Analysis Of Variance
Analysis Of Variance
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The fundamental technique is a partitioning of the total sum of squares (abbreviated SS) into components related to the effects used in the model. For example, we show the model for a simplified ANOVA with one type of treatment at different levels.
So, the number of degrees of freedom (abbreviated df) can be partitioned in a similar way and specifies the chi-square distribution which describes the associated sums of squares.
See also Lack-of-fit sum of squares.
The F-test is used for comparisons of the components of the total deviation. For example, in one-way, or single-factor ANOVA, statistical significance is tested for by comparing the F test statistic
where
and
to the F-distribution with I − 1,nT − I degrees of freedom. Using the F-distribution is a natural candidate because the test statistic is the quotient of two mean sums of squares which have a chi-square distribution.
When the data do not meet the assumptions of normality, the suggestion has arisen to replace each original data value by its rank (from 1 for the smallest to N for the largest), then run a standard ANOVA calculation on the rank-transformed data. Conover and Iman (1981) provided a review of the four main types of rank transformations. Commercial statistical software packages (e.g., SAS, 1985, 1987, 2008) followed with recommendations to data analysts to run their data sets through a ranking procedure (e.g., PROC RANK) prior to conducting standard analyses using parametric procedures.
This rank-based procedure has been recommended as being robust to non-normal errors, resistant to outliers, and highly efficient for many distributions. It may result in a known statistic (e.g., Wilcoxon Rank-Sum / Mann-Whitney U), and indeed provide the desired robustness and increased statistical power that is sought. For example, Monte Carlo studies have shown that the rank transformation in the two independent samples t test layout can be successfully extended to the one-way independent samples ANOVA, as well as the two independent samples multivariate Hotelling's T2 layouts (Nanna, 2002).
So, the number of degrees of freedom (abbreviated df) can be partitioned in a similar way and specifies the chi-square distribution which describes the associated sums of squares.
See also Lack-of-fit sum of squares.
The F-test is used for comparisons of the components of the total deviation. For example, in one-way, or single-factor ANOVA, statistical significance is tested for by comparing the F test statistic
where
and
to the F-distribution with I − 1,nT − I degrees of freedom. Using the F-distribution is a natural candidate because the test statistic is the quotient of two mean sums of squares which have a chi-square distribution.
When the data do not meet the assumptions of normality, the suggestion has arisen to replace each original data value by its rank (from 1 for the smallest to N for the largest), then run a standard ANOVA calculation on the rank-transformed data. Conover and Iman (1981) provided a review of the four main types of rank transformations. Commercial statistical software packages (e.g., SAS, 1985, 1987, 2008) followed with recommendations to data analysts to run their data sets through a ranking procedure (e.g., PROC RANK) prior to conducting standard analyses using parametric procedures.
This rank-based procedure has been recommended as being robust to non-normal errors, resistant to outliers, and highly efficient for many distributions. It may result in a known statistic (e.g., Wilcoxon Rank-Sum / Mann-Whitney U), and indeed provide the desired robustness and increased statistical power that is sought. For example, Monte Carlo studies have shown that the rank transformation in the two independent samples t test layout can be successfully extended to the one-way independent samples ANOVA, as well as the two independent samples multivariate Hotelling's T2 layouts (Nanna, 2002).
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