Common variance

In the case in which the errors are independent of each other their covariances will be zero, and if they also have the same variance, then D = oH, with the constant being the common variance and I being the identity matrix. In this case, the same 0 minimizing (Eq. 3.3) would also minimize (Eq. 3.2) and the OLSE can therefore be seen as a particular case of the WLSE. 

The main difference between factor analysis and principal component analysis is the way in which the variances of Eq. (8.20) are handled. Whereas the interest of FA is directed on the common variance var Xij)comm and both the other terms are summarized as unique variance... 

The criterion D is a measure of divergence among the models, obtained from information theory. The quantity nt is the prior probability associated with model / after the nth observation is obtained o2 is the common variance of the n observations y(l), y( 2), , y(n — 1), y(n) a2 is the variance for the predicted value of y(n + 1) by model i. When we have two models, D simplifies to... 

A second reason is that the use of local variance versus global variance can result in markedly different bands. The separate calculations of variance at levels throughout the range of standards produces a wider confidence interval at lower values as seen in Kurtz method. If a common variance is used as the variance estimate then a lower confidence interval is calculated at each point as is probably the case in Wegscheider s method. 

Intuitively, we assume that individual data points are generated by a normal distribution. A common variance is assumed for model error as in a conventional regression analysis. 

To cany out a Lagrange multiplier test of the hypothesis of equal variances, we require the separate and common variance estimators based on the restricted slope estimator. This, in turn, is the pooled least squares estimator. For the combined sample, we obtain... 

Up to now the technique of calculations in analysis of variance has been analyzed in more detail. Now let us briefly consider the analysis of variance theory. Let us consider the model for a one-way analysis of variance. Here it is assumed that the columns of data are J-random samples from J-independent normal populations with means i, i2,...,P, and common variance a2. The one-way analysis of variance technique will give us a procedure for testing the hypothesis H0 F.i=p.2=---=F-j against the alternative Hj at least two ij not equal. The statistical model gives us the structure of each observation in the IxJ matrix ... 

The results from FA for the other, smaller particle size fractions are qualitatively comparable the extracted factors have a similar pattern (Tab. 7-8 and 7-9). The dependence of the eigenvalues for the extracted factors on the particle diameter (Fig. 7-22) illustrates that the part of common variance described by the sea spray factor decreases with decreasing particle size and that the part of common variance described by the anthropogenic factor increases continuously with decreasing particle size. 

In preparing the data matrix of the Elster sediments for FA, the data set from sampling points on the left and the right sides of the river were treated separately. By application of FA to the matrix of 6 features and 14 samples, two factors were extracted which describe 74.3% of the common variance (Tab. 8-9). 

A mixture model implicitly assumes that some fraction. (p) of the population has one set of typical values of response, and that the remaining fraction (1 — p) has another set of typical values. In this model, the only difference initially allowed in the typical values between the two groups was the maximal fractional reduction in seizure frequency after treatment with pregabalin, that is, EmaxA EmaxB- Values for these two parameters and the mixing fraction p were estimated. Random interindividual variability effects r i and r 2 were assumed to be normally distributed with zero means and common variance co. The estimation... 

Investigation 11 dealt with transport models for the fractionation performance of sieve trays. Data from three laboratories were analyzed. Two to six variables were observed in each experiment. These observations were treated in Eq. (7.1-1) as independent events u, with equal weights Wu for all values reported from a given laboratory. Thus, the observations from Laboratory i were reduced to a column of values piu with a common variance an. The resulting covariance matrix was diagonal, as in Table 7.2c, and the Y array was block-rectangular with block widths mb = 1. 

Psychometric procedures like Spearman—Brown provide psychological windows on human variation akin to the microscope in biology and the telescope in astronomy. With an average inter-item correlation of 0.20, a mere 40 item scale can generate a composite whose common (reliable) variance is 91%. This is how psychometrics distils dimensions of common variance for submission to construct validation procedures. 

Among the more important points illustrated here is the abstract nature ofg. When heterogeneous collections of cognitive tests are aggregated, they form a general factor relatively free of any particular content or product, which accounts for approximately 50% of their common variance. This occurs when pictorial, quantitative and verbal item types are administered individually or in a group, orally or by paper and pencil. Because it is general, g can be assessed in many... 

Because the number of correlations among the n elements is n(n—1)/2, the sum of the item covariances increases more rapidly as a function of n than the sum of the item variances. In standard test batteries, such as the Wechsler, the Stanford—Binet and the British IQ scales, which have large numbers of items, the item covariances account for about 90% of the total variance. Hence most of a typical test s variance attributable to individual differences in performance results from the correlations, or common variance, among its various elements. 

It is also possible mathematically, by means of factor analysis, to express these elements common variance, not in terms of the various elements themselves, but in terms of one or more linearly independent (i.e. uncorrelated) hypothetical sources of variance (Carroll 1993,1997, Jensen 1998). 

In our opinion, therefore, nonparametric methods should be chosen when assumptions (such as normality for the t test) are clearly not met and the sample sizes are so small that there is very little confidence about the properties of the underlying distribution. The nonparametric method discussed in this section is a test of a shift in the distribution between two populations with a common variance represented by two samples, and it will always be valid when comparing two independent groups. 

Principal component analysis is used to reduce the information in many variables into a set of weighted linear combinations of those variables it does not differentiate between common and unique variance. If latent variables have to be determined, which contribute to the common variance in a set of measured variables, factor analysis (FA) is a valuable statistical method, since it attempts to exclude unique variance from the analysis. 

The M nonzero eigenvalues of both matrices Qa and Qb coincide (M being the minimum rank between the two Q matrices) and twice the sum of the square root of these eigenvalues X is the interset common variance Vab ... 

Being the maximum theoretical value of the common variance,... 

In practice, at first one may wish to try to obtain a separate estimate of the variance on each occasion, i.e., obtain estimates of coj, w2, etc. If the variance terms are approximately equal (in general if the ratio of the largest to smallest variance component is less than four, the variances are treated as equivalent) then one can assume that Wj = w2 =. .. (a , or that there is a common variance between occasions, and reestimate the model. If, however, there is a trend in the variances over time then one may wish to treat as a function of time. Alternatively, one may wish to examine whether IOV can be explained by any covariates in the data set. For most data sets, such complex IOV models cannot be supported by the data and these complications will not be explored any further. 

If tii,n2..riff are mutually uncorrelated random variables with a common variance a, ... 

Consider a case with equal numbers of patients per group and where the common variance is known. Suppose that the means from the treatment groups have been divided... 

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