Repeated Measures: Choose by Outcome Type, and by Population or Patient

อ่านฉบับภาษาไทย (Thai version)

Abstract

Repeated measurements on the same patients call for a model chosen by outcome type first. A repeated continuous outcome points to repeated-measures analysis of variance (RM-ANOVA), the classic choice, or to a linear mixed-effects model (LMM) when more flexibility is needed, because RM-ANOVA leans on a sphericity assumption and handles missing visits poorly. A repeated binary or count outcome rules RM-ANOVA out: generalized estimating equations (GEE) give a population-averaged effect, and a generalized linear mixed model (GLMM) gives a subject-specific effect, read as each patient's trajectory and the differences between patients. A repeated categorical outcome needs one more split, binary, ordinal or nominal, which sets the link: logistic, ordinal or multinomial GEE, or the matching mixed model. Pain rated none, mild, moderate or severe at 4 visits is ordinal, so it takes ordinal GEE or an ordinal mixed-effects model, never ordinary logistic regression. This article gives both decision tables, two short rules to remember, and the common slips.

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Pain rated at four visits

A rheumatology clinic follows adults with knee osteoarthritis after they start a new pain treatment. At each of 4 visits, the clinician rates the patient's pain as none, mild, moderate or severe. The same patients are measured again and again, and the first method that comes to mind for repeated measurements is repeated-measures analysis of variance (RM-ANOVA).

That instinct names the right method for one kind of outcome only. Measurements from the same patient are correlated, because each patient carries their own level from visit to visit, and every method below allows for that; the article on correlation in repeated measures shows why it matters. Which method fits depends first on the type of outcome, and the choice is easier to remember when it is sorted that way.

Sort by outcome type first

Four rows cover the common cases, and the outcome type picks the row. A continuous outcome, such as a symptom score, goes to the first two rows; a binary outcome, such as responder or not, or a count, such as the number of flare-ups since the last visit, goes to the last two.

The methods in the table, briefly. RM-ANOVA compares mean levels across visits and arms. A linear mixed-effects model (LMM) is a linear regression with random effects, patient-level deviations such as each patient's own starting level (a random intercept), which carry the correlation between one patient's visits [1].

Generalized estimating equations (GEE) fit a regression for the population mean at each visit and correct the standard errors for the correlation [2]. A generalized linear mixed model (GLMM) adds random effects to a logistic, Poisson or other generalized linear model.

For a binary or count outcome the table asks one more thing: which effect is wanted. A population-averaged effect is the average effect at the population level, the shift in the average outcome across all patients. A subject-specific effect is read at the level of the patient: each patient's trajectory over the visits and the differences between patients [3].

Repeated data: the common method by outcome type

RM-ANOVA is the classic, with two limits

RM-ANOVA is the classic method for repeated continuous data. It separates the variation between patients from the variation within patients and compares arms and visits on that basis. It has two limits [4].

• Covariance and sphericity. It relies on an assumption about how the visits covary, called sphericity: the variance of the difference between two visits is the same for every pair of visits. Scores that spread out as follow-up lengthens break it.
• Missing data. It needs every visit for every patient, so a patient with one missing visit drops out of the analysis. A mixed model handles missing visits more flexibly, because it uses every measurement each patient contributes.

When either limit bites, the second row of the table applies. The LMM models each patient's own line through random effects, and the covariance between visits becomes part of the model rather than an assumption to be met [1]. The mixed model series guide works through those choices.

When the outcome is binary

Now let the outcome be binary: at each visit the patient is a responder or not. Three verdicts follow.

• RM-ANOVA is not suitable. It models the mean of a continuous outcome, and a yes or no response at each visit is not one.
• GEE when the question wants a population-averaged effect.
• GLMM when the question wants a subject-specific effect.

The choice between the last two matters. On the odds-ratio scale the two effects are different quantities and give different numbers from the same data, with the subject-specific odds ratio further from 1 [5]. For a continuous outcome modelled on its own scale they coincide, which is why the continuous rows of the table do not split by question. The article on conditional versus marginal readings explains the gap.

Repeated categorical data: first ask which kind

When the repeated outcome is categorical, first separate which kind of categorical it is, because the model differs [6].

• Binary: two categories, such as yes or no.
• Ordinal: more than two categories with a natural order, such as mild, moderate and severe.
• Nominal: more than two categories with no order, such as A, B and C.

Each kind has its own link, the scale on which the model works: the logit for a binary outcome, a cumulative logit (the log odds of being at or below each level) for an ordinal one, and a multinomial logit for a nominal one. Within each kind, the population-averaged versus subject-specific split stays as it was. For the single-visit versions of these models, see ordinal logistic regression and multinomial versus ordinal regression.

Repeated categorical outcomes: kind by question

Same patients, repeated ordinal outcome

Put the two steps together for the case that is easiest to get wrong: the same patients are measured repeatedly, and the outcome is ordinal.

• Do not reach for RM-ANOVA. The outcome is not continuous, even when its levels are labelled 0, 1, 2 and 3.
• If the question wants a population-averaged effect, use the GEE family: ordinal GEE.
• If the question wants a subject-specific effect, use the mixed-effects or GLMM family: an ordinal mixed model.

Worked example: pain severity at four visits

Back to the clinic. Pain severity is rated none, mild, moderate or severe at each of 4 visits in the same patients. The choice takes three steps and no arithmetic: the outcome type fixes the kind of model, and the question fixes the family.

1. Step 1. Name the outcome type

   Four ordered levels, measured repeatedly on the same patients: a repeated ordinal categorical outcome. RM-ANOVA and ordinary logistic regression are out.

