Data analysis concept: An ethereal bridge of confidence intervals spanning a city skyline made of data visualizations.

Decoding Inference: How to Make Smarter Decisions with Interval Data

Elliot Brynn in Business & Economy August 2025 • 5 min read.

"Navigate the complexities of partial identification and gain a sharper edge in your data analysis."

In today's data-saturated world, we often face situations where information isn't precise. Instead of pinpoint values, we might have a range of possibilities. This is common in economic models, scientific studies, and even everyday decision-making. Imagine trying to estimate the impact of a new marketing campaign when you can only track results within a certain margin of error, or predicting economic growth when various indicators offer conflicting signals.

This challenge, known as partial identification, requires a different approach than traditional statistical analysis. Instead of seeking a single, definitive answer, we work with an interval – a range within which the true value is likely to lie. The key is to understand the properties of this interval and how to make reliable inferences from it. This article unpacks the concept of inference with interval-identified sets, drawing from recent research to provide a practical guide for anyone dealing with imprecise data.

We'll explore the power properties of inference, focusing on how to construct and compare confidence intervals (CIs) when dealing with data that provides a range of possible values rather than a single point. We will explore recent research of Federico A. Bugni, Mengsi Gao, Filip Obradović and Amilcar Velez that sheds light on the best strategies for this type of analysis, offering insights into making smarter decisions even when the picture isn't perfectly clear.

Confidence Intervals: Your Toolkit for Interval Data Analysis

Data analysis concept: An ethereal bridge of confidence intervals spanning a city skyline made of data visualizations.

Confidence intervals (CIs) are a cornerstone of statistical inference. They provide a range of values within which the true parameter of interest is likely to fall, given a certain level of confidence. When dealing with interval-identified data, the construction and interpretation of CIs become particularly important. Recent research highlights three primary methods for constructing CIs, each with its own strengths and weaknesses:

These methods, often called CIL, CI2, and CI3, were first proposed by Imbens and Manski (2004) and Stoye (2009). It's helpful to think of them as different lenses through which you can view your data, each offering a slightly different perspective on the range of plausible values. Each method assumes that the researcher uses bounds estimators to construct the CIs and that these estimators are ordered, meaning that the lower bound estimate is always less than or equal to the upper bound estimate. Let’s look closer:

CIL: First introduced by Imbens and Manski, this method is easy to implement and provides a solid foundation for inference with interval data.
CI2: Stoye refined this method to improve upon CIL. This confidence interval calibrates critical values taking into account that the underlying problem is bivariate. CI2 is generally considered to be improvement to CIL.
CI3: Stoye developed CI3 so that inference on a CI does not require superefficiency conditions.

The good news is that all three methods are proven to be valid and exact under a broad set of assumptions. This means that, over the long run, they will provide coverage at the desired confidence level. However, this doesn't mean they are identical in practice.

Making the Right Choice: A Path Forward

Navigating the world of inference with interval-identified sets can feel daunting, but by understanding the properties of different confidence intervals and estimators, you can make informed decisions that lead to more robust conclusions. Whether you're analyzing economic trends, evaluating policy impacts, or simply trying to make sense of uncertain data, the tools and insights discussed here will empower you to extract meaningful knowledge from even the most challenging datasets. By combining statistical rigor with a healthy dose of critical thinking, you can unlock the power of interval data and gain a sharper understanding of the world around you.

About this Article -

This article was crafted using a human-AI hybrid and collaborative approach. AI assisted our team with initial drafting, research insights, identifying key questions, and image generation. Our human editors guided topic selection, defined the angle, structured the content, ensured factual accuracy and relevance, refined the tone, and conducted thorough editing to deliver helpful, high-quality information.See our About page for more information.

This article is based on research published under:

DOI-LINK: https://doi.org/10.48550/arXiv.2407.20386,

Title: On The Power Properties Of Inference For Parameters With Interval Identified Sets

Subject: econ.em

Authors: Federico A. Bugni, Mengsi Gao, Filip Obradovic, Amilcar Velez

Published: 29-07-2024

Everything You Need To Know

What is partial identification in the context of data analysis?

Partial identification arises when data provides a range of possible values rather than a single, precise point. This is common in scenarios like economic modeling or evaluating marketing campaigns, where data isn't perfectly clear. Instead of a definitive answer, you work with an interval, understanding the range within which the true value likely lies. This approach requires different strategies than traditional statistical analysis.

What are confidence intervals (CIs), and why are they important in interval data analysis?

Confidence intervals (CIs) are a range of values within which the true parameter of interest is likely to fall, given a certain level of confidence. In interval data analysis, CIs are crucial because they provide a way to make inferences when your data is imprecise. By constructing and interpreting CIs, you can understand the plausible range of values for your analysis, even when you don't have exact point estimates. Various methods exist for constructing CIs when dealing with interval-identified data, each offering a slightly different perspective on the data.

What are the main methods for constructing confidence intervals (CIs) for interval-identified data?

Three primary methods exist for constructing CIs: CIL, CI2, and CI3. CIL, proposed by Imbens and Manski, is a foundational method. CI2, refined by Stoye, builds upon CIL and considers that the underlying problem is bivariate. CI3, also developed by Stoye, ensures inference on a CI does not require superefficiency conditions. All three methods are valid and exact under a broad set of assumptions, ensuring reliable coverage at the desired confidence level.

How do the methods for constructing confidence intervals (CIL, CI2, and CI3) differ from each other?

While all three methods (CIL, CI2, and CI3) provide valid CIs, they differ in their approach. CIL, the first method, offers a basic framework. CI2 improves upon CIL by calibrating critical values with the underlying problem being bivariate, leading to potential improvements. CI3, developed to eliminate the need for superefficiency conditions, offers another distinct approach. Each method utilizes bounds estimators and the estimators must be ordered (lower bound estimate is always less than or equal to the upper bound estimate). Choosing the right method may depend on the specific characteristics of your data and the research question, as each offers a slightly different lens through which to view the data.

How can understanding these methods help improve decision-making with imprecise data?

By understanding the properties of different confidence intervals and estimators, you can make better decisions even when data isn't perfectly clear. Each method offers a different perspective on the range of plausible values. By using CIL, CI2, and CI3, you can extract meaningful knowledge from challenging datasets, whether analyzing economic trends, or evaluating policy impacts. Combining statistical rigor with critical thinking allows you to better understand the world around you. This approach empowers you to make smarter decisions by accounting for uncertainty and embracing the range of possibilities.