The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded — TEI-tagged view

Shinichi Nakagawa, P. Johnson, Holger Schielzeth · 2017 · statistics · doi:10.1098/rsif.2017.0213 · TEI xml · tagged edition · index · concepts · authors
Where this record's text came from. 2026-08-23 schema-0.3 pass: every quote verified as an exact substring of the staged text and anchored at merge time; concept terms verified present; every edge endpoint an existing node.
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Other fields define these differently the same word, defined another way in another discipline
overdispersion 2 fields
Tags on this paper every tag the record carries, whether or not the words occur in the text below — a number is how many papers in the corpus carry that tag; a tag with no number is unique to this paper
marginal R 22 conditional R 22 observation-level variance distribution-specific variance median26 mean109 formal-model9
Method, as described: derivation plus comparison: three routes to the same variance term, checked against each other and against exact solutions where those exist, with worked examples in R

Our tag layer for this paper

Cross-disciplinary analysis
content-tags & facets: phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text
phenomenonone of the most-reported numbers in science, with no single definition once the model has random effects— “The coefficient of determination R 2 quantifies the proportion of variance explained by a statistical model and is an important summary statistic of biological interest. However, estimating R 2 for generalized linear mixed models (GLMMs) remains challenging.”
claim-typemethod— “As shown in the electronic supplementary material, appendix S2, ln(1 + 1/ λ ) (lognormal approximation), 1/ λ (delta method approximation) and (trigamma function) give similar results when λ is greater than 2. Our recommendation is to use the trigamma function for obtaining whenever this is possible.”
method — evidence— “A comparison among equations (6.7)–(6.9) is also shown in electronic supplementary material, appendix S3 (it turns out equation (6.8) gives the best approximation). Simulation will give the most accurate conversions when no exact solutions are available.”
object of study — evidence— “The coefficient of determination R 2 quantifies the proportion of variance explained by a statistical model and is an important summary statistic of biological interest. However, estimating R 2 for generalized linear mixed models (GLMMs) remains challenging.”
research question — evidence— “However, obtaining for specific distributions is not always possible, because in many families of GLMMs, (observation-level variance) cannot be clearly separated into (overdispersion variance) and (distribution-specific variance). It turns out that binomial and Poisson distributions are special cases where can be usefully calculated, because…
contribution — evidence— “As shown in the electronic supplementary material, appendix S2, ln(1 + 1/ λ ) (lognormal approximation), 1/ λ (delta method approximation) and (trigamma function) give similar results when λ is greater than 2. Our recommendation is to use the trigamma function for obtaining whenever this is possible.”
mechanismA NAMING RULE - WHEN A MODIFIER IN A NAME IS LICENSED: `In such a case, the ICC should not be called `ADJUSTED`. For an ICC value to be adjusted for a source of variance, that variance must be more than 0 and omitted from the ICC calculation.` A published condition on the use of one word inside a statistic's name - the same shape as buck2011's rule for when two fields may share an acronym, one level down— “In such a case, the ICC should not be called ‘adjusted’ ( sensu [ 2 ]). For an ICC value to be adjusted for a source of variance, that variance must be more than 0 and omitted from the ICC calculation.”mechanisms[0]
mechanismAND ONE NAME FOR TWO QUANTITIES, KEPT APART BY A SUBSCRIPT: `MARGINAL AND CONDITIONAL R2 differ only in whether the random effect variance is included in the numerator` - the marginal is the proportion explained by the fixed effects, the conditional by fixed AND random. Two numbers a reader will see reported as `R2`— “where represents the marginal R 2 , which is the proportion of the total variance explained by the fixed effects, represents the conditional R 2 , which is the proportion of the variance explained by both fixed and random effects, and is the variance explained by fixed effects [ 11 ]. As marginal and conditional R 2 differ only in whether the…mechanisms[1]
