Defining States From Scientific Questions

Start With the Scientific Question

A multi-state analysis describes how people occupy states over time and how their recorded histories distinguish one condition from another. The states are not supplied automatically by the data: investigators choose them. That choice should follow the scientific question. A useful state definition captures distinctions that matter to the question, while avoiding distinctions that the study cannot measure or does not need.

For example, a study of whether a disease progresses may need to distinguish disease severity. A study of health-care use may instead need to distinguish where a patient receives care. The same person can be described under either scheme, but the resulting states answer different questions. Defining states is therefore a scientific and measurement decision, not merely a convenient way to label records.

Definition: A state is a scientifically specified category describing the condition or status of an individual at a particular time. For a chosen target population and set of observation times, a state scheme is mutually exclusive if an individual can be assigned to no more than one state at a time, and exhaustive if every individual in scope can be assigned to one of the states.

Let \(\Omega\) denote the set of person-time situations that the study aims to classify, and let \(C_1,\ldots,C_K\) be the proposed state categories. A valid classification has no overlap and no gap:

C_j \cap C_k = \varnothing \quad (j \ne k), \qquad \bigcup_{j=1}^{K} C_j = \Omega.

If \(X_i(t)\) denotes the state assigned to person \(i\) at time \(t\), then these requirements mean that \(X_i(t)\) has exactly one value in the state set whenever person \(i\) is in scope at time \(t\). The definition must specify enough detail to make that assignment reproducible: what evidence determines the state, how competing criteria are resolved, and what happens when information is missing or ambiguous.

States Are Not the Same as Measurements or Events

A state summarizes a status at a time. A measurement is evidence that may be used to classify that status. An event is something that occurs, such as a diagnosis, a procedure, or a change in treatment. The distinction matters because measurements may fluctuate, and events may not by themselves tell you the person’s current status. A study should specify whether a state is based on a current assessment, a prior history, a threshold, or a combination.

The classification rule also needs a reference time. “Has a positive test” could mean positive at today’s assessment, positive at any assessment so far, or positive on a confirmatory test within a defined window. These definitions are not interchangeable. The first describes current test status; the second includes history; the third adds a confirmation requirement. The question and the available measurement schedule determine which is suitable.

State categories should normally be defined using information available at the time of classification. Defining a state by a later outcome—for example, assigning someone at baseline to “eventual progressor”—builds future information into the baseline label. That changes the question and can make a prospective description misleading.

Worked Example: Defining Disease Status for a Progression Question

Hypothetical study. Investigators assess 120 adults with a chronic inflammatory disease every three months. Their question is: “How does current disease activity change over follow-up?” At each visit, a prespecified clinical assessment classifies activity as active or inactive. The records also indicate whether a participant has ever received the diagnosis.

A suitable scheme for the stated question has two states: active disease at the visit and inactive disease at the visit. The operational rule is the visit-specific clinical assessment, not treatment history or whether the person has ever had the disease. The states are mutually exclusive because the assessment assigns one of the two activity classifications at a visit. They are exhaustive only if the assessment is available and valid for every visit in scope; visits with unavailable assessments need a stated missing-data rule rather than an implicit third clinical status.

Suppose, at one visit, 72 participants are classified active and 48 inactive. The counts sum to the 120 assessed participants:

$$ 72 + 48 = 120. $$

Thus the observed proportions at that visit are \(72/120=0.600\) active and \(48/120=0.400\) inactive. These arithmetic checks support the partition for this visit; they do not establish that the classification is clinically valid. A separate review of the assessment criteria and their use is still needed.

If the scientific question instead concerned long-term disease history, “ever diagnosed” might be relevant. It is not a substitute for current activity: a participant can have a past diagnosis while currently inactive. Including both variables as if they were competing current states would create overlap unless the categories were redesigned to resolve it.

Make the Categories Operational

A state label such as “severe” is not yet a usable definition. The study needs criteria that identify who belongs in that category. Criteria might use a validated score and a stated cutoff, an adjudicated clinical diagnosis, a documented location of care, or another prespecified rule. If clinical judgment is involved, the protocol should explain how assessors apply it. If a person could satisfy multiple labels, the rule must resolve that overlap or the labels must be revised.