2. Step 2. A population-level question

   If the interest is the average effect at the population level, for example how the spread of pain levels shifts across all treated patients, use ordinal GEE.

3. Step 3. A patient-level question

   If the interest is each patient's trajectory over the 4 visits and the differences between patients, use an ordinal mixed-effects model.

Result: Pain severity at 4 visits is a repeated ordinal outcome: ordinal GEE for the population-average effect, an ordinal mixed-effects model for trajectories and differences between patients.

Common pitfalls

• "RM-ANOVA will do for a repeated binary outcome."

  RM-ANOVA is built for a continuous outcome; a responder or not at each visit is not one.

  Fix: Use GEE for a population-averaged effect and a GLMM for a subject-specific effect.

• "Code the pain levels 0 to 3 and run RM-ANOVA."

  Numbering ordered levels does not make them continuous, and measuring the same patients repeatedly does not change that.

  Fix: Treat the outcome as repeated ordinal: ordinal GEE for a population-averaged effect, an ordinal mixed model for a subject-specific effect.

• "Ordinary logistic regression handles a repeated ordinal outcome."

  Ordinary logistic regression handles two categories; ordered levels need an ordinal model.

  Fix: Fit an ordinal model, with GEE or a mixed model according to whether the question is population-averaged or subject-specific.

• "Unordered categories can go into an ordinal model."

  A nominal outcome such as A, B or C has no order for an ordinal model to use.

  Fix: Use multinomial GEE for a population-averaged effect or a multinomial mixed-effects model for a subject-specific effect.

Glossary

repeated measures (การวัดซ้ำ)

Measurements of the same outcome taken on the same patient more than once.

RM-ANOVA (การวิเคราะห์ความแปรปรวนแบบวัดซ้ำ)

Repeated-measures analysis of variance: the classic method for a repeated continuous outcome; it relies on sphericity and needs every visit for every patient.

sphericity (ความเท่ากันของความแปรปรวนของผลต่างระหว่างครั้งที่วัด)

Equal variance of the difference between every pair of visits.

LMM (แบบจำลองผสมเชิงเส้น)

Linear mixed-effects model: a linear regression with patient-level random effects; the more flexible choice for a repeated continuous outcome.

random intercept (ค่าตัดแกนสุ่ม)

A patient-level shift in the starting level, shared by all of that patient's measurements.

GEE (สมการประมาณค่าวางนัยทั่วไป)

Generalized estimating equations: a regression for the population mean at each visit, with standard errors corrected for the correlation; gives a population-averaged effect.

GLMM (แบบจำลองผสมเชิงเส้นวางนัยทั่วไป)

Generalized linear mixed model: a logistic, Poisson or other generalized linear model with random effects; gives a subject-specific effect.

population-averaged effect (ผลระดับประชากร)

The average effect at the population level: the shift in the average outcome across all patients.

subject-specific effect (ผลระดับบุคคล)

The effect read at the level of the patient: each patient's trajectory and the differences between patients.

ordinal outcome (ผลลัพธ์เรียงอันดับ)

More than two categories with a natural order, such as none, mild, moderate and severe.

nominal outcome (ผลลัพธ์หลายหมวดที่ไม่มีลำดับ)

More than two categories with no order, such as A, B and C.

cumulative logit (logit สะสม)

The link of an ordinal model: the log odds of being at or below each level.

References

1. Laird NM, Ware JH. Random-effects models for longitudinal data. Biometrics. 1982;38(4):963-974. https://doi.org/10.2307/2529876
2. Liang KY, Zeger SL. Longitudinal data analysis using generalized linear models. Biometrika. 1986;73:13-22. https://doi.org/10.1093/biomet/73.1.13
3. Zeger SL, Liang KY, Albert PS. Models for longitudinal data: a generalized estimating equation approach. Biometrics. 1988;44(4):1049-1060. https://doi.org/10.2307/2531734
4. Gueorguieva R, Krystal JH. Move over ANOVA: progress in analyzing repeated-measures data and its reflection in papers published in the Archives of General Psychiatry. Arch Gen Psychiatry. 2004;61(3):310-317. https://doi.org/10.1001/archpsyc.61.3.310
5. Hubbard AE, Ahern J, Fleischer NL, Van der Laan M, Lippman SA, Jewell N, et al. To GEE or not to GEE: comparing population average and mixed models for estimating the associations between neighborhood risk factors and health. Epidemiology. 2010;21(4):467-474. https://doi.org/10.1097/EDE.0b013e3181caeb90
6. Fitzmaurice GM, Laird NM, Ware JH. Applied longitudinal analysis. 2nd ed. Wiley; 2011. https://doi.org/10.1002/9781119513469

Key takeaways

• Choose the repeated-measures method by outcome type first: continuous → RM-ANOVA or, when more flexibility is needed, an LMM; binary or count → GEE for a population-averaged effect, a GLMM for a subject-specific effect.
• RM-ANOVA is the classic for repeated continuous data, but it relies on sphericity and handles missing visits less flexibly than a mixed model.
• Remember: repeated continuous → RM-ANOVA / LMM; repeated categorical → GEE / GLMM, then choose the link and model by binary, ordinal or nominal.
• Pain severity rated none, mild, moderate or severe at 4 visits is a repeated ordinal outcome: ordinal GEE for the population-average effect, an ordinal mixed-effects model for trajectories and differences between patients.
• Ordinal repeated data is not ordinary logistic regression: ordinary logistic is for a binary outcome, and ordered levels need an ordinal model.

Related in the wiki: [[mixed-model-series-guide]] [[mixed-model-conditional-vs-marginal-covariance]]