Multifactor ANOVA, also called N-way ANOVA, determines how a response is affected by any number of categorical factors as well as their interactions. For example, you might measure plant height in response to fertilizer type, watering frequency, light exposure, and soil pH. In this example, fertilizer is one factor, watering is another, light is the third, and soil pH is the fourth. Read elsewhere to learn about when to use multifactor ANOVA, entering data, and interpreting the results.

Are your data sampled from Gaussian distributions?

Multifactor ANOVA assumes that your data are sampled from populations that follow a Gaussian (normal) distribution. While this assumption is not too important with large samples due to the Central Limit Theorem, it becomes important with small sample sizes, especially when those samples are unequal across groups.

Prism can test for violations of this assumption using the Residuals tab. Check the QQ plot to see if points fall reasonably close to the diagonal line, and run the four normality tests (D'Agostino-Pearson, Anderson-Darling, Shapiro-Wilk, and Kolmogorov-Smirnov) to assess whether residuals are approximately normally distributed. However, keep in mind that normality tests have limited power with small samples and may be overly sensitive with large samples.

If your data don't come from Gaussian distributions, you have several options. Your best choice is often to transform the values to make the distributions more Gaussian. For example, taking the logarithm or the reciprocal of the values may help address departures from normality. Another option is to use ANOVA anyway, recognizing that it's fairly robust to violations of normality when you have large samples. Finally, you could use nonparametric tests, though good nonparametric alternatives for multifactor designs are limited. For single-factor comparisons, consider the Kruskal-Wallis test instead of ANOVA.

Do the populations all have the same standard deviation?

ANOVA also assumes that all populations have the same standard deviation (and thus the same variance). This assumption matters less when all groups have the same or nearly the same number of observations, but becomes more important when sample sizes differ substantially across treatment combinations.

Prism can test for equality of variance on the Residuals tab. The Spearman's rank correlation test for heteroscedasticity examines whether cells with larger values tend to have larger variation. The P value from this test answers an important question: if the populations really do have the same variance, what's the chance you'd randomly select samples whose variances differ as much as those you observed? A small P value suggests the variances are genuinely different, so a large P value is typically what you're looking for.

However, as with all statistical tests, don't base your conclusion solely on a P value. Also look at the homoscedasticity plot to see if there's a clear pattern in the spread of residuals. Think about data from other similar experiments. If you have plenty of previous evidence that variances are really equal, you might reasonably interpret the ANOVA results as usual unless the QQ plot shows an obvious problem. Some statisticians recommend ignoring tests for equal variance altogether when sample sizes are equal or nearly equal, since ANOVA is quite robust in balanced designs.

In some experimental contexts, finding different variances may be as scientifically important as finding different means. If the variances truly differ, then your populations are fundamentally different regardless of what ANOVA concludes about differences between means.

If you conclude that variances are truly unequal, consider transforming your data to stabilize the variance. For single-factor designs, you can use Welch's ANOVA, which doesn't assume equal variance. Unfortunately, this option isn't available for multifactor designs, so you'll need to either transform the data or acknowledge the limitation when interpreting your results.

What do the P values from the ANOVA tell you?

Multifactor ANOVA tests several null hypotheses simultaneously. For each factor (main effect), the null hypothesis states that the means of all levels of that factor are equal when averaged across all other factors. For each interaction, the null hypothesis states that there's no interaction - in other words, that the effect of one factor is the same across all levels of the other factor or factors.

The F statistic for each test is calculated as the ratio of variance attributable to that effect divided by the residual (error) variance. When this ratio is large, it suggests the effect is real given the amount of random variation in your data. As the F statistic increases, the corresponding P value decreases. If the F statistic is large enough (resulting in the P value being smaller than the specified value of alpha) you can reject the null hypothesis for that effect.

Here's a critical point: if an interaction is significant, be very cautious about interpreting main effects alone. A significant interaction means the effect of one factor genuinely depends on the level of another factor. In this situation, statements like "Factor A increases the response" or "Factor B has no effect" may be misleading or incomplete. Instead, use simple effect comparisons to understand how factors combine. For example, comparing the levels of Factor A separately at each level of Factor B.

Are the data unpaired (are groups independent)?

Multifactor ANOVA works by comparing differences among group means with the pooled standard deviations of the groups. This approach assumes all observations are independent. Or in other words that each observation comes from a different subject or experimental unit, with no relationship between observations in different groups.

When the same subjects are measured multiple times or subjects are paired or matched to each other based on particular characteristics, this is called a "repeated measures" or "paired" design. If your data are matched or involve repeated measures then you need repeated-measures ANOVA. Use the dedicated analyses for repeated measures one-way ANOVA, repeated measures two-way ANOVA, or three-way ANOVA if one or more of your factors involves repeated measures. The current multifactor ANOVA in Prism doesn't support repeated measures designs.

