# Practical approach for analysis of variance between groups - ANOVA

I have three independent variables

1. Continent (2 levels)
2. Region (4 levels)
3. Animal (2 levels)

and three dependent variables

1. weight (Kg)
2. size (m3)

a possible table could be like this one:

I would like to test the hypothesis that the two animals in the different continents (and then in the different regions) have similar weights and similar sizes. Basically, my theory is that American animals 1 and 2 are different from their European equivalents. However, I also suspect that within America (or EU), the animals are different depending on the regions (North, East, South and West).

Data have been checked for normality, so then I thought to use one of the analyses of variance available. My options are:

• use twice the 2 ways ANOVA, considering the three independent variables and one dependent variable per time. However, I think my data do not respect the independence of observations condition, but I am not sure I completely understood the meaning of this requirement.
• Because of the point above, I thought to use a Two-way Repeated Measures ANOVA.

However, I am not even sure that using ANOVA is the best statical approach to my case.

May you suggest a good approach?

Your data is completely nested. There are two Continents (A/E), each containing four Regions (N/E/S/W). It's crucial to note that the North in A is NOT the same as the North in E. In addition, within each region, there are two animal species. Again, the effects of species are NOT the same cross regions. This is a three-level nested model, which is very complicated in general.

Fortunately, your model has no random effects. Based on your descriptions, you may simplify the model by considering 16 distinct effects of species (Continents $$*$$ Regions $$*$$ Animals = $$2*4*2=16$$), and attribute all randomness to the error term. Therefore, you carry out one-way ANOVA twice for Weight and Size, separately. Let $$Y_{ij}$$ denote the Weight of the $$j$$th unit of the $$i$$th "species". Formally, we write $$Y_{ij}=\mu_i+e_{ij},\qquad e_{ij}\overset{iid}{\sim}N(0,\sigma^2_i),\qquad \begin{cases}i=1,...,16\\j=1,...,n_{i}\end{cases}.$$ About $$\mu_i$$'s, see the end of this answer. You may test for homogeneity first and replace all $$\sigma_i^2$$ by $$\sigma^2$$ if the data supports equality of variances. Then the questions of interest can be translated to hypotheses.

1. "American animals 1 and 2 are different from their European equivalents": $$H_0\colon \mu_1+\mu_3+\mu_5+\mu_7=\mu_9+\mu_{11}+\mu_{13}+\mu_{15}\quad \text{versus}\quad H_1\colon \ne\\ H_0\colon \mu_2+\mu_4+\mu_6+\mu_8= \mu_{10}+\mu_{12}+\mu_{14}+\mu_{16}\quad \text{versus}\quad H_1\colon \ne$$
2. "within America (or EU), the animals are different depending on the regions (North, East, South and West)": $$H_0\colon \mu_1+\mu_2=\mu_3+\mu_4=\mu_5+\mu_6=\mu_7+\mu_8 \quad \text{versus}\quad H_1\colon \ \text{not all equal}\\ H_0\colon \mu_9+\mu_{10}=\mu_{11}+\mu_{12}=\mu_{13}+ \mu_{14}=\mu_{15}+\mu_{16}\quad \text{versus}\quad H_1\colon \ \text{not all equal}$$

In total, the hypotheses impose 8 restrictions (8 equal signs) on the parameters, and I have checked that they are linearly independent (not conflicting with each other). So you may carry out an F test for them simultaneously.

If not all effects are observed, then it's possible some hypotheses are not testable because of a lack of data.

Effects Continents Regions Animals
$$\mu_1$$ America North A1
$$\mu_2$$ America North A2
$$\mu_3$$ America East A1
$$\mu_4$$ America East A2
$$\mu_5$$ America South A1
$$\mu_6$$ America South A2
$$\mu_7$$ America West A1
$$\mu_8$$ America West A2
$$\mu_9$$ EU North A1
$$\mu_{10}$$ EU North A2
$$\mu_{11}$$ EU East A1
$$\mu_{12}$$ EU East A2
$$\mu_{13}$$ EU South A1
$$\mu_{14}$$ EU South A2
$$\mu_{15}$$ EU West A1
$$\mu_{16}$$ EU West A2
• This is a very good answer! I really appreciate it. I have a further question: Can I use the same approach in case the regions within Countries are different? for example, America contains (2 levels) "North America" and "South America", and Europe contains (3 levels) like "Scandinavia", "Central E", and "Mediterranean"? Feb 18 at 6:17
• @Strobila Thanks! I’m glad you found it helpful. "Can I use the same approach in case the regions within Countries are different?" Yes, you can; the numbers are irrelevant. This approach's ground is that every level (Continents/Regions) above individuals/units are completed nested and fixed.
– Min
Feb 18 at 8:08
• @Strobila To understand fixedness, let's consider its opposite: randomness. If within each continent, there are thousands of regions/habitats, and you randomly pick some small number of regions as representatives of the underlying population. Then not all randomness goes into the pure error: you have sampling errors. With data in hand, whether or not to treat effects as random depends on your research questions. Are you interested in only those regions, or are you trying to use them as a sample to infer a much larger population?
– Min
Feb 18 at 8:13
• thank you again for the brilliant explanations. "Are you interested in only those regions, or are you trying to use them as a sample to infer a much larger population?" I am interested in those particular Countries and in those particular regions. Feb 18 at 8:22
• @Strobila That's a good question. A nested ANOVA can test "on average, whether animals in regions of America different from those in EU". That is, it aggregates the micro-level (animal 1 and 2) data to the macro level. It's called the aggregation method. If you are only interested in macro-level propositions, then nested ANOVA is easiest. But you are going to compare animal 1 across continents, rather than animal 1 and 2 combined. At least for the basic nested ANOVA, it's impossible to distinguish animals 1 and 2 at the continent level.
– Min
Feb 18 at 8:43