How does the F-statistic work to support comparison of group means in an ANOVA?In the sum-of-squares perspective, what role does the MS-between and MS-within play in comparing group means?Begin by...

How does the F-statistic work to support comparison of group means in an ANOVA?In the sum-of-squares perspective, what role does the MS-between and MS-within play in comparing group means?Begin by writing the F-statistic using these two MS terms.After discussing this approach, write the F-statistic as it would be defined for model comparisons.How does this work.In this equation, define the part of the equation that applies to model fit, and the part that applies to model complexity.Which version of the F-test makes more sense to you?Why?Can you think of why the model comparison F-test conceptualization was developed later than the sum-of-squares conceptualization F-test?

Solution

David answered on Dec 20 2021
Analysis of Variance (ANOVA) is a systematic procedure of splitting the total variance present
in a set of observations, as measured by arithmetic mean of the squares of deviations of the
observations from their arithmetic mean, into components associated with defined sources of
variations according to the nature of classification of data.
With the help of this procedure, it will be possible for us to perform certain test of hypothesis.
Suppose, there are ‘g’ no. of groups to be compared. The means of the response variable for
these groups be µ1, µ2,…,µg. The sample means are y1-bar, y2-bar,…,yg-bar.
The Analysis of Variance (ANOVA) is a F-test of H: µ1 = µ2 =… = µg vs. K: At least two of the
population means are unequal
If H is rejected it implies that either all the population means differ or some differs or only two
of them differ.
The assumptions of the test are as follows:
1. The population distribution of the response variable is normally distributed for all the
groups
2. The standard deviation for the population distribution of the response variable is the same
for each group (denote the common Standard Deviation by σ)
3. The samples from each group are independent random samples.
In ANOVA the F-statistic is defined as:
F = Between groups estimate of variance/Within groups estimate of variance
Between groups estimate of variance uses the variability between each sample mean yi-bar and
the overall sample mean y-bar. The within groups estimate of variance uses the variability within
each group of the sample observations about their separate means – the observations from the
first group from y1 –bar, the observations from the second group from y2 –bar and so on.
Therefore, the F-statistics compares the between group variability to the within group variability.
We accept H if the between group variability is insignificant with respect to within group
variability.
Now, Between groups estimate of variance = ∑ni(yi-bar – y-bar)
2
(g-1) = SSB/g-1 = MSB
MSB is the Mean between sum of squares
Here ni = sample size for the i-th group, i = 1,2,…,g
Degrees of freedom = g – 1 = number of groups – 1
Within groups estimate of variance = ∑∑(yij – yi-bar)
2
(n-g) = SSW/n-g = MSW
MSW is the mean within sum of squares
Here, n = ∑ni
yij = j-th observation of the i-th...
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