# What is the minimum sample size for factorial anova?

#### Nina_joon

##### New Member
Hi,
I have 10 people participated in my study, I believe the parameters of our study qualify it for factorial anova. I would like to know minimum accepted sample size for this test, please.

#### j58

##### Active Member
Theoretically, 10 subjects is enough to perform a 2x2 factorial ANOVA, but you won't have much statistical power. Also, you can't evenly allocate 10 subjects to four treatment groups, so you can't validly use classical sums of squares to analyze your data. You'll have to use a regression, or linear model, formulation of ANOVA for your analysis.

#### GretaGarbo

##### Human
The minimum sample size is 2. You can have one factor with 2 levels (1 and 0). You can't do significance test but you can estimate effect. This is the absolutely most common design globally.

Next is you can have 3 factors in four runs like:
Code:
+1 +1 +1

-1 +1 -1

+1 -1 -1

-1 -1 +1
You can not do significance tests but you can estimate the effects. Skip one factor and only estimate main effects and you can do significance tests.

And you can have 7 factors in 8 runs. (But I am to lazy to write down the design).
If all variables are quantitative you can put the two extra in the center and use that as estimates of error and do significance tests.

And, j58 does not know your standard deviation or your effect size. If the std is 0.00001 and the effect is 100000, then your power will be pretty good.

#### Nina_joon

##### New Member
Theoretically, 10 subjects is enough to perform a 2x2 factorial ANOVA, but you won't have much statistical power. Also, you can't evenly allocate 10 subjects to four treatment groups, so you can't validly use classical sums of squares to analyze your data. You'll have to use a regression, or linear model, formulation of ANOVA for your analysis.
I have 3 groups separated based on their background: in 2 of them, I have 3 participant and in the last one, I have 4 participants. Does this mean that I cannot use factorial anova?

#### j58

##### Active Member
By definition, in a factorial ANOVA, you have at least 2 factors, with at least 2 levels each, and you randomize subjects to every combination of the levels of the factors. So, if factor A has levels a1 and a2, and factor B has levels b1 and b2, then in a factorial ANOVA you have subjects randomized to four groups: a1/b1, a1/b2, a2/b1, a2/b2. If you have more than 2 groups or groups with more than 2 levels, then there will be more than 4 groups, and subjects will be assigned to all of them. Does this describe your experiment? If so, what are the factors, and how many levels does each factor have? It sounds from your last post, that you have only one factor, "background," which has three levels. If that is the case, factorial ANOVA is inapplicable; but 1-way ANOVA might be appropriate. Based on your descrition, your design is not balanced, but that does not matter in 1-way ANOVA.

#### Nina_joon

##### New Member
By definition, in a factorial ANOVA, you have at least 2 factors, with at least 2 levels each, and you randomize subjects to every combination of the levels of the factors. So, if factor A has levels a1 and a2, and factor B has levels b1 and b2, then in a factorial ANOVA you have subjects randomized to four groups: a1/b1, a1/b2, a2/b1, a2/b2. If you have more than 2 groups or groups with more than 2 levels, then there will be more than 4 groups, and subjects will be assigned to all of them. Does this describe your experiment? If so, what are the factors, and how many levels does each factor have? It sounds from your last post, that you have only one factor, "background," which has three levels. If that is the case, factorial ANOVA is inapplicable; but 1-way ANOVA might be appropriate. Based on your descrition, your design is not balanced, but that does not matter in 1-way ANOVA.
Thanks.
Here is the detailed description of my study:
1- I have 3 groups with different background
2- Each group participated in 3 different phases to annotate some documents
Now, we want to know whether any of these groups performed better than the other in any of phases. we want to know if there is any difference based on the background as well as phases.

So, based on your description we cannot use factorial anova?

#### j58

##### Active Member
If any participant in the study participated in more than one "phase," then your experiment does not have a factorial design, and factorial ANOVA is not applicable. You have more of a repeated-measures design. My default method to analyze such designs, because of its flexibility, is the (generalized) linear mixed model, but there are other approaches that may be appropriate, such as MANOVA.

#### Nina_joon

##### New Member
If any participant in the study participated in more than one "phase," then your experiment does not have a factorial design, and factorial ANOVA is not applicable. You have more of a repeated-measures design. My default method to analyze such designs, because of its flexibility, is the (generalized) linear mixed model, but there are other approaches that may be appropriate, such as MANOVA.
Thanks.

#### GretaGarbo

##### Human
By definition, in a factorial ANOVA, you have at least 2 factors, with at least 2 levels each, and you randomize subjects to every combination of the levels of the factors. So, if factor A has levels a1 and a2, and factor B has levels b1 and b2, then in a factorial ANOVA you have subjects randomized to four groups: a1/b1, a1/b2, a2/b1, a2/b2. If you have more than 2 groups or groups with more than 2 levels, then there will be more than 4 groups, and subjects will be assigned to all of them.
J58, what I was referring to when talking about 3 factors in 4 runs is called a "fractional factorial". So you don't have to run all permutations. It is common for example to run a 2^(8-3) that is 8 factors in 32 runs.

The OP asked for the minimum, not what was resonable.

#### j58

##### Active Member
@GretaGarbo - since the OP stated the number of subject she had, I interpreted her question to mean "is this enough subjects," rather than literally asking about the theoretical minimum.

#### Nina_joon

##### New Member
If any participant in the study participated in more than one "phase," then your experiment does not have a factorial design, and factorial ANOVA is not applicable. You have more of a repeated-measures design. My default method to analyze such designs, because of its flexibility, is the (generalized) linear mixed model, but there are other approaches that may be appropriate, such as MANOVA.
Hi @j58.