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    Sample size for testing a factor model’s fit by the RMSEA

    How many participants are needed to test the fit of a confirmatory factor model by the root mean square error of approximation, from the model’s degrees of freedom and the RMSEA under the null and the alternative. Covers dropout.

    Engine version 1.0.0 · Reviewed 9 October 2026

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    https://statwisely.com/methods/measurement-cfa-rmsea/

    When to use it

    • Planning a confirmatory factor analysis of a questionnaire’s structure, so that close fit can be told apart from poor fit.

    Not for

    • Exploratory factor analysis, which has rules of thumb rather than a power calculation, and models too complex to converge in the sample planned.

    Notation

    NN
    Number of participants
    dd
    Degrees of freedom of the model
    ε0, ε1\varepsilon_0,\ \varepsilon_1
    RMSEA under the null and under the alternative
    λj\lambda_j
    Noncentrality of the fit statistic, (N − 1) d ε_j²
    χd2(λ)\chi^{2}_{d}(\lambda)
    Noncentral chi-square on d degrees of freedom
    α\alpha
    Significance level: the chance of a false positive
    1−β1-\beta
    Power: the chance of detecting the effect if it is real
    qq
    Expected dropout

    Formulas

    Power of the RMSEA test

    1−β=P ⁣(χd2(λ1)>c),P ⁣(χd2(λ0)>c)=α1-\beta = P\!\left(\chi^{2}_{d}(\lambda_1) > c\right), \qquad P\!\left(\chi^{2}_{d}(\lambda_0) > c\right) = \alpha

    The likelihood-ratio fit statistic is noncentral chi-square with λ = (N − 1)dε². For a test of close fit (ε₁ > ε₀) c is the upper α quantile under ε₀; for not-close fit (ε₁ < ε₀) the lower tail is used [1].

    Dropout

    nrecruit=⌈n1−q⌉n_{\text{recruit}} = \left\lceil \frac{n}{1 - q} \right\rceil

    Applied last, to the participants actually recruited.

    Parameters

    Model degrees of freedomdd

    Values: At least 1

    p(p + 1)/2 minus the free parameters for p items: 20 items on two correlated factors give 169. More degrees of freedom need fewer participants.

    RMSEA under the nullε0\varepsilon_0

    Values: 0 or more

    0.05 for tests of close and not-close fit; 0 for exact fit.

    RMSEA under the alternativeε1\varepsilon_1

    Values: 0 or more, not ε₀

    0.08 to reject close fit when fit is poor; 0.01 to reject not-close fit when fit is good.

    Significance levelα\alpha

    Values: Between 0 and 1; usually 0.05

    The false-positive rate you accept. A smaller α, for example to allow for several primary outcomes, needs a larger sample.

    Power1−β1-\beta

    Values: Between 0 and 1; usually 0.80 or 0.90

    The chance of a significant result if the assumed effect is real. Funders and ethics committees commonly expect at least 0.80.

    Expected dropoutqqOptional

    Values: 0% up to, but not including, 100%

    The share of participants expected to provide no outcome, from comparable studies. Recruiting more restores the numbers, not the comparability of the groups: if dropout is related to the outcome, the estimate can still be biased.

    Assumptions

    • Multivariate normal items, maximum-likelihood estimation and a model that is identified and converges; non-convergence and improper solutions in small samples are not part of this power.
    • Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

    Worked examples

    Test of close fit

    d = 40, ε₀ = 0.05 against ε₁ = 0.08: N = 252, as in MacCallum, Browne and Sugawara’s table.

    Sample size: 252

    Protocol statement

    To distinguish an RMSEA of 0.05 from one of 0.08 with 80% power at a significance level of 0.05, using the RMSEA test of a factor model’s fit on 40 degrees of freedom (MacCallum, Browne and Sugawara), the required sample size is 252 participants. Calculated with StatWisely (statwisely.com) using: N: noncentral χ² power for RMSEA, λ = (N-1)dε², H₀ ε₀ against ε₁ (MacCallum-Browne-Sugawara). These are planning estimates; confirm the final design with a qualified statistician.

    Open this example in the calculator

    Test of not-close fit

    ε₀ = 0.05 against ε₁ = 0.01: N = 307 (their table, with N rounded differently, gives 306).

    Sample size: 307

    Open this example in the calculator

    Test of exact fit

    ε₀ = 0 against ε₁ = 0.05: N = 277.

    Sample size: 277

    Open this example in the calculator

    Twenty items on two factors

    d = 169: N = 94. Power alone would allow this, but a model of this size usually needs more participants to converge reliably.

    Sample size: 94

    Open this example in the calculator

    Validation

    • Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
    • An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.
    • The sizes reproduce MacCallum, Browne and Sugawara’s published table for the test of close fit.

    How to cite

    StatWisely. Sample size for testing a factor model’s fit by the RMSEA. Method document, engine version 1.0.0, reviewed 2026-10-09. https://statwisely.com/methods/measurement-cfa-rmsea/

    Cite the original references for the formulas your design uses as well.

    References

    1. 1.MacCallum RC, Browne MW, Sugawara HM. Power analysis and determination of sample size for covariance structure modeling. Psychological Methods. 1996;1(2):130–149.