Question

Does this describe the difference between Between-Treatments (Groups) and Within-Treatments (Groups) variation? Why? or Why not?

The concept of a between-treatments estimate of Ơ2 and showed how to compute it when the sample sizes were equal. This estimate of Ơ2 is called the mean square due to treatments and is denoted MSTR. The numerator is called the sum of squares due to treatments and is denoted SSTR. The denominator, k – 1, represents the degrees of freedom associated with SSTR. The mean square due to treatments can be computed using formula:

MSTR = SSTR / k -1

The concept of a within-treatments estimate of Ơ2 and showed how to compute it when the sample sizes were equal. This estimate of Ơ2 is called the mean square due to error and is denoted MSE. The numerator is called the sum of squares due to error and is denoted SSE. The denominator of MSE is referred to as the degrees of freedom associated with SSE. The formula for MSE can be:

Answer 1

BETWEEN-TREATMENTS(GROUPS) AND WITHIN-TREATMENTS VARIATION :

Between group variation is used in ANOVA (analysis of variance) to measure variation between separate groups of interest. Unlike within group variation, where the focus is on the differences between a population and its mean, between group variation is concerned with finding how the means of groups differ from each other. For example, let’s say you had four groups, representing drugs A B C D, with each group composed of 20 people in each group and you’re measuring people’s cholesterol levels. For between-group variation, you’ll look at variances in cholesterol levels for people in group A, group B, group C and group D simultaneously.

The formula for between-group variation is:

and is called the sum of squares between groups, or SS(B). This measures the interaction between the groups or samples. If the group means   don’t differ greatly from each other and the grand mean , the SS(B) will be small.

Note that for k groups, there will be k-1 degrees of freedom. The between groups variance is the variation, or SS(B), divided by its degree of freedom. We sometimes refer to the between groups variance as s_b²

Within-group variation (sometimes called error group or error variance) is a term used in ANOVA tests. It refers to variations caused by differences within individual groups (or levels). In other words, not all the values within each group (e.g. means) are the same. These are differences not caused by the independent variable. Each sample is looked at on its own. In other words, no interactions between samples are considered. For the example given above, for within-group variation, you’ll look at variances in cholesterol levels for people in group A, without considering groups B,C, and D. Then you would look at cholesterol levels for people in group B, without considering groups A,C, and D. And so on.

The formula for the within-group variation is given by

where is the j-th observation of the i-th class and is the mean of the j-th group.

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