Homogeneity
This article focuses on homogeneity of processes. A process is homogeneous, when its distributional characteristics—such as means, variances, autocorrelations, etc.—are invariant across individuals in a population. Homogeneity is one of the two fundamental requirements for ergodicity, which determines whether we can meaningfully generalize findings from the population level to the individual level and vice versa.
In practice, many processes are likely to be non-homogeneous: Individuals tend to differ in how they think, feel, and behave, both on average as well as in the patterns of their fluctuations or changes over time. It is important to consider how and in what way processes may differ from person to person, as this has implications for how and when to measure a process, for analyzing the data, and the generalizability of your findings.
In this article, you will find: 1) a description of homogeneity and its relevance for your study; and 2) a discussion on its relation to stationarity.
1 Homogeneity in your study
Homogeneity implies that each individual in a population can be described using the same distributional characteristics and the same statistical model. This means that all individuals can be characterized by the same means or trends over time, but also the same variances, skewness, [autocorrelations], and so on. However, this is rarely the case for psychological processes. Individuals are often characterized by their own mean levels, patterns of variability or change, and relationships between variables over time. The presence of such individual differences lie at the heart of the [within/between distinction] and are often what longitudinal research seeks to capture and understand.
Thinking about how homogeneous, or, conversely, heterogeneous, the process is that you are interested in, is important and can be a research question in itself.
1.1 A spectrum of individual variation
In theory, homogeneity implies complete invariance across individuals. In reality, homogeneity might exists on a spectrum ranging from (strict) homogeneity to heterogeneity (Liu et al., 2024). Some processes may be highly similar across individuals, while others may differ substantially, not just in degree but also in form. For example, everyone may respond similar to a stressor, but some individuals show stronger or more prolonged reactions than others; a difference that can be considered quantitative in nature. In other cases, individual differences may be qualitative: One person may cope with stress by seeking social support, whereas others turn to exercise.
This raises important questions you can ask yourself: Which aspects of the process do I assume are homogeneous and which aspect do I expect to be heterogenous? When I expect heterogeneity, is this of a qualitative or quantitative nature? Based on such considerations, you can make more informed decisions about how to gather your data, and how to analyze these.
1.2 Temporal resolution and homogeneity
If and the extent to which a process appears homogeneous is also likely to depend on how it is measured and in particular on the temporal lens that is used. For example, the more you zoom in on a process (i.e., when you sample more densely in time), the more individual nuances you are likely to uncover. Sampling less densely (or aggregated data), may obscure important individual differences, leading to a false sense of homogeneity. What may seem homogeneous from one temporal design, may, with more fine-grained measurement, turn out to be less homogeneous (Adolph et al., 2008). Thus, the temporal resolution of your data can fundamentally shape what you observe and what conclusions you draw about the presence of heterogeneity, and it is therefore important to think about this when setting up your ILD study.
Thomas is interested in the development of object permanence in young infants. He conducts a longitudinal study in which he observes the behavior of a group of infants once per week for several weeks using a simple hiding task: A toy is placed under a cloth, and he records whether the infant attempts to retrieve it. When analyzing the data, Thomas observes each child exhibits a common developmental trajectory. That is, from one week on to the next, each child exhibits a sudden change, transitioning from not searching for the hidden toy, to consistently retrieving the hidden toy. While the pattern looks uniform, the timing of the shift does differ across infants. Based on this, Thomas concludes that the developmental process is homogeneous, except for individual differences in onset age.
Just like Thomas, Zainab studies the same process using the same task, but she collects data multiple times a day instead of once a week. When analyzing the data, she finds that it tells a different story. Like Thomas, Zainab also observes that infants develop object permanence at different ages, but her data shows that the developmental trajectories differ between infants. For example, while some infants show a sudden change in behavior, others show intermittent attempts throughout the day before the skill stabilizes. Based on her denser sampling, Zainab concludes that the developmental process is more heterogeneous than Thomas assumed, differing not only in onset age, but also in form of change.
This example is in part based on a study conducted by Adolph et al. (2008).
2 Homogeneity vs. stationarity
Homogeneity is central to the concept of ergodicity, which requires both homogeneity and stationarity to hold; only then can results be generalization from the group to the individual, or vice versa. When a process is heterogeneous, we cannot assume that what holds true for one person, will apply to another person, nor that population-level statistics reflect what is happening at the individual level (Molenaar, 2004).
While both homogeneity and stationarity concern invariance in distributional properties, they apply to different dimensions of the data:
- stationarity refers to invariance over time
- homogeneity refers to invariance across individuals
Importantly, these properties do not always co-occur: A processes can be stationary but not homogeneous, or the other way around, a process might be homogeneous but not stationary.
For example, classical conditioning may be considered a homogeneous process in the sense that most individuals exhibit similar patterns of learning. For most people, the association of a conditioned stimulus with an unconditioned stimulus typically results in a conditioned response. For example, if someone learns to associate a sound with a mild electric shock, they are likely to develop a fear response to the sound, much like others would. However, this is a learning process, characterized by changes over time; hence, the process is not stationary, and therefore, ergodicity does not hold.
