Lecture 5: Factorial ANOVA
January 27, 2026Multifactorial Analysis of Variance (ANOVA)
Analysis of variance is one of the most used statistical frameworks. It can tackle from simple to extremely complex data and designs.
In multifactorial ANOVAs, we consider more than one factor (but usually not more than 3 factors due to the complexity of interpretation; even though mathematically it can be done). Based on the fictional example above, consider now that we are interested in studying how fish grow according to combinations of say temperature and food amount. Now levels (groups) will be made of combinations of temperature level (low, intermediate and high) and food (say low and high). You can use as many groups (levels) per factor, assuming you have enough observations (number of individual fish).
The ANOVA fictional example in this lecture are based on equal number of observations per combination of groups (levels or treatments; remember we use these terms interchangeably). In the fictional diet example, there were 5 individuals in each of the 4 combinations of diet (yes/no) and exercise (yes/no). In the gene expression study, there were 2 individuals in each of the 12 combinations of strain (2 strains) and brain region (6 regions). For balanced designs, we say that the design is fully orthogonal because there is no variation that is shared between factors (a concept we will see in a few lectures; under ANCOVA). For fully orthogonal designs, can use either Type I Sum-of-Squares (Type I SS) or III Sum-of-Squares (Type III SS); they give exactly the same results. When factors are not fully orthogonal, then we use the Type III SS (Sum-of-Squares). We will learn about Type III in the ANCOVA module; the concepts there will be applicable to multifactorial ANOVAs that are not balanced, i.e., non-orthogonal).
A whiteboard overview of two-way anovas: main effects and interactions Although interaction plots are covered in our lecture (as everything we watch in videos from others that I post), I find it important to understand these concepts from different perspectives as there are many ways to learn and better understand the same concept. Remember that our learning styles are diverse and different ways to understand the same concept is relevant.
Normal QQ plots to assess the normality assumption
This video is also covered in Tutorial 3 and provides a good explanation of QQ plots to assess whether we can assume normality (important assumption for ANOVA) for our observed data. QQ plots were covered in BIOL322 but if they haven’t been covered in your stats course or don’t remember them well, here is a refresher. They are widely used to assess normality based on data visualization.
Lecture
Slides will be posted here prior to the lecture