What is Multifactor ANOVA?

Multifactor ANOVA (also called N-way ANOVA) is a flexible approach to analysis of variance that can handle any number of factors in a single analysis. While Prism offers dedicated analyses for one-way, two-way, and three-way ANOVA, Multifactor ANOVA provides a unified framework that can handle all of these designs plus more complex experiments with four or more factors.

When to use Multifactor ANOVA:

•You want to use a standard approach for ANOVA with any number of factors (multifactor ANOVA allows you to perform one-way, two-way, three-way, or higher order designs using the same data structures and the same analysis parameters dialogs)

•You have four or more factors (not available in other Prism ANOVA analyses)

•You want a flexible approach that works with the Multiple Variables data table

•You need to test main effects and interactions up to three-way interactions

•Your design is completely randomized (all groups are independent)

Key features:

•Works with Prism's Multiple Variables data table format

•Handles any number of factors (categorical grouping variables)

•Tests main effects for all factors

•Tests two-way interactions between all pairs of factors

•Tests three-way interactions between all combinations of three factors

•Does not test four-way or higher-order interactions

•Currently supports ordinary (non-repeated measures) designs only

When to use the dedicated ANOVA analyses instead of multifactor ANOVA

•Repeated measures designs: If any of your factors involve repeated measurements on the same subjects, use the dedicated one-way, two-way, or three-way repeated measures ANOVA

•Lognormal ANOVA: If your data are sampled from a lognormal distribution (Multifactor ANOVA assumes normal distribution). Alternatively transform your lognormal variables with a log transform prior to including the data into the analysis

•Welch or Brown-Forsythe ANOVA: If your groups have substantially unequal variances (one-way only)

•Nonparametric tests: If your data don't meet normality assumptions and transformations don't help (Kruskal-Wallis, Friedman)

•Familiarity or continuity of existing results: This isn't really a great reason to avoid using multifactor ANOVA as - over time - you will learn to use this analysis and its results as easily as any other ANOVA that you've used historically with Prism. But still, those other analyses will be there if you need them!

Understanding Multifactor Designs

What is a factor?

A factor is a categorical variable (grouping variable) that defines different experimental conditions. Each factor has two or more levels (groups).

Examples of factors in biological research:

•Treatment (Control, Drug A, Drug B)

•Genotype (Wild-type, Knockout, Heterozygote)

•Gender (Male, Female)

•Age group (Young, Middle-aged, Old)

•Tissue type (Liver, Kidney, Heart, Brain)

•Time point (0h, 6h, 12h, 24h)

•Diet (Standard, High-fat, High-protein)

•Environmental condition (Standard, Heat stress, Cold stress)

 

Example multifactor design:

Suppose you're studying plant growth with these factors:

•Factor 1 - Fertilizer: None, Organic, Synthetic (3 levels)

•Factor 2 - Watering: Low, Medium, High (3 levels)

•Factor 3 - Light exposure: Shade, Partial sun, Full sun (3 levels)

•Factor 4 - Soil pH: Acidic, Neutral, Alkaline (3 levels)

This is a 3 × 3 × 3 × 3 design with 81 total treatment combinations. Multifactor ANOVA can analyze all four factors as well as all of their interactions simultaneously.

What Multifactor ANOVA Tests

Multifactor ANOVA partitions the total variation in your data and tests multiple hypotheses:

Main Effects

For each factor, ANOVA tests whether the levels of that factor differ, averaging across all other factors.

Example questions:

•Is there a fertilizer effect? (averaging across all watering, light, and pH combinations)

•Is there a watering effect? (averaging across all fertilizer, light, and pH combinations)

•Is there a light effect? (averaging across all other factors)

•Is there a pH effect? (averaging across all other factors)

 

Two-Way Interactions

For each pair of factors, ANOVA tests whether the effect of one factor depends on the level of the other factor.

Example questions:

•Does the fertilizer effect depend on watering level? (Fertilizer × Watering)

•Does the light effect depend on pH? (Light × pH)

•Do fertilizer and light interact? (Fertilizer × Light)

 

With 4 factors, there are 6 possible two-way interactions:

•Fertilizer × Watering

•Fertilizer × Light

•Fertilizer × pH

•Watering × Light

•Watering × pH

•Light × pH

 

Three-Way Interactions

For each combination of three factors, ANOVA tests whether the two-way interaction between two factors depends on the level of a third factor.

Example questions:

•Does the Fertilizer × Watering interaction differ across light levels?

•Does the Light × pH interaction differ across fertilizer types?

