The DOI matches its Crossref record; 58 of the 58 references it cites were found in a registry or an index; 5 of the 44 sources its claims cite were checked for support.
Analysis of variance—why it is more important than ever
Open accessAbstract
Analysis of variance (ANOVA) is an extremely important method in exploratory and confirmatory data analysis. Unfortunately, in complex problems (e.g., split-plot designs), it is not always easy to set up an appropriate ANOVA. We propose a hierarchical analysis that automatically gives the correct ANOVA comparisons even in complex scenarios. The inferences for all means and variances are performed under a model with a separate batch of effects for each row of the ANOVA table. We connect to classical ANOVA by working with finite-sample variance components: fixed and random effects models are characterized by inferences about existing levels of a factor and new levels, respectively. We also introduce a new graphical display showing inferences about the standard deviations of each batch of effects. We illustrate with two examples from our applied data analysis, first illustrating the usefulness of our hierarchical computations and displays, and second showing how the ideas of ANOVA are helpful in understanding a previously fit hierarchical model.
Overview
AI snapshot
Written from the paper’s full text.
TL;DR
Gelman argues that ANOVA remains central because it is essentially the batching of model parameters into variance-components groups, and that hierarchical modeling makes correct ANOVA comparisons automatic in complex designs. The paper proposes finite-population variance summaries and graphical displays, illustrating them with split-plot, nested industrial, Web connect-time, and election-poll examples.
Key findings
- Treating every row of an ANOVA table as a batch of exchangeable effects with its own estimated variance avoids the classical need to manually choose error terms; the split-plot Latin square example has 9 variance components, and the nested machine-treatment example has 3 (treatment, treatment × machine, treatment × machine × measurement).
- The paper introduces finite-population standard deviations \(s_m\) and a graphical ANOVA display with point estimates and 50% and 95% intervals; in the split-plot Latin square, this display shows simultaneous uncertainty for all variance components rather than testing each under a null model.
- In the Web connect-time example, a \(4 \times 45 \times 2 \times 25 \times 2\) factorial of log connect times, the multilevel display shows destination as the largest source of variation, company as large on its own and in interactions, while classical ANOVA finds all main effects and almost all interactions statistically significant but does not directly compare their magnitudes.
- In the 1988 election logistic example using seven CBS polls, ethnicity is the most important demographic factor and state explains substantial variation; ethnicity × state × region interactions are surprisingly large.
Methods
- The paper is methodological and applied: it formulates ANOVA as a hierarchical regression in which each row \(m\) has coefficients \(\beta^{(m)}_j \sim N(0,\sigma^2_m)\), with classical method-of-moments estimation for balanced designs and Bayesian Gibbs sampling for more general models.
- Data examples include a \(5 \times 5 \times 2\) split-plot Latin square; a nested industrial experiment with 4 treatments, 20 machines, and 6 measurements per machine (120 data points); Web connect times from 45 locations to 4 destinations for 2 companies over 25 hours and 2 weeks; and seven CBS News polls from the week before the 1988 U.S. presidential election with demographic, state, and region indicators.
- Analysis focuses on estimating variance components and finite-population standard deviations, with simulation-based uncertainty intervals and graphical displays; the election example uses Bayesian posterior simulation via Bugs linked to R.
Limitations
- The paper states that its exchangeable-effects model is not universally appropriate and discusses nonexchangeable, spatial, time-series, and nonadditive interaction structures as extensions beyond the basic framework.
- The paper notes that with a uniform prior the posterior for \(\sigma_m\) is improper when a batch has one degree of freedom, so those coefficients are essentially estimated by maximum likelihood; near-zero variance components are expected and point estimates can understate uncertainty.
- Discussants (Tjur, McCullagh, Hox and Hoijtink) argue that the model must be specified first, that automatic correctness is not guaranteed, that exchangeability may be inappropriate for ordered or structured factors, and that the approach is mainly descriptive with uninformative priors; Gelman’s rejoinder concedes that model specification comes first and that the automatic step can be inappropriate in particular cases.
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Claims the paper makes
5 of 44 cited sources checked · 3 supported · 0 flagged · accuracy not yet measured
The sources with a verdict support this claim; the rest weren’t checked yet, have no full text we could read, or their verdicts aren’t rated yet.
In the paper
“More complex structures, such as time-series and spatial models [see Ripley (1981) and Besag and Higdon (1999)], or negative intraclass correlations, cannot be additively decomposed in a natural way into exchangeable components.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
What the sources say
- Not checked — no full text of the sourcecites Ripley (1981) source
- Supportedcites Besag and Higdon (1999) source
The citation follows the examples of time-series and spatial models. The paper is an example of the latter: its ‘Plot-based priors for fertility effects’ section says the authors adopt intrinsic autoregressions to model spatial variation in field experiments. The citation need not establish the separate statement about negative intraclass correlations. (LLM, conf=0.87)
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesEvery source the paper cites for this claim was checked and supports it.
