Research story · Counts and rounded answers

What do rounded answers hide?

Self-reported counts pile up on round numbers, and these spikes distort the estimates. In our 2025 article, we modeled the spikes instead of ignoring them and proposed a new class of regression models, to study the age at which people start smoking.

At a glance

Period: 2020 to 2025
Methods: GAITD regression (Yee and Ma, 2024); doubly constrained reduced-rank VGLMs, proposed in our article
Co-authors: Thomas W. Yee (University of Auckland), Chenchen Ma
Data: US National Health and Nutrition Examination Survey (NHANES)
Key paper: The Annals of Applied Statistics 2025

The question

Ask people at what age they started smoking, or how much they smoke, and many will answer with a round number. Large health surveys used in addiction studies are full of this digit preference, known as heaping: spikes at multiples of 5 and 10 that reflect how people remember and report rather than what they did. Its opposite, seeping, makes some values rarer than they should be. Both distort the estimates, and methods and software to deal with them have long been lacking.

What people report

Underlying distribution and excess on round numbers

Schematic illustration with simulated data. Bars at multiples of 5 are in ochre.

A visit to Auckland 2020

In February and March 2020 I was a visiting fellow at the Department of Statistics of the University of Auckland, to collaborate on methods for count distributions that are generally altered, inflated and truncated.

GAITD regression for heaped counts 2024–2025

Generally Altered, Inflated, Truncated and Deflated (GAITD) regression was introduced in an earlier article (Statistical Science, 2024 (opens in a new tab)). It allows selected values of a count, such as multiples of 5 and 10, to be inflated or deflated with respect to an underlying distribution, and models them within the regression.

In our article, we identified three generic problems of simple addiction studies affected by heaping and seeping, and solved them with GAITD regression (The Annals of Applied Statistics, 2025 (opens in a new tab); presented earlier at NZSA 2024). In conjunction, we proposed doubly constrained reduced-rank vector generalized linear models (DRR-VGLMs), which give structure to reduced-rank regression through linear constraints and further simplify the dimension reduction.

Applied to a recent data set of the US National Health and Nutrition Examination Survey, the models describe the distribution of the age at which people start smoking and its association with tobacco consumption and smoking duration. They address questions such as: is starting to smoke earlier associated with higher consumption later in life? Among those who quit, is starting later associated with smoking for a shorter time?

Publications in this line

2025

Heaping and seeping, GAITD regression and doubly constrained reduced-rank vector generalized linear models, in smoking studies (opens in a new tab)
T. Yee, L. Frigau, C. Ma
The Annals of Applied Statistics, 19(4): 3045–3070
Applies GAITD regression and doubly constrained reduced-rank VGLMs to self-reported counts that pile up on round numbers, to study the age at which people start smoking in a large US health survey.
doi:10.1214/25-AOAS2065 (opens in a new tab)
Cite
```
@article{yee2025heaping,
  author    = {Yee, T. and Frigau, L. and Ma, C.},
  title     = {Heaping and seeping, GAITD regression and doubly constrained reduced-rank vector generalized linear models, in smoking studies},
  journal   = {The Annals of Applied Statistics},
  year      = {2025},
  volume    = {19},
  number    = {4},
  pages     = {3045--3070},
  doi       = {10.1214/25-AOAS2065},
  issn      = {1932-6157},
}
```

2024

Heaping and seeping, GAITD regression and doubly constrained reduced rank vector generalized linear models, in smoking studies

T. Yee, L. Frigau, C. Ma

NZSA 2024 - Presenters and Abstracts booklet, p. 38

Cite

@inproceedings{yee2024heaping,
  author    = {Yee, T. and Frigau, L. and Ma, C.},
  title     = {Heaping and seeping, GAITD regression and doubly constrained reduced rank vector generalized linear models, in smoking studies},
  booktitle = {NZSA 2024 - Presenters and Abstracts booklet},
  year      = {2024},
  pages     = {38},
}