mechanismAND A RENAMING WHERE THE NOTATION DELIBERATELY DOES NOT FOLLOW: `We refer to this generalized version ... as `THE OBSERVATION-LEVEL VARIANCE` here rather than the residual variance (but we keep the notation). Note that the observation-level variance should not be confused with the variance associated with OLRE, which ... can be considered to be A PART OF it.` The name changes, the symbol does not, and the sentence that renames it also warns which neighbouring term it is not— “However, as we will show below, we can always obtain the GLMM version of (on the latent scale) directly. We refer to this generalized version of as ‘the observation-level variance’ here rather than the residual variance (but we keep the notation ). Note that the observation-level variance, , should not be confused with the variance associated…mechanisms[2]
mechanismAND THE EARLIER DEFINITION WORKED ONLY FOR TWO SPECIAL CASES, SAID PLAINLY: `obtaining for specific distributions is not always possible, because in many families of GLMMs, (observation-level variance) cannot be clearly separated into (overdispersion variance) and (distribution-specific variance). It turns out that binomial and Poisson distributions are special cases where can be usefully calculated ... This is not the case for other families.` The general method was a special case and the authors are the ones who found out— “However, obtaining for specific distributions is not always possible, because in many families of GLMMs, (observation-level variance) cannot be clearly separated into (overdispersion variance) and (distribution-specific variance). It turns out that binomial and Poisson distributions are special cases where can be usefully calculated, because…mechanisms[3]
mechanismAND A DISTINCTION THE AUTHORS WITHDRAW FOR ONE FAMILY AND KEEP FOR ANOTHER: `In the past, we distinguished between ICC on the latent scale and on the observation scale. Such a distinction turns out to be strictly appropriate only for binomial distributions but not for Poisson distributions (and probably also not for other non-Gaussian distributions).` A retraction with its scope stated, including the part they are only probably right about— “In the past, we distinguished between ICC on the latent scale and on the observation scale [ 2 ]. Such a distinction turns out to be strictly appropriate only for binomial distributions but not for Poisson distributions (and probably also not for other non-Gaussian distributions). This is because the property of what we have called the…mechanisms[4]
mechanismAND THEY NAME WHAT THEY USED TO CALL IT: `note that we would have called 1/lambda THE DISTRIBUTION-SPECIFIC VARIANCE` - the same quantity, the old name, and the new one, in a parenthesis. The corpus has papers that rename things and papers that correct things; this one prints the substitution table— “As with the observation-level variance for the log-Poisson model (which is 1/ λ and changes with λ ; note that we would have called 1/ λ the distribution-specific variance; [ 2 , 3 ]), the observation-level variance of the binomial distribution changes as p changes (see electronic supplementary material, appendix S5), suggesting these two…mechanisms[5]
mechanismAND THE NAMING CHOICE MOVES EVERY REPORTED NUMBER, IN A KNOWN DIRECTION: `the MINIMUM VALUE of 1/(p(1-p)) IS 4, which is larger than pi^2/3 = 3.29, meaning that the use of 1/p(1-p) IN R2 and ICC for binary data will always produce larger values than those using pi^2/3.` Which term you call the denominator decides whether every R2 in a literature is larger or smaller - not a range, a DIRECTION, with a floor. And Browne et al. observed exactly that— “Further, the minimum value of 1/( p (1 – p )) is 4, which is larger than π 2 /3 ≈ 3.29, meaning that the use of 1/ p (1 – p ) in R 2 and ICC for binary data will always produce larger values than those using π 2 /3. Consequently, Browne et al . [ 14 ] showed that ICC values (or variance partition coefficients, VPCs) estimated using π 2 /3…mechanisms[6]
mechanismAND TWO OF THEIR OWN EARLIER RECOMMENDATIONS CONTRADICT EACH OTHER: `In our earlier papers, we proposed to use the exponential of the intercept ... WE ALSO SUGGESTED that it is possible to use the mean of observed values. Unfortunately, these two recommendations are often inconsistent with each other.` Two ways to compute one input, from one paper, that disagreemechanisms[7]
mechanismAND THE REASON IS THAT ONE OF THEM ESTIMATES A DIFFERENT LOCATION ENTIRELY: `In fact, exp(beta0) gives the median value of y rather than the mean ... Thus, the use of exp(beta0) will often OVERESTIMATE, providing SMALLER ESTIMATES OF R2 AND ICC ... when lambda is small, the difference can be substantial so the choice of the method needs to be reported for reproducibility.` A methods choice promoted to a reporting requirement because the two answers differ
Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher.