In the paper
“For example, in the Internet example in Section 7.1 the coefficients in any batch of two-way or higher-level interactions have a natural gridded structure that is potentially more complex than the pure exchangeability of additive components [see Aldous (1981)].”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
What the sources say
- Supportedcites Aldous (1981) source
The Introduction defines arrays whose rows and columns are exchangeable separately, and Section 5 identifies these grids as partially exchangeable under separate permutations of their two indices. This supports the cited distinction between gridded structure and ordinary exchangeability; the Internet example is the citing authors’ application of that distinction. (LLM, conf=0.86)
Strength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passes- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“The multilevel model allows more accurate inferences-the usual partial pooling or Bayesian rationale [see Park, Gelman and Bafumi (2004)]”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3. ANOVA is not just for linear models.
What the sources say
- Supportedcites Park, Gelman and Bafumi (2004) source
The Discussion (p. 9) says multilevel modeling outperformed complete- and no-pooling estimates and explains why the authors prefer its partial pooling. The comparison with election outcomes (p. 8) reports lower mean absolute errors for the multilevel estimates. The paper uses a Bayesian approach but says the multilevel framework is the key to this result. (LLM, conf=0.92)
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Supported
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Now we consider two examples where classical nonhierarchical regression cannot be used to automatically get the correct answer.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2.1.2. Comparing two treatments.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The simplest regression uses four predictors-one constant term and three treatment indicators-with 116 residual degrees of freedom.This model gives the wrong residual variance: we want the between-machine, not the betweenmeasurement, variance.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2.2.2. A simple hierarchical design.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“As a result, any given survey will have few or no data in many categories.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The information in the multilevel display in Figure 5 is not simply contained in the mean squares of the classical ANOVA table in Figure 4.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7. Examples.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We can now immediately see that the lowest-level variation is more important in variance than any of the factors except for the main effect of the destination.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7. Examples.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Estimation of these complex models was made possible by the unusual size of the survey datasets (over 700,000 respondents).”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2. Variance components as a focus of scientific research.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Our general solution to the ANOVA problem is simple: we treat every row in the table as a batch of “random effects”; that is, a set of regression coefficients drawn from a distribution with mean 0 and some standard deviation to be estimated from the data.”
- Also stated in: § 3.2. Batching of regression coefficients. · plain
“But, more importantly, the variance components and standard errors are estimated from the data, without any need to specify comparisons based on the design.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.2. Batching of regression coefficients.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: Each batch of regression coefficients is modeled as exchangeable normal effects with mean zero and its own variance.
- Also stated in: § 3.2. Batching of regression coefficients. · plain
- Probably not a substantial claim. 66
- It may state two separate claims as one. 53
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“The hierarchical regression analysis is based on the model of exchangeable effects within batches, as expressed in model (7), which is not necessarily the best analysis in any particular application.For example, Besag and Higdon (1999) recommend using spatial models (rather than exchangeable row and column effects) for data such as in the split-plot experiment described previously.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.3. Getting something for nothing?
What the sources say
- Partly supportedcites Besag and Higdon (1999) sourceExperimental — accuracy not yet measured
Besag and Higdon develop spatial models for agricultural field experiments and say that a model's invariance to arbitrary row and column effects is undesirable (section 2.3.2, page unavailable). This supports the general preference for spatial modelling, but the passages do not specifically recommend it over exchangeable row and column effects for the split-plot experiment. The statement about the citing authors’ hierarchical regression model is not attributed to Besag and Higdon. (LLM, conf=0.78)
Strength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- The authors state this as an assumption: Each batch of regression coefficients is modeled as exchangeable normal effects with mean zero and its own variance.
- Partly supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“For balanced designs, however, simple formulas exist.We do not go through all the literature here [see, e.g., Cornfield and Tukey (1956), Green and Tukey (1960) and Plackett (1960)].A summary is given in Searle, Casella and McCulloch [(1992), Section 4.2].The basic idea is that, in a balanced design, the effect estimates β(m) j in a batch m are simply averages of data, adjusted to fit a set of”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4.1. Classical inference.
What the sources say
- Partly supportedcites Cornfield and Tukey (1956) sourceExperimental — accuracy not yet measured
The Summary (p. 1) says the paper provides formulas for factorial analyses and describes them as not very complex, supporting the point that simple formulas exist. The Description of Situations Treated (pp. 11, 13) discusses averaging and normalization of contributions, but does not establish the claimed rule that estimates of effects in a batch are averages of observed data adjusted to satisfy linear constraints. (LLM, conf=0.82)
- Not checked — no full text of the sourcecites Green and Tukey (1960) source
- Not checked — no full text of the sourcecites Searle, Casella and McCulloch [(1992), Section 4.2] source
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
- Partly supported
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Figure 1 shows three attempts to fit the split-plot data with aov, only the last of which worked.”
- Also stated in: § 2.2.2. A simple hierarchical design. · plain
“Three attempts at running the aov command in S-Plus.Only the last gave the correct comparisons.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2.2.1. A split-plot Latin square.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § 2.2.2. A simple hierarchical design. · plain
In the paper
“As we show in Section 2 of this paper, least-squares regression solves some ANOVA problems but has trouble with hierarchical structures [see also Gelman (2000)].”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 1. Is ANOVA obsolete?
