# Combining Experiments with Systematic Errors

## TL;DR

This paper analyzes methods for combining multiple datasets with shared systematic uncertainties, demonstrating their equivalence or differences depending on error type and application, which is crucial for accurate data analysis at particle colliders.

## Contribution

It clarifies the theoretical relationship between the full matrix and extra parameter techniques for systematic error treatment in data fitting, including conditions for their equivalence or bias.

## Key findings

- For additive uncertainties, the methods are theoretically equivalent.
- For multiplicative errors, the matrix fit can be biased if applied to data points.
- Applying the multiplicative factor to the function avoids bias in the fit.

## Abstract

We consider fits to two or more datasets for which results from the sa me experiment share a common systematic uncertainty in addition to their individ ual statistical errors. This is important in extracting the maximum information from a set of similar bu t different experiments (or the same experiment under different conditions) an alysing similar but different datasets, as happens at the LHC and other particle colliders. There are two techniques in use: using the full matrix and using extra paramneters, and we show, for a completely general fit, that for an addit ive uncertainty they are in principle equivalent even though in practice the det ails differ. For a multiplicative error the matrix fit is equivalent to the extra parameter fit if the factor is applied to the data points but not if it is applied to the function: the former leads to biassed estimates and the latter avoids them.