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Incorporating published univariable associations in diagnostic and prognostic modeling
- Thomas P A Debray^{1}Email author,
- Hendrik Koffijberg^{1},
- Difei Lu^{2},
- Yvonne Vergouwe^{1, 2},
- Ewout W Steyerberg†^{2} and
- Karel G M Moons†^{1}
https://doi.org/10.1186/1471-2288-12-121
© Debray et al.; licensee BioMed Central Ltd. 2012
Received: 12 January 2012
Accepted: 26 June 2012
Published: 10 August 2012
Abstract
Background
Diagnostic and prognostic literature is overwhelmed with studies reporting univariable predictor-outcome associations. Currently, methods to incorporate such information in the construction of a prediction model are underdeveloped and unfamiliar to many researchers.
Methods
This article aims to improve upon an adaptation method originally proposed by Greenland (1987) and Steyerberg (2000) to incorporate previously published univariable associations in the construction of a novel prediction model. The proposed method improves upon the variance estimation component by reconfiguring the adaptation process in established theory and making it more robust. Different variants of the proposed method were tested in a simulation study, where performance was measured by comparing estimated associations with their predefined values according to the Mean Squared Error and coverage of the 90% confidence intervals.
Results
Results demonstrate that performance of estimated multivariable associations considerably improves for small datasets where external evidence is included. Although the error of estimated associations decreases with increasing amount of individual participant data, it does not disappear completely, even in very large datasets.
Conclusions
The proposed method to aggregate previously published univariable associations with individual participant data in the construction of a novel prediction models outperforms established approaches and is especially worthwhile when relatively limited individual participant data are available.
Background
Recent medical literature has shown an increasing interest in clinical prediction models obtained from cross-sectional studies (diagnostic models) as well as case-control, cohort and randomized controlled data (prognostic models) [1–5]. Such models combine multiple predictors or markers that are independently associated with the presence (in case of diagnosis) or future occurrence (in case of prognosis) of a particular outcome. Typically, logistic regression is used to model these binary outcomes. Alternatively, Cox proportional hazards regression may be applied to account for the time-to-event.
The development of a novel prediction model requires a dataset with a sufficient amount of participants to obtain accurate associations and to make reliable predictions. Also, larger numbers of participants increase the statistical power when selecting predictive subject characteristics to be included in predictive models. Although numerous prediction models are constructed from a single dataset, it is possible to increase the amount of evidence available by incorporating information from the literature.
The availability of individual participant data (IPD) is commonly recommended as gold standard for combining existing information with newly collected data [6, 7]. However, this situation is often unfeasible due to practical constraints [8, 9], for instance when studies were conducted several years ago. Fortunately, numerous papers contain baseline population characteristics from which univariable predictor-outcome associations can be derived. Consequently, these associations represent an appealing source of evidence when developing a novel prediction model [5, 10–17].
Greenland and Steyerberg have recently proposed adaptation methods to incorporate previously published univariable predictor-outcome associations as prior evidence in a regression analysis [18, 19]. These methods combine the result of a univariable meta-analysis with the results of a univariable and multivariable logistic regression analysis on the IPD. Although these quantitative approaches may considerably improve the quality of a model’s regression coefficients and its resulting performance, they are not yet frequently used in practice [20, 21].
Here we present an improved alternative to the methods proposed by Greenland and Steyerberg that aims to further increase the accuracy and precision of the multivariable associations estimated using external evidence. This method improves upon the variance estimation component by reconfiguring the adaptation process in established theory and making it more robust. We present two variants of our method and test their performance in a simulation study. We illustrate the proposed methods’ application in a clinical example involving the prediction of peri-operative mortality after elective abdominal aortic aneurysm surgery [22].
Methods
This method is intended to address the specific situation where IPD have been collected to evaluate the effect of a number of predictors on a dichotomous outcome using logistic regression analysis. Here, univariable and multivariable associations (logistic regression coefficients) are estimated and denoted as β _{u} and β _{m}. Particularly, two sources of associations are assumed to be available, namely the IPD of the study at hand ( I ) and aggregated data from the literature ( L ). The univariable and multivariable associations estimated in the derivation data are denoted as ${\widehat{\beta}}_{\mathrm{u}|\text{I}}$ and ${\widehat{\beta}}_{\mathrm{m}|\text{I}}$. For the literature, only univariable associations are available ( ${\widehat{\beta}}_{\mathrm{u}|\text{L}}$ ). It is assumed that the study at hand and the studies forming the literature are both random samples from a common underlying patient population.