What the sources say
- Not matchedcites Gelman (2000)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This suggests a model with 24 predictors: one constant, three treatment indicators, and 20 machine indicators.But these predictors are collinear, so we must eliminate four of the machine indicators.Unfortunately, the standard errors of the treatment effects in this model are estimated using the within-machine variation, which is still wrong.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2.2.2. A simple hierarchical design.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Formulas (3) and ( 4) give the same estimated treatment effects but imply different standard errors and different ANOVA F-tests.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2.2.2. A simple hierarchical design.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A claim is checked only against the cited paper’s full text, never its abstract, and none is available for its sources.
In the paper
“The data are consistent with a standard deviation of 0, but it could also be as high as 10 or 20.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.4. Classical and Bayesian interpretations.
What the sources say
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Company has a large effect on its own and, perhaps more interestingly, in interaction with to, from, and in the three-way interaction.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7. Examples.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The ethnicity * state * region interactions are surprisingly large: the differences between African-Americans and others vary dramatically by state.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Figure 6 displays the ANOVA table, which shows that ethnicity is by far the most important demographic factor, with state also explaining quite a bit of variation.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The ANOVA reveals that all the main effects and almost all the interactions are statistically significant.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7. Examples.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For example, the effects of from * hour have a relatively high estimated standard deviation but a relatively low mean square (see, e.g., to * week).”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7. Examples.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Similar patterns were identified for other quality dimensions measured in the survey.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2. Variance components as a focus of scientific research.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In the application areas in which we have worked, interest has lain in questions of the form, “How important are the effects of factor X?,” rather than “Does factor X have an effect?””
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 5. Estimation and hypothesis testing.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The problem becomes even more difficult if the design is unbalanced.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2.2.2. A simple hierarchical design.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Unfortunately, in complex problems (e.g., split-plot designs), it is not always easy to set up an appropriate ANOVA.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > Abstract
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, the struggles of Section 2.2 are avoided, and (hierarchical) linear regression can indeed be used to compute ANOVA automatically, once the rows of the table (the sources of variation) have been specified.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.2. Batching of regression coefficients.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Each batch of regression coefficients is modeled as exchangeable normal effects with mean zero and its own variance.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We include this not to disparage S-Plus in any way but just to point out that ANOVA can be done in many ways in the classical linear regression framework, and not all these ways give the correct answer.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2.2.1. A split-plot Latin square.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“There will be more shrinkage for batches of effects whose standard deviations σ m are near zero, which will occur for factors that contribute little variation to the data.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.4. Classical and Bayesian interpretations.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Each batch of regression coefficients is modeled as exchangeable normal effects with mean zero and its own variance.
In the paper
“The second part of making the hierarchical ANOVA work is that the information from the design is encoded in the design matrix of the linear regression [as shown by Nelder (1965a, b) and implemented in the software Genstat].”
- Also stated in: § 3.3. Getting something for nothing? · plain
“The automatic encoding is particularly useful in incomplete designs where there is no simple classical analysis.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.3. Getting something for nothing?
What the sources say
- Not matchedcites Nelder (1965a, b)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § 3.3. Getting something for nothing? · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In classical ANOVA, you (sometimes) need to know the design in order to select the correct analysis, as in the examples in Section 2.2.But the hierarchical analysis does it automatically.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.3. Getting something for nothing?
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Each batch of regression coefficients is modeled as exchangeable normal effects with mean zero and its own variance.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The two variances can be given the same point estimate-in classical unbiased estimation E(s 2 m |σ 2 m ) = σ 2 m , and in Bayesian inference with a noninformative prior distribution (see Section 4.2) the conditional posterior mode of σ 2 m given all other parameters in the model is s 2 .The superpopulation variance has more uncertainty, however.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.4. Classical and Bayesian interpretations.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In some contexts (e.g., obtaining inference for the 50 U.S. states) the finite population seems more meaningful, whereas in others (e.g., subject-level effects in a psychological experiment) interest clearly lies in the superpopulation.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.4. Classical and Bayesian interpretations.
Tests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may be too vague to check against evidence. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As we discuss in Section 6, we believe that much of the literature on fixed and random effects can be fruitfully reexpressed in terms of finite-population and superpopulation inferences.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.4. Classical and Bayesian interpretations.
Tests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may be too vague to check against evidence.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The superpopulation standard deviation characterizes the uncertainty for predicting a new coefficient from batch m, whereas the finite-population standard deviation describes the existing J m coefficients.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.4. Classical and Bayesian interpretations.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“From the hierarchical model the coefficient estimates will be pulled toward zero, with the amount of shrinkage determined by the estimated variance components.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.2. Batching of regression coefficients.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Each batch of regression coefficients is modeled as exchangeable normal effects with mean zero and its own variance.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“To see the difference between the two variances, consider the extreme case in which J m = 2 [and so (df ) m = 1] and a large amount of data is available in both groups.Then the two parameters β (m) 1 and β (m) 2 will be estimated accurately and so will s 2 m = (β (m) 1β (m) 2 ) 2 /2.The superpopulation variance σ 2 m , on the other hand, is only being estimated by a measurement that is”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.4. Classical and Bayesian interpretations.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“First, as with the classical analyses, we require that the rows of the ANOVA be specified by the modeler.In the notation of ( 5) and ( 6), the user must specify the structuring or batching of the linear parameters β.In the classical analysis, however, this is not enough, as discussed in Section 2.2.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.3. Getting something for nothing?
