Two-Way ANOVA Calculator

Type one line for each level of factor A. Inside a line, separate the cells with a vertical bar (|): one cell for each level of factor B. The numbers inside a cell, separated by commas or spaces, are the replicates: the observations made under that exact combination of the two factors. Every cell needs the same number of values. A 2 × 3 design with four replicates per cell looks like this:

Lines that start with # are ignored. If your data are in Excel, this is the layout of the input range of Anova: Two-Factor With Replication: each line here is one block of consecutive rows there (the number of rows in a block is Excel's "Rows per sample"), and each | separated cell is one column.

The ANOVA splits the variation of the data into four parts: what factor A explains, what factor B explains, what their combination adds beyond the two (the interaction), and the error, which is the variation of the replicates around their own cell mean. Each effect is compared with the error by an F ratio.

With a levels of A, b levels of B and r replicates in every cell (n = abr observations):

Partial η² is SS_effect / (SS_effect + SS_error). The results also show the critical F for your α: the value F must exceed for p to fall below α.

The ToothGrowth data set that ships with R (Crampton 1947; Bliss 1952) records the tooth length of 60 guinea pigs that received vitamin C as orange juice (OJ) or as ascorbic acid (VC) at three doses, ten animals per combination. Enter OJ as the first line (A1), VC as the second (A2) and the doses 0.5, 1 and 2 mg/day as the three cells (B1, B2, B3), or press Load ToothGrowth example.

The calculator returns this table, which matches R's summary(aov(len ~ supp * factor(dose), data = ToothGrowth)):

• Dose has by far the largest effect (partial η² = 0.77): tooth length rises with every step in dose, and the three doses differ pairwise (Tukey HSD, p < 0.001).
• Orange juice gives longer teeth than ascorbic acid on average (20.66 against 16.96, p = 0.000231), but the interaction is significant too (p = 0.0219), so that average hides the real pattern.
• At 0.5 mg orange juice is 5.25 units ahead and at 1 mg 5.93 units ahead (t(54) = 3.23 and 3.65, p = 0.0021 and 0.0006, before any correction for three comparisons), but at 2 mg the two supplements are equal (difference −0.08, p = 0.96). In the interaction plot the two lines converge at the highest dose.
• Checks: the Brown-Forsythe test across the six cells gives W = 1.709, p = 0.148 (df 5, 54), and a Shapiro-Wilk test of the residuals gives W = 0.985, p = 0.669, so neither assumption is in doubt.

Reported in APA style: there were significant main effects of supplement, F(1, 54) = 15.57, p < .001, partial η² = .22, and of dose, F(2, 54) = 92.00, p < .001, partial η² = .77, and a significant supplement × dose interaction, F(2, 54) = 4.11, p = .022, partial η² = .13.

The plot draws one line per level of factor A through the cell means at each level of factor B. Parallel lines mean that the effect of A is the same at every level of B (no interaction). Lines that converge or fan out mean that the size of the effect changes, and lines that cross mean that its direction changes. The F test for A × B decides whether the departure from parallel is larger than the noise between replicates.

Look at the interaction first. When it is not significant, read the main effects from the level means in the margins of the cell means table. When it is significant, a main effect is an average over levels where the effect differs, and it can even vanish: enter 5, 6, 7 | 9, 10, 11 on one line and 9, 10, 11 | 5, 6, 7 on the next, and both main effects come out exactly 0 (all four margins equal 8) while the interaction has F = 48 (p = 0.000121). Then describe the cell means, or test the effect of one factor separately at each level of the other, and correct for the number of comparisons with Bonferroni or Tukey HSD.

• Independent observations. Each value comes from a different subject or unit. Measuring the same subject in several cells breaks this (see the repeated measures question below).
• Roughly normal residuals. The residuals are the values minus their cell means. With about ten or more replicates per cell moderate skew does little harm. Check them with the Shapiro-Wilk test and read normality tests explained.
• Equal variances in every cell. The results list the cell standard deviations and run a Brown-Forsythe test across all cells (it needs at least 3 replicates per cell). With the same number of replicates in every cell the F tests tolerate moderately unequal variances; if the spread differs a lot, try a log or square-root transformation, or look closer with the Levene test calculator.
• A numeric outcome and a balanced layout. Ordinal outcomes and unequal cell counts call for other methods.

When the assumptions clearly fail, a transformation is the simplest fix. For a rank-based or robust two-way analysis use an R package such as ARTool or WRS2; the nonparametric calculators here handle one factor at a time, for example Kruskal-Wallis.

If every cell holds a single value, put one number in each cell (for example 45.2 | 47.9 | 52.1). There is no replicate to estimate the interaction separately, so the calculator fits the additive model: the interaction residual becomes the error term, with (a − 1)(b − 1) degrees of freedom, and only the two main effects are tested. This is Excel's Anova: Two-Factor Without Replication and the standard analysis of a randomized complete block design, where one factor is the treatment and the other a blocking factor. It is valid only when the two factors do not interact.

Example (made-up yields of four varieties, A, in three fields, B):

gives F(3, 6) = 8.0964 (p = 0.015686) for A, F(2, 6) = 58.7668 (p = 0.000115) for B, and an error sum of squares of 5.445 with 6 degrees of freedom.

With the same number of replicates in every cell the effects are orthogonal: SS_A + SS_B + SS_AB + SS_error equals the total sum of squares exactly, and Type I, II and III sums of squares give the same table. With unequal cell counts they differ. Type I is sequential and depends on the order of the factors (R's anova() and summary(aov()) use it), Type II tests each main effect after the other main effect and ignores the interaction, and Type III (the default in SPSS) tests each effect after all the others. The right choice depends on the question you are asking, so this calculator refuses an unbalanced grid instead of choosing a type for you. Use general linear-model software for unbalanced data.

In Python, first run import statsmodels.api as sm and from statsmodels.formula.api import ols. Make numeric codes categorical (factor(dose) in R, C(dose) in Python); otherwise the dose is fitted as a slope and the degrees of freedom differ. The sums of squares, F ratios and p-values agree with this calculator for balanced data.

A significant main effect with three or more levels only says that some levels differ. To find which, run Tukey HSD on the level means of that factor, using the error mean square and error degrees of freedom from the ANOVA table (the calculator has a "Means + MSE and df" mode; for the dose in the example that is the means 10.605, 19.735 and 26.1 with 20 observations each, MSE 13.1871 and 54 df). When the interaction is significant, compare the levels of one factor within each level of the other instead.

Report an effect size with every test: partial η² here, or use the effect size calculator and read effect size explained. To plan how many replicates you need before collecting data, see statistical power and sample size explained. Critical values are in the F table, and which statistical test to use helps when two factors are not what you have.