Previous simulations have however shown that the original unweighted method (c = 1 in expression 3) has a similar performance.
Concerns and proposed solutions
Although it is possible to assume that estimated associations from the literature and IPD at hand are independent, i.e. $\mathrm{Cov}({\widehat{\beta}}_{\mathrm{u}|\text{L}},{\widehat{\beta}}_{\mathrm{m}|\text{I}})=\mathrm{Cov}({\widehat{\beta}}_{\mathrm{u}|\text{L}},{\widehat{\beta}}_{\mathrm{u}|\text{I}})=0$, the remaining assumption that $\mathrm{Cov}({\widehat{\beta}}_{\mathrm{m}|\text{I}},{\widehat{\beta}}_{\mathrm{u}|\text{I}})=\mathrm{Var}\left({\widehat{\beta}}_{\mathrm{u}|\text{I}}\right)$ seems unrealistic. Particularly, this assumption requires that the univariable and multivariable association in the IPD at hand are strongly correlated and neglects $\mathrm{Var}\left({\widehat{\beta}}_{\mathrm{m}|\text{I}}\right)$, as $\mathrm{Cov}({\widehat{\beta}}_{\mathrm{m}|\text{I}},{\widehat{\beta}}_{\mathrm{u}|\text{I}})=\rho ({\widehat{\beta}}_{\mathrm{m}|\text{I}},{\widehat{\beta}}_{\mathrm{u}|\text{I}})\phantom{\rule{0.3em}{0ex}}\mathrm{Var}\left({\widehat{\beta}}_{\mathrm{m}|\text{I}}\right)\phantom{\rule{0.3em}{0ex}}\mathrm{Var}\left({\widehat{\beta}}_{\mathrm{u}|\text{I}}\right)$. Consequently, expression 2 may yield biased variance estimates of adapted multivariable associations. Although it is even possible that $\hat{\mathrm{Var}}\left({\widehat{\beta}}_{\mathrm{m}|\text{L}}\right)$ becomes negative when $\hat{\mathrm{Var}}\left({\widehat{\beta}}_{\mathrm{m}|\text{I}}\right)<\hat{\mathrm{Var}}\left({\widehat{\beta}}_{\mathrm{u}|\text{I}}\right)$, this is unlikely to happen because adjustment of logistic regression coefficients is expected to result in a loss of precision [24].
The probabilistic adaptation from univariable to multivariable association $\mathcal{N}({\mu}_{\delta},{\sigma}_{\delta}^{2})$ can be estimated from the IPD at hand using bootstrap sampling [25]. This procedure applies repeated sampling with replacement of subjects from the derivation dataset. Hence, it allows generating numerous datasets (bootstrap samples) where the adaptation can be estimated. Unfortunately, the bootstrap procedure may become unstable when the effective sample size is small, and yield regression coefficients with extreme values [26–28]. This, in turn, may strongly affect the quality of estimated adaptations and result in poor estimates of β _{m|L }. For this reason, we propose to shrink the adaptation by implementing a Bayesian prior for the univariable and multivariable associations of the IPD at hand. Recently, Gelman et al. proposed a weakly default prior distribution that is based on the Cauchy distribution and assumes a probability of 70.48% for associations between -5 and 5. This distribution is less conservative than the uniform prior distribution (which assumes higher probabilities for extreme associations), and yields estimates that make more sense and have predictive performance better than maximum likelihood estimates [29]. The weakly informative prior distribution for generalized linear modeling was recently implemented in R, and is available in the package arm.
Finally, the summary of univariable associations from the literature $\mathcal{N}({\mu}_{\mathrm{u}|\text{L}},{\sigma}_{\mathrm{u}|\text{L}}^{2})$ is originally estimated by applying a fixed effects meta-analysis [30, 31]. Because this estimate may be unstable when few studies are available, Steyerberg et al. proposed using the univariable associations from the literature (published as ${\widehat{\beta}}_{\mathrm{u}|\text{L}}$ ) and the IPD at hand (estimated as ${\widehat{\beta}}_{\mathrm{u}|\text{I}}$) [19]. When the homogeneity assumptions made by the adaptation method are violated, it is possible to assume random effects to further improve the robustness of estimated associations.
