Mean square value, ANOVA

Now that we added the assumption that the errors follow a normal distribution to our hypotheses, we can return to the ANOVA and use the mean square values to test if the regression equation is statistically significant. When Pi = 0, that is, when there is no relation between X and y, it can be demonstrated that the ratio of the MSr and MSr mean squares... 

ANOVA table for the linear and quadratic models for the apparent density. The SS (sum of squares) and MS (mean square) values are multiplied by 10,000. df stands for degrees of freedom... 

Half normal plots give a visual indication of which factors are statistically significant. The technique is useful when there are few or no degrees of freedom available for a residual mean square in ANOVA or regression. Half normal plots are therefore useful when the experimental design is saturated and all effects are of interest. The half normal plot is a type of probability plot where a numerical value for the factor effects is plotted on the vertical axis against the expected normal order statistics on the horizontal axis (see Grove and Davis, 1997). 

The OUTSTAT= output data set pvalue contains the p-value in the PROB variable. If you have multiple predictor variables, you need to use the PROC GLM ODS data set OverallANOVA to get the overall model p-value from the ProbF variable. These output data sets contain other variables, such as the degrees of freedom, sum of squares, mean square, and F statistic, if you need them for an ANOVA table presentation. 

If we denote the residual mean square in the ANOVA by s2, the 95% confidence limits for a (denoted by A, the notation for the true—as opposed to the estimated— value for this parameter) are calculated as... 

Some authors [29,34] present the statistical interpretation method as an ANOVA table. A general example for a 2 full factorial design is given in Table 3.19. The sums of squares (55x) are obtained with the effect values (Ex) and the number of experiments in the design (N). The mean square... 

It is readily apparent that the HORRAT ratio of 2.89 indicates that the method is not sufficiently precise for inter-laboratory work. However, inspection of the ANOVA table shows a large value for the mean square due to the samples, MSs. This is an indication that they may not have been homogeneous and that the method may in fact be satisfactory. This demonstrates a major advantage in partitioning the variances between laboratory, sample and error rather than the traditional within- and between-laboratories method. 

This calculated F value is then compared to the critical value, Fa(1) VlV2, where Vj = regression DF = 1, and v2 = residual DF = n — 2. The residual mean square is often written as sfx, a representation denoting that it is the variance of Y after taking into account the dependence of Y on X. The square root of this quantity — that is, Syj—is called the standard error of estimate (occasionally termed the standard error of the regression). The ANOVA calculations are summarized in Table 2.2. 

Table IV shows the overall analysis of variance (ANOVA) and lists some miscellaneous statistics. The ANOVA table breaks down the total sum of squares for the response variable into the portion attributable to the model, Equation 3, and the portion the model does not account for, which is attributed to error. The mean square for error is an estimate of the variance of the residuals — differences between observed values of suspensibility and those predicted by the empirical equation. The F-value provides a method for testing how well the model as a whole — after adjusting for the mean — accounts for the variation in suspensibility. A small value for the significance probability, labelled PR> F and 0.0006 in this case, indicates that the correlation is significant. The R2 (correlation coefficient) value of 0.90S5 indicates that Equation 3 accounts for 91% of the experimental variation in suspensibility. The coefficient of variation (C.V.) is a measure of the amount variation in suspensibility. It is equal to the standard deviation of the response variable (STD DEV) expressed as a percentage of the mean of the response response variable (SUSP MEAN). Since the coefficient of variation is unitless, it is often preferred for estimating the goodness of fit.

Table 4.5.6 Statistics for the main effects and one significant interaction in the ANOVA of the concentration of boron. These include the residual sums of squares, the degrees of freedom (df) for the residual, the residual mean square, an F-ratio calculated using the appropriate error term, and the p-value corresponding to the F-ratio...

In our example of three treatment groups there are three pairwise comparisons of interest. Therefore, each pairwise comparison will be tested at an a level of 0.05/3 = 0.01667. This a level will require defining a critical value from the t distribution with 12 (that is, 15 - 3) df that cuts off an area of 0.00833 (half of 0.01667) in the right-hand tail. Use of statistical software reveals that the critical value is 2.77947. From inspection of the ANOVA table presented as Table 11.4 the within-samples mean square (mean square error) can be seen to be 1. The final component needed for the MSD is ... 

The quantity represented by the letter "q" is determined from a table of values used just for this test. Two characteristics are needed to determine the appropriate value of q each time that it is used. These characteristics are represented by the letters "a" and "v." The letter a represents the number of groups, which in this example is 3. The letter v represents the df, which in this test is the df associated with the within-samples mean square. In this case, the value of v is 12, as calculated for and shown in the ANOVA summary table in Table 11.4. From the table of q values for Tukey s test (provided in Appendix 5) the value of q associated with an... 

