# Lecture 4: Estimators and Factorial ANOVA

January 19, 2023 (2nd week of classes)**Properties of estimators and Multifactorial Analysis of Variance (ANOVA)**

In the first part of this lecture, we will review concepts related to property of estimators and the role of degrees of freedom in statistical analyses. These concepts are critical to statistical analyses and not well understood by practitioners.

In the second part of the lecture, we will quick review ANOVA and then introduce a more complex designs based on multifactorial ANOVA.

Analysis of variance is one of the most used statistical frameworks. It can tackle from simple to extremely complex data and designs. In this lecture we will quickly review the simplest form of ANOVA design and introduce more complex designs based on multiple factors.

In simple ANOVAs (one-factorial OR one-way ANOVA; these terms mean the same thing) we have one factor (say temperature) and one response variable say (fish growth). We measure fish growth according to say 3 groups (levels) of temperatures; say low, intermediate and high.

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).

We start the lecture by covering the principles of unbiased estimators (mean and variance), which are key to statistics, including ANOVA and most of the techniques covered in this course.

**ANOVA - the theoretical basis, by S. Taylor**

It provides a good visual aid to better understand simple ANOVA.

**Watch for your own interest; no need if you don’t want to**

**One-Way ANOVA (Analysis of Variance): Introduction**

It provides a quick summary of the main principles involving ANOVA.

**Watch for your own interest; no need if you don’t want to**

**Lecture**