Overview of approaches
No meta-analysis | Greenland/Steyerberg | Improved adaptation method | |||
---|---|---|---|---|---|
adaptation method | Variant 1 | Variant 2 | |||
Step 1 | Estimate associations in IPD | ||||
Implemented | Yes | Yes | Yes | Yes | |
Association type | m | u+m | u+m | u+m | |
Prior distribution | none | none | none | weakly informative | |
Step 2 | Summarize univariable associations | ||||
Implemented | No | Yes | Yes | Yes | |
Source | - | I+L | I+L | I+L | |
Pooling Method | - | random effects | random effects | random effects | |
Step 3 | Estimate adaptation from univariable to multivariable association | ||||
Implemented | No | Yes | Yes | Yes | |
Assumptions | - | (1)+(2) | (1) | (1) | |
Estimation procedure | - | analytic | bootstrap | bootstrap | |
Prior distributions | - | none | none | weakly informative | |
Step 4 | Apply adaptation to summary estimate from the literature and estimate β _{m|L} | ||||
Implemented | No | Yes | Yes | Yes |
Simulation study
Results simulation study
No meta-analysis | Greenland/Steyerberg | Improved adaptation method | Improved adaptation method | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
adaptation method | (no prior) | (weakly informative prior) | ||||||||||||||
N _{ I } | N _{ L } | σ _{ h } | ρ(x _{ 1 }, x _{2}) | PB | MSE | coverage | PB | MSE | coverage | (*) | PB | MSE | coverage | PB | MSE | coverage |
100 | 500 | 0 | 0 | 15.07% | 0.613 | 89.0% | 8.87% | 0.219 | 89.2% | 8 | 1.3 e+12% | 1.8 e+23 | 97.8% | -1.98% | 0.065 | 89.6% |
200 | 500 | 0 | 0 | 6.58% | 0.186 | 90.0% | 2.34% | 0.063 | 90.8% | 1 | 18.13% | 3.671 | 94.4% | -1.44% | 0.043 | 89.0% |
500 | 500 | 0 | 0 | 3.65% | 0.061 | 90.4% | 1.00% | 0.024 | 90.0% | 0 | 2.21% | 0.026 | 91.0% | -0.54% | 0.021 | 89.0% |
1000 | 500 | 0 | 0 | 1.31% | 0.028 | 90.2% | 0.84% | 0.014 | 91.2% | 0 | 1.34% | 0.014 | 90.6% | -0.11% | 0.013 | 90.0% |
100 | 500 | 0 | 0.50 | 20.39% | 0.888 | 91.2% | 5.75% | 0.166 | 94.4% | 7 | -80.77% | 3.9 e+04 | 98.4% | 1.41% | 0.048 | 96.2% |
200 | 500 | 0 | 0.50 | 8.22% | 0.226 | 91.0% | 1.63% | 0.037 | 93.0% | 0 | 4.55% | 0.091 | 94.2% | 0.32% | 0.031 | 93.6% |
500 | 500 | 0 | 0.50 | 1.89% | 0.073 | 87.6% | 0.45% | 0.019 | 92.2% | 0 | 0.89% | 0.020 | 90.8% | -0.32% | 0.019 | 91.4% |
1000 | 500 | 0 | 0.50 | 0.88% | 0.031 | 92.2% | 0.33% | 0.011 | 93.8% | 0 | 0.55% | 0.012 | 92.8% | -0.19% | 0.011 | 93.8% |
100 | 500 | 0.20 | 0 | 10.89% | 0.440 | 92.4% | 5.17% | 0.140 | 90.4% | 8 | -3.7 e+02% | 5.6 e+04 | 98.0% | -4.02% | 0.056 | 89.8% |
200 | 500 | 0.20 | 0 | 6.54% | 0.177 | 92.0% | 3.81% | 0.060 | 91.6% | 1 | -11.08% | 0.801 | 95.6% | -0.18% | 0.039 | 91.6% |
500 | 500 | 0.20 | 0 | 1.23% | 0.049 | 93.8% | 0.34% | 0.024 | 92.2% | 0 | 1.53% | 0.026 | 92.2% | -1.13% | 0.022 | 90.8% |
1000 | 500 | 0.20 | 0 | 0.94% | 0.029 | 89.2% | 0.89% | 0.017 | 90.4% | 0 | 1.42% | 0.018 | 90.4% | 0.02% | 0.016 | 89.8% |
100 | 2000 | 0 | 0 | 47.95% | 4.9 e+01 | 93.2% | 37.63% | 4.3 e+01 | 86.2% | 21 | 1.6 e+12% | 1.5 e+23 | 98.2% | -1.09% | 0.058 | 89.6% |
200 | 2000 | 0 | 0 | 5.60% | 0.184 | 90.2% | 3.31% | 0.058 | 89.8% | 1 | 54.36% | 2.1 e+02 | 94.2% | -0.12% | 0.036 | 88.2% |
