Two-Way ANOVA with Replication Calculator

Two-Way ANOVA (With Replication)

Factor A Name (Rows)

Factor B Name (Columns)

Factor A Levels (Rows)

Factor B Levels (Cols)

Significance Level (α) Default hypothesis rejection threshold is 0.05 (5%)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Full Factorial Experimental Design

What Is Two-Way ANOVA with Replication?

A Two-Way ANOVA with Replication (also known as a full factorial ANOVA) evaluates the simultaneous effects of two independent categorical factors on a continuous outcome variable when there are **multiple observations (replicates, $n \ge 2$) per experimental cell**.

Unlike the design without replication, having replicates allows the model to test not only the two independent **Main Effects (Factor A and Factor B)**, but also the critical **Interaction Effect ($A \times B$)**—determining whether the effect of one factor depends on the level of the other.

When to Use It

$a \times b$ Factorial Experiments: Evaluating two interventions simultaneously with balanced replicates per treatment combination.
Interaction Hypothesis Testing: Assessing whether treatments produce synergistic or antagonistic interaction outcomes ($p_{AB} \le 0.05$).
Controlled Agricultural & Clinical Trials: Combining drug dosages and patient demographics across multi-subject cohorts.
Independent Cells with $n \ge 2$: When each factor combination contains distinct experimental units.

The Factorial Partitioning Formulas

$$SS_{\text{tot}} = \sum x_{ijk}^2 - CM, \quad CM = \frac{G^2}{abn}$$ $$SS_A = \frac{\sum A_i^2}{bn} - CM, \quad SS_B = \frac{\sum B_j^2}{an} - CM$$ $$SS_{AB} = \frac{\sum AB_{ij}^2}{n} - CM - SS_A - SS_B$$ $$SS_{\text{error}} = \sum_{ijk} (x_{ijk} - \bar{x}_{ij\cdot})^2 = SS_{\text{tot}} - (SS_A + SS_B + SS_{AB})$$ $$F_A = \frac{MS_A}{MS_{\text{error}}}, \quad F_B = \frac{MS_B}{MS_{\text{error}}}, \quad F_{AB} = \frac{MS_{AB}}{MS_{\text{error}}}$$

Where $a$ is the number of Factor A levels, $b$ is the number of Factor B levels, $n$ is the number of replicates per cell, and $df_{\text{error}} = ab(n - 1) = N - ab$.

Worked Example

Scenario: 2 Diets ($a=2$) $\times$ 3 Exercise levels ($b=3$) with $n=3$ subjects per cell ($N=18$), testing at $\alpha=0.05$.

Step 1: Compute Totals & Correction Term ($CM$):
Grand Total $G = 333 \implies CM = \frac{333^2}{18} = 6160.50$.
$\sum x_{ijk}^2 = 6495.00 \implies SS_{\text{tot}} = 6495 - 6160.50 = 334.50$.

Step 2: Calculate Factor Sums of Squares:
$SS_A = 40.50, \quad SS_B = 243.00, \quad SS_{AB} = 3.00$.
$SS_{\text{error}} = 334.50 - (40.50 + 243.00 + 3.00) = 48.00$.

Step 3: Mean Squares & F-Statistics ($df_e = 12, MS_e = 4.0$):
$F_A = \frac{40.50 / 1}{4.0} = 10.13 \implies p_A = 0.0079 \text{ (Sig Diet Main Effect)}$
$F_B = \frac{243.00 / 2}{4.0} = 30.38 \implies p_B < 0.0001 \text{ (Sig Exercise Main Effect)}$
$F_{AB} = \frac{3.00 / 2}{4.0} = 0.38 \implies p_{AB} = 0.6951 \text{ (No Interaction)}$

Factorial Analysis Complete

Two-Way ANOVA Results

Primary Factorial Outcome

Significant Main Effects Detected

Factor A (p = 0.0079) and Factor B (p < 0.0001) are statistically significant (α = 0.05).

Factor A (F) 10.13

Factor B (F) 30.38

Interaction A×B (F) 0.38

Interaction p-val 0.6951

Error MS (MS_error) 4.0000

Total Sample (N) 18 (n=3/cell)

ANOVA Summary Table

Source of Variation	Sum of Squares (SS)	df	Mean Square (MS)	F-Statistic	p-value	Partial η²

Two-Way ANOVA with Replication Calculator