One-factor ANOVA

Leaves were dark-adapted therefore, there is no detectable level of zeaxanthin. Concentrations are nmol pigment (mol chi a — b), the P value from one factor ANOVA is displayed below each column. V-A-Z = xanthophyll pool (violaxanthin, antheraxanthin, zeaxanthin) EPS = epoxidation state. Reprinted with permission from P. J. Ralph et al. [76]. 

When the ANOVA shows that the means are not equal, there are several post hoc that can be used to determine where the differences lie, such as the Tukey or the Student-Newman-Keuls test. However, the use of these tests is beyond the scope of this chapter. In addition to simple one-way or one-factor ANOVA described above, other types are available to analyse more complex situations involving several factors. Again, these are beyond the scope of this chapter. 

Spreadsheet 2.5. (a) Input to and (b) output from the one-factor ANOVA example. 

A common example where ANOVA can be applied is in interlaboratory trials or method comparison. For example, one may wish to compare the results from four laboratories, or perhaps to evaluate three different methods performed in the same laboratory. With inter-laboratory data, there is clearly variation between the laboratories (between sample/treatment means) and within the laboratory samples (treatment means). ANOVA is used in practice to separate the between-laboratories variation (the treatment variation) from the random within-sample variation. Using ANOVA in this way is known as one-way (or one factor) ANOVA. 

A one-factor ANOVA is the appropriate analysis here assuming that the data are normal The only factor of interest is the dose of drug given. There are three levels of this factor 10, 20, and 30 mg. Following convention, the results of an ANOVA are displayed in an ANOVA summary table such as the model in Table 11.3. In the following calculations the values are presented without their units of measurement (mmHg) simply for convenience. At the end of the calculations, however, it is very important to remember that the numerical terms represent values measured in mmHg. The calculations needed are as follows. 

There are times when the required assumptions for ANOVA, a parametric test, are not met. One example would be if the underlying distributions are non-normal. In these cases, nonparametric tests are very useful and informative. For example, we saw in Section 11.3 that a nonparametric analog to the two-sample ttest, Wilcoxon s rank sum test, makes use of the ranks of observations rather than the scores themselves. When a one-factor ANOVA is not appropriate in a particular case a corresponding nonparametric approach called the Kruskal-Wallis test can be used. This test is a hypothesis test of the location of (more than) two distributions. 

Use ANOYA if you want to know if there is significant difference among a number of instances of a factor. Always use ANOVA for more than one factor. ANOVA data must be normally distributed and homoscedastic. Use a Mest for testing pairs of instances. The data must be normally distributed but need not be homoscedastic. (Sections 3.8, 4.2)... 

An efficient way of determining whether there is a significant difference is to do a one-factor ANOVA. The means and 95% confidence intervals are calculated (see chapter 2) and plotted in figure 4.2. 

The Kruskal-Wallis model is the nonparametric analog of a one-factor ANOVA model. It is used to compare multiple groups of one factor. For example, suppose one wants to evaluate the antimicrobial effects of five different hand soaps the Kruskal-Wallis model could be employed for this evaluation. 

Kruskal-Wallis test is the technique tests the null hypothesis that several populations have the same median. It is the nonparametric equivalent of the one-factor ANOVA. The test statistic is ... 

There are various other ways of examining the variate in question in this case. Let us first examine simple one-way ANOVA of the variate by sex as in Table 16.16. In neither of the two cases was there any indication of significant treatment differences at any reasonable level. Because the two sexes did not show any pretreatment differences based on the two-factor analysis of the covariate, let us combine the two sexes and analyze the data by one-way ANOVA as in Table 16.17. In this case, because of the increased sample sizes for combining the two sexes, there was indication of some treatment differences (p = 0.0454). Unfortunately, this analysis assumes that because there were no pretreatment differences between the two sexes, that pattern will hold during the posttreatment period. That often may not be the case because of biological reasons. 

In Excel, there are three options for ANOVA one factor, two factor without replication, and two factor with replication. The different options in the Data Analysis Tools menu are shown in spreadsheet 2,6,... 

Choosing the one-factor option (which must have replicate data, although not necessarily the same number of repeats for each instance of the factor), the output includes an ANOVA table shown in spreadsheet 2,5, In the output table, SS stands for sum of squares, df is degrees of freedom, and MS is... 

FIGURE 10.2 (CONTINUED) and unattached larvae. N = six (6) replicates (dishes) were done for all treatments. The results of the assay are expressed as percentage settlement of the seawater (untreated) control. Data are mean + S.E. Treatments lacking error bars indicate 100% settlement in all replicates. Statistical analysis of the data (separate one-factor analysis of variance ANOVA for each of Figure 10.2A and 10.2B, followed by Tukey s post-hoc comparison among means) showed that only extracts from D. pulchra significantly deterred settlement (at both natural and twice natural concentrations). 