500 | 2000 | 0 | 0 | 2.36% | 0.064 | 87.2% | 1.10% | 0.017 | 89.2% | 0 | 2.31% | 0.020 | 91.4% | -0.07% | 0.015 | 88.8% |
1000 | 2000 | 0 | 0 | 1.17% | 0.027 | 90.0% | 0.58% | 0.009 | 90.2% | 0 | 1.16% | 0.010 | 89.2% | -0.03% | 0.009 | 87.4% |
100 | 2000 | 0 | 0.50 | 20.05% | 0.856 | 89.6% | 5.68% | 0.139 | 92.0% | 11 | 3.5 e+12% | 1.3 e+23 | 98.4% | 1.67% | 0.045 | 95.4% |
200 | 2000 | 0 | 0.50 | 6.99% | 0.206 | 90.8% | 2.67% | 0.035 | 92.2% | 1 | 5.94% | 0.120 | 93.8% | 2.02% | 0.029 | 92.2% |
500 | 2000 | 0 | 0.50 | 2.44% | 0.063 | 90.8% | 0.75% | 0.011 | 92.8% | 0 | 1.18% | 0.011 | 92.0% | 0.45% | 0.010 | 92.2% |
1000 | 2000 | 0 | 0.50 | 1.62% | 0.032 | 89.4% | 0.26% | 0.007 | 91.6% | 0 | 0.45% | 0.007 | 91.6% | 0.02% | 0.007 | 91.4% |
100 | 2000 | 0.20 | 0 | 16.17% | 0.654 | 92.6% | 7.67% | 0.201 | 89.8% | 16 | 1.5 e+03% | 3.9 e+04 | 98.2% | -2.66% | 0.046 | 91.0% |
200 | 2000 | 0.20 | 0 | 6.63% | 0.177 | 93.0% | 3.74% | 0.057 | 89.2% | 1 | 13.89% | 0.754 | 94.8% | 0.26% | 0.037 | 88.8% |
500 | 2000 | 0.20 | 0 | 2.33% | 0.056 | 92.8% | 1.23% | 0.021 | 89.6% | 0 | 2.46% | 0.023 | 89.4% | -0.08% | 0.019 | 88.6% |
1000 | 2000 | 0.20 | 0 | 2.02% | 0.027 | 92.2% | 1.07% | 0.014 | 87.4% | 0 | 1.62% | 0.015 | 86.6% | 0.37% | 0.013 | 85.8% |
No meta-analysis (classical approach)
Results demonstrate that the classical approach to logistic regression, ignoring published univariable evidence from previous studies, considerably overestimates multivariable associations, particularly when the IPD at hand is very small. Although the percentage bias and MSE of ${\widehat{\beta}}_{1}$ decreases in larger datasets, it does not completely disappear. Similar to previous research, we found that the bias of estimated regression coefficients increases when collinearity occurs and effective sample sizes are small [33]. The coverage of the 90% confidence interval was adequate for all scenarios considered.
Greenland/Steyerberg adaptation method
The multivariable associations estimated with the Greenland/Steyerberg Adaptation method were far more accurate than those estimated with the classical approach, especially when little actual data were available. Estimated associations remain, however, too extreme compared to the associations from the reference model. The coverage of the 90% confidence interval was good for most scenarios, although we observed over-coverage when collinearity was present, and under-coverage when the literature studies were very large and heterogeneous. Unfortunately, we also noticed that some estimates for $\mathrm{Var}\left({\widehat{\beta}}_{\mathrm{m}|\text{L}}\right)$ were negative when IPDs were small, and particularly when the literature studies were large (such that $\mathrm{Var}\left({\widehat{\beta}}_{\mathrm{u}|\text{L}}\right)$ becomes negligible). Finally, the presence of heterogeneity in the literature associations did not influence the accuracy of estimated associations. This finding can however be explained by the fact that heterogeneity was only introduced in the spread of the literature associations.
