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Elementary statistics help assists you to understand the basic concepts based on statistics. Here in this article we will discuss about the Analysis of variance (ANOVA) and Multiple Analysis of variance (MANOVA). Various concepts relating to ANOVA and MANOVA especially partitioning of sum of squares and multiple factor ANOVA are been discussed here.
What is ANOVA?
ANOVA (Analysis of Variance) is a systematic procedure for achieving the total variation present in a set of observable quantities. With the help of this technique it is possible to perform certain test of hypotheses and to provide estimates for components of variation. In ANOVA we see whether there is any difference between groups on some variable or not. For both parametric and non parametric test ANOVA can be performed.
Different Types of ANOVA
One-way repeated measures
Two-way repeated measures
One-way between groups
Two-way between groups
How is The Analysis Performed?
ANOVA calculates the Group mean.
It calculates the mean for all the groups combined - the Overall Mean.
It calculates Within Group Variation.
After which we compute between group variation i.e, the deviation of group means from overall mean
Lastly for testing F statistics is computed. F statistics is defined as the ratio of between group variation to within group variation.
A significant difference exists between groups if the variation between groups is highly and significantly greater than variation within groups.
We will now discuss the two main points. They are as follows:
1. The Partitioning of the Sum of Squares.
2. Multiple Factor ANOVA.
1. The Partitioning of the Sum of Squares:
Partitioning the Sum of Squares
If we recall that a measure of the variation in a data set can be built using deviations from the mean: xi-xbar .By simply adding all the deviations we get the following result:
The error sum of squares is SSE and between sum of squares is SSB. The mean square error and mean square between are denoted by MSE and MSB.
MSE=SSE/ (the corresponding degrees of freedom) and MSB= SSB/ (the corresponding degrees of freedom).
The expected value of the MSE (Mean Square Error) is the common variance α
What is Tested in ANOVA? The Reason Behind ANOVA:
We test if all the treatment means are equal against the alternative that they are not equal. If all treatment means are identical (null hypothesis H_0 true), then the sample treatment means should be relatively similar and hence the SSB should be relatively small. On the other hand, if some treatment means differ (false), then the SSB should be relatively large.
How is It Tested? Significance Testing:
The test statistic is given by F= MSB/ MSE.
A "large" value of F provides evidence against the null hypothesis of equal treatment means. "Large" is determined by the level of significance and the distribution.
Apart from all these you need to know what is independent and dependent variable before you perform the ANOVA. Independent variables are actually the factors in ANOVA measured on categorical scale. The variable that we want to measure is the dependent variable.
2. Multiple Factor ANOVA
Purpose and How to Detect Significant Factors:
The ANOVA is used to detect significant factors in a multi-factor model. In the multiple factor model, there is one dependent or response variable and one or more factor (independent) variables. This model helps the experimenter to set the value for each of the factor variables and then measure the response variable. The no of factors are referred to as levels of a factor. The difference between Balanced and unbalanced designs is that in the former the level combinations have equal number of observations while in the latter one we have varying number of observations.
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Examples of Multiple-Factor ANOVA
With two-factor analysis of variance, there are two study factors (we'll call them factor A with a level and factor B with b levels) and we study all (a takes b) combinations of levels. The two-way table containing mean , SD, sample size can be used to display the data.
In an ANOVA multi-factor analysis of variance, we look at interactions along with main effects. The joint effect of two factors ,the effect of one factor differs according as the level of another factor is the interaction