ANOVA

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ANOVA stands for Analysis of Variance, and is a procedure that compares the variability within the given groups of data to the variability between the given groups of data. The main purpose of ANOVA is to check if there is a significant change in the mean between data sets.

Contents

Introduction

The phrase treatment is used in reference to an individual group or data set.

The response for each of the treatments is the random variable being assessed (X), or the data points of the set.

For a given set of treatments and responses:

ANOVA Table.PNG

The total observations is n = (1 + 2 + ... + k)

The Anova model is the sum of the mean response for the ith treatment μi and the individual error component ϵij.

Xij= μi + ϵij

Both μi and ϵij are normally distributed. However, X has normal distribution with mean μi and deviation σ, and ϵ has mean 0.

Testing a Hypothesis with ANOVA

The only hypothesis test that we assess is the one which looks to prove that at least two of the means in a set of means are different. Before we can do this, we have to fill in the data in the ANOVA table below.

ANOVA Table

The ANOVA table is useful to find the F-statistic as it simplifies the process of data gathering and calculation.

ANOVA Table 2.PNG

A = k – 1 (where k is the number of groups/treatments)
B = n – k (where n is the number of observations)
H = A + B
E = C / A
F = D / B
G = E/F

C, D and I are all formulas discussed below.

Total Sum of Squares (Total amount of variation in a global sample)

SStot.PNG

Treatment Sum of Squares (The variation between the group’s means)

SStr.PNG

Error Sum of Squares (The variation within the group)

SSer.PNG

The mean squared values are both unbiased estimators of the σ2 value, however the MSTr value will only be unbiased when all of the means are equal.

MStr.PNG

MSer.PNG

These values culminate with the calculation of the Fishers F-distribution value (F).

F = (MSTr)/(MSEr)

This value follows a distribution F ~ F(k-1,n-k).

Hypothesis Test

The null hypothesis (H0) is that there is no change in the means for the treatments.

H0: μ1 = μ2 = μ3 = ⋯

The alternative hypothesis is that they are not all equal.

Ha: Not all the means are equal.

To test whether the means are all equal we check if the F-distribution for the observed values is greater than the F-distribution value from the F-distribution chart.

F0 > f(dfTr,dfEr;1-α)

This is the destination of ANOVA analysis, as we compare the F value for the set of data to determine if the hypothesis that all the means are equal is true or not.

We find that not all of the averages are equal if this inequality is true (that is, if the f value found through looking at the observations is greater than the f value from the table, we reject the hypothesis that all means are equal).

ANOVA assumptions

For the given hypothesis to be valid, the central assumptions are:

  • Each group the data is normally distributed
  • The standard deviation is the same for all distributions
  • The groups sample spaces are all independent and random

The assumption of normality and constant variance can be checked by plotting the residuals and finding a random distribution, not one which follows any pattern.

End

This is the end of this topic. Click here to go back to the main subject page for Numerical Methods & Statistics.

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