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“But we would argue that such situations will be common in ANOVA settings, for two reasons.First, when studying the many rows of a large ANOVA table, we expect (in fact, we hope) to see various nearzero variances at higher levels of interaction.”
- Also stated in: § 3.4. Classical and Bayesian interpretations. · plain
“Our second reason for expecting to see near-zero variance components is that, as informative covariates are added to a linear FIG. 2. Illustration of the difficulties of point estimation for variance components.Pictured is the marginal posterior distribution for a hierarchical standard deviation parameter from Rubin (1981) and Gelman, Carlin, Stern and Rubin [(1995), Chapter 5].The simplest point”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3.4. Classical and Bayesian interpretations.
What the sources say
- Not matchedcites Gelman (1996)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § 3.4. Classical and Bayesian interpretations. · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This is not a problem, however, if a multilevel model is fit, as is done automatically in our ANOVA procedure: each factor or set of interactions in the model, corresponding to a row in the ANOVA table, is automatically given a variance component.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In either case, including contrasts adds another twist in that defining a superpopulation for predictive purposes now requires specifying a distribution over the contrast variable (e.g., in the Latin square example, if the rows are labeled as -2, -1, 0, 1, 2, then a reasonable superpopulation might be a uniform distribution on the range [-2.5, 2.5]).”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“However, for effects that are unique or in pairs [i.e., batches for which (df ) m = 1], the posterior density for the corresponding σ m is improper, with infinite mass in the limit σ j → ∞ [Gelman, Carlin, Stern and Rubin (1995), Exercise 5.8], and so the coefficients β (m) j in these batches are essentially being estimated via maximum likelihood.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4.2. Bayesian inference.
What the sources say
- Not checked — no full text of the sourcecites Gelman, Carlin, Stern and Rubin (1995) source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“This simulation has a parametric bootstrap or Bayesian flavor and is motivated by the approximate equivalence between repeated-sampling and Bayesian inferences [see, e.g., DeGroot (1970) and Efron and Tibshirani (1993)].”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4.1. Classical inference.
What the sources say
- Not matchedcites DeGroot (1970)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked — no full text of the sourcecites Efron and Tibshirani (1993) source
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Given σ , the parameters β (m) j have a multivariate normal distribution (in Bayesian terms, a conditional posterior distribution; in classical terms, a predictive distribution).”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4.1. Classical inference.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Hypothesis testing is in general a more difficult problem than estimation because many different possible hypotheses can be considered.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In addition, using careful formatting, we can display this in no more space than is required by the classical ANOVA table.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 5. A new ANOVA table.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the election survey example, one can add state-level predictors such as previous Presidential election results.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“A crucial difficulty in classical ANOVA and, more generally, in classical linear modeling, is identifying the correct variance components to use in computing standard errors and testing hypotheses.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In our ANOVA formulation (5) or (6), the regression error terms are just the highest-level interactions, β (M) j , and so the distributions (7) include the likelihood as well as the prior distribution.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4.2. Bayesian inference.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Conversely, the ideas of ANOVA are extremely powerful in modeling complex data of the sort that we increasingly handle in statistics-hence the title of this paper.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Load-bearingStrength:boosted· not yet calibratedTests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may be too vague to check against evidence.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Analysis of variance is fundamentally about multilevel modeling: each row in the ANOVA table corresponds to a different batch of parameters, along with inference about the standard deviation of the parameters in this batch.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Conditional on the simulation for σ , we can now estimate the finite-population standard deviations s m .As discussed in Section 3.5, the data provide additional information about these, and so our intervals for s m will be narrower than for σ m , especially for variance components with few degrees of freedom.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4.1. Classical inference.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Our alternative ANOVA table presents, for each source of variation m, the estimates and uncertainties for s m , the standard deviation of the coefficients corresponding to that row of the table.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 5. A new ANOVA table.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Since the data are on the logarithmic scale, the standard deviation parameters can be interpreted directly.For example, s m = 0.20 corresponds to a coefficient of variation of exp(0.2) -1 ≈ 0.2 on the original scale, and so the unlogged coefficients exp(β (m) j ) in this batch correspond to multiplicative increases or decreases in the range of 20%.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7. Examples.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Most simply, the distributions for the effects in each batch can be generalized beyond normality (e.g., using t or mixture distributions), and the variance parameters can themselves be modeled hierarchically, as discussed immediately above.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, the classical hierarchical ANOVA computations reduce to estimating the expected mean squares EMS m (and thus EV m ) in terms of the estimated variance components σ m .For nonbalanced designs, this can be complicated compared to the Bayesian computation as described in Section 4.2.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4.1. Classical inference.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In some relatively simple balanced designs, the hypotheses can be tested independently; for example, the split-plot Latin square allows independent testing of row, column and treatment effects at the between-and within-plot levels.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, as we discuss in this paper, assigning probability distributions for all variance components automatically gives the correct comparisons and standard errors.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“A more efficient updating reparameterizes into vectors γ , α and τ , which are defined as follows:”
- Also stated in: § 4.2. Bayesian inference. · plain
“The auxiliary parameters α are given a uniform prior distribution, and then this reduces to the original model [see Boscardin (1996), Meng and van Dyk (1997), Liu, Rubin and Wu (1998), Liu and Wu (1999) and Gelman (2004)].”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4.2. Bayesian inference.