Maria is interested in the development of motor skills in young infants, specifically their ability to stand. She conducts a longitudinal study in which she tracks infants over several weeks, observing whether and when they manage to stand independently.
Over the course of the study, Maria observes a common developmental trajectory: Eventually all infants begin to stand on their own. From this outcome-focused perspective, the process Maria is studying appears homogeneous (i.e., by the end of the study, all infants have developed the same ability). However, the process is not stationary, as it involves a clear change over time.
Alternatively, a process can be stationary but heterogeneous. For example, mood over a short time span might be stationary, but not homogeneous. For many individuals, their mood fluctuates around a stable mean over time, however, these stable mood levels can differ remarkably between individuals; some people may consistently report higher levels of positive affect, while others consistently report more neutral or negative affect on average. Moreover, they may also differ in their variability and skewness.
Peter is interested in how people’s mood fluctuates in daily life. Using a smartphone app, he collects data from individuals about their affect five times a day for one week. When inspecting the data, Peter finds that, for most individuals, mood fluctuates from moment to moment, but these fluctuations center around a stable personal mean. Additionally, while the variance remains stable over the course of the study for each individual, individuals differ in the amount of variance. Some people their mood can be characterized by a more positive mood with large fluctuations, while others show a more negative mood with little variability, and vice versa.
In other words, mood appears to be stationary within the short time span of the study: While individual fluctuate over time, there patterns of fluctuations do not change during the study. However, the average mood levels and the amount of variance differ substantially between individuals. Peter concludes that mood fluctuations in daily life are stationary, but not homogeneous across individuals in his sample.
Arguably, more often than not, processes might be neither homogeneous nor stationary. Understanding whether a process is consistent over time, across individuals, both, or neither, fundamentally shapes how you measure and analyze the process you are interested in and what kind of inferences you can draw.
Omar is studying the emotional adjustment of individuals who have lost a close loved one. To better understand individual trajectories, Omar conducts an ILD study in which participants report their levels of grief daily over the course of three months following their loss. When analyzing the data, Omar finds that grief does not follow a stable pattern over time. For some individuals, symptoms gradually decrease over time, for others, they remain elevated, and for some individuals, symptoms even worsen over time.
This indicates that the process Omar is interested in, is non-stationary: Emotional responses to grief change across the study period. At the same time, Omar also observes that the form and timing of these changes vary greatly between individuals. While some individuals show intense emotional distress immediately after the loss of their loved one, other people are characterized by delayed reactions. And while some individuals recover quickly, others continue to experience prolonged grief. These individual differences imply that the process Omar is interested in is also non-homogeneous in addition to non-stationary.
Ultimately, complex psychological process often involve both change over time and person-to-person variability. Capturing these nuances requires ILD research and analyses that can account for both these temporal dynamics and changes as well as individual differences therein.
3 Think more about
To explore whether a process is homogeneous, you need ILD from at least two individuals in order to compare their distributional characteristics. However, even if two individuals are characterized by the same means, variances, autocorrelations, etc., this does not imply that the broader population is characterized by that same process. Hence, a more robust assessment of homogeneity requires ILD from a sufficiently large sample of individuals.
A large sample raises the question of how to analyze your data: Should you use N=1 or N>1 analyses? Choosing between these approaches depends in part on how much you expect the process you are interested in to differ from person to person, and in turn, determines how much heterogeneity you will be able to detect. For example, N=1 analyses allows for individually tailored models that can capture [idiographic] patterns that are characteristic of a specific individual; this makes the analyses particularly useful when you expect substantial heterogeneity. By contrast, N>1 analyses can, in addition to enabling inferences at the population level, also provide insights into individual patterns. But the methods used within this approach often assume that individual differences are drawn from a common distribution, or that there is a limited number of subgroups that are characterized by different patterns.
Therefore, when selecting an analysis approach, it is important to consider the degree and nature of heterogeneity in the process you are interested in and how this aligns with the assumptions your methods impose about individual differences.
4 Takeaway
Homogeneity is a fundamental concept if you want to generalize results across persons. Yet, individuals often tend to differ in the means, temporal trends, and patters of their processes. Such differences may be subtle or substantial, quantitative or qualitative. Considering whether, and to what extent, a process is heterogeneous has important implications for how to best measure and analyze the process. But the reverse is also true: How you measures and analyze a process will also determine to what extent you are able to detect heterogeneity across people. Hence, thinking carefully about these reciprocal influences is essential for planing your study and for drawing valid conclusions from it.
5 Further reading
We have collected various topics for you to read more about below.
- Intra-individual versus inter-individual correlation
- [Within/between distinction]
- Ecological fallacy
- Common data types: Cross-sectional, time series, panel, and intensive longitudinal data
- [Idiographic versus nomothetic]
- N=1 versus N>1
- [Population-focused, process-focused, or both]
References
Citation
@article{hoekstra2025,
author = {Hoekstra, Ria H. A.},
title = {Homogeneity},
journal = {MATILDA},
number = {2025-05-23},
date = {2025-05-23},
url = {https://matilda.fss.uu.nl/articles/homogeneity.html},
langid = {en}
}