Improved adaptation method (no prior)
When no shrinkage was applied for the associations of the IPD at hand, estimated multivariable associations had the largest error, particularly when few data were available. Regression coefficients in bootstrap samples were often non-identifiable (results not shown), resulting in unstable estimates and over-coverage of multivariable regression coefficients. When the size of the IPD at hand increased, this approach performed similar to the improved adaptation method with a weakly informative default prior and the approach proposed by Greenland and Steyerberg.
Improved adaptation method (weakly informative prior)
Results demonstrate that estimated associations were most accurate when a weakly informative prior was used during estimation of the adaptation. Even when the rule of thumb that logistic models should be used with a minimum of 10 outcome events per predictor variable is clearly violated, this approach yielded superior estimates of b_{1} that were very similar to estimates obtained from large amounts of IPD. Finally, we observed over-coverage of the 90% confidence interval when collinearity was present, and under-coverage when the literature studies were very large and heterogeneous with the IPD at hand.
Application
Calculation of adapted associations in the application
Female sex | MI | CHF | Ischemia | |
---|---|---|---|---|
Adaptation ${\widehat{\mathbf{\mu}}}_{\mathbf{\delta}}$ ; ${\widehat{\mathbf{\sigma}}}_{\mathbf{\delta}}^{\mathbf{2}}$ | ||||
Greenland/Steyerberg Adapt. method | 0.02; 0.13 | -0.76; 0.07 | -0.74; 0.05 | -0.72; 0.08 |
Improved Adapt. method (no prior) | 0.04; 0.39 | -0.69; 0.15 | -0.67; 0.16 | -0.72; 0.41 |
Improved Adapt. method (weakly informative prior) | 0.05; 0.12 | -0.65; 0.07 | -0.63, 0.05 | -0.67; 0.11 |
Univariable association ${\widehat{\mathbf{\mu}}}_{\mathbf{u}}$ ; ${\widehat{\mathbf{\sigma}}}_{\mathbf{u}}^{\mathbf{2}}$ | ||||
Greenland/Steyerberg Adapt. method | 0.35; 0.03 | 1.02; 0.07 | 1.58; 0.12 | 1.52; 0.10 |
Improved Adapt. method (no prior) | 0.35; 0.03 | 1.02; 0.07 | 1.58; 0.12 | 1.52; 0.10 |
Improved Adapt. method (weakly informative prior) | 0.34; 0.03 | 1.00; 0.07 | 1.52; 0.11 | 1.48; 0.09 |
Multivariable association ${\widehat{\mathbf{\mu}}}_{\mathbf{m}}$ ; ${\widehat{\mathbf{\sigma}}}_{\mathbf{m}}^{\mathbf{2}}$ | ||||
No meta-analysis | 0.30; 0.75 | 0.74; 0.32 | 1.04; 0.35 | 0.99; 0.38 |
Greenland/Steyerberg Adapt. method | 0.36; 0.16 | 0.26; 0.14 | 0.84; 0.17 | 0.80; 0.18 |
Improved Adapt. method (no prior) | 0.38; 0.42 | 0.33; 0.22 | 0.91; 0.28 | 0.80; 0.51 |
Improved Adapt. method (weakly informative prior) | 0.39; 0.15 | 0.35; 0.14 | 0.90; 0.16 | 0.81; 0.21 |
No meta-analysis (classical approach)
The poor quality of estimated associations can be illustrated by their substantial variance. The predictor ‘Female Sex’ is a good example, since the 90% confidence interval of its multivariable association was estimated as [−1.30,2.00].
Greenland/Steyerberg adaptation method
The Greenland/Steyerberg Adaptation method yielded notably different multivariable associations. For instance, whereas the classical approach estimated a multivariable association of 0.74 (OR_{adj} = 2.10) for the predictor ‘History of MI’, this estimate was shrunk to 0.26 (OR_{adj} = 1.20) by the adaptation method. Here, the considerable difference in univariable associations between the individual dataset and the literature is a major cause of shrinkage. Finally, the variance of multivariable associations was much smaller when published evidence from the literature was incorporated.
Improved adaptation method (no prior)
We noticed a substantial increase in the variance of estimated adaptations due to the occurrence of non-identifiability in some of the bootstrap samples. These findings illustrate the need for a prior distribution that shrinks the associations of the individual dataset and thereby robustifies the adaptation.
Improved adaptation method (weakly informative prior)
Multivariable associations were similar but not equal to those estimated with the Greenland/Steyerberg Adaptation method. For instance, the multivariable association of the predictor ‘History of MI’ was shrunk to a lesser extent by both variants of the improved adaptation method. Furthermore, the variance of estimated adaptations and multivariable associations decreased considerably by implementing a weakly informative prior distribution.