What the sources say
- Not matchedcites Boscardin (1996)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked — no full text of the sourcecites Meng and van Dyk (1997) source
- Not matchedcites Liu, Rubin and Wu (1998)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § 4.2. Bayesian inference. · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This makes sense when comparing the two companies and 25 hours, but the “to” sites, the “from” sites and the weeks are sampled from a larger population, and for these generalizations, the superpopulation variances would be relevant.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7. Examples.
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In summary, it is easily possible for a factor to be “fixed” according to some of the definitions above and “random” for others.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3. "When a sample exhausts the population, the corresponding variable is fixed;
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“From a classical perspective of inference for variance components, these cases can be considered as examples of the effectiveness of automatically setting up hierarchical models with random effects for each row in the ANOVA table.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7. Examples.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For example, the five rows, columns and possibly treatments in the Latin square are ordered, and systematic patterns there could be modeled, at the very least, using regression coefficients for linear trends.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:hedged· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For values of m in which J m is large (i.e., rows of the ANOVA table corresponding to many linear predictors), σ m is essentially estimated from data.When J m is small, the flat prior distribution implies that σ is allowed the possibility of taking on large values, which minimizes the amount of shrinkage in the effect estimates.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4.2. Bayesian inference.
Strength:hedged· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
In the paper
“This relates to the classical result that shrinkage estimation dominates least squares when estimating three or more parameters in a normal model [James and Stein (1961)].”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4.2. Bayesian inference.
What the sources say
- Not matchedcites James and Stein (1961)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Strength:boosted· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
None of the sources cited for this claim have been checked yet.
In the paper
“It turns out that different-in fact, incompatible-definitions are used in different contexts.[See also Kreft and de Leeuw (1998), Section 1.3.3,for a discussion of the multiplicity of definitions of fixed and random effects and coefficients, and Robinson (1998) for a historical overview.]”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 6. Fixed and random effects.
What the sources say
- Not checked — no full text of the sourcecites Kreft and de Leeuw (1998) source
- Not matchedcites Robinson (1998)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In our ANOVA table, the inferences for all the variance components are simultaneous, in contrast to the classical approach in which each variance component is tested under the model that all others, except for the error term, are zero.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 5. A new ANOVA table.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The mean square is the variance explained per parameter, which is not directly comparable to the parameters s 2 m and σ 2 m , which represent underlying variance components.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 5. A new ANOVA table.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In this terminology (which we would apply at any level of the hierarchy in a multilevel model), varying effects occur in batches, whether or not the effects are interesting in themselves (definition 2), and whether or not they are a sample from a larger set (definition 3).”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3. "When a sample exhausts the population, the corresponding variable is fixed;
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“After subtracting batch-level regression predictors, the additive effects for the factor levels in each batch could be modeled as exchangeable.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Definitions 4 and 5 do not arise for us since we estimate all batches of effects hierarchically, with the variance components σ m estimated from data.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3. "When a sample exhausts the population, the corresponding variable is fixed;
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This corresponds to analysis of covariance or contrast analysis in classical ANOVA.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Our model (7) for the linear parameters corresponds to the default inferences in ANOVA, based on computations of variances and exchangeable coefficients within each batch.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the ANOVA context such a model is potentially more difficult to set up since it should ideally reflect the structure of the variance components (e.g., if two sets of main effects are large, then one might expect their interaction to be potentially large).”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For balanced designs, then, variance components can be estimated by starting at the bottom of the table (with the highest-level interaction, or residuals) and then working upwards, at each step using the appropriate variance components from lower in the table in formulas ( 10) and ( 11).In this way the variance components σ 2 m can be estimated noniteratively.Alternatively, we can compute the”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4.1. Classical inference.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As discussed in Section 3.5, uncertainties can be much lower for finite-population variances s 2 m than for superpopulation variances σ 2 m , and it is through finitepopulation variances that we connect to classical ANOVA, in which it is possible to draw useful inferences for even small batches (as in our split-plot Latin square example).”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“As described by Gelman and Little (1997) and Bafumi, Gelman and Park (2002), this inferential procedure works well and outperforms standard survey estimates when estimating state-level outcomes.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
What the sources say
- Not matchedcites Gelman and Little (1997)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites Bafumi, Gelman and Park (2002)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Load-bearingStrength:boosted· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“with only five treatment levels, nonzero superpopulation variances are inherently difficult to estimate”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 6. Finite-population and superpopulation summaries.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“ANOVA may be “more important than ever” but it is intended to add perspective to, not to take the place of, inference for quantities of substantive interest.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4. ANOVA as a supplement to inferences about quantities of interest.
