Why two-way designs exist
A two-way ANOVA crosses two factors in one experiment: teaching method by time of day, drug by sex, fertiliser by soil. The design buys three things one-way studies cannot offer. It tests both factors at once on the same subjects, halving the data collection of two separate studies. It measures each factor while the other varies, which is how the real world will use the result. And it alone can detect an interaction: the possibility that a factor's effect depends on the other factor's level, which is often the finding that changes practice.
Cell means (three students per cell): method A1 scores 5 in the morning, 7 in the evening; A2 scores 9 and 13. SS partitions exactly: SSA = 75, SSB = 27, SSAB = 3, SSE = 8, total 113. F values 75.00, 27.00 and 3.00; the interaction's p ≈ 0.122 is not significant. Reading the cells: A2 beats A1 by 4 points in the morning and 6 in the evening. The advantage is real in both slots and roughly constant, which is what "no interaction" looks like in numbers.
The interaction-first reading order
Always interpret the interaction before the main effects. A significant interaction means the main effects are averages over a non-uniform pattern: "method A2 is better by 5 points" could hide "better by 11 in the evening and worse by 1 in the morning", two claims with opposite advice. Plot the cell means: parallel lines say no interaction; lines that converge, diverge or cross say the interaction is the story, and simple-effects tests (one factor examined at each level of the other) tell it in detail. The two-way ANOVA calculator on this site enforces the balanced design this clean SS partition needs and leads its output with the interaction verdict.
Deleting a "non-significant" factor and rerunning as one-way without thinking. Pooling changes the error term and every remaining test; sometimes that is the right refit, but it is a modelling decision to state, not a reflex.
Frequently asked questions
What is an interaction, in one example?
A drug that lowers blood pressure in men but raises it in women: the drug's effect flips with sex. Averaged over both sexes it looks like no effect, which is why the interaction term exists.
Does a two-way ANOVA need equal group sizes?
The clean sums-of-squares partition does: with balance, the effects are independent and SSA + SSB + SSAB + SSE = SST exactly. Unbalanced data needs a stated SS type (I, II or III) and software that supports it.
How is a two-way ANOVA different from running two one-way ANOVAs?
Two one-ways cannot see an interaction at all, and each ignores the other factor's variation, which stays in its error term. The crossed design estimates everything jointly and more precisely.