A practical state-definition checklist is:

Target: Specify the population and the time points or periods to which the classification applies.
Meaning: State what scientific or clinical condition each category represents.
Evidence: Identify the measurements, records, or criteria used to assign the category.
Boundary rules: Specify cutoffs, tie-breaking rules, and the treatment of uncertainty or missing information.
Coverage: Check that all in-scope situations have one assignment and no situation has two.

These decisions should be documented before interpreting state histories. Otherwise, apparently simple labels can conceal different rules across sites, assessors, or time periods. A classification can be consistent in its wording but inconsistent in its application.

Worked Example: Defining States for a Care-Pathway Question

Hypothetical study. A hospital network wants to describe where patients with an acute respiratory condition are receiving care at a daily midnight assessment. The records cover 80 patients on a particular day. The scientific question concerns current care setting, not disease severity.

The investigators define three states: hospital ward, intensive care unit, and outside the hospital. The rule uses the patient’s recorded location at midnight. To make the categories exclusive, “intensive care unit” takes precedence over the general hospital label; a patient in the ICU is not also counted as being on a ward. “Outside the hospital” includes patients who are at home or in another non-hospital setting, if those locations are within the study’s scope.

Suppose the records show 42 patients on a ward, 11 in the ICU, and 27 outside the hospital. The classification check is:

$$ 42 + 11 + 27 = 80. $$

The corresponding proportions are \(42/80=0.525\), \(11/80=0.1375\), and \(27/80=0.3375\); they sum to \(1.000\). This verifies that the observed records have one of the three labels each. It does not verify that the location records are accurate, or that “outside the hospital” is sufficiently detailed for every care-pathway question.

If the aim were to study service use outside the hospital, combining home and a skilled nursing facility might conceal an important distinction. If the aim were only to contrast hospital with non-hospital care, the combined category could be adequate. The right level of detail depends on what the investigators need to distinguish.

Choose the Right Level of Detail

Too few states can combine scientifically different situations. Too many can create categories that are difficult to measure reliably, sparse in the observed data, or irrelevant to the question. A useful design asks, for each proposed distinction: Would a different answer to the scientific question be possible depending on which category a person occupies? Can the distinction be measured consistently at the required times? If the distinction does not matter to the question, or cannot be classified credibly, extra detail may create complexity without useful information.

State definitions also determine which contrasts the analysis can express. If early and advanced disease are combined as “disease present,” the resulting scheme cannot describe differences between those levels. Conversely, dividing a broad state into several subcategories permits more detailed description only if the data support those distinctions. This is a trade-off in the definition, not a technical correction that can be made after interpretation.

Worked Example: Comparing Two Valid Disease Classifications

Hypothetical study. At a scheduled assessment, 100 patients are classified using a clinical score. The study’s broad question is whether patients are currently well or have symptoms. The detailed scores show 58 patients without symptoms, 26 with mild symptoms, and 16 with severe symptoms.

For the broad question, a two-state scheme is possible: no symptoms and symptoms present. Its symptom category combines mild and severe cases, so the count is \(26+16=42\). The observed proportion with symptoms is:

\frac{26+16}{100}=\frac{42}{100}=0.420.

For a question about symptom severity, a three-state scheme is more informative: no symptoms, mild symptoms, and severe symptoms. The proportions are \(58/100=0.580\), \(26/100=0.260\), and \(16/100=0.160\). They sum to \(1.000\), as expected for an exhaustive classification.

Both schemes can be valid for their respective questions, provided the clinical criteria distinguish the categories reliably. The broad scheme cannot answer how many patients have severe symptoms separately. The detailed scheme can answer that question, but requires reliable severity classification and is unnecessary if only the overall presence of symptoms matters.

