Components of variance

The component-of-varlance analysis Is based upon the premise that the total variance for a particular population of samples Is composed of the variance from each of the Identified sources of error plus an error term which Is the sample-to-sample variance. The total population variance Is usually unknown therefore. It must be estimated from a set of samples collected from the population. The total variance of this set of samples Is estimated from the summation of the sum of squares (SS) for each of the Identified components of variance plus a residual error or error SS. For example ... 

Determine the components of variance that should be built Into the statistical design Proper stratification of the study area will allow Identification and quantification of several sources of variation. The sources of variation that can be controlled by the sampling are determined by the particular sampling design and by the pattern of sample collection superimposed over the area. An analysis of variance of the data provides estimates of the components of variance ... 

Knowing the listed analytical components of variance allows the determination of the homogeneity component. 

The sponsoring laboratory may have all the fun it wants within its own walls by using nested factorials, components of variance, or anything else that the workers believe will help in the fashioning of a test procedure. At some time the chosen procedure should undergo the kind of mutilation that results from the departures from the specified procedure that occur in other laboratories [Youden (1961b)]. Comment. 

The most powerful tool to minimise the component of variance due to error in the laboratory is the discipline which recognised accreditation schemes bring. They encompass all the likely areas which produce mistakes, documented procedures, training, checking procedures, control of samples, monitoring conditions, formal audits and perhaps above all calibration. The general quality movement has produced pressures to make laboratory accreditation commonplace and as more laboratories reach this status it must be expected that reproducibility will improve. In the current economic climate, a problem is finding sufficient laboratories able to devote sufficient time to precision trials. 

To obtain some information on the magnitude of turbidity measurement uncertainty, the analysis of variance (ANOVA) method [11] was used to identify individual random effects in measurement so that they could be properly taken into account. The first estimate was the within-instrument component of variance (that is the variance of observations made on the same instrument) which was denoted as si. The second estimate, si, was the pooled estimate of variance obtained... 

One may immediately imagine further subdividing the last type of variability. For example, one might wish to distinguish intraindividual variability due to different aspects of kinetics and separate all such variability from that due to measurement error. The problem with so doing is only that most data are insufficiently detailed and complete to allow these components of variance to be estimated separately. The two-way division we have proposed appears to suffice for most applications and data sets. 

The principal additions to this Edition are a substantial enlargement of Chapter I and two new chapters, Chapter XIII on balanced incomplete blocks amd Chapter XIV on confounding. Further additions are the components of variance for unequal column size in Chapter VII (d), the exact formula for the residual variance about a regression line in Chapter IX (h), the Doolittle method of computation in multiple regression in Chapter X (d), and the partitioning of sums of squares in Chapter XII (c). 

Source of Variance Sum of Squares Degrees of Freedom Mean Squares Components of Variance... 

In the present instance we are more interested in the actual averages for the three foundries, and are using the analysis of variance to test the significance of the apparent difference between the averages. In the more general case, of the analysis between and within batches, as discussed in the two previous sections, we probably wish to calculate the two components of variance. The Within Foundries Mean Square as before estimates the Within Foundries variance, i.e. 

Such an example was discussed in Qiapter VII (t). Table 11.1 below gives the degrees of freedom and components of variance. 

There are now gts individuals, and each can be uniquely determined by specifying particular ues of g, t, and s. We thus have three factors suppliers, grades of polder and temperature of processing. The technique of the analysis of variance for three factors is given later. The degrees of freedom and the components of variance are reproduced in Table 11.2. 

For the testing of the main effects, there are a mmiber of possibilities — (i) All three interactions are non-significant. Strikmg them out in the table of components of variance in Table 11.2, and pooling thw degrees of freedom wift the residual, we obtain Table 11,3, —... 

The components of variance are reproduced in Table 11.8 below, along with the respective degrees of freedom. 

The variance can then be analysed into the components given in Table 11.21. Table 11.20 is derived from Table 11.21 by pooling the sums of squares and degrees of freedom of the G and G X S terms. The component of variance conesponding to the G effect in Table 11.21 is more of the nature of an interaction than a main effect, but we will symbolically use G (rather than g) as a suffix to the component of variance to denote its rather different character. 

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