Two-factors 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. 

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

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

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. 

Other forms of ANOVA The simple ANOVA set out above tests for the presence of one unknown factor that produces (additive) differences between outwardly similar groups of measurements. Extensions of the concept allow one to simultaneously test for several factors (two-way ANOVA, etc.). The... 

By means of Two-way ANOVA two factors can be studied simultaneously. The model... 

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. 

Table lