Analysis of variance (ANOVA) is a statistical test used to analyze the differences between group means. The name comes from the way ANOVA partitions the observed variance into different parts to find out what the source of the variance is. ANOVA has several variations, including one-way, two-way, and three-way ANOVA.

## One-Way ANOVA

One-way ANOVA is used to compare the means of two or more independent groups. For example, to test if different dosage levels of a drug have different effects on systolic blood pressure, researchers could divide participants into 3 groups, give each group a different drug dosage, and then compare the mean systolic blood pressure between groups. One-way ANOVA tests if there are any statistically significant differences between the means of the groups.

## Two-Way ANOVA

Two-way ANOVA is used when there are two independent variables, and the goal is to test the interaction between them on the dependent variable. For example, researchers may want to test both drug dosage (variable 1) and patient weight (variable 2) on systolic blood pressure (dependent variable). Two-way ANOVA not only tests for differences between groups for each independent variable, it also tells you if there is an interaction effect between the two independent variables on the dependent variable.

## Three-Way ANOVA

Three-way ANOVA, sometimes called factorial ANOVA, is an extension of two-way ANOVA with an additional independent variable added. The same logic applies – the interactions between the 3 independent variables are statistically tested to see if there is an interaction effect on the dependent variable.

For example, a researcher may want to test drug dosage (variable 1), patient weight (variable 2), and patient gender (variable 3) on systolic blood pressure (dependent variable). Three-way ANOVA will tell the researcher if there are any statistically significant interactions between the 3 independent variables on systolic blood pressure.

## The Formal Name for 3 Way ANOVA

The formal name for 3 way ANOVA is factorial ANOVA. The term “factorial” comes from the fact that multiple factors, or independent variables, are included in the analysis. So a 3 way ANOVA may also be called a 3 factor ANOVA or a factorial ANOVA with 3 factors. But in casual conversation, researchers will often simply refer to it as a 3 way ANOVA.

## Assumptions of 3 Way ANOVA

There are several assumptions that must be met to reliably use a 3 way ANOVA:

- The dependent variable should be measured on a continuous scale (e.g. systolic blood pressure)
- The independent variables should consist of 2 or more categorical groups (e.g. drug dosage, weight categories, gender)
- There should be independence of observations between groups
- There should be no significant outliers
- The dependent variable should be approximately normally distributed in each group
- There should be homogeneity of variances between groups

## Running a 3 Way ANOVA

To run a 3 way ANOVA, the following steps are taken:

- State the null and alternative hypotheses
- Null hypothesis: There are no significant interaction effects between the independent variables on the dependent variable
- Alternative hypothesis: There is a significant interaction effect between the independent variables on the dependent variable

- Calculate the F ratio, which represents the variance between groups divided by the variance within groups
- Find the p-value associated with the F ratio based on the degrees of freedom
- If the p-value is less than the significance level (usually 0.05), reject the null hypothesis and conclude there is a significant interaction effect
- Use post-hoc testing to identify where the significant differences lie if the overall F test is significant

## Interpreting the Results of 3 Way ANOVA

Interpreting a 3 way ANOVA involves examining the main effects of each independent variable and any interaction effects between variables. Here are some key things to look for:

- A main effect means one of the independent variables has a significant effect on the dependent variable, ignoring the effects of other independent variables.
- An interaction effect occurs when the effect of one independent variable depends on the level of another. For example, the effect of drug dosage may depend on the weight category.
- Higher order interactions are possible with 3 or more variables. For example, there may be an interaction between drug dosage, weight, and gender.
- If a higher order interaction is significant, main effects and lower order interactions are not interpretable on their own.
- Post-hoc testing clarifies which groups or interactions differ if the overall F test indicates significance.

## Reporting 3 Way ANOVA Results

The results section of a paper reporting a 3 way ANOVA should include:

- A statement of the independent and dependent variables
- The sample size in each group
- The F ratio, degrees of freedom, and significance level
- An indication of which main effects and interactions were statistically significant
- Relevant means and standard deviations for each group
- Clarification from post-hoc testing of where differences lie, if applicable
- Comments on effect sizes and observed power, if sample size is small

For example: A 3 way ANOVA was conducted to examine the effect of drug dosage, weight category, and gender on systolic blood pressure in 240 patients. There was a significant interaction effect between drug dosage and weight category, F(4, 216) = 3.14, p = .015, but no other interactions were statistically significant. Post-hoc Tukey tests revealed…

## Advantages of 3 Way ANOVA

Some advantages of using a 3 way ANOVA include:

- It allows you to test more complex experimental designs with 3 independent variables
- You can examine interaction effects between variables
- It controls for Type 1 error inflation that would occur by running multiple two-way ANOVAs
- As a parametric test, it is more powerful than equivalent non-parametric tests

## Disadvantages of 3 Way ANOVA

Some limitations of the 3 way ANOVA include:

- It has more stringent assumptions that may be violated, such as normality and homogeneity of variance
- Higher order designs require larger sample sizes
- Significant results may be difficult to interpret if there are multiple interaction effects
- It cannot determine cause-and-effect relationships between variables

## Alternatives to 3 Way ANOVA

Some alternatives to the 3 way ANOVA include:

- Non-parametric tests like the Friedman test if assumptions are severely violated
- Robust ANOVA methods that are less sensitive to assumption violations
- Multiple regression analysis for exploring predictive relationships between continuous variables
- Loglinear analysis for categorical dependent variables
- Structural equation modeling for modeling complex variable relationships

## Example of 3 Way ANOVA

Here is an example of running and interpreting a 3 way ANOVA:

Researchers want to test a new drug used to treat high blood pressure. They recruit 180 patients and divide them into groups based on drug dosage (0mg, 10mg, 20mg daily), weight category (normal, overweight, obese), and gender (male, female). The dependent variable is systolic blood pressure measured after 8 weeks of treatment.

The results of the 3 way ANOVA indicate there was a significant main effect of drug dosage, F(2, 162) = 28.32, p

This indicates that:

- Overall, higher drug dosages were associated with lower systolic blood pressure
- The effect of drug dosage on reducing systolic blood pressure depended on the weight category of the patient
- Gender did not have a significant effect on treatment results

Follow up Tukey tests showed that obese patients had a greater reduction in systolic blood pressure on the higher 20mg dose compared to normal weight patients. This suggests that optimal drug dosage depends on patient weight.

## Conclusion

In summary, a 3 way ANOVA, also called factorial ANOVA, is used to test the interaction effects between 3 independent variables on a continuous dependent variable. It offers advantages over multiple two-way ANOVA tests by controlling for inflated error rates. Interpreting significant interactions requires careful post-hoc testing. While a powerful technique, the 3 way ANOVA does have assumptions and limitations to be aware of when drawing conclusions.