Factor, ANOVA

Let us now investigate whether there is any major sex difference in the effect on the variate by a two-factor ANOVA as in Table 16.18. 

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. 

Spreadsheet 2.6. (a) Excel Data Analysis ToolPak menu showing ANOVA options, (b) Single-factor ANOVA menu. 

Spreadsheet 2.8. Excel menu for two-factor ANOVA with replication. 

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 ... 

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. 

An exploratory analysis was performed using a four-factor ANOVA model, with treatment, period, and sequence as fixed factors and subject within sequence as random factor. The results from the ANOVA were used to calculate the back-transformed 90 % confidence intervals (Cl) for the differences between the fed and fasted condition in the log-transformed exposure measurements (Cmax, AUCo-t and AUCo-cc). For Cmax the difference between fasting and fed conditions was found to be statistically significant while this was not the case for the AUC parameters. 

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). 

Analysis of Variance (ANOVA) A collection of statistical procedures for analysis of responses from experiments. Single-factor ANOVA allows comparison of more than two means of populations. 

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)... 

Yes, for single-factor ANOVA. No, for two-factor ANOVA. (Sections 4.6, 4.9)... 

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... 

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. 

Statistical analysis was made using a computer-assisted single-factor ANOVA and Newman-Keuls test multiple comparison. 

With respect to the data of the short-term stability study carried out at +37 °C (Table 6.7), an F-test (single-factor ANOVA) was used to determine whether or not a significant difference existed among the trimethyllead concentrations determined after 5, 10 and 15 days. According to this test, the... 

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. 

A statistical analysis was performed on the collected data. The 0. 05 level of significance was utilized. A two-factor ANOVA statistic was used to compare bare hand and outer glove microbial count differences between the zero- and 3-hour samples times. 

The concept underlying Phase 2 of the study is illustrated in Figure 2. The results from Phase 2 are presented in Table 3. A two-factor ANOVA model was used to compare times and product configurations. Figure 3 graphically displays the data obtained from the surfaces directly exposed to the inoculated ground beef for each product configuration. 

A two-factor ANOVA model was used to compare times and product configurations. 

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. 

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 ... 

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