When matching is effective at controlling for experimental variability, repeated-measures ANOVA will be more powerful than ordinary ANOVA because it accounts for the correlation between related measurements.

Are the "errors" independent?

The term "error" refers to the difference between each observed value and the mean predicted by the model for that combination of factors. The results of multifactor ANOVA only make sense when this scatter is truly random. In other words when whatever factor caused a value to be too high or too low affects only that single value and doesn't create correlations between values.

Prism can't test this assumption for you: you must think carefully about your experimental design. For example, imagine you have six values in each treatment combination, but these six measurements came from just two subjects with triplicate measurements from each. In this case, some unknown factor might cause all three measurements from one subject to be consistently high or low. This is pseudoreplication, and ordinary multifactor ANOVA would not be appropriate.

True replicates are independent experimental units. If your "replicates" come from the same subject, the same cell culture preparation, or the same experimental run, they aren't independent. In such cases, you should average technical replicates before analysis, or use repeated measures or mixed model approaches that account for the hierarchical structure of your data.

Are you relying only on looking at P values? DON'T!

Multifactor ANOVA compares means across groups. It's entirely possible to get a tiny P value even when the distributions for different groups overlap considerably. In some research contexts, like assessing the usefulness of a diagnostic test, you might care more about the overlap of distributions than about differences between means. ANOVA won't directly tell you about this overlap.

Additionally, ANOVA tests whether means differ but doesn't automatically tell you how much they differ or whether the difference matters practically. A statistically significant result means you have good evidence of a real difference, but that difference could be tiny and biologically meaningless. Always examine effect sizes (percentage of variation explained, partial η², Cohen's f) in addition to P values to assess both whether an effect exists and how large or important it is.

Don't confuse multifactor ANOVA with one-way or two-way ANOVA with multiple groups

Don't mix up multifactor ANOVA with one-way ANOVA that happens to have multiple groups. With multifactor ANOVA, you have three or more grouping variables (factors) - perhaps fertilizer type, watering frequency, and light exposure. With one-way ANOVA, you have just one grouping variable (perhaps treatment type). If there are five different treatments, you still need one-way ANOVA, not five-way ANOVA.

Similarly, two-way ANOVA with four levels of one factor and three levels of another is still two-way ANOVA (two factors), not seven-way ANOVA. The number of "ways" refers to the number of factors (grouping variables) you're studying, not the total number of groups or levels across all factors.

How many factors should you include?

While multifactor ANOVA can theoretically handle any number of factors, practical considerations limit how many you should include in a single analysis.

Sample size requirements grow exponentially as you add factors. Three factors with three levels each creates 27 treatment combinations. Four factors with three levels each produces 81 combinations. Five factors with three levels each yields 243 combinations. If you want adequate replication (say, 5-10 observations per combination for reasonable statistical power), you quickly need impractically large sample sizes.

Interpretation also becomes increasingly complex as you add factors. More factors means more main effects and interactions to understand and explain. A four-factor design produces 14 different effects to test: four main effects, six two-way interactions, four three-way interactions, and potentially one four-way interaction. Three-way and higher-order interactions are notoriously difficult to interpret in meaningful biological terms.

Consider carefully whether all your factors are truly necessary for answering your research question. Include factors that are of primary scientific interest and factors needed to control for known confounding variables. If you have many potential factors, consider using screening designs to identify which factors are most important before conducting a full factorial experiment with all combinations.

Finally note again that higher-order interaction terms not only increase the required sample size for your experiment, but are also notoriously difficult to interpret. Currently for multifactor ANOVA, Prism only supports up to three-way interactions (interactions between three different factors in the ANOVA model).

Does multifactor ANOVA test higher-order interactions?

Multifactor ANOVA in Prism allows you to test all main effects (the effect of each individual factor), all two-way interactions (how each pair of factors combines), and all three-way interactions (how each triplet of factors combines).

However, four-way and higher-order interactions are not tested. These very high-order interactions are extremely difficult to interpret, rarely significant in practice, require very large sample sizes to detect with adequate power, and often don't correspond to meaningful biological mechanisms. The variation that would be attributed to four-way and higher interactions is instead pooled into the residual (error) term.

If you have a compelling reason to test four-way or higher-order interactions, consult with a statistician about alternative approaches or specialized software that can handle these more complex models.

Are you using the correct data table format?

Multifactor ANOVA requires a Multiple Variables data table in Prism. This is different from the Column or Grouped tables used for one-way and two-way ANOVA, respectively.

In a Multiple Variables table, each row represents one observation (one subject, sample, or experimental unit). Each column represents one variable. You need one column for your response variable (the continuous outcome you're measuring) and additional columns for your grouping variables (the categorical factors that define your treatment groups).