As noted in Section 7.5, one-factor independent groups ANOVA can also be used in cases where the independent groups f-test is appropriate. The term independent groups is derived in exactly the same way as was independent groups f-test, in that independent groups of subjects are employed. The term one-factor relates to the fact that, in our ongoing example, there is only one factor that is of interest that factor is type of treatment administered. A factor is an influence that one wishes to study it is of interest to know whether the factor is a systematic source of influence on, and therefore a systematic source of variance in, the data collected in a study. An equivalent designation is not necessary in the case of the f-test, since it can only be used when there is just one factor of interest. 

The simple fact that there are three treatment groups means that an independent groups f-test cannot be employed that test can only handle two treatment groups. In this case, a one-factor independent groups ANOVA is appropriate. From now on, the one-factor part of the name will be left off, since our examples focus on designs where only one source of influence is being investigated. 

Table 2.1 summarizes the single factor ANOVA calculations. The test for the equality of means is a one-tailed variance ratio test, where the groups MS is placed in the numerator so as to inquire whether it is significantly larger than the error MS ... 

Since the function of the one-way ANOVA is so similar to the of the two-sample /-test, it is no great surprise that the factors that govern its outcome are also virtually identical. They are shown in Figure 13.2. In considering this diagram, we need to keep a close eye on the term variation . There are two types. 

When more than one factor is present and the factors are crossed, a multifactor ANOVA is appropriate. Both main effects and interactions between the factors may be estimated. In a graphical representation of the results of an ANOVA the points are scaled so that any levels that differ by more than the difference exhibited in the distribution of the residuals are significantly different. 

ANOVA can be extended to situations where the experimental units (in our context, study participants) are classified on a number of factors. When they are classified on the basis of one factor, it is referred to as a one-way ANOVA. The result of partitioning the total variance into its components, in this case among and within samples defined by one factor, is displayed in Table 11.2. 

One-way ANOVA an ANOYA in which a single factor is varied. (Section 4.4)... 

In this chapter we briefly show how an ANOVA is performed for the simplest case of a single factor (so-called one-way ANOVA) and then for a two-way ANOVA. ANOVA is available (albeit in a restricted form) in Excel, and in most other statistical packages. Although we shall show you how to do a one-way ANOVA by hand, the chapter will concentrate on the interpretation of ANOVA output from software applications. 

In a one-way ANOVA we have instances of the factor being investigated, with replicate results for each instance. For example... 

Excel offers three flavors of ANOVA via its Analysis ToolPak ANOVA Single Factor ANOVA Two Factor with Replication and ANOVA Two Factor without Replication. The first is what we have just seen and accepts a data matrix set out as described, with variables in columns and repeats in rows. In this case the Grouped By Columns radio button is checked. Column headers in the first row can be included, which helps with interpreting the output. A value of a, the probability at which the null hypothesis will be rejected, must be specified for an / -test, with 0.05 being the default. Thus the F-value is tested at the 95% probability level. The output looks like that in table 4.3, except that the terms within groups and between groups are used, and there are two extra columns. One has the... 

Excel only caters for one- and two-way ANOVA. In two-way ANOVA there are two factors being considered. For example, we may be interested in the effect of changing the catalyst and the temperature in a synthesis. One factor is the catalyst (e.g., Zn or Li) and the other is temperature (e.g., 50, 70, or 90°C). In two-way ANOVA in Excel the distinction is made between measurements that are repeated and those for which only a single measurement is made, at each combination of factors. The layout for this example is given in table 4.5. 

Analysis of variance (ANOVA) is also a common parametric statistic for comparing data from more than two groups [2]. There are a number of variants of this model, depending upon the number and combination of groups, categories, and levels one desires to evaluate. Common ones include one-factor, two-factor, and three-factor designs, as well as crossover and nested designs. 

Analysis of the data for the different handwashing and gloving regimens was carried out using a two-factor ANOVA model. Both the time and product factors as well as the product versus factor interaction term were significant p < 0.05). The interaction is significant because each product began at the same baseline microbial level, but the levels were different from one another at the 3-hour study completion time. 

By means of a one-way ANOVA, the effect of one factor can be investigated at different levels. In our example, the effect of a laboratory on the results of determinations was tested. In many applications, several factors have to be evaluated simultaneously. For example, apart from the laboratory effects, influences by the operator and the quality ofthe instrumentation are to be expected. These effects can be studied by using the two- or multiway analysis of variance. 

In addition, measurements could be repeated for all factor combinations, as was shown for one-way ANOVA (see. Table 2.12). All sums must then be indexed over the repetitions for computation of the means (Eqs. 2.52-2.54). 

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