Variance between groups

Since a series of t-tests is cumbersome to carry out, and does not answer all questions, all measurements will be simultaneously evaluated to find differences between means. The total variance (relative to the grand mean xqm) is broken down into a component Vi variance within groups, which corresponds to the residual variance, and a component V2 variance between groups. If Hq is true, Vi and V2 should be similar, and all values can be pooled because they belong to the same population. When one or more means deviate from the rest, Vj must be significantly larger than Vi. 

Variance within groups Variance between groups Total variance... 

Equation 6.1 Ratio of interest = effect variance (between-group variance)... 

If no excess between-group variance is found, stop testing and pool all values, because they probably all belong to the same population. If significant excess variance is detected, continue testing. 

In general, one maximizes between-cluster Euclidean distance or minimizes within-cluster Euclidean distance or variance. This really amounts to the same. As described by Bratchell [6], one can partition total variation, represented by T, into between-group (B) and within-group components (W). 

A difficulty with Hansch analysis is to decide which parameters and functions of parameters to include in the regression equation. This problem of selection of predictor variables has been discussed in Section 10.3.3. Another problem is due to the high correlations between groups of physicochemical parameters. This is the multicollinearity problem which leads to large variances in the coefficients of the regression equations and, hence, to unreliable predictions (see Section 10.5). It can be remedied by means of multivariate techniques such as principal components regression and partial least squares regression, applications of which are discussed below. 

If input data are differentiated for subpopulations, the between-group variance should be examined before pooling the data. 

Traditionally, the determination of a difference in costs between groups has been made using the Student s r-test or analysis of variance (ANOVA) (univariate analysis) and ordinary least-squares regression (multivariable analysis). The recent proposal of the generalized linear model promises to improve the predictive power of multivariable analyses. 

The appropriate test when comparing more than two means is analysis of variance (ANOVA). The essential process in ANOVA is to split up, or decompose, the overall variance in the data. This variability is due to differences between the means due to the treatment effect (between-group variance) and that due to random variability between individuals within each group (within-group variance, sometimes called unexplained or residual variance), hence the name analysis of variance. ... 

When duplicate or split samples are sent for analysis, the repeatability and reproducibility can be calculated from an ANOVA of the data with the laboratories as the grouping factor. If the between-groups mean square is significantly greater than the within-groups mean square, as determined by an F test, then the variance due to laboratory bias can be computed as described in chapter 2. 

The visual estimation of differences between groups of data has to be proved using multivariate statistical methods, as for example with multivariate analysis of variance and discriminant analysis (see Section 5.6). 

Comparing Between-Group Variance and Within-Group Variance... 

Between-group variance can be called the effect variance, and within-group variance can be called the error variance. The effect variance is directly associated with the treatment administered, while the error variance is due to chance alone. The larger the effect variance when compared with the error variance, the more likely it is that compelling evidence of systematic variation will be revealed by inferential statistical analysis. Conversely, the smaller the effect variance when compared with the error variance, the less likely it is that compelling evidence of systematic variation will be revealed. 

F = effect variance = between treatment groups variance error variance within treatment groups variance... 

Divide the between-groups variance by the within-groups variance to give the test statistic F. 

From these results, it is clear that the random sampling error (the between-group variance) is statistically significantly different compared with the random analytical error (the within-groups variance). 

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