Variance techniques

H. F. Gollob, A statistical model which combines features of factor analytic and analysis of variance techniques. Psychometrika, 33 (1968) 73-111. 

The analysis of variance techniques of Section IV,A have been seen to provide information about the overall goodness of fit or about testing the importance of the contribution of certain terms in the model toward providing this overall fit of the data. Although these procedures are quite useful, more subtle model inadequacies can exist, even though the overall goodness of fit is quite acceptable. These inadequacies can often be detected through an analysis of the residuals of the model. 

Electrochemical applications are grouped with respect to each test, and are presented in combination with pertinent theory for immediate illustration. Only a modicum of understanding of critical normal-, and chi-square tables, and analysis of variance techniques, is required as a necessary background. 

As we have already seen, there will be settings where the pattern of differences between treatment groups does not conform to proportional hazards, where the hazard ratio is not a constant, single value. Such situations are best handled by using an alternative model to incorporate baseline factors. The accelerated failure time model is an analysis of variance technique which models the survival time itself, but on the log scale ... 

The technique known as analysis of variance (ANOVA)2) uses tests based on variance ratios to determine whether or not significant differences exist among the means of several groups of observations, where each group follows a normal distribution. The analysis of variance technique extends the t-test used to determine whether or not two means differ to the case where there are three or more means. 

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 analysis of variance technique for testing equality of means is a rather robust procedure. That is, when the assumption of normality and homogeneity of variances is slightly violated the F-test remains a good procedure to use. In the one-way model, for example, with an equal number of observations per column it has been exhibited that the F-test is not significantly effected. However, if the sample size varies across columns, then the validity of the F-test can be greatly affected. There are various techniques for testing the equality of k variances Oi, 02,..., crj,. We discuss... 

As shown in the above works, an optimal feedback/feedforward controller can be derived as an analytical function of the numerator and denominator polynomials of Gp(B) and Gn(B). No iteration or integration is required to generate the feedback law, as a consequence of the one step ahead criterion. Shinnar and Palmor (52) have also clearly demonstrated how dead time compensation (discrete time Smith predictor) arises naturally out of the minimum variance controller. These minimum variance techniques can also be extended to multi-variable systems, as shown by MacGregor (51). 

Unlike W plasma etch back process, the typical W CMP process usually removes the adhesion layer such as Ti/TiN or TiN during the primary polish. As a result, during the over polish step there is some oxide loss. Since the oxide deposition, planarization CMP (oxide CMP), and tungsten CMP steps are subsequent to each other, the oxide thickness profile could become worse further into the process flow. Therefore, the across-wafer non-uniformity of the oxide loss during W CMP process is one of the very important process parameters needs to be optimized. To determine the effect of the process and hardware parameters on the polish rate and the across-wafer uniformity, designed experiments were run and trends were determined using analysis of variance techniques. Table speed, wafer carrier speed, down force, back pressure, blocked hole pattern, and carrier types were examined for their effects on polish rate and across-wafer uniformity. The variable ranges encompassed by the experiments used in this study are summarized in Table I. 

The NOEC and the LOEC are the usual calculations reported from chronic toxicity tests. The NOEC is the highest concentration in which the measured effect is not statistically different from that of the control. The LOEC is the lowest concentration at which a statistically significant effect occurred. These concentrations are based on the most sensitive effect parameters, that is, hatchability, growth, and reproduction. The statistical procedure for these calculations combines the use of analysis of variance techniques and multiple comparison tests. In some cases, the maximum acceptable toxic concentration (MATC) is reported from chronic toxicity results. The MATC is a concentration [x) that is within the range of the NOEC and LOEC NOEC a < LOEC. The first-effect concentration can be expressed as the geometric mean of the two terms. 

In contrast, analysis of variance techniques (ANOVA) is superior to quantify the discriminatory power with respect to binders and inactives [149]. This optimization using ANOVA on distributions of docking scores for actives and inactives starts from the empirical function implemented in the docking program ProPose. However, as sometimes there is no reliable quantification of nonbinding and no structural information for weak ligands available, a correct calibration remains difficult in these cases. 

Testing for hnearity of regression using an analysis-of-variance technique. The test statistic is the F-statistic. 

The mechanical test results were analysed using the analysis of variance technique (ANOVA) with the SAS software. 

Although this procedure can be used to test the equivalence of sample variances (as shown below), it has much greater importance because of its application to what is known as the analysis of variance technique used in experimental design. This will be discussed later. 

Analysis of variance techniques can be used to study differences between glass samples, without reference to quantitative elemental standards. 

The computational techniques for all the designs that are available may be found in texts on statistical methods or experimental design (Cochran and Cox, 1950 Kempthorne, 1952 Anderson and Bancroft, 1952). Some of the examples to be presented later will be of the types referred to, whereas others will be of more complex nature. It is hoped that the examples, together with some reading in appropriate references, will provide information concerning the uses (and limitations) of analysis of variance techniques. 

Values connected by horizontal lines do not differ significantly from each other. Significance was determined by the analysis of variance technique and by using the error mean squares derived therefrom to derive the standard errors of the means these in turn were used to apply Duncan s Multiple Range Test. Data from Angeles (1966). 

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