Why the Partition Requirement Matters

Mutual exclusivity and exhaustiveness are not formalities. They ensure that a person-time observation contributes to exactly one state. To see why, define \(I_j(i,t)=1\) when person \(i\) at time \(t\) belongs to \(C_j\), and \(I_j(i,t)=0\) otherwise. For a valid state scheme:

\sum_{j=1}^{K} I_j(i,t)=1.

The equation follows directly from the two requirements. Because the categories are mutually exclusive, at most one indicator can equal 1, so the sum is at most 1. Because the categories are exhaustive, at least one indicator equals 1, so the sum is at least 1. Together, these imply that the sum equals 1. If categories overlap, the sum can exceed 1; if there is a gap, it can equal 0. Either outcome means the proposed labels do not define exactly one state for every in-scope observation.

For a sample of \(n\) in-scope observations, summing that identity across observations gives a second check:

\sum_{j=1}^{K} n_j = n,

where \(n_j\) is the number assigned to state \(j\). A failure of this equality is a signal to investigate overlap, missing classifications, exclusions, or data-processing errors. Passing it is necessary but not sufficient: a consistently applied set of categories can still fail to represent the scientific concept of interest.

Diagnostics, Alternatives, and Limitations

Several checks help reveal a weak definition before results are interpreted. Tabulate the number assigned to each category at each assessment time. Look for unclassified records, duplicate assignments, and sudden changes in classification that could reflect a revised measurement rule rather than a change in the underlying condition. Review borderline records and compare assignments across assessors or sites where relevant. Also ask whether a category is defined by information that would only be known later.

The main alternative is not a different statistical estimator; it is a different representation of the scientific information. A single broad state scheme is easier to interpret but may hide distinctions. A more detailed scheme can preserve them but requires more reliable measurement and can make the analysis harder to communicate. In some studies, it is sensible to report a primary, scientifically justified classification and use a coarser or finer definition as a sensitivity analysis. Such alternatives should be described as different definitions of the target, not as interchangeable labels for an identical question.

State definitions have limitations. A category compresses underlying variation: two people assigned to the same state may differ substantially. Measurement error can place a person in the wrong category, and a classification rule can behave differently across populations or settings. A state scheme also cannot recover a distinction that was never recorded. These limitations should be acknowledged when explaining what the resulting analysis can and cannot say.

Common pitfall: A researcher proposes “mild disease,” “severe disease,” and “disease present” as three separate states because all three labels appear in the records. This is not an exclusive classification: a person with severe disease also has disease present. The correction is to define nonoverlapping categories, such as “no disease,” “mild disease,” and “severe disease,” with explicit criteria for each, or to choose one variable that matches the scientific question.

A good state scheme is therefore neither the longest list of available labels nor the simplest list that can be counted. It is the smallest clearly measured set of categories that preserves the distinctions needed for the study question, with explicit rules for assigning every in-scope observation once.

Check Your Understanding

For each question, identify the key issue in the proposed classification. Brief solutions follow.

A study classifies a patient at each visit as “receiving treatment” or “has severe disease.” Why might these fail to be mutually exclusive?
A care study uses “ward” and “ICU” as categories but omits patients at home. What property of the state scheme fails if home patients are in scope?
At an assessment, 30 people are classified as mild, 12 as severe, and 8 as both. What is wrong with treating the first three labels as exclusive states?
A study asks whether symptoms are present and combines all symptom severity levels. What information does this definition deliberately discard?
For 50 in-scope observations, state counts are 18, 21, and 9. What should be checked before interpreting the state proportions?

Solutions. (1) Treatment status and disease severity are different dimensions; a patient can receive treatment while having severe disease. The categories overlap unless the rules form a single exclusive classification. (2) Exhaustiveness fails because some in-scope patients have no category. Add an appropriate outside-hospital category or explicitly narrow the target population. (3) The labels overlap: the 8 severe cases counted in “both” are also included in mild under the stated assignments. Revise the definitions to create disjoint categories and apply them consistently. (4) The scheme discards the distinction between mild and severe symptoms; it can describe symptom presence but not severity separately. (5) The counts sum to \(18+21+9=48\), not 50. Check for two unclassified observations, exclusions, missing data, or a counting error before calculating or interpreting proportions.