For ANOVA employed in regression, three primary sum-of-squares values are needed the total sum of squares, SSj, the sum of squares explained by the regression SSr, and the sum of squares due to the random error, SSe- The total sum of squares is merely the sum of squares of the differences between actual y,- observations and the y mean ... 

The ANOVA (Table 5.6) confirms the superiority of the quadratic model. The new model accounts for 99.37% of the toted variance, against only 80.63% for the linear model. The value of MSR/MSr increases to 471.4, from 29.14 for the linear model. Since introducing the P2 parameter in the model transfers 1 degree of freedom from the residual mean square to the regression mean square, the new MSR/MSr value should be compared with F2,6 (which is 5.14 at the 95% level), rather than with F17. In any case, from these results we may conclude that now we have a highly significant fit. 

Test rule For Levene s test we conduct an analysis of variance (ANOVA) of the absolute deviations from each sample average. Details on the ANOVA procedure are given in another chapter of this handbook. If the observed mean square ratio exceeds the appropriate critical value of the F statistic, we reject the hypothesis that all variances are equal. 

ANOVA calculations show that the mean squares for the between-days and within-days variations are 111 and 3.25 respectively. Hence P= 111/3.25 = 34. The critical value of Pj g is 4.066 (P = 0.05), so the mean concentrations differ significantly. The sampling variance is given by (111 - 3.25)/3 = 35.9. 

This is two-way ANOVA without replication. The between-row (i.e. between-solution) mean square is 0.00370 (3 d.f.) the between-column (i.e. between-method) mean square is 0.00601 (2 d.f.) and the residual mean square is 0.00470 (6 d.f.). The between-solution mean square is less than the residual one, so is not significant. Comparison of the between-method and residual mean squares gives F = 0.00601/0.00470 = 1.28. The critical value of p2,6 (P= 0.05) is 5.14, so the between-method variation is not significant. 

Again, a two-way ANOVA experiment without replication. The between-soil, between-day and residual mean squares are respectively 4.67 (4 d.f.), 144.8 (2 d.f.) and 26.47 (8 d.f.). The between-soil mean square is less than the residual mean square, so there are no significant differences between soils. Comparing the between-day and residual mean squares gives F = 144.8/26.47 = 5.47. The critical value of 2,8 is 4.46, so this source of variation is significant at P = 0.05. The actual probability (Excel) is 0.0318. 

Another two-way ANOVA experiment without replication. (Replication would be needed to study possible interaction effects.) The between-compound, between-molar ratio and residual mean squares are respectively 4204 (3 d.f.), 584 (2 d.f.) and 706 (6 d.f.). Thus molar ratios have no significant effect. Comparing the between-compound and residual mean squares gives F = 4204/706 = 5.95. The critical value of p3 6 is 4.76 (P= 0.05), so this variation is significant. (Pis given by Excel as 0.0313.) Common sense should be applied to these and all other data - diphenylamine seems to behave differently from the other three compounds. 

In the following ANOVA table we report rounded values, it is, therefore, not possible to reproduce the F-values exactly as ratios of the mean sums of the respective line and the residual mean sum of squares (according to Section 2.3) ... 

As we know from the section on ANOVA (analysis of variance see Section 2.3) in univariate cases, where only one feature is investigated, the sum of the squares of deviations of all n measuring values from the total mean is split into a part determined by... 

Assuming a normal multivariate distribution, with the same covariance matrices, in each of the populations, (X, X2,..., Xp) V(7t , 5), the multivariate analysis of variance MANOVA) for a single factor with k levels (extension of the single factor ANOVA to the case of p variables), permits the equality of the k mean vectors in p variables to be tested Hq = jl = 7 2 = = where ft. = fl, fif,..., fVp) is the mean vector of p variables in population Wi. The statistic used in the comparison is the A of Wilks, the value of which can be estimated by another statistic with F-distribution. If the calculated value is greater than the tabulated value, the null hypothesis for equality of the k mean vectors must be rejected. To establish whether the variables can distinguish each pair of groups a statistic is used with the F-distribution with p and n — p — k + i df, based on the square of Mahalanobis distance between the centroids, that permits the equality of the pairs of mean vectors to be compared Hq = jti = ft j) (Aflfl and Azen 1979 Marti n-Alvarez 2000). 

The scatterplots facilitate the interpretation of the variability observed for each factor tested. Whereas there is no significant difference between the positions 1 to 5, the difference between transects I to IV can be clearly seen. The values in transect I are lower than those in the other three transects. An ANOVA (based on type III sums of squares) was used to investigate in greater detail some of the factors that could have an effect on the concentration of B (Table 4.5.6). This means that the contribution of each factor is measured after removing the effects of all of the other factors. In this table, there is a decomposition of the variability in concentration of B according to the... 

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