Why the Choice Needs a Rule
When a repeated-measures effect has more than two conditions, the usual univariate F test relies on sphericity: the variances of the pairwise difference scores must be equal. If that condition is doubtful, a sphericity correction reduces the test’s degrees of freedom. Greenhouse–Geisser (GG), Huynh–Feldt (HF), and lower-bound (LB) corrections use different values of epsilon, so they can yield different p-values for the same observed F statistic.
The practical question is not which correction gives the smallest p-value. The choice should follow a defensible analysis policy, informed by the study plan and the estimated degree of sphericity departure. A correction does not change the observed F statistic; it changes the reference F distribution used to calculate the p-value.
For the usual univariate test of a single within-subject effect, the sphericity epsilon satisfies \(1/(k-1) \leq \epsilon \leq 1\). The GG estimate is generally cautious when sphericity is violated. HF is designed to reduce that conservatism in some circumstances, but its estimate can be too optimistic, particularly with small samples. The LB value is the minimum possible epsilon, \(1/(k-1)\), and therefore gives the strongest degrees-of-freedom reduction among these choices.
What the Corrections Change
Suppose the uncorrected test is \(F(df_1,df_2)=F_{\text{obs}}\). The three corrected tests are formed in the same way; only the epsilon value differs. For example, using GG epsilon \(\epsilon_{GG}\) gives:
The corresponding p-value is the probability, under that F distribution, of a value at least as large as \(F_{\text{obs}}\). The same construction applies to HF and LB. Because an F distribution depends on both degrees of freedom, a correction can change the p-value even though the observed F statistic is identical. Usually, reducing the degrees of freedom makes the upper-tail p-value larger, but the amount depends on the statistic and both degrees of freedom.
A Practical Decision Framework
A sound choice combines the analysis plan with the epsilon estimates and test results. A common historical guideline is to use GG when the estimated epsilon is relatively low and to consider HF when it is high (often a threshold near 0.75 is cited). This is a convention, not a theorem that guarantees the best error-rate behavior in every design. The estimates are sample-based, and no single cutoff replaces a prespecified analysis plan.
When a protocol or analysis plan names a correction, follow that plan unless there is a clear, documented reason not to. If no method was prespecified, report the choice and its rationale; for example, a cautious GG analysis may be a reasonable default when sphericity is in doubt. The LB result can be useful as a conservative comparison or sensitivity analysis, but it is not automatically the preferred primary result. Do not select whichever row happens to put the p-value on the desired side of \(\alpha\).
Confirm the effect, the repeated conditions, \(df_1\), \(df_2\), the observed F statistic, and the significance level \(\alpha\).
Check that the GG and HF values are within the allowable bounds and compare them with the lower-bound value \(1/(k-1)\).
Use the correction named in the plan, or explain the principled rule used when the plan did not specify one. Do not make the choice based on which p-value is significant.
Give the effect, the correction, corrected degrees of freedom, F statistic, p-value, and the conclusion relative to \(\alpha\).
Worked Examples
Worked Example: Applying a Prespecified Greenhouse–Geisser Rule
In a hypothetical study, 24 participants each complete a task under four repeated conditions. The analysis plan specifies GG correction if sphericity is not assumed. The uncorrected within-subject test has \(df_1=3\), \(df_2=69\), and \(F_{\text{obs}}=2.80\). The estimated GG epsilon is 0.62; the HF estimate is 0.72.
There are \(k=4\) conditions, so the lower-bound epsilon is \(1/(4-1)=0.3333\). Both estimates fall within the permissible range from 0.3333 to 1. The plan specifies GG, so the choice does not depend on whether HF would produce a smaller p-value.
Corrected degrees of freedom:
The corrected result is \(F(1.86,42.78)=2.80\). Its p-value is the upper-tail probability \(P(F_{1.86,42.78}\geq 2.80)\), calculated from that F distribution. The observed statistic remains 2.80; it is not multiplied by epsilon. A complete report would give this p-value and compare it with the prespecified \(\alpha\). The HF degrees of freedom, if shown as a sensitivity comparison, would be \(0.72(3)=2.16\) and \(0.72(69)=49.68\), but they do not replace the planned primary result.
Worked Example: Why the Lower-Bound Test Can Be More Conservative
Suppose a hypothetical three-condition repeated-measures test has uncorrected \(df_1=2\), \(df_2=40\), and \(F_{\text{obs}}=4.00\). For illustration, suppose the LB correction is used. With \(k=3\), the lower-bound epsilon is \(1/(3-1)=0.5\).
