Analysis of variance, one-way

In our first step up the ladder of complexity we will stick with just one experimental factor, but consider cases where that factor has more than two levels. 

As an example, we want to improve the chemical synthesis of a drug. It is already known that finely divided platinum is a good catalyst for the reaction, but we want to investigate the potential use of other related metals. We therefore perform the synthesis of the drug under fixed conditions of pressure, temperature, etc., but vary the catalyst added. The metals investigated are platinum, palladium, iridium and rhodium and additionally an alloy of palladium and iridium. Five replicate 

When results from several treatments are to be compared, multiple t-tests should not be used - this leads to increased risk of false positives. 

Instead we need a single test that will consider the whole data set and deliver a single verdict. The appropriate test is called the one-way analysis of variance . The one-way part of the name reflects the fact that there is still only one factor under investigation (in this case, which catalyst to use). 

CH13 ANALYSES OF VARIANCE - GOING BEYOND f-TESTS 

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A comparison of two or more means can be made with a one-way analysis of variance. This tool compares... 

FIGURE 11.3 One-way ANOVA (analysis of variance). One-way analysis of variance of basal rates of metabolism in melanophores (as measured by spontaneous dispersion of pigment due to G,.-protein activation) for four experiments. Cells were transiently transfected with cDNA for human calcitonin receptor (8 j-ig/ml) on four separate occasions to induce constitutive receptor activity. The means of the four basal readings for the cells for each experiment (see Table 11.4) are shown in the histogram (with standard errors). The one-way analysis of variance is used to determine whether there is a significant effect of test occasion (any one of the four experiments is different with respect to level of constitutive activity). 

One-way analysis of variance, 229-230, 230f—231f Operational model derivation of, 54-55 description of, 45—47, 46f function for variable slope, 55 for inverse agonists, 221 of agonism, 47f orthosteric antagonism, 222 partial agonists with, 124, 220-221 Opium, 147 Orphan receptors, 180 Orthosteric antagonism... 

Estimate the sampling variance (S s) by one-way analysis of variance (2 x 6n) using the average value of each well (estimation variance between each portion and each sample). 

One-way analysis of variance is based on a linear model like the following ... 

The measurement scheme of One-way analysis of variance is given in Table 5.1 for i = 1... m levels of the factor a (in analytical practice frequently a factor is studied only on two levels to compare, e.g., two laboratories, two operators, two different techniques, etc). 

One-way analysis of variance (ANOVA) to test the significant effect of the degrading impact on each soil characteristic was performed using the computer software, SPSS 10.0.5J (SPSS Japan Inc., Tokyo). The Dunnett T3 test was chosen as the post-hoc test. 

Data analysis was reduced to a separate one-way analysis of variance on the data from individual laboratories in order to examine the difference between types of sampling bottle on a single (common) hydrowire, and to determine the influences of the three types of hydrowire using a single type of sampling bottle (modified GO-FLO). Samples were replicated so that there were, in all cases, two or more replicates to determine the lowest level and analytical error. 

One-way analysis of variance/covariance/regression and preplanned and post hoc group comparisons. 

What is presented here is the workhorse of toxicology—the one-way analysis of variance. Many other forms exist for more complicated experimental designs. 

Analysis of Variance (ANOVA). Keeping in mind that the total variance is the sum of squares of deviations from the grand mean, this mathematical operation allows one to partition variance. ANOVA is therefore a statistical procedure that helps one to learn whether sample means of various factors vary significantly from one another and whether they interact significantly with each other. One-way analysis of variance is used to test the null hypothesis that multiple population means are aU equal. 

In this setting there is a technique, termed one-way analysis of variance (one-way ANOVA), which gives an overall p-value for the simultaneous comparison of all of the treatments. Suppose, for example, we have four treatment groups with means Pi, p2> P 3 P4. This procedure gives a p-value for the null hypothesis ... 

In summary, there is not much to be gained in using one-way analysis of variance with multiple treatment groups. A simpler analysis structuring the appropriate pairwise comparisons will more directly answer the questions of interest. One final word of caution though undertaking multiple comparisons in this way raises another problem, that of multiplicity. For the time being we will put that issue to one side we will, however, return to it in Chapter 10. 

For continuous data there is a procedure within the one-way analysis of variance methodology that is able to focus on this we would be looking for a trend across the treatment groups. 

Significantly different (p < 0.05) by one-way analysis of variance (ANOVA) with the Least Significant Difference post hoc test using JMP program from SAS, Cary, NC on a Macintosh. Reproduced with permission from (Ischiropoulos et al., 1992a). 

Having calculated the level of significance can be obtained from appropriate tables. The Wilcoxon signed rank test is the non-parametric equivalent of the paired t-test. The Kruskal-Wallis test is another rank sums test that is used to test the null hypothesis that k independent samples come from identical populations against the alternative that the means of the populations are unequal. It provides a non-parametric alternative to the one-way analysis of variance. 

Figure 2. Effects of local administration of TTX, alone and in combination with 10 pM imetit, on 100 mM potassium-evoked release of ACh from the cortex of freely moving rats. At 40 (Si) and 140 (S2) min the perfusion medium was changed from 4 to 100 mM KC1 for 10 min after equilibration. TTX was added 20 min, and imetit 10 min before S2 stimulation to the perfusion medium. Both remained throughout the S2 stimulation. Shown are means S.E. of (n) experiments. The presence of significant treatment effects was determined by one-way analysis of variance followed by Scheffe s test. P < 0.001 vs. control.

Each data point is subscripted, first to identify its column location and second to identify its row location. Thus, X32 (read X three two) is the data point in the third column and second row. Each column may be regarded as size I random sample drawn from the normal population. This matrix might represent the example of one-way analysis of variance given earlier. The columns would be the four catalysts and the rows would simply identify the succesive runs made at identical conditions. 

In one-way analysis of variance, we have several groups for which we wish to test equality of means. To apply the standard methods, we must assume that each group is normally distributed and that the population variance Ox is constant among the groups. In other words, one-way analysis of variance is used in situations when we want to test the J population means. The procedure is based on the assumption that... 

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 ... 

Data were expressed as the mean standard error of the mean (SEM). Differences between means were determined using one-way analysis of variance (ANOVA) followed by the Tukey-Kramer post hoc comparison and two-sided t test. For comparing percentages, nonparametric tests were also applied (Mann-Whitney, Kruskal-Wallis). Differences were considered significant when p < 0.05. 

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