Prism offers several types of ANOVA to handle different experimental designs and data characteristics. When deciding on which analysis to perform, be sure to make you choice based on:
1.How many factors you have
2.Whether measurements are independent or repeated/matched
3.What distribution your data were sampled from
4.Whether your data meet the assumption of equal variances
By Number of Factors
Before going into some of the details of the specialized versions of ANOVA in Prism, it's worth noting that the Multifactor ANOVA in Prism can handle any number of factors, meaning you only need to learn one set of analysis parameters that you can then apply to one-way, two-way, or three-way, or even more complex design ANOVAs.
•When to use: One factor with three or more levels
•Example: Comparing blood pressure across four different drug treatments
•What it tests: Whether the means of all groups are equal
•When to use: Two factors, each with two or more levels
•Example: Testing three drugs in both males and females
•What it tests:
oMain effect of Factor 1 (e.g., drug effect)
oMain effect of Factor 2 (e.g., gender effect)
oInteraction between factors (does the drug effect differ between genders?)
•When to use: Three factors
•Example: Testing three drugs in males and females at multiple time points
•What it tests: Three main effects, three two-way interactions, and one three-way interaction
Note: The specialized three-way ANOVA in Prism only supports up to 2×2×K designs (meaning two of your factors can only have exactly two levels, while the other may have many levels). The Multifactor ANOVA in Prism does not have such limitations, and can perform three-way ANOVA with any number of levels in each factor.
When to use: Any number of factors, including designs with four or more factors. Also supports more flexible three-way ANOVA designs (greater than 2×2×K designs) than the named "three-way ANOVA" in Prism. Currently only supports ordinary (non-repeated-measures) designs.
Example: Testing drugs × gender × age group × disease severity
What it tests: All main effects and interactions up to three-way
Note: Uses Multiple Variables data table format for all designs (can perform one-, two-, three-, or N-way ANOVA from the same data structure eliminating the need to learn different data arrangements for different analysis designs).
By Experimental Design and Matching
Ordinary ANOVA (Completely Randomized Design)
•When to use: Each subject/observation belongs to only one group
•Example: Different patients receive different drugs (each patient gets only one drug)
•Data structure: Independent groups with no matching
•When to use: Same subjects measured multiple times, or subjects matched into sets
•Examples:
oMeasuring blood pressure in the same patients before, during, and after treatment
oLittermates assigned to different treatments (matched by litter)
oRunning the same experiment multiple times with all treatments in parallel
•Advantage: More powerful because it accounts for subject-to-subject variability
•Important: Matching must be based on characteristics other than the outcome variable
Mixed Effects Models
Mixed effects models are a special type of regression model that can account for both "fixed" and "random" effects. In the context of ANOVA, repeated measures are defined by using "random" factors. Importantly, when there are no missing values in the data (each subject has a measurement for each appropriate combination of factors), then standard ANOVA approaches are applicable. However, if data are missing, then special mixed effects model approaches are required to analyze the data. When performing a repeated measures design ANOVA, Prism offers the choice to analyze the data using the classic ANOVA approach, using the more complex mixed effects (GLM) approach, or to allow Prism to decide automatically.
•When to use: Some factors are repeated measures and others are not, particularly when there are missing data
•Example: Testing three drugs (between-subjects) at four time points (within-subjects), with each subject being measured at every time point. Particularly helpful when a given subject is missing measurements at a given time point
•Also called: Mixed-model ANOVA
•Advantage: Can handle missing data when using Prism's mixed model approach
•When to use: You have a hierarchical structure in your data
•Example: Patients within clinics, where different clinics use different protocols
•Structure: One factor is "nested" within another
Note: Requires special consideration in model specification. Currently, Prism only supports "Nested one-way ANOVA" in which you have technical replicates for subjects or groups that are each nested within a single factor of interest.
By Distribution Assumption
Ordinary (Parametric) ANOVA
•Assumes: Data sampled from a normal (Gaussian) distribution
•Tests: Whether population means are equal
•Most common: This is the default ANOVA in most situations
Lognormal ANOVA
•Assumes: Data sampled from a lognormal distribution
•Tests: Whether population geometric means are equal
•When to use: Positive-valued data with right-skewed distributions (common in biology)
•How it works: Prism log-transforms your data automatically, performs ANOVA, then back-transforms results
•Results show: Geometric means and ratios of geometric means (rather than differences in means)
Welch's ANOVA and Brown-Forsythe ANOVA
•Assumes: Normal distribution BUT does not assume equal variances across groups
•When to use: Groups have substantially different standard deviations
•Available for: One-way ANOVA only
•Advantage: Valid when homogeneity of variance assumption is violated
Nonparametric Alternatives
When your data don't meet ANOVA assumptions (and transformations don't help), consider:
•Kruskal-Wallis test: Nonparametric alternative to one-way ANOVA
•Friedman test: Nonparametric alternative to repeated measures one-way ANOVA
Note: These tests don't assume a specific distribution, but they do assume the distributions have similar shapes (equal dispersions)
Key ANOVA Concepts
The ANOVA Table
Every ANOVA produces an ANOVA table that partitions variation into components. Understanding this table is key to interpreting results.
Components of the ANOVA table:
Source of VariationWhat it representsBetween groups (or factor effects)Variation due to differences between group meansWithin groups (or residual/error)Variation within each group due to individual differencesTotalTotal variation in all the data combined
Columns in the ANOVA table:
Sum of Squares (SS): Quantifies the amount of variation from each source
Degrees of Freedom (df): Number of independent pieces of information
Mean Square (MS): SS divided by df - represents variance from each source
F-ratio: MS(between) / MS(within) - the test statistic
P-value: Probability of observing this F-ratio (or larger) if groups are truly equal
Interpreting the F-ratio:
F ≈ 1: Between-group and within-group variation are similar (no effect)
F >> 1: Between-group variation is much larger than within-group variation (likely effect)
Read more about the ANOVA table →
Understanding Interactions
When you have two or more factors, ANOVA can test for interactions between them. An interaction occurs when the effect of one factor depends on the level of another factor.
What is an interaction?
No interaction example: A drug lowers blood pressure by 10 points in both men and women. The drug effect is the same regardless of gender - no interaction.
Interaction example: A drug lowers blood pressure by 20 points in men but only 5 points in women. The drug effect depends on gender - this is an interaction.
Why interactions matter:
Interactions are often the most interesting finding in a multifactor experiment. They tell you that:
Effects are not simple or additive
You cannot fully understand one factor without considering the other
Subgroup analyses may be warranted
Visualizing interactions:
The best way to understand interactions is graphically:
Parallel lines = No interaction (effects are additive)
Non-parallel lines = Interaction present (effects are not additive)
Crossing lines = Strong interaction (effect reverses across groups)
Types of interactions in multifactor designs:
Two-way interaction: Between two factors (A × B)
Three-way interaction: Between three factors (A × B × C)
Higher-order interactions: Four-way and beyond (increasingly difficult to interpret)
Read more about interactions in two-way ANOVA →
