Variance within 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... 

It is possible to compare the means of two relatively small sets of observations when the variances within the sets can be regarded as the same, as indicated by the F test. One can consider the distribution involving estimates of the true variance. With sj determined from a group of observations and S2 from a second group of N2 observations, the distribution of the ratio of the sample variances is given by the F statistic ... 

Analysis of variance (ANOVA) tests whether one group of subjects (e.g., batch, method, laboratory, etc.) differs from the population of subjects investigated (several batches of one product different methods for the same parameter several laboratories participating in a round-robin test to validate a method, for examples see Refs. 5, 9, 21, 30. Multiple measurements are necessary to establish a benchmark variability ( within-group ) typical for the type of subject. Whenever a difference significantly exceeds this benchmark, at least two populations of subjects are involved. A graphical analogue is the Youden plot (see Fig. 2.1). An additive model is assumed for ANOVA. 

However, some undetermined factor produces a negative bias, particularly in samples nos. 3, 6, 9, and 10 this points to either improper handling of the fixed-volume dispenser, or clogging. The reduction of the within-group variance from VVV to VWV is, to a major part, due to the elimination of... 

Data Evaluation The Bartlett test (Section 1.7.3 cf. program MULTI using data file MOISTURE.dat) was first applied to determine whether the within-group variances were homogeneous, with the following intermediate results A = 0.1719, B = -424.16, C = 1.4286, D = 70, E = 3.50, F = 1.052, G = 3.32. 

To further analyze the relationships within descriptor space we performed a principle component analysis of the whole data matrix. Descriptors have been normalized before the analysis to have a mean of 0 and standard deviation of 1. The first two principal components explain 78% of variance within the data. The resultant loadings, which characterize contributions of the original descriptors to these principal components, are shown on Fig. 5.8. On the plot we can see that PSA, Hhed and Uhba are indeed closely grouped together. Calculated octanol-water partition coefficient CLOGP is located in the opposite corner of the property space. This analysis also demonstrates that CLOGP and PSA are the two parameters with... 

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). 

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. 

Within this group of structures, none of the different combinations of factors gave an R-value significantly worse - at the 5% level of probability - than the best. Thus there is no justification for choosing any one in preference to any other, and so the mean was taken as the best structure. The standard deviations of the parameters of this model were calculated from the overall variance within a structure factor set. That is, the contribution to uncertainty due to errors in the structure factor set have been omitted. Thus these standard deviations are almost certainly underestimates, for whilst the structure factors of Yokouchi et al might be significantly better than those of the other authors, they are certain to contain some error. 

As mentioned, hierarchical cluster analysis usually offers a series of possible cluster solutions which differ in the number of clusters. A measure of the total within-groups variance can then be utilized to decide the probable number of clusters. The procedure is very similar to that described in Section 5.4 under the name scree plot. If one plots the variance sum for each cluster solution against the number of clusters in the respective solution a decay pattern (curve) will result, hopefully tailing in a plateau level this indicates that further increasing the number of clusters in a solution will have no effect. 

In discriminant analysis, in a manner similar to factor analysis, new synthetic features have to be created as linear combinations of the original features which should best indicate the differences between the classes, in contrast with the variances within the classes. These new features are called discriminant functions. Discriminant analysis is based on the same matrices B and W as above. The above tested groups or classes of data are modeled with the aim of reclassifying the given objects with a low error risk and of classifying ( discriminating ) another objects using the model functions. 

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. 

We need to become familiar with the topic of analysis of variance, often abbreviated ANOVA, in order to test the null hypothesis (H0) p1 = p2 = " = where k is the number of experimental groups, or samples. In the ANOVA, we assume that a = a2 = = ol, and we estimate the population variance assumed common to all k groups by a variance obtained using the pooled sum of squares (within-groups SS) and the pooled degree of freedom (within-groups DF) ... 

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