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Individual treatment effects do not occur as parameters in this model, but they may be estimated by prediction”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > University of Chicago
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Our procedure gives an appropriate answer in a wide range of classical problems”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 1. General comments.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. It may be too vague to check against evidence. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This finding has implications for quality improvement, suggesting that interventions to improve quality of care might have to be directed to health care providers in an area rather than trying to identify and improve lower-performing health plans.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2. Variance components as a focus of scientific research.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
- Rests on an assumption: Health care is largely provided by doctors and hospitals that are organizationally independent of health plans and may contract with multiple plans.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It certainly contradicts the notion implied in Sections 3.2 and 3.3 that using hierarchical modeling, so to speak, automagically leads to a correct model.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > Utrecht University
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We suspect that statisticians are generally unaware of the many conflicting definitions of the terms “fixed” and “random””
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > FIG. A.
Strength:hedged· not yet calibratedTests✓ Asserted: passes✗ Checkable: fails✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“the simple decomposition of Gelman's equation (1) only applies when there are unbiased estimators with independent errors, which is not likely to be the case for complex models with unbalanced datasets.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > Method of moments.
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“we agree with Zaslavsky that posterior predictive checking is the best approach, since it allows a hypothesis about any subset of parameters to be tested while accounting for uncertainty in the estimation of the other parameters in the model.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 5. Estimation and hypothesis testing.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✗ Checkable: fails✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This is the way that researchers in psychology use ANOVA, but they are often ill-served by the classical framework of mean squares and F-tests.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > FIG. A.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“A block factor cannot be transformed into a treatment factor because the blocks are unlabelled.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > University of Chicago
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The transition from ANOVA to variance components modeling shifts attention from decomposition of the variance of the sample as in classical ANOVA to decomposition of the variance of the population.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > Additional information
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Scale-invariant improper priors typically yield degenerate posteriors in variance components models.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > Utrecht University
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“each exchangeable variance-components model is a trivial representation in the space of symmetric matrices that is closed under deletion of levels.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > University of Chicago
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“an already overloaded expression that has at least five other existing interpretations in the statistical literature!”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > FIG. A.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“our basic idea is to recognize that the parameters in a model are not simply a long undifferentiated vector but can be usefully grouped into batches, which in fact are already specified in the classical ANOVA table.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > FIG. A.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The ideas of the analysis of variance also help us to include finite-population and superpopulation inferences in a single fitted model, hence unifying fixed and random effects.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“ANOVA is more important than ever because it represents a key idea in statistical modeling of complex data structures-the grouping of predictor variables and their coefficients into batches.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Tests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may be too vague to check against evidence.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“ANOVA decomposition requires one further ingredient in the form of an inner product on the observation space.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > University of Chicago
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“Gelman correctly notes that inference can be made for both finite-population and superpopulation variances, and that the distinction between these two targets of inference corresponds to the distinction between fixed and random effects.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 3. Miscellaneous comments.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These examples illustrate that despite their relative unfamiliarity in many fields, variance components can be interpreted to nonstatisticians in a scientifically meaningful way.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2. Variance components as a focus of scientific research.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. It may be too vague to check against evidence. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“if you are already fitting a statistical model, its parameters can probably be grouped into batches, and it is probably interesting to compare the magnitude of the variation of the parameters in each batch.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4. ANOVA as a supplement to inferences about quantities of interest.
Tests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. It may be too vague to check against evidence. The paper may give no reason why it holds. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Posterior predictive checking can also be applied to the multilevel model as a whole to test assumptions such as additivity, linearity and normality.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 5. Estimation and hypothesis testing.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“One particularly interesting class of generalizations of classical ANOVA involves the nonadditive structures of interactions.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 7.2. A multilevel logistic regression model with interactions: political opinions.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“other studies have shown substantial geographical variation in their practice patterns.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2. Variance components as a focus of scientific research.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“in any specific application an applied researcher will typically be interested in particular treatment effects, or comparisons of treatment effects, rather than in variance components.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 4. ANOVA as a supplement to inferences about quantities of interest.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. It may be too vague to check against evidence. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The subspace A ⊂ R n determined by a factor is of a very special type: it is also a ring, closed under functional multiplication, with 1 as identity element.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > University of Chicago
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“we find the summary in terms of within-batch standard deviations to be more relevant than the usual ANOVA table of sums of squares, mean squares and F-tests.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 1. General comments.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The term “random effect” has no clear (let alone “formal”) definition”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > FIG. A.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. It may be too vague to check against evidence. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Treating the treatment effects as fixed versus random requires knowledge about the actual design of the study, and a decision on how we should view these treatments.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > Utrecht University
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“such batches should not necessarily be modeled exchangeably”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2. The model comes first.
