Survey Methodology
The inverse-variance trap: A simple ratio fix for combining skewed data

by Anton Grafström, Wilmer Prentius and Åsa RanlundNote 1

  • Release date: June 29, 2026
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Abstract

Combining estimates from independent surveys via inverse-variance weights can lead to negative bias when unknown variances are estimated and the target variable is non-negative and positively skewed. In such cases, strong positive correlations typically arise between the estimators and their corresponding variance estimators, causing standard linear combinations with inverse-variance weights to exhibit negative bias. We introduce a strikingly simple method to reduce bias: replace the standard weight with the ratio of the estimator to the variance estimator. Under a linear model linking the two, we show that the new ratio-weighted estimator is approximately unbiased, whereas the conventional inverse-variance combination exhibits downward bias. Through simulations, we demonstrate that the new method brings both the bias and the mean squared error closer to the optimum for a wide range of different target variables. As our method uses only standardly reported summary statistics, it can be immediately adopted to reduce this widespread bias and improve the reliability of scientific findings in various fields.

Key Words: Combining survey information; Design-based sampling; Environmental surveys; Linear combination estimator.

Table of contents

How to cite

Grafström, A., Prentius, W. and Ranlund, Å. (2026). The inverse-variance trap: A simple ratio fix for combining skewed data. Survey Methodology, 52(1), 117-128. Aavailable at: http://www.statcan.gc.ca/pub/12-001-x/2026001/article/00009-eng.pdf.

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ISSN : 1492-0921

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Published by authority of the Minister responsible for Statistics Canada.

© His Majesty the King in Right of Canada, as represented by the Minister of Industry, 2026

Use of this publication is governed by the Statistics Canada Open Licence Agreement.

Catalogue No. 12-001-X

Frequency: Semi-annual

Ottawa

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