The corrected degrees of freedom are:
Thus the corrected test is \(F(1,20)=4.00\). For an F distribution with numerator degrees of freedom 1, this is equivalent to a two-sided t test with \(t=\sqrt{4.00}=2.00\) and 20 degrees of freedom. Its p-value is approximately 0.0593. At \(\alpha=0.05\), this corrected test is not statistically significant.
For comparison, the uncorrected \(F(2,40)=4.00\) has upper-tail probability \((40/(40+2(4)))^{20}\), approximately 0.0261. The two conclusions differ because they use different reference distributions, not because the observed effect or F statistic changed. This comparison does not establish that LB should be selected whenever it changes a conclusion; the correction still needs a principled rationale.
Worked Example: Comparing Corrected Results Without Choosing by Significance
A hypothetical analysis of three repeated conditions gives \(F_{\text{obs}}=3.12\), with uncorrected degrees of freedom \(df_1=2\) and \(df_2=46\). The estimates are \(\epsilon_{GG}=0.62\) and \(\epsilon_{HF}=0.99\); the LB epsilon is 0.50. Suppose the analysis plan calls for GG as the primary correction and asks for HF and LB as sensitivity results.
The corrected degrees of freedom are:
Each row uses \(F_{\text{obs}}=3.12\), with its own degrees of freedom to obtain the upper-tail p-value. For the HF row, for example, the p-value is \(P(F_{1.98,45.54}\geq3.12)\), approximately 0.054. It is not approximately 0.052: the tail probability must be calculated using the stated corrected degrees of freedom. The LB p-value is approximately 0.091. The GG value should likewise be obtained from \(F(1.24,28.52)\), not inferred by copying either of the other rows.
At \(\alpha=0.05\), the HF and LB results are not statistically significant. The primary conclusion must be based on the planned GG result, with its p-value calculated from the GG degrees of freedom. The sensitivity rows show how the inference responds to alternative corrections; they do not give permission to report HF as primary merely because its p-value is smaller. If a report includes all three, it should label their roles clearly and avoid presenting them as three independent tests.
Why the Choice Matters—and What It Cannot Fix
A correction can affect a decision near the significance threshold, so reporting only “corrected” without naming the method is inadequate. Readers need to know which epsilon was applied and which degrees of freedom generated the p-value. When corrected and uncorrected conclusions differ, state that difference plainly and identify the corrected result used for the primary inference.
The correction addresses the univariate test’s degrees-of-freedom calibration under departures from sphericity. It does not repair dependence between different participants, compensate for a poorly defined outcome, or make every other model assumption true. In particular, repeated observations from one participant are not independent observations; the experimental unit for the between-participant independence assumption is the participant. Also, epsilon correction does not turn an observational comparison into a causal conclusion.
A useful short justification names the policy and the evidence: “We used the prespecified GG correction; the estimated GG epsilon was 0.62, and the effect was tested with corrected degrees of freedom.” If the method was selected after examining the data, disclose that fact rather than implying it was chosen in advance.
Common Mistakes
- Choosing the smallest p-value. This uses the outcome to select the method and can inflate the chance of a false positive. Use a prespecified rule or state a transparent, defensible rationale.
- Multiplying F by epsilon. Epsilon adjusts both degrees of freedom, not the observed F statistic.
- Applying epsilon to only one degree of freedom. The same epsilon multiplies both \(df_1\) and \(df_2\).
- Treating HF as automatically preferable when its estimate is near 1. A large estimate may support less correction under a declared policy, but it does not by itself override the analysis plan.
- Calling LB universally “the correct” correction. LB is the most conservative bound on epsilon, not a universal replacement for GG or HF.
Check Your Understanding
Answer each question by identifying the correction logic, not by guessing which p-value is most convenient.
- For an effect with four repeated conditions and uncorrected degrees of freedom \(df_1=3\), \(df_2=57\), what corrected degrees of freedom result from \(\epsilon=0.70\)?
- Does a sphericity correction change the observed F statistic, the degrees of freedom, or both?
- Why is choosing HF only because its p-value falls below 0.05 a poor analysis practice?
- For three repeated conditions, what is the lower-bound epsilon, and what does it represent?
- Name two items that should be included when reporting a corrected repeated-measures test.