Strength:hedged· not yet calibratedTests✓ Asserted: passes✗ Checkable: fails✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The boundary problem is not an issue with this approach since the reference distribution is determined by simulation rather than by asymptotics, and indeed no prior distribution is required for the variance component being tested.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > Utrecht University
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“A joint prior distribution on the variance components avoids the discontinuity associated with near-zero estimated variance components.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > University of Chicago
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“His prior distribution is uninformative, and there are neither competing models nor different sources of prior information that are evaluated using posterior probabilities.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > Utrecht University
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“The effects of measured characteristics of individual members or plans are fairly small, and the effects of an individual's characteristics are of little interest because the primary objective of the survey is to evaluate health care systems, not the predictors of an individual's reported experiences.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > 2. Variance components as a focus of scientific research.
What the sources say
- Not checked — no full text of the sourcecites Zaslavsky, Zaborski and Cleary (2004) source
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As Gelman notes, in a Bayesian framework the estimation of variance components is relatively automatic; attention can be focused on defining sensible models rather than on constructing designs that can be analyzed easily.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > Additional information
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“By contrast, a Taguchi-type model in which the variance depends on one or more factor levels is a representation, but not a trivial representation.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > University of Chicago
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“if you are already fitting a complicated model, your inferences can be better understood using the structure of that model.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > FIG. A.
Tests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. It may be too vague to check against evidence. The paper may give no reason why it holds.
In the paper
“Given the small sample size, maximum likelihood estimation is not attractive, but MLwiN includes a fully Bayesian inference option.”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > Utrecht University
What the sources say
- Not matchedcites Goldstein et al. (1998)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- Not matched
In the paper
“all models discussed by Gelman for these data can be analyzed using the software MLwiN [Goldstein et al. (1998)].”
§ ANALYSIS OF VARIANCE-WHY IT IS MORE IMPORTANT THAN EVER 1 > Utrecht University
What the sources say
- Not matchedcites Goldstein et al. (1998)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
The authors state this as an assumption.
Each batch of regression coefficients is modeled as exchangeable normal effects with mean zero and its own variance.
“We shall use a hierarchical formulation in which each batch of regression coefficients is modeled as a sample from a normal distribution with mean 0 and its own variance σ 2 m :”
“Model (7) corresponds to exchangeability of each set of factor levels, which is a form of partial exchangeability or invariance of the entire set of cell means [see Aldous (1981)].”
The authors state this as an assumption.
All variance components are nonnegative.
“the restriction that all the variance components be nonnegative.”
The paper relies on this without support.
The Census adult population counts for each demographic-state category are accurate.
“From the Census, we get the adult population N j for each category j .”
The paper relies on this without support.
Numerous examples exist in which one or more factors have levels that are not exchangeable, and inference for a specific treatment or classification contrast is the primary purpose of the experiment.
“However, numerous examples exist in which one or more factors have levels that are not of this type, where inference for a specific treatment contrast or a specific classification contrast is the primary purpose of the experiment.”
The authors state this as an assumption.
The four treatments in the example exhaust all treatments of interest.
“the multilevel analysis approach outlined above makes sense if we assume that the four treatments exhaust all treatments we are interested in.”
The authors state this as an assumption.
There is a population of treatments, or variations in implementation can lead to different outcomes.
“If we assume that there is a population of treatments, or that variations in implementation can lead to different outcomes, we can structure the data as a three-level model”
The paper relies on this without support.
Health care is largely provided by doctors and hospitals that are organizationally independent of health plans and may contract with multiple plans.
“The latter is largely provided by doctors and hospitals that are organizationally independent of the plans and might contract with multiple plans”
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
The treatment-trend predictor changes its second weight
MinorThe finite-population linear contrast assigns the second treatment a weight of −1, but the predictor given for its superpopulation counterpart assigns it +1. Those are different treatment patterns, rather than two interpretations of the same linear contrast.
“The finite-population contrast is -2 • β 1 + (-1) • β 2 + 0 • β 3 + 1 • β 4 + 2 • β 5”
“the superpopulation contrast is the appropriately scaled coefficient of (-2, 1, 0, 1, 2) included as a treatment-level predictor in the multilevel model.”
Figure 3 labels finite-population deviations with the wrong symbol
MinorThe paper defines s_m as the finite-population standard deviation and σ_m as the superpopulation standard deviation. The Figure 3 caption calls σ_m finite-population, reversing that distinction.
“The bars display 50% and 95% intervals for the finite-population standard deviations σ m .”
“For each row m of an ANOVA table, there are two natural variance parameters to estimate: the superpopulation standard deviation σ m and the finite-population standard deviation s m as defined in (8).”
State indicators are included and then said to be unnecessary
MinorThe model description says that the design matrix includes indicators for all 50 states, then says no state main effects are needed. State indicators are state main-effect terms, so the two descriptions conflict as written.
“We also include in X indicators for the 50 states and for the four regions of the country (northeast, midwest, south and west).”
“Since the states are nested within regions (which is implied by the design matrix of the regression), no main effects for states are needed.”
