Analysis of Variance: Statistical test of significance developed by Sir Ronald fisher. It is also called the F-Ratio or ANOVA for ANalysis Of Variance. The test is designed to establish whether or not a significant (nonchance/nonrandom) difference exists among several sample means. Statistically, it is the ratio of the variance occurring between the sample means to the variance occurring within the sample groups.and from Wikipedia
A large F-Ratio, that is when the variance between is larger than the variance within, usually indicates a nonchance/nonrandom significant difference -- that is a difference created by the introduction of the independent variable.
Analysis of variance (ANOVA) is a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts.
The initial techniques of the analysis of variance were pioneered by the statistician and geneticist R. A. Fisher in the 1920s and 1930s, and is sometimes known as Fisher's ANOVA or Fisher's analysis of variance, due to the use of Fisher's F-distribution as part of the test of statistical significance.
One-way ANOVA is used to test for differences among three or more independent groups.
Multivariate analysis of variance (MANOVA) is used when there is more than one dependent variable.
Multivariate Analysis of Variance
Multivariate analysis of variance (MANOVA) is an extension of analysis of variance (ANOVA) methods to cover cases where there is more than one dependent variable and where the dependent variables cannot simply be combined.
As well as
* identifying whether changes in the independent variables have a significant effect on the dependent variables,
* the technique also seeks to identify the interactions among the independent variables and the association between dependent variables, if any.
And from manova page here
The main objective in using MANOVA is to determine if the response variables, are altered by the observer’s manipulation of the independent variables. Therefore, there are several types of research questions that may be answered by using MANOVA:
1) What are the main effects of the independent variables?
2) What are the interactions among the independent variables?
3) What is the importance of the dependent variables?
4) What is the strength of association between dependent variables?
5) What are the effects of covariates? How may they be utilized?
Assumptions
Normal Distribution: - The dependent variable should be normally distributed within groups. Overall, the F test is robust to non-normality, if the non-normality is caused by skewness rather than by outliers. Tests for outliers should be run before performing a MANOVA, and outliers should be transformed or removed.
Linearity: - MANOVA assumes that there are linear relationships among all pairs of dependent variables, all pairs of covariates, and all dependent variable-covariate pairs in each cell. Therefore, when the relationship deviates from linearity, the power of the analysis will be compromised.
Homogeneity of Variances: - Homogeneity of variances assumes that the dependent variables exhibit equal levels of variance across the range of predictor variables. Remember that the error variance is computed (SS error) by adding up the sums of squares within each group. If the variances in the two groups are different from each other, then adding the two together is not appropriate, and will not yield an estimate of the common within-group variance. Homoscedasticity can be examined graphically or by means of a number of statistical tests.
Homogeneity of Variances and Covariances: - In multivariate designs, with multiple dependent measures, the homogeneity of variances assumption described earlier also applies. However, since there are multiple dependent variables, it is also required that their intercorrelations (covariances) are homogeneous across the cells of the design. There are various specific tests of this assumption.
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