Least significant difference method

The least significant difference method described above is not entirely rigorous it can be shown that it leads to rather too many significant differences. However, it is a simple follow-up test when ANOVA has indicated that there is a significant difference between the means. Descriptions of other more rigorous tests are given in the references at the end of this chapter. 

Show that the percentage of water recovered differs significantly at different depths. Use the least significant difference method described in Section 3.9 to find the causes of this significant result. 

If significant differences are indicated in ANOVA, we are often interested in the cause. Is one mean different from the others Are all the means different Are there two distinct groups that the means fall into There are several methods to determine which means are significantly different. One of the simplest is the least significant difference (LSD) method. In this method, a difference is calculated that is judged to be the smallest difference that is significant. The difference between each pair of means is then compared with the least significant difference to determine which means are different. 

CAI = cadmium availability index (in pg kg ) determined by 1 mol NH4C3 extraction method of Krishnamurti et al. (1995b). LSD = least significant difference at p = 0.01 and p = 0.05. Without application of Idaho phosphate fertilizer. With application of Idaho phosphate fertilizer. 

The scale of operations, accuracy, precision, sensitivity, time, and cost of methods involving redox titrations are similar to those described earlier in the chapter for acid-base and complexometric titrimetric methods. As with acid-base titrations, redox titrations can be extended to the analysis of mixtures if there is a significant difference in the ease with which the analytes can be oxidized or reduced. Figure 9.40 shows an example of the titration curve for a mixture of Fe + and Sn +, using Ce + as the titrant. The titration of a mixture of analytes whose standard-state potentials or formal potentials differ by at least 200 mV will result in a separate equivalence point for each analyte. 

Experience shows that such methods can result in significantly different dead volumes for the same GPC unit. There are at least two reasons for these observations (1) the peak shape is changed by the laminar flow with a parabolic... 

Analysis of variance (ANOVA) tests whether one group of subjects (e.g., batch, method, laboratory, etc.) differs from the population of subjects investigated (several batches of one product different methods for the same parameter several laboratories participating in a round-robin test to validate a method, for examples see Refs. 5, 9, 21, 30. Multiple measurements are necessary to establish a benchmark variability ( within-group ) typical for the type of subject. Whenever a difference significantly exceeds this benchmark, at least two populations of subjects are involved. A graphical analogue is the Youden plot (see Fig. 2.1). An additive model is assumed for ANOVA. 

Thus, we may obtain a best value fork from a modification of the normal least squares procedure. As k , F hkl will only significantly differ from Fm when Fm is small and Fumi is large, i.e. the reflections carrying the most information about k are those very reflections that are poorly observed. Hence, this is not a reliable method. 

We have unified experimental and theoretical DFT data to illustrate the significance of this difference between boron and aluminum, and to show the consistency of data from different laboratories and different methods. The results are shown in Fig. 6. For B-ZSM-5, proton transfer takes place at proton affinities of the probe molecule of at least 854 kJ/mol, which is somewhat higher than that reported for A1 zeolites (821 kJ/mol) [235]. 

Analysis of variance was the statistical model used with preplanned comparison testing for significant differences by the least square means method. 

Structure of the Vanadyl Ion Complexes.— The structures of at least three different compounds containing vanadyl ion have been determined by X-ray methods. It is significant that V03 crystallizes in a highly distorted rutile (TiC ) structure, in which there is one conspicuously short V-O bond (1.76 A.) in each VOe unit.8 Thus there seems to be a greater driving force to... 

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