisch et al., 1994	MI	75.0	<=60 dB: 1.0 (reference), 61-65 dB: 1.2 (0.8-1.7); 66-70 dB: 0.9 (0.6-1.4); 71-75dB: 1.1 (0.7-1.7); 76-80 dB: 1.5 (0.8-2.8) Number of cases and controls in analysis: 645 and 3390, respectively.
Babisch et al., 1994	MI	80.0	<=60 dB: 1.0 (reference), 61-65 dB: 1.2 (0.8-1.7); 66-70 dB: 0.9 (0.6-1.4); 71-75dB: 1.1 (0.7-1.7); 76-80 dB: 1.5 (0.8-2.8) Number of cases and controls in analysis: 645 and 3390, respectively.
Hao et al., 2022	MI	75.3	NA
Magnoni et al., 2021	MI	75.0	From Table 2. Lden < 65 dB: HR = 1 (ref); 65-69 dB: HR = 0.994 (0.951-1.040); 70-74 dB: HR = 1.005 (0.958-1.053); = 75 dB: HR = 0.999 (0.951-1.050). (N= 1,087,110)
Pyko et al., 2023 // Recalculated	MI	75.0	Model 2: 40 db: 1.00 (0.92-1.08), 45 dB: 1.00 (0.95-1.04); 50 dB: 0.97 (0.94-1.01); 55 dB: 0.98 (0.95-1.01); 60 dB: 1.00 (0.97-1.03); 65 dB: 1.02 (0.97-1.06); 70 dB: 1.06 (0.99-1.14); 75 dB: 1.14 (0.99-1.31) Size of studybase and number of cases in analysis provided in column AH.

Storke Sensitivity

Studies with risk estimates >= 75 dB
Study	subtype	Tranformed exposure, Lden	Categorical RR (95% CI) adjusted for SES, lifestyle
Hao et al., 2022	Stroke	75.3	NA
Magnoni et al., 2021	Ischemic stroke	75.0	NA
Cai et al., 2018 // EPIC-OXFORD	Cerebrovascular disease	76.5	NA
Cai et al., 2018 // EPIC-OXFORD	Cerebrovascular disease	81.5	NA