 

With 4 factors, there are 4 possible three-way interactions:

•Fertilizer × Watering × Light

•Fertilizer × Watering × pH

•Fertilizer × Light × pH

•Watering × Light × pH

 

Four-Way and Higher Interactions

Multifactor ANOVA in Prism does not test four-way or higher-order interactions. These interactions are:

•Extremely difficult to interpret biologically

•Rarely significant in practice

•Require very large sample sizes to detect reliably

•Often better understood by focusing on simpler effects

The variation from four-way and higher interactions is pooled into the residual error term.

 

Interpreting Interactions

Interactions are often the most scientifically interesting findings in multifactor experiments. The examples below provide some guidance in how to think about them.

Two-Way Interaction

A two-way interaction means the effect of one factor depends on the level of another factor. Continuing with the example of an experimental design investigating fertilizer, watering level, light conditions, and soil pH levels on plant growth, we can see what it would mean for there to be a two-way interaction or not.

If there is no two-way interaction between fertilizer and watering:

•Fertilizer increases plant height by 10 cm regardless of watering level

•The fertilizer effect is constant across watering conditions

•Effects are additive

 

If there is a two-way interaction between fertilizer and watering:

•Fertilizer increases height by 20 cm with high watering

•Fertilizer increases height by only 5 cm with low watering

•The fertilizer effect depends on watering level

•Effects are not simply additive

Using interaction plots is a great way to visualize whether or not a two-way interaction may exist in your data. Interaction plots work by plotting the means of your response variable with the levels of one factor on the X axis of the graph with separate lines for each level of the other factor. If the lines are parallel, then there's no interaction between these factors. If the lines are not parallel or if they cross, then may be an interaction between these factors in the data.

Three-Way Interaction

A three-way interaction is a logical extension of a two-way interaction. In a two-way interaction, the effect of one factor depends on the levels of a second factor. In a three-way interaction, the effect of a two-way interaction between two factors depends on the levels of a third factor.

Example:

•At acidic pH: Fertilizer × Watering interaction is strong

•At neutral pH: Fertilizer × Watering interaction is weak

•At alkaline pH: No Fertilizer × Watering interaction

The result of this sort of interaction is that you cannot fully understand the Fertilizer × Watering interaction without also considering pH. If you look only at one level of pH (or if you average across all levels of pH) to try and interpret the two-way interaction between Fertilizer and Watering, you may be mislead by the results.

Similar to two-way interactions, you can also create interaction plots for three-way interactions, but it's a bit more complex. In this case, you would need to create separate interaction plots for each level of the third factor. If the pattern of interactions changes across the different graphs, this may suggest the presence of a three-way interaction. This visual interpretation already gets a bit complex, and generally represents the limit of what is often meaningful in biological research.

Before worrying too much about a "significant" three-way interaction, conduct your own reality check on whether or not the interaction is experimentally important or relevant. Statistical significance does not automatically indicate biological or practical relevance, particularly with larger sample sizes when even trivial interactions can achieve statistical significance.

If a three-way interaction is experimentally or biologically relevant (only you can decide this), then when that interaction is statistically significant:

•Simple two-way interactions may be misleading

•Consider analyzing subsets of data separately

•Focus interpretation on specific factor combinations of interest

•Consultation with a statistician may be helpful

Data Requirements

Multifactor ANOVA requires a Multiple Variables data table in Prism. With this table format:

•Each row represents one observation (subject, sample, experimental unit)

•Each column represents one variable

•One column contains your response (outcome) variable - the continuous measurement you want to analyze

•Additional columns contain your grouping variables (factors) - categorical variables defining your groups

Example table structure:

In this example:

•Response variable: Height (continuous) - the measured height of different plants (one per row) growing under different conditions (described by values of other variables)

•Grouping variables (factors): Fertilizer, Watering, Light, pH (all categorical)

Variable Types

Response (Y) variable:

•Must be continuous

•Examples: height, weight, concentration, temperature

•Should be normally distributed within each group (ANOVA assumption)

•All observations should use the same units

Grouping variables (factors):

•Must be categorical

•Can have two or more levels (groups)

•Examples: treatment groups, genotypes, time points (treated as categories), locations

•Can be text or numeric, but variable should be classified as categorical in the data table

•Should be clearly defined with meaningful labels

Note about numeric factors: If you have a numeric variable like dose (0, 10, 20, 50 mg) or time (0, 2, 4, 8 hours), you can treat it as a factor in ANOVA. To do so, you must first change the variable type assignment from continuous to categorical. By doing this, the ANOVA will ignore the ordering and spacing of values. As an alternative, consider multiple linear regression which uses the numeric information directly.

