Start With the Research Aim
A study may record responses from people, products, hospitals, or assessors. Those entities do not carry an automatic fixed or random label. The classification depends on what the analysis is intended to describe: the particular levels included in the study, or a broader population of possible levels from which those observed levels are treated as a sample.
This distinction matters because it sets the scope of the conclusions. A study comparing a named set of clinics can support conclusions about those clinics; a study intended to describe variation among clinics in a defined region needs a target population of clinics. The data might look similar in both studies, but the inferential target differs.
The word “random” refers to the factor’s levels being viewed as a sample from a target population; it does not simply mean that something in the study was randomized. Random assignment to treatments and sampling of factor levels answer different questions. Also, the same entity can appear in different roles: a person might be a subject in one study or a judge in another. Classify the factor named by the research question, not the entity in isolation.
A Practical Way to Classify Common Factors
For subjects, items, sites, and judges, use the same reasoning. First name the levels actually observed. Then state exactly which levels or population the conclusions are meant to cover. Finally, decide whether the observed levels themselves are the focus or stand in for a broader population.
For example, identify the participating patients, products, clinics, or assessors. Do not confuse those levels with the measurements recorded on them.
Specify the named people or locations, or define the population from which levels could have been sampled.
If conclusions concern the named levels, treat the factor as fixed. If the levels represent a defined population and conclusions concern that population, treat it as random.
A common random-effects representation for one grouping factor is \(Y_j=\mu+u_j+\epsilon_j\), where \(Y_j\) is a response for level \(j\), \(\mu\) is a common mean, \(u_j\) is that level’s deviation from the mean, and \(\epsilon_j\) is residual variation. In the usual random-intercept formulation, the \(u_j\) values are modeled as independent draws from a population with mean zero and variance \(\sigma^2_u\); residuals are modeled as independent of those effects, with mean zero and a common variance. These are model assumptions, not facts guaranteed by calling a factor random. The classification decision is about the target; the study design and subject-matter knowledge must still support the model assumptions.
Subjects: The People Observed
Worked Example: The Same Participants, Two Different Targets
A hypothetical study records a fatigue score after a standardized task for 40 patients at a rehabilitation center. The patients are identified as P01 through P40. Consider two possible research aims using these same observations.
Aim A: describe these 40 patients. The target is precisely P01 through P40. The subject levels are the specific people of interest, so subjects are treated as fixed for this aim. A conclusion about the average response is limited to these named participants; it is not automatically a conclusion about all patients with similar diagnoses.
Aim B: describe variation among eligible patients at the center. Suppose the 40 patients were sampled from a defined population of eligible patients and the aim is to generalize to that population. Subjects are treated as random levels for this aim. The target population must be described, for example, “patients meeting the center’s eligibility criteria during the stated recruitment period.” “Patients in general” is too vague to define the scope of the conclusion.
The recorded fatigue scores do not decide between these classifications. Nor does the fact that the patients were randomly assigned to task order, if that occurred: assignment concerns the task order, while this decision concerns whether the subject levels represent a population.
A related distinction is between the people who supply measurements and the factor those measurements concern. If each participant rates several products, the participants are not necessarily the product factor; they may be subjects, judges, or both, depending on the study’s structure and question. For this tutorial’s classification, ask separately what the intended target is for each named kind of level.
Items: Products, Prompts, or Other Materials
Worked Example: Product-Specific Claims or a Wider Item Population?
A hypothetical sensory panel obtains ratings for six coded tea blends: Blend A through Blend F. The blends have the following mean ratings on a nine-point scale: 6.2, 6.8, 5.9, 7.1, 6.5, and 6.3. Their average is
This arithmetic summarizes the six observed blends; it does not by itself determine whether “blend” is fixed or random.
Aim A: compare these six named blends. The six blends are the complete set of product levels of interest. Treat the item factor as fixed: the intended comparisons concern A through F, not untested blends.
Aim B: learn about blends from a defined formulation process. Suppose the six blends were selected to represent a clearly specified population of possible blends from that process, and the aim is to describe variation across that population. Treat items as random levels for this target. The population might be defined by ingredient limits and production conditions. Without such a definition, “all possible tea blends” is not a sufficiently precise target.
The calculated mean of 6.47 is the average of the six observed means, in rating-scale points. Whether that summary answers the research question depends on the target; the numerical value alone cannot justify generalization to other blends.
Sites and Judges: Define the Reach of Generalization
Worked Example: Named Clinics or a Population of Clinics?
A hypothetical evaluation records a recovery outcome for patients treated at four clinics: North, East, Central, and West. The investigators must decide what their site-related conclusion is intended to cover.
Aim A: compare these four clinics. The target consists of North, East, Central, and West as named locations. Site is fixed, and the interpretation is specific to these clinics under the study conditions.
Aim B: describe site-to-site variation across clinics in the province. Suppose these four clinics were selected from a defined provincial population of eligible clinics, and the aim is to generalize to that population. Site is random for this aim. The target is the eligible clinics in that province, not all healthcare settings everywhere.
If the four clinics were instead chosen for convenience and differ systematically from the clinics the researchers want to discuss, simply labeling site random does not make the sample representative. The sampling and recruitment process must support the intended generalization.
Worked Example: A Named Panel or a Population of Judges?
A hypothetical comparison of three food samples uses ratings from five trained assessors: J1, J2, J3, J4, and J5. Each gives one rating per sample.
Aim A: report the opinions of these five assessors. The named panel is the target. Judge is fixed, and conclusions refer to J1 through J5. The fact that the assessors were trained does not change the target.
Aim B: estimate how trained assessors would rate the samples as a group. If the five assessors were sampled from a defined population of trained judges and the intended conclusion concerns that population, judge is random. A suitable target might be trained assessors meeting stated experience and eligibility criteria in a specified setting.
The role matters: if the question is about ratings from judges, the factor is judge. If the question is about the people who received a treatment, the same type of person may instead be a subject. State the factor’s role and its target rather than using a label such as “people” alone.
Why a Defined Target Is Essential
For a random factor, the population of levels gives the generalization a precise boundary. For example, “variation among the 12 sampled sites” describes the observed sites; “variation among eligible sites in a specified health network” refers to a wider target. Those are different statements, even if both analyses use the same 12 observations. A population should be defined by relevant features such as location, eligibility, time period, or the process that produces levels.
The basic logic can be expressed with the random-intercept notation above. If a level’s mean is \(\mu+u_j\), and the target population has \(E(u_j)=0\), then the mean level response over that population is
This identity explains why a random factor is described in relation to a population of levels: the level deviations are understood relative to that population. It does not establish that a particular sample represents the population, or that the model assumptions hold. Those require justification from the design and context.
Common Mistake
A useful reporting habit is to pair the classification with its target: “subjects were treated as random levels representing eligible patients at this center,” or “the four named clinics were treated as fixed levels.” This wording makes the scope of the analysis visible and helps prevent conclusions from extending beyond the research aim.
Check Your Understanding
For each situation, identify the factor status and state the target that supports your decision.
- A study compares ratings for eight specifically selected photographs and intends to make claims only about those photographs. How should the photograph factor be treated?
- Researchers sample patients from a defined registry to describe variation among eligible patients in that registry. What is the subject target?
- A report concerns outcomes at three named hospitals only. Is hospital a fixed or random factor for this aim, and why?
- Five judges were randomly assigned to different rating orders. Does that assignment alone make judge a random factor? Explain.
- Give one precise target population for a study treating laboratory items as random levels. What information would make that population definition more useful?