Multi-factor Anova: degrees of freedom

Source: http://blog.sina.com.cn/s/blog_60a751620101gkja.html

In brain science, research is usually done by means of repeated measurements. That is to design different tasks, let the same subjects do it again, to find out the difference. This is not only to save money (think if 4 conditions, each condition to find 20 people, you have to give 80 people the trial fee!) On the other hand, the individual differences are also controlled as much as possible, because both EEG/ERP and fMRI are signals of noisy enrichment, and individual differences are very large.

Figure 1 points to the decomposition of the test room and the subject in the test.

Of course, repeated measurement also has its drawbacks, such as the retention effect, latent effect and learning effect, here does not unfold. The repeated measurement of variance analysis can be said to be the most commonly used tool. For example: to study different emotional images, in the recall and re-recognition of the task at each electrode theta wave difference. That is an emotion (happy, sad and Neutral) x task (recall, re-recognition) x electrode (4 electrodes) experiment, that is, 3 factor variance analysis. If the subjects were divided into two groups, complete the recall and re-recognition task, which is more complicated, and became a hybrid design of the 3-factor variance analysis. Sadly, flipping through a lot of textbooks focuses on single-factor Anova, or a completely randomized design of variance analysis. For our common repeated measurement multivariate analysis of variance, to maintain the collective silence. Read a lot of research reports, found that students are writing f (x, y) =???, p=???, usually the degree of freedom and x-y write wrong. Here I summarize the commonly used experimental design how to determine the degree of freedom, hope that everyone's practice is helpful.

(Note that this handout does not include factors that do not interact with each other, nor do they include situations where freedom is to be corrected!) )

Decomposition tips:
Variance Analysis Multi-factor
First look at the design of the idea
Repetition or not is the key
total degrees of freedom for projects

Count the subjects first.
Determining the degree of freedom of the subjects

Mix design Be careful
Decomposition of the inter-group factors

There is freedom in the subjects
The total minus test room can be drawn
Set the factors in the test
The error is naturally not mistaken

Core idea:
Gross position degrees of freedom decomposition into the subjects (between subject) degrees of freedom and in-test (within subject) degrees of freedom

1. Repetitive Measurement single factor variance analysis
Number of subjects: N Group internal factors: a
Total degrees of freedom: an-1
Degree of freedom of the subjects: n-1
Internal degrees of freedom of the subjects: N (A-1)
* Factor: A-1
* Factor X Test Room: (n-1) (A-1)
Result of the build:
F (A-1, (N-1) (A-1))

2. Repeat Measurement 2 factor variance Analysis
Number of subjects: N Group internal factors: a, B
Total degrees of freedom: abn-1
Degree of freedom of the subjects: n-1
Internal degrees of freedom of the subjects: N (ab-1)
* Factor A:a-1
* Factor b:b-1
* Interaction: (A-1) (b-1)
* Factor Ax was tried: (n-1) (A-1)

* Factor BX Test room: (n-1) (A-1)

* Factor ax Factor BX Test room: (n-1) (A-1)
Result of the build:
F (A-1, (N-1) (A-1))
F (B-1, (n-1) (b-1))

F ((A-1) (B-1), (n-1) (A-1) (b-1))

3. 2-factor variance analysis of mixed designs (between existing and intra-group)
Number of subjects: N inter-group factors: a group of factors: B
Total: abn-1
trial Room: an-1
* inter-group factor: A-1
* Subjects: A (n-1)
Under test: An (b-1)
* Intra-group factor: B-1
* Intra-group x group: (A-1) (b-1)
* ERROR: A (b-1) (n-1)
result of the build:
F (A-1,a (n-1))
F (B-1,a (b-1) (n-1))

4. repeated measurement of 3 factor variance analysis , which is very common in brain science research!
Number of subjects: N Group factor: A,b,c
Total: abcn-1
trial Room: n-1
in the test: N (abc-1)
* Factor A:a-1
* Factor b:b-1
* Factor c:c-1
* Interactive AB: (A-1) (b-1)
* Interactive BC: (b-1) (c-1)
* Interactive AC: (A-1) (c-1)
* Interactive ABC: (A-1) (b-1) (c-1)
* Factor Ax was tried: (A-1) (n-1)
* Factors ABX The Trial Room: (A-1) (b-1) (n-1)
* Factors abcx The Trial Room: (A-1) (b-1) (c-1) (n-1)