Sample Size Considerations

Minimum requirements:

•At least 2 observations per treatment combination (cell) required

•Recommended minimum 3-5 observations per cell

•More observations per cell = more power to detect effects

Sample size grows exponentially with additional factors:

•2 factors with 3 levels each = 9 cells (3 × 3)

•3 factors with 3 levels each = 27 cells (3 × 3 × 3)

•4 factors with 3 levels each = 81 cells (3 × 3 × 3 × 3)

•5 factors with 3 levels each = 243 cells (3 × 3 × 3 × 3 × 3)

Practical implications: With 4 factors (3 levels each) and 5 replicates per cell, you need 405 total observations. This can quickly become impractical.

Strategies for complex designs:

•Consider whether you really need all factors in one analysis

•Use fractional factorial designs (test main effects and selected interactions only)

•Focus on factors of primary interest; analyze secondary factors separately

•Increase replication for key factor combinations of interest

•Accept lower power for higher-order interactions

Balanced vs. Unbalanced Designs

Balanced design: All treatment combinations have the same sample size

•Advantages: Simpler interpretation, clearer estimation of effects, maximum statistical power

•Preferred whenever possible

Unbalanced design: Treatment combinations have different sample sizes

•Common reasons: Missing data, attrition, unequal availability of subjects, observational data

•Still valid: Multifactor ANOVA can handle unbalanced designs

•Note: Prism uses Type III sums of squares, which handles unbalanced designs appropriately

•Caution: Severely unbalanced designs (some cells with very few observations) can reduce power and affect interpretation

Assumptions of Multifactor ANOVA

Like all ANOVA methods, Multifactor ANOVA makes several assumptions:

1.Independence: observations must be independent of each other. Each measurement should come from a different experimental unit.

2.Normality: data should be sampled from normally distributed populations.

3.Homogeneity of variance: all groups should have equal variance (homoscedasticity)

Multifactor ANOVA vs. Multiple Linear Regression

Both Multifactor ANOVA and Multiple Linear Regression can analyze the same data with multiple categorical predictors. However, there are a number of important differences between these two analyses.

Coding of categorical variables:

•ANOVA (effects coding): Compares each group to the overall grand mean; estimates sum to zero

•Regression (dummy/reference coding): Compares each group to a reference group; one group is the baseline

Parameter interpretation:

•ANOVA: Effect of being in a particular group relative to the average across all groups

•Regression: Effect of being in a particular group relative to the reference group

Output format:

•ANOVA: The primary output is the ANOVA table with F-tests for each factor and interaction

•Regression: The primary output are the regression coefficients and t-tests for each parameter. However, Prism will also report an ANOVA table when performing multiple linear regression

Multiple comparisons:

•ANOVA: Integrated multiple comparison tests with family-wise error control

•Regression: Requires custom contrasts or post-hoc calculations (not available directly/automatically in Prism)

When to use Multifactor ANOVA:

•You want to test overall effects of factors

•You want traditional ANOVA output format

•You plan to do multiple comparisons among groups

•Your collaborators/field expects ANOVA results

•You think in terms of "group means" rather than "regression coefficients"

When to use Multiple Linear Regression:

•You want to compare groups to a specific control/reference group

•You have a mix of categorical and continuous predictors

•You want regression coefficients and confidence intervals

•You need more control over model specification

•You're more comfortable with regression framework

Critical Point: For balanced designs with categorical predictors only, ANOVA and regression will give you identical P-values and R² values. They're mathematically equivalent but present results differently.

When Multifactor ANOVA May Not Be Appropriate

Multifactor ANOVA is a very powerful and very useful tool. But it's certainly not always going to be the best choice for your data or your experimental conditions. The following are some scenarios in which you may want to consider alternatives to multifactor ANOVA in Prism:

1.Repeated measures design

oIssue: Currently, multifactor ANOVA in Prism does not support repeated measures, and assumes all observations are independent

oAlternative: Use dedicated repeated measures ANOVA or mixed models for one-way or two-way ANOVA

2. Ordered factors (dose, time)

oIssue: ANOVA treats ordered categories as unordered and ignores spacing between levels

oAlternative: Consider using multiple linear regression instead

3.Too many factors

oIssue: With 5+ factors, the model becomes complex and difficult to interpret; requires very large sample sizes to achieve reasonable power to detect effects

oAlternative: Prioritize key factors or consider using fractional factorial designs

4.Hierarchical/nested structure

oIssue: If factors are nested (e.g., students within classrooms within schools), standard ANOVA is inappropriate

oAlternative: Nested ANOVA (available only as nested one-way ANOVA)