Pooled variance

XI.4.4.2 The standard deviation for repeatability for each sample was calculated from pair-wise (repeat pairs) variances pooled across the laboratories. The standard deviation for reproducibility was calculated from the variance of the mean values of each pair. This variance is equal to the sum of two variances, the variance a, due to differences between laboratories and the variance due to repeatability error a, divided by the number of replicates... 

It is unclear, however, how many degrees of freedom are associated with f(a, v) since there are two sets of independent measurements. If the variances sa and sb estimate the same O, then the two standard deviations can be factored out of equation 4.19 and replaced by a pooled standard deviation. Spool, which provides a better estimate for the precision of the analysis. Thus, equation 4.19 becomes... 

To begin with, we must determine whether the variances for the two analyses are significantly different. This is done using an T-test as outlined in Example 4.18. Since no significant difference was found, a pooled standard deviation with 10 degrees of freedom is calculated... 

Since Fgxp is larger than the critical value of 7.15 for F(0.05, 5, 5), the null hypothesis is rejected and the alternative hypothesis that the variances are significantly different is accepted. As a result, a pooled standard deviation cannot be calculated. 

A second way to work with the data in Table 14.7 is to treat the results for each analyst separately. Because the repeatability for any analyst is influenced by indeterminate errors, the variance, s, of the data in each column provides an estimate of O rand- A better estimate is obtained by pooling the individual variances. The result, which is called the within-sample variance (s ), is calculated by summing the squares of the differences between the replicates for each sample and that sample s mean, and dividing by the degrees of freedom. 

Phospholipase C, 24 pK0, 125, 144-145 Polar metabolites, 165 Polymorphisms, 4 Polypharmacology, 190-192 Pooled variance, 228 Populations, 226-228, 232 Positive agonism, 49 Potency... 

Subcase b3 When ni and tij are not equal the degrees of freedom are calculated as / = wi + W2 - 2 for the variance of the difference. Up to this point a random pick of statistics textbooks " shows agreement among the authors. The pooled variance is given in Eq. (1.14) where the numerator is the sum of the squares of the residuals, taken relative to Xmean, i or Xmean,2. as appropriate. Some authors simplify the equation by dropping the -1 in n - 1, under the assumption n 1 (Eq. 1.16), something that might have been appropriate to do in the precomputer era in order to simplify the equations and lessen the calculational burden. 

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. 

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. 

Results The uncertainties associated with the slopes are very different and n = H2, so that the pooled variance is roughly estimated as (V + V2)/2, see case c in Table 1.10 this gives a pooled standard deviation of 0.020 a simple r-test is performed to determine whether the slopes can be distinguished. (0.831 - 0.673)/0.020 = 7.9 is definitely larger than the critical /-value for p - 0.05 and / = 3 (3.182). Only a test for H[ t > tc makes sense, so a one-sided test must be used to estimate the probability of error, most likely of the order p = 0.001 or smaller. 

The six data sets do not differ in variance (Bartlett test) or in means (ANOVA), so there is no way to group them using the multiple range test. This being so, the data were pooled for compounds A and B, yielding the two columns at right (data in CU-Assay2.dat). 

In eq. (33.3) and (33.4) x, and Xj are the sample mean vectors, that describe the location of the centroids in m-dimensional space and S is the pooled sample variance-covariance matrix of the training sets of the two classes. 

The use of a pooled variance-covariance matrix implies that the variance-covariance matrices for both populations are assumed to be the same. The consequences of this are discussed in Section 33.2.3. 

A simple two-dimensional example concerns the data from Table 33.1 and Fig. 33.9. The pooled variance-covariance matrix is obtained as [K K -1- L L]/(n, + 3 - 2), i.e. by first computing for each class the centred sum of squares (for the diagonal elements) and the cross-products between variables (for the other... 

Computation of the cross-product term in the pooled variance-covariance matrix for the data of Table 33. 

When all are considered equal, this means that they can be replaced by S, the pooled variance-covariance matrix, which is the case for linear discriminant analysis. The discrimination boundaries then are linear and is given by... 

As stated earlier, LDA requires that the variance-covariance matrices of the classes being considered can be pooled. This is only so when these matrices can be considered to be equal, in the same way that variances can only be pooled, when they are considered equal (see Section 2.1.4.4). Equal variance-covariance means that the 95% confidence ellipsoids have an equal volume (variance) and orientation in space (covariance). Figure 33.10 illustrates situations of unequal variance or covariance. Clearly, Fig. 33.1 displays unequal variance-covariance, so that one must expect that QDA gives better classification, as is indeed the case (Fig. 33.2). When the number of objects is smaller than the number of variables m, the variance-covariance matrix is singular. Clearly, this problem is more severe for QDA (which requires m < n ) than for LDA, where the variance-covariance matrix is pooled and therefore the number of objects N is the sum of all objects... 

Figure 65-1 shows a schematic representation of the F-test for linearity. Note that there are some similarities to the Durbin-Watson test. The key difference between this test and the Durbin-Watson test is that in order to use the F-test as a test for (non) linearity, you must have measured many repeat samples at each value of the analyte. The variabilities of the readings for each sample are pooled, providing an estimate of the within-sample variance. This is indicated by the label Operative difference for denominator . By Analysis of Variance, we know that the total variation of residuals around the calibration line is the sum of the within-sample variance (52within) plus the variance of the means around the calibration line. Now, if the residuals are truly random, unbiased, and in particular the model is linear, then we know that the means for each sample will cluster... 

Because of the many decisions regarding inclusion or exclusion of studies, different meta-analyses might reach very different conclusions on the same topic. Even after the studies are chosen, there are many other methodologic issues in choosing how to combine means and variances (e.g., what weighting methods should be used). Pooled analysis should report both relative risks and risk reductions as well as absolute risks and risk reductions (Sinclair and Bracken, 1994). 

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