# IPD meta-analysis of observational studies

For observational studies several arguments that meta-analysis from published data are in general insufficient to calculate a pooled estimate are discussed in Blettner et al (1999): Traditional reviews, meta-analyses and pooled analyses in epidemiology.

Provided that individual patient data are available, we have proposed A new strategy for meta-analysis of continuous covariates in observational studies (S&R 2011).

We ...provide a summary estimate of the functional relationship between a continuous covariate and the outcome in a regression model, adjusting for confounding factors. Our procedure comprises three steps. First, we determine a confounder model. Ideally, the latter should include the same variables across studies, but this may be impossible. Next, we estimate the functional form for the continuous variable of interest in each study, adjusted for the confounder model. Finally, we combine the individual functions by weighted averaging to obtain a summary estimate of the function. Fractional polynomial methodology and pointwise weighted averaging of functions are the key components.

Reproduced from the Abstract of S&R (2011) with permission from John Wiley & Sons Ltd.

# IPD meta analysis of treatment effect functions from randomized trials

Beside of obvious applications in epidemiology such as the assessment of continuous risk factors our meta-analysis strategy can also be used to investigate continuous variables for potential treatment heterogeneity in randomized trials.

If IPD data from several RCTs are available, we can estimate continuous treatment effect functions in each of the trials and use the meta-analysis approach to derive an averaged continuous treatment effect function (TEF) for a potential treatment modifier. We propose using the MFPI approach to derive a TEF separately in each study and combine these TEFs by using our meta-analysis approach. Obviously, other approaches (for example splines) may be used to derive a continuous effect function and other approaches to combine functions across several studies may be used.

For more details about our first project combining MFPI with the meta-analysis approach see Kasenda et al (2014) Investigation of continuous effect modifiers in a meta-analysis on higher versus lower PEEP in patients requiring mechanical ventilation – protocol of the ICEM study.

The medical paper presenting results is published (Kasenda et al (2016) Multivariable fractional polynomial interaction to investigate continuous effect modifiers in a meta-analysis on higher versus lower PEEP for patients with ARDS). Using data from eight randomized controlled trials in breast cancer, S&R (2022: Investigating treatment-effect modification by a continuous covariate in IPD meta-analysis: an approach using fractional polynomials, illustrate several methodological issues to average TEF functions. To improve reporting of available data and all steps of the analysis we introduce a three-part profile called MethProf-MA.

# IPD meta-analysis of non-linear exposure-outcome relationships: A comparison of two methods

Exposure-outcome relationships describe how the risk of an outcome (e.g. new onset of disease) varies with a risk factor of interest measured on a continuous scale (e.g. age, body mass index (BMI) or systolic blood pressure).

In single studies, the practice of using “categorized” continuous risk factors in an analysis is heavily criticized. Categorization means that the continuous nature of a risk factor is replaced by two or more groups (e.g. BMI below 20, 20-25, 25-30 or above 30). The exposure-outcome relationship is taken to be unvarying within these groups. Critical decisions for such an analysis include how many groups to use and where to put the dividing lines.

Alternatively, it is often assumed that a linear relationship describes the exposure-outcome relationship adequately. However, in many cases, the assumption is clearly seen to be wrong, that is, the true relationship is non-linear. Seriously wrong conclusions may result from false assumptions of linearity.

To overcome such problems, various flexible methods of quantifying an exposure-outcome relationship have been developed. In single studies several spline function procedures (Perperoglou et al 2019) and fractional polynomials can be used. The latter approach uses observations to assess whether there is strong evidence for a non-linear function, and if not, a linear function is accepted.

Difficulties caused by categorization or an incorrect assumption of linearity apply as much in meta-analyses of multiple studies as they do in analysis of single studies.

Two methods, called *metacurve* (Sauerbrei and Royston 2011) and *mvmeta *(White 2009, Gasparrini et al 2012), have been proposed to combine continuous functions of a risk factor across several studies. White et al (2019) compare the two methods via theoretical arguments and via a small simulation study. The main portion of the paper concerns the comparison of results by meta-analysis of individual participant data from the Emerging Risk Factors Collaboration and concentrates on assessing the influence of BMI (exposure) on the risk of new-onset coronary heart disease and all-cause mortality (outcomes) (>80 cohorts, >18000 events).

The paper adopts a two-stage approach. In the first stage, the authors estimate the exposure-outcome relationship using fractional polynomials in each study, allowing for generally accepted confounders. The selected exposure-outcome relationships are non-linear and exhibit great variability across studies. In the second stage, they use *metacurve* and *mvmeta* to obtain averaged functions to describe the exposure-outcome relationships across studies.

The authors identify minor differences between results from the two methods. Overall, however, the exposure-outcome relationships derived agree well with earlier analyses. For coronary heart disease, the authors detect a small increase in risk at the lowest levels of BMI, whereas for all-cause mortality, they identify a steep U-shape. They conclude that provided individual participant data are available, both *metacurve* and *mvmeta* are suitable methods for assessing the effect of a continuous risk factor on an outcome of interest.