Citations
Cites
An Introduction to the Bootstrap
Efron, B., & Tibshirani, R. J. (1994) · doi:10.1201/9780429246593 · Cited by 40,500
Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2003) · doi:10.1201/9780429258480 · Cited by 27,276
Self-citationAn Introduction to the Bootstrap
Efron, B., & Tibshirani, R. J. (1993) · doi:10.1007/978-1-4899-4541-9 · Cited by 21,230
Spline Models for Observational Data
Wahba, G. (1990) · doi:10.2307/2938687 · Cited by 7,398
Spline Models for Observational Data
Wahba, G. (1990) · doi:10.1137/1.9781611970128 · Cited by 4,184
Bayesian Analysis of Binary and Polychotomous Response Data
Albert, J. H., & Chib, S. (1993) · Journal of the American Statistical Association · doi:10.1080/01621459.1993.10476321 · Cited by 3,160 · cited 3×
“, Gelman, Carlin, Stern and Rubin (1995)] or data augmentation [see Albert and Chib (1993) and Liu (2002)].”
BackgroundMethodIntroducing Multilevel Modeling
Kreft, I., & de Leeuw, J. (1998) · doi:10.4135/9781849209366 · Cited by 2,997
Variable Selection via Gibbs Sampling
George, E. I., & McCulloch, R. E. (1993) · Journal of the American Statistical Association · doi:10.1080/01621459.1993.10476353 · Cited by 2,828
BAYES AND EMPIRICAL BAYES METHODS FOR DATA ANALYSIS
Leonard, T., Carlin, B., & Louis, T. (1996) · Statistics and computing · doi:10.1023/a:1018577817064 · Cited by 2,562
Ripley, B. D. (1981) · Wiley Series in Probability and Statistics · doi:10.1002/0471725218 · Cited by 2,052
Flagged first, then by citations
Research works printed but not linked: 11
- Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Addison–Wesley, Reading, MA.
- DeGroot, M. H. (1970). Optimal Statistical Decisions. McGraw-Hill, New York.
- Cochran, W. G. and Cox, G. M. (1957). Experimental Designs, 2nd ed. Wiley, New York.
- Cochran, W. G. and Cox, G. M. (1957). Experimental Designs, 2nd ed. Wiley, New York.
- Snedecor, G. W. and Cochran, W. G. (1989). Statistical Methods, 8th ed. Iowa State Univ. Press, Ames, IA.
- Boscardin, W. J. (1996). Bayesian analysis for some hierarchical linear models. Ph.D. dissertation, Dept. Statistics, Univ. California, Berkeley.
- James, W. and Stein, C. (1961). Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1 361–379. Univ. California Press, Berkeley.
- LaMotte, L. R. (1983). Fixed-, random-, and mixed-effects models. In Encyclopedia of Statistical Sciences (S. Kotz, N. L. Johnson and C. B. Read, eds.) 3 137–141. Wiley, New York.
- Montgomery, D. C. (1986). Design and Analysis of Experiments, 2nd ed. Wiley, New York.
- Robinson, G. K. (1998). Variance components. In Encyclopedia of Biostatistics (P. Armitage and T. Colton, eds.) 6 4713–4719. Wiley, Chichester.
- Wu, C. F. J. and Hamada, M. (2000). Experiments: Planning, Analysis, and Parameter Design Optimization. Wiley, New York.
Cited by
Sorted by citations
Items 101–107 of 107
Improving estimation in genetic models using prior information
Eilertsen, E. M. (2016) · Norsk Epidemiologi · doi:10.5324/nje.v26i1-2.2017
Luna, L. S. D. (2016) · SSRN Electronic Journal · doi:10.2139/ssrn.2960860
Neubauer, E. F. (2016) · Open Access Victoria University of Wellington | Te Herenga Waka (Figshare) · doi:10.26686/wgtn.17014715
(2015) · Oxford University Press eBooks · doi:10.1093/acprof:oso/9780199672547.003.0001
(2015) · Oxford University Press eBooks · doi:10.1093/acprof:oso/9780199672547.002.0005
(2015) · Oxford University Press eBooks · doi:10.1093/acprof:oso/9780199672547.002.0004
(2015) · Oxford University Press eBooks · doi:10.1093/acprof:oso/9780199672547.002.0009
Metadata
- Authors
- Gelman, A.
- Corresponding author
- Gelman, Andrew
- ORCID
- Gelman, Andrew
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 2005-02-01
- Volume
- 33
- Issue
- 1
- Pages
- 1-53
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- OpenAlex
- W217328182
- Cited by
- 861
- References
- 53
- Citations by year
- 2026: 4 · 2025: 39 · 2024: 60 · 2023: 53 · 2022: 50 · 2021: 80 · 2020: 67 · 2019: 69 · 2018: 55 · 2017: 55 · 2016: 49 · 2015: 70 · 2014: 42 · 2013: 37 · 2012: 46
- FWCI
- 12.07
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Quality Engineering
- Funders
- National Science Foundation (9987748, 9708424, 9796129, 0084368)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Social Sciences › Decision Sciences › Management Science and Operations Research › Optimal Experimental Design Methods
- Keywords
- analysis of variance, hierarchical models, split-plot design, variance components, fixed and random effects, exploratory data analysis, confirmatory data analysis
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-02, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-02 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (gpt-6-sol).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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