Result of the build:
F (A-1, (A-1) (n-1))
F (B-1, (b-1) (n-1))

F ((A-1) (B-1), (A-1) (b-1) (n-1))
F ((A-1) (b-1) (c-1), (A-1) (b-1) (c-1) (n-1))

5. 3-factor variance analysis of mixed designs (between existing and intra-group)
If the study effect of a factor processing is very strong, can not adopt the program 4, usually adopts the scheme 5来 avoids the learning effect, therefore, this program is also very common in the brain science research!
Number of subjects: N inter-group factors: a group of factors: B,c
Total: abcn-1
trial Room: an-1
* Inter-group factor A:a-1
* Subjects: A (n-1)
internal degrees of freedom of the subjects: an (bc-1)
* Intra-group factor b:b-1
* Intra-group factor c:c-1
* Interactive AB: (A-1) (b-1)
* Interactive BC: (b-1) (c-1)
* Interactive AC: (A-1) (c-1)
* Interactive ABC: (A-1) (b-1) (c-1)
* ERROR: A (bc-1) (n-1)
result of the build:
F (A-1,a (n-1))
F (B-1,a (bc-1) (n-1))

6. attention and the difference between completely randomized designs

examples are as follows:
number of subjects: ABN inter-group factor: A, B
if the ABN is randomly divided into a A-b two factors, it constitutes a completely random design of the 2-factor square difference analysis.
The decomposition sequence and the repeated measurement are also different, and there is no test and the test within this step, it becomes the direct processing and decomposition within the processing.
Total degrees of freedom: abn-1
processing room: ab-1
* Factor A:a-1
* Factor b:b-1
* Interactive AB: (A-1) (b-1)
* In Process: AB (n-1)
result of the build:
F (A-1,ab (n-1))
F (B-1,ab (n-1))

7.If the ABCN is randomly divided into a,b,c three factors, it constitutes a completely randomized design of 3-factor variance analysis .
direct processing and in-process decomposition.
Total: abcn-1
processing room: abc-1
* Factor A:a-1
* Factor b:b-1
* Factor c:c-1
* Interactive AB: (A-1) (b-1)
* Interactive BC: (b-1) (c-1)
* Interactive AC: (A-1) (c-1)
* Interactive ABC: (A-1) (b-1) (c-1)
* In Process: ABC (N-1)
result of the build:
F (A-1,ABC (n-1))
F (B-1,ABC (n-1))

Given the above degrees of freedom, we have to prove a point: "The same experimental factors and the amount of each treatment under the test, completely random than the mixing design of high degrees of freedom, mixed design than repeated measurement of freedom high." Take 3 factor variance analysis as an example:

Completely random: A and B

F (A-1,ABC (n-1))
F (B-1,ABC (n-1))

Hybrid Design: Inter-Group A and Group B

F (A-1,a (n-1))
F (B-1,a (bc-1) (n-1))

Repeat measurements: A and B

F (A-1, (A-1) (n-1))
F (B-1, (b-1) (n-1))

As you can see, for a factor: ABC (N-1) >a (n-1) > (A-1) (n-1);

For B Factor: ABC (N-1) >a (bc-1) (n-1) > (b-1) (n-1);

So the above view is set up. It can also be seen that for mixed designs: A (n-1) <=a (bc-1) (n-1), which is, in the case of a=b, the conditions in the group are greater than those between groups.

here is the MATLAB command to do Anova , note that the MATLAB software comes with the command only to provide a completely random design variance analysis , to do the repeated measurement of variance analysis needs to mathworks official website download

1. Fully randomized design single-factor variance analysis

Anova1

2. Fully randomized design two factor variance analysis

Anova2

3. Fully randomized design multivariate variance analysis

Anovan

4. Repetitive measurement single factor variance analysis

Anova_rm

5. Repeat measurement two factor variance analysis

Rm_anova2

6. Repeated measurement of three-factor variance analysis

RMAOV33

Click the command to download directly

The following is a general roadmap for statistical analysis of experimental data

Multi-factor Anova: degrees of freedom