mechanisms[8]
mechanismAND THREE ROUTES REACH ONE QUANTITY AND AGREE ONLY ABOVE A THRESHOLD: `ln(1 + 1/lambda) (LOGNORMAL APPROXIMATION), 1/lambda (DELTA METHOD APPROXIMATION) and (TRIGAMMA FUNCTION) GIVE SIMILAR RESULTS WHEN LAMBDA IS GREATER THAN 2. Our recommendation is to use the trigamma function.` Three names, one target, and a condition under which the choice stops mattering
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mechanisms[9]
extends the authors' own earlier definitions of R2_GLMM and ICC_GLMM (Nakagawa & Schielzeth 2013; Nakagawa, Johnson & Schielzeth), which covered only binomial and Poisson families - 'It turns out that binomial and Poisson distributions are special cases'
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claim_relations[0]
contradicts their own earlier distinction between ICC on the latent scale and on the observation scale, which 'turns out to be strictly appropriate only for binomial distributions but not for Poisson distributions'
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claim_relations[1]
sense`marginal R2` - the proportion of total variance explained by the FIXED effects only; `conditional R2` - by fixed AND random. They `differ only in whether the random effect variance is included in the numerator`
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senses[0]
sense`observation-level variance` (this paper) - the generalized residual term, explicitly renamed `rather than the residual variance` while KEEPING THE NOTATION, and not to be confused with the variance of an observation-level random effect
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senses[1]
sense`distribution-specific variance` - what 1/lambda would have been called in the authors' earlier papers; kept as a separate concept for binomial distributions and declared not analogous
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senses[2]
sense`repeatability R` (ECOLOGY AND EVOLUTION) - a type of intra-class correlation where the grouping factor is individuals measured repeatedly
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senses[3]
sense`adjusted ICC` - and the condition on the word: an ICC may only be called adjusted if the variance it is adjusted for is `MORE THAN 0 AND OMITTED FROM THE CALCULATION`
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senses[4]
controlderiving the same observation-level variance three independent ways - delta method, lognormal approximation, trigamma function - and comparing them — held
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— rules out a variance term that is an artefact of one derivation; they agree when lambda > 2 and the trigamma is recommended where it applies
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controls[0]
controlcomparing their own bias-correction approximation against McCulloch et al.'s and Zeger et al.'s, and against simulation — failed
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— rules out recommending their own formula without testing it - and the test says somebody else's is better
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controls[1]
controlrequiring that the method used to obtain lambda be reported — partial
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— rules out two papers reporting 'R2' from the same data and the same model with different numbers, because they estimated lambda differently
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controls[2]
weightthe authors' own earlier definitions of R2_GLMM and ICC_GLMM (Nakagawa & Schielzeth 2013; Nakagawa, Johnson & Schielzeth), which covered only binomial and Poisson families - 'It turns out that binomial and Poisson distributions are special cases' — 0.95
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claim_relations[0]
weighttheir own earlier distinction between ICC on the latent scale and on the observation scale, which 'turns out to be strictly appropriate only for binomial distributions but not for Poisson distributions' — 0.9
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claim_relations[1]
notemarginal R 2 —distinct-from→ conditional R 2 — 'As MARGINAL AND…' One name for two numbers, kept apart by a subscript a reader may not carry across papers
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concept_edges[0]
relationmarginal R 2 —distinct-from→ conditional R 2 — the two differ ONLY in whether the random-effect variance is in the numerator - and both are reported as 'R2'
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concept_edges[0]
weightmarginal R 2 —distinct-from→ conditional R 2 — 0.95
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noteobservation-level variance —distinct-from→ distribution-specific variance — 'ARE ANALOGOUS…' The authors previously called one of these by the other's name - 'note that we would have called 1/lambda THE DISTRIBUTION-SPECIFIC VARIANCE' - and the edge records the distinction they drew afterwards
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concept_edges[1]
relationobservation-level variance —distinct-from→ distribution-specific variance — the two observation-level variances are analogous to each other and the distribution-specific variance is not - which is why the term was renamed
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concept_edges[1]
weightobservation-level variance —distinct-from→ distribution-specific variance — 0.95
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concept_edges[1]
