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The Block Anova allows you to study more than 2 paired samples.

The Block Anova, also called repeated measures or randomized blocks in English, is a special case of the Anova that can be applied to study:

More than 2 matched samples, where the test can only be 2
Samples of which a number of measurements have been made several times.

## The principle

In the same way as the Anova, we compare the average of the different groups of samples. The statement of the subject is identified as:

• n: The number of blocks, either the number of k measures, or the number of k groups of data matched. It is relative to the conditions we want to test.
• K: The number of measures. It is representative of the element we want to test.

Example:

• Case of matched data: one wishes to study the efficiency of an additive to reduce the consumption of cars. We carry out tests without and with additive and compare on 5 different car brands. We have thus n = 2 groups (before-after) with each K = 5 cars.
• Case of repeated measurements: we want to study the endurance of car tires. For this reason, K = 5 different tyre marks and the test on each tyre is carried out n = 10 times, consisting of rolling a car until the wear indicator is reached.

Since we are in a particular case of the Anova, the variance is broken down so as to highlight the one due to the K measurements. It is a Anova To two factors with K measures as the first factor and n blocks, the second factor. The principle is based on comparing the average of a multitude of groups with the general average. We calculate:

• Intergroup Variance K, SST: difference between the average of each K group and the general average.
• Variance Intragroup N, SSB: The difference between the value of each block and the general average.
• residual Variance, SSE: difference between the total variance and the sum of the intergroup variances (SST + SSB).
• Total Variance, TSS: difference between the value of each individual and the general average.

## Copie de Tableau ANOVA en blocs

Source of varianceSum of SquaresDOFMean SquaresFp-value
Intergroup (K measures)SSTK - 1MST$\frac{\mathbf{CME}}{\mathbf{CMR}}$p-value
Intragroup (n blocks)SSBn - 1MSB
ErrorSSE(n - 1) * (K - 1)MSE
TotalTSSn * K - 1

## Step 1: Assumptions

The purpose of the test is strictly the same as for the one-way Anova. The initial hypothesis is:

H0 : μ1 = μ2 = … = μk

H 1 : At least 2 averages are different

## Step 2: Calculate the sum of squares of the treatment-SST

The difference between the general average and the average of each of the K groups is calculated. In other words, at the most this value will be greater than the average of each of the K groups differ from each other.

• n: Number of measurement blocks
• μkgroup : Average of each of the groups of K measures
• General μ: average of all measures

## Step 3: Calculate the sum of the squares of the blocks – SSB

The Variance explained by the blocks is calculated. Thus, at the most this one will be large, the more the variance brought by the K measures will be low and therefore at least our samples will be different.

• K: Number of measures per block
• μnblocs : Average of each of the blocks
• General μ: Average of all measures

## Step 4: Calculate Total square sum – TSS

The sum of the total squares represents the difference of each of our measures vis-a-vis the general average.

## Step 5: Deduct the sum of squares from residues – SSE

The residuals are inferred by calculating the difference between the total variance and the group variances. In other words, it is the variability that is not known explained by the variance of the n blocks or the K measurements.

SSE = TSS – SST – SSB

## Step 6: Calculate the number of degrees of freedom

The degrees of freedom are representative of the level of knowledge we can draw from our test. In our case, we get the following elements:

• Nb of dof Intra: dofSST = K-1
• Number of dof Intergroup: dofSSB = N – 1
• Residual dof :dofSSE = (n – 1) * (K – 1)
• Total dof : dof TSS = n * K-1

## Step 7: Calculate the average squares

The average squares represent the “weight” that can be given to the different values of variances. They are calculated by making the connection with the dof.

• MST = SST / dofSST
• MSB = SSB/ dofSSB

## Step 8: Practical value

The test statistic represents the ratio between the variance explained by our K measurements and the residual variance. It is calculated in the following way:

F = MST/MSB

## Step 9: Calculating the critical value

The test statistic follows a Fisher’s law at (K-1, (n-1) * (K-1)) degrees of freedom. You choose the desired risk value, typically 5%, and then determine the critical value using Excel using the following formula:

F. INV (1- α; dofSST ; dofSSE)

## Step 10: Calculate the P-Value

To validate the significance of the test, we calculate the p-value which also follows a Fisher’s law. For this Excel one uses the following formula:

p-value = F.DIST (practical value; dofSST ; dofSSE )

## Step 11: Interpretation

Test directionResultStatistical conclusionPractical conclusion
BilateralPractical value ≥ Critical valueWe reject H0The samples have averages that differ at the given level of risk α.
Practical value < Critical valueWe retain H0The samples have averages that do not differ at the given level of risk α.
ResultStatistical conclusionPractical conclusion
p-value > αWe retain H0Our data series are identical or close with a risk of being wrong with p-value%
p-Value < αWe reject H0Our data series are statistically different with a risk of being wrong with p-value%

## Source

D. R. Shupe (1995) – Inferential statistics: An Introduction to the analysis of variance

D. C. Howell (2001)-Statistical Methods for psychology

G. W. Cobb (1998) – Introduction to design and analysis of experiments

R. Rafiq (2013) – population comparison, parametric tests