Analysis of variance table

The resultant analysis-of-variance tables will remain exactly the same. However,... 

The following quantities are needed for the analysis of variance table. 

Analysis of variance table for check clearing data. 

Set up the analysis of variance table as shown. The number of degrees of freedom is one less than the level of each factor For interactions, the number of degrees of freedom is the product of the degrees of freedom for each factor. For replication, the number of degrees of freedom is given by the number of pairs tested. The mean square is the sum of squares divided by the degrees of freedom. 

Thus, there is no reason to believe that one fertilizer promotes growth more than another. Generally speaking, analysis-of-variance problems are not solved in the form used in this example. A standard form called the analysis-of-variance table has been developed, and it is particularly useful for more complex problems ... 

The analysis-of-variance table is constructed from the following quantities calculated from the data ... 

Two-way analysis of variance (and higher classifications) leads to the presence of interactions. If, for example, an additive A is added to a lube oil stock to improve its resistance to oxidation and another additive, B, is added to inhibit corrosion by the stock under load or stress, it is entirely possible that the performance of the lube oil in a standard ball-and-socket wear test will be different from that expected if only one additive has present. In other words, the presence of one additive may adversely or helpfully affect the action of the other additive in modifying the properties of the lube oil. The same phenomenon is clearly evident in a composite rocket propellant where the catalyst effect on burning rate of the propellant drastically depends on the influence of fine oxidizer particles. These are termed antagonistic and synergistic effects, respectively. It is important to consider the presence of such interactions in any treatment of multiply classified data. To do this, the two-way analysis of variance table is set up as shown in Table 1.24. 

Now that the sum of squares of experimental error SSE =12.470 with the degree of freedom f=11 has been calculated, we can offer the analysis of variance table... 

Statistically significant differences between car tires, tire positions and tire types are clearly evident from analysis of variance. Table 2.60 indicates the highest wear out of tires on car IV and the lowest on car I. This does not present an error but is simply the upper and lower limit of tire wear-out in this research. 

Factorial design can be applied to any number of factors and levels. Frequently, in exploratory work, only two levels of each factor are chosen for the factorial design. Two factors and two levels constitute a 22 factorial design and permit setting up a 2 x 2 analysis-of-variance table with one factor on columns and another on rows. By determining the row effect and the column effect, it is possible to determine, for example in a reactor, whether a variation in temperature or pressure affects the reaction yield. A choice of substantially different values of temperature and pressure would be desirable to ensure conclusions which would be based on a wide range of factors. 

We then form a table of this analysis of variance (Table 12.9). 

These observations may be summarized conveniently in an analysis-of-variance table-. Table 26-7 illustrates this type of table for the above case. The overall variance (total mean square) Sj(N — 1) contains contributions due to variances within as well as between classes. The variation between classes contains both variation within classes and a variation associated with the classes themselves and is given by the expected mean square aj + not. Whether not is significant can be determined by the F test. Under the null hypothesis, = 0. Whether the ratio... 

The following analysis-of-variance table for the above table is obtained by summing ... 

A simpler case, but one frequently encountered in analytical chemistry, is that in which the strata are equal in size. To judge the effect of stratification, suppose that a variance-analysis study has been carried out. If there are k strata containing n observations in each, the analysis-of-variance table will be identical to Table 26-7. 

A quantitative measure of the robustness of an analytical procedure is the different levels of the intermediate precision and their comparison (analysis of variances) (Table 4). Of course, this approach only addresses random effects, which depend on the extent... 

The above calculations are often set out in the form of an Analysis of Variance Table as shown below... 

Analysis of Variance Table for Linear Multiple Regression... 

Type the following anova(Norris.lm.l) System responds with Analysis of Variance Table Response y Terms added sequentially (first to last) Df SumofSq Mean Sq F Value Pr(F) X 1 4255954 4255954 5436386 0 Residuals 34 27 1 ... 

Table 8 Expected Mean Squares for the Two-Factor with Interaction Analysis of Variance Table Shown in Table 7...

The problem now is to decide if the model as a whole is significant, and if so, which coefficients are significant. This is most easily done in an analysis of variance table (ANOVA). 

Analysis of variance (Tables 4, 5, and 6) shows that the statistical significance for the responses of the percentage of hydrolysis is appropriate because a high determination coefficient (R ) of 0.98542, 0.90664, and 0.95227 was obtained for olive, canola, and soybean, respectively. In this part of the study, RSM was used as an approach for determining the region where the percentage of hydrolysis is maximized for the oils tested... 

The analysis of variance table can therefore be set up as described previously. Table 9.8 Analysis of Variance of Solubility Data for a Hydrophobic Drug... 

As can be seen, the = 0.905, and the analysis of variance table portrays the model as highly significant in explaining the sum of squares, yet inadequate with all the data in the model. In Figure 7.8, we can see that the... 

The full regression model presented in Table 7.10 has a partial analysis of variance table, provided here ... 

Let us start with an ANOVA in case of a single factor, termed a one-way analysis of variance. Table 2.12 demonstrates the general scheme of the measurements of this type of ANOVA. 

The assumption of a normal distribution of the errors allows us to put confidence limits on the fit of the line to the data. This is carried out by the construction of an analysis of variance table (the basis of many statistical tests) in which a number of sums of squares are collected. The total sum of squares (TSS), in other words the total variation in y, is given by summation of the difference between the observed y values and their mean. 

By putting the results in the form of an analysis of variance table (Table IX), it is noted that what has been accomplished is a subdivision of the... 

The simple analysis of variance (Table XI) indicates that large differences in flavor were evident among the five fats tested as well as large differences in the judges scores within fats. These facts are evident from a consideration of the large F values, both of which are highly significant at P = 0.01. 

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