Start with the observations, group sizes and measurement units. Then read the estimates and their uncertainty alongside the tests. Each part of the output answers a different question.

What does the omnibus test say?

ANOVA compares between-group and within-group mean squares through an F statistic. Its null hypothesis is equality of all population group means; it does not specify which pair differs. See NIST's ANOVA table.

A p-value describes incompatibility with the specified null model, under its assumptions. It is neither the probability that the hypothesis is true nor a measure of biological importance. A large p-value does not establish equivalence. The ASA statement explains why conclusions need more than a threshold.

What do Tukey comparisons add?

Tukey comparisons address the family of pairwise mean differences, with adjustment for multiple comparisons. Their confidence intervals concern differences, with a family-wise coverage target. The balanced procedure and unequal-size adjustment are described in the R TukeyHSD documentation.

In Moleculytics, a difference is the first group mean minus the second. A negative value therefore indicates a lower first mean. Check the group order and outcome units. The workspace uses Tukey for equal group sizes and Tukey–Kramer for unequal sizes. It evaluates the full pairwise family regardless of the omnibus decision. Displaying fewer figure brackets does not reduce that family or change its adjustment.

Effect size and uncertainty belong together

Eta-squared describes the between-group share of total sample variation. Omega-squared adjusts for bias; it is still an estimate, not a guarantee of biological relevance. See Lakens' effect-size primer. Moleculytics reports raw omega-squared, including negative estimates, and does not assign “small” or “large” labels. It currently provides no confidence interval for either effect-size estimate.

For a 95% confidence procedure, repeated samples under the model would produce intervals covering the target about 95% of the time. That is a property of the procedure, not a 95% probability assigned to the fixed parameter after seeing this interval. See NIST on confidence intervals.

The figure's individual mean intervals are different from the simultaneous intervals for pairwise differences. SD describes spread; SEM describes estimated precision of a mean. Always name the error bars you display; use the comparisons table to read pairwise inference.

A synthetic result to practise with

The workspace's synthetic example has Control [10, 12, 14], Low dose [12, 14, 16], and High dose [14, 16, 18]. It produces F(2, 6) = 3 and p = 0.125, with eta-squared = 0.5 and raw omega-squared approximately 0.308. At alpha 0.05 the omnibus test does not reject equal means. The sample effects do not establish a biological finding or equivalence. These numbers are for illustration, not experimental evidence.

Load the example in the workspace to inspect the observations and complete comparison family. Report the design, independent n, estimates, intervals, test statistics and exclusions together.

Respect unavailable results

This beta withholds some low-df and extreme-tail Tukey outputs and marks near-threshold numerical uncertainty. An unavailable probability is not zero, a significance decision, or a validated upper bound. Read the validation limits, including the known high-offset precision failure, before using results in a report.

References