If your data are currently in a Column or Grouped table, you'll need to reorganize them into a Multiple Variables table before you can run multifactor ANOVA. See Entering data for Multifactor ANOVA for detailed guidance on setting up your table correctly.

Are all your factors categorical?

Multifactor ANOVA requires that all grouping variables (factors) are categorical. This means that they must define distinct groups or conditions rather than representing continuous measurements.

Appropriate factors include treatment groups (Control, Drug_A, Drug_B), genotypes (WT, Het, KO), locations (Site_A, Site_B, Site_C), or time points when you're treating them as distinct categories (0h, 6h, 12h, 24h). These all represent discrete, qualitative differences between groups.

Variables that are not appropriate for multifactor ANOVA include continuous predictors like age, weight, or dose when measured on a continuous scale, or time when you want to analyze it as a continuous progression rather than discrete categories. If you have continuous predictors, use multiple linear regression instead. Multiple regression can also handle a mix of categorical and continuous predictors, but uses a different coding method for categorical variables than ANOVA.

There's a special case worth mentioning: ordered categorical factors like dose levels (0, 10, 25, 50 mg) or time points. You can include these in ANOVA, but be aware that ANOVA treats them as unordered categories and ignores the numeric relationship between levels. If the ordering or spacing matters for your research question, consider using regression to model the predictor as continuous.

Is the response variable continuous?

Multifactor ANOVA requires a continuous response variable: an outcome measured on an interval or ratio scale where numerical differences are meaningful.

Appropriate response variables include physical measurements (height, weight, length), biochemical measurements (concentration, expression level), temporal measurements (time, rate, frequency), or really any numeric measurement where the intervals between values have consistent meaning.

ANOVA is not appropriate for categorical outcomes (diseased/healthy, positive/negative). These outcome variables require different tests such as Chi-square tests or logistic regression. Binary outcomes (yes/no, survived/died) also need logistic regression. Count data with many zeros may violate ANOVA assumptions and might need Poisson regression.

Fixed effects vs. random effects

Prism performs Type III ANOVA, which is a fixed-effects ANOVA. This tests for differences among the means of the specific groups (levels) you collected data from.

Fixed effects are appropriate when you deliberately selected specific groups to study (like specific drug doses or particular genotypes), you want to draw conclusions about these specific groups rather than a broader population, or your groups represent all the levels you care about (or at least the complete set of interest).

Random effects would be appropriate in a different scenario: if you randomly selected groups from a large population of possible groups and want to generalize your findings to all possible groups, not just the ones in your study. For example, if you randomly selected several clinics from all clinics in a region and want to make inferences about clinics in general, not just your specific sample, you'd want random effects.

Different calculations are needed for random effects models. Prism's multifactor ANOVA currently assumes all factors are fixed effects, but will be expanded in the future to support random effects as well.

Are you testing the right hypothesis?

Multifactor ANOVA tests whether group means differ, but this isn't the same as testing whether groups are identical in all respects, whether the effect is large enough to matter, whether one specific group is "best," or what mechanism causes the differences.

What ANOVA tells you: whether there's statistical evidence that your factors affect the outcome, which factors and interactions show significant effects, and how much variance is explained by each effect. What ANOVA doesn't directly tell you: which specific groups differ from each other (you need multiple comparisons for this), how large the differences are in practical terms (check effect sizes and confidence intervals), whether differences are biologically or clinically meaningful (this requires scientific judgment beyond statistics), or why groups differ (ANOVA detects patterns but doesn't explain mechanisms).

After finding significant effects in ANOVA, use multiple comparisons tests to identify which specific groups differ. Examine effect sizes to assess whether differences are large enough to care about. Think about the biological context to determine whether statistically significant differences are also scientifically meaningful.

Have you checked assumptions before interpreting results?

Before trusting your ANOVA results, check assumptions using the Residuals tab.

Create the three diagnostic plots: the residual plot (should show random scatter around zero), the homoscedasticity plot (should show no trend in spread), and the QQ plot (points should fall close to the diagonal line). Run the diagnostic tests if you want objective measures: Spearman's test for heteroscedasticity checks for unequal variance, and the four normality tests assess whether residuals follow a normal distribution.

Look for problems in these diagnostics: patterns in residuals (curved or funnel-shaped instead of random), outliers sitting far from the main cluster of points, non-normal distribution where the QQ plot deviates substantially from the diagonal line, or unequal spread across groups visible in the homoscedasticity plot.

If assumptions are violated, consider transforming your data (log or square root transformations often help), acknowledge the limitations and proceed with appropriate caution, or use alternative analysis methods better suited to your data structure.