Discussion
The incorporation of previously published univariable associations from single diagnostic or prognostic test, predictor or marker studies, into the development of a novel prediction model is both feasible and beneficial. A simple method for this purpose was proposed by Greenland and Steyerberg using the change from univariable to multivariable association observed in the IPD to adapt the univariable associations from the literature. We present an improved adaptation method and demonstrate its additional value in a simulation study. Particularly when the individual dataset is relatively small, this method estimates multivariable associations with a smaller MSE, and obtains better coverage of their 90% confidence intervals. Major performance gain is obtained by shrinking the associations from the individual dataset when calculating the adaptation. When no shrinkage was applied (no prior), non-identifiability occurred in some of the bootstrap samples and estimated adaptations were no longer normally distributed. Since we know that extreme associations are very rare in medical sciences, the use of a weakly informative default prior is justified [29], resulting in improved accuracy and precision of the adaptation and hence also the multivariable associations under study.
Several issues must be considered when evaluating these findings: Firstly, performance was evaluated here through the estimation of an association in a small prediction model. Our method may perform better in larger models where correlations between univariable and multivariable associations may be less strong, but this remains untested. Secondly, advanced Bayesian approaches for summarizing the evidence from the literature were not considered. Although these approaches might further improve the accuracy and coverage of multivariable associations, they are less readily compared with meta-analytical models and require more modeling expertise.
Third, the assumption that studies from the literature are exchangeable with the data at hand might not always hold. Simulations showed an under-coverage of the estimated 90% confidence interval when comparability between the considered associations was low, indicating that incorporating strongly heterogeneous evidence from the literature into prediction modeling remains problematic. In those scenarios, the change from univariable to multivariable association in the IPD at hand may no longer be representative for associations from the literature. Evidently, the incorporation of strongly heterogeneous evidence (for example indicated by the I ^{2} statistic) from the literature into the development of a novel prediction model remains questionable [34, 35]. In addition, aggregating published results may not be desirable if publication bias is present or suspected. Fortunately, the use of random effects when summarizing the associations from the literature seems to counter this problem to some extent.
Fourth, we did not consider the situation in which multivariable (rather than univariable) associations are available from the literature. Although their incorporation may be difficult due to the diversity of considered predictors, it could further improve the quality of estimated associations. The synthesis process of associations from the literature should then account for differences in model specification and included associations. Future research will investigate how these challenges can be assessed [36].
Finally, our simulation study only evaluated the performance of estimated multivariable predictor-outcome associations. Although Steyerberg et al. showed that improved estimates may increase the quality of the prediction model [19], this relation was not assessed here. It is possible that all adaptation methods perform similar in a prediction task. However, we showed that the Improved Adaptation Method with a weakly informative prior may further reduce the bias of multivariable associations when datasets are small. It may be clear that for strong predictors, this improvement may have a meaningful impact when making predictions. Additional research is needed to evaluate the extent to which improved predictor-outcome associations result in an improved model performance.
Conclusions
Our study demonstrates that the MSE in multivariable associations of a novel prediction model is largest when external evidence, in this case previously published univariable predictor-outcome associations, is ignored. Although this error decreases with increasing amount of IPD, it does not disappear completely, even in very large datasets. Therefore, it is valuable to incorporate any existing univariable evidence from the literature unless this evidence is strongly heterogeneous. Even when the individual dataset is relatively large compared to the literature, the proposed method will still result in an estimate closer to the underlying multivariable association than the standard method ignoring the literature. The improved and original adaptation methods are robust approaches for this purpose. Whereas the latter method is simpler to apply, the former is more vigorous in small datasets and provides the most stable estimates.
Author’s contributions
TD performed the statistical analyses and drafted the manuscript. DL contributed in the statistical models. HK and YV supervised the analyses and advised on several modeling issues. Finally, ES and KM provided critical feedback and streamlined the manuscript during the final stage. All authors read and approved the final manuscript.
Funding
We gratefully acknowledge the financial support by the Netherlands Organization for Scientific Research (9120.8004 and 918.10.615 and 916.11.126).
Notes
Declarations
Acknowledgements
We gratefully acknowledge Dr Rene Eijkemans for statistical advice regarding the adaptation methods.
Authors’ Affiliations
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