notemedian —distinct-from→ mean — 'In fact, exp(beta0) gives the median value of y rather than the mean of y, assuming a Poisson distribution. Thus, the use of exp(beta0) will often overestimate, providing smaller estimates of R2 AND ICC ... when lambda is small, the difference can be substantial so the choice of the method needs to be reported for reproducibility.' A textbook distinction with a reporting requirement attached to it
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concept_edges[2]
relationmedian —distinct-from→ mean — exp(beta0) estimates the MEDIAN where the formula requires the MEAN, so using it systematically shrinks R2 and ICC
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concept_edges[2]
weightmedian —distinct-from→ mean — 0.95
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concept_edges[2]
outcomecontrol — held
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controls[0]
outcomecontrol — failed
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controls[1]
outcomecontrol — partial
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controls[2]
limitationTHEIR OWN APPROXIMATION IS NOT THE BEST ONE, AND THEY SAY SO: `A comparison among equations (6.7)-(6.9) is also shown in electronic supplementary material, appendix S3 (it turns out equation (6.8) gives the best approximation). Simulation will give the most accurate conversions when no exact solutions are available.` Equation 6.7 is theirs; 6.8 is McCulloch et al.'s
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limitations[0]
limitationAND A NEIGHBOURING STATISTIC IS ADMITTED INTO THE FAMILY WITH A HEDGE IN PARENTHESES: the trigamma function has been used for `heritability (which can be seen as a type of ICC although in a strict sense, it is not)`. A kindhood claim and its denial in eleven words - the corpus's `is-a` cannot carry that, so it is here in prose
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limitations[1]
limitationand one field renames the statistic outright: `a type of ICC is often referred to as repeatability R, where the grouping factor is often individuals that have been phenotyped repeatedly` - ecology and evolution's name for what the rest of the literature calls an intra-class correlation
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limitations[2]
Content tags
content-observations: finding · quantity · variable · definition · key-term — grounded in the full text
finding
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content_tags.findings[0]
finding
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content_tags.findings[1]
finding
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content_tags.findings[2]
finding
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content_tags.findings[3]
finding
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content_tags.findings[4]
finding
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content_tags.findings[5]
quantity1/(p(1-p)) has a minimum of 4; pi^2/3 = 3.29 —
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content_tags.quantities[0]
quantitylambda greater than 2 —
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content_tags.quantities[1]
quantityequation (6.8), McCulloch et al.'s, not the authors' own (6.7) —
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content_tags.quantities[2]
variableoverdispersion
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content_tags.variables[0]
definitioncoefficient of determination
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content_tags.definitions[0]
definitionadjusted ICC
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content_tags.definitions[1]
Concepts
terms this paper carries, each with the sentence it was cut from — the layer that joins one paper to another. Terms with no quote yet are on CONCEPTS.html.
conceptmarginal R 2 — not defined in this paper's own words; variance explained by the fixed effects alone
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concepts[0]
conceptconditional R 2 — not defined in this paper's own words; variance explained by fixed and random effects together
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concepts[1]
conceptobservation-level variance — not defined in this paper's own words; the generalized residual term, renamed here and keeping its old notation
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concepts[2]
conceptdistribution-specific variance — not defined in this paper's own words; the term the authors previously used for the same quantity in Poisson models, and still use for binomial ones
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concepts[3]
conceptmedian — not defined in this paper's own words; what exp(beta0) actually estimates, against the mean that the formula needs
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concepts[4]
conceptmean — not defined in this paper's own words; the quantity required, and the reason the two earlier recommendations disagree
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concepts[5]
Statistics — cross-field join keys
field-invariant statistical terms, used in their statistical sense (gated on a number / symbol / test). A p-value or a standard deviation means the SAME thing in physics, biology, psychology and sociology — so these are the strongest cross-field join keys. Grounded verbatim.
intra-class correlation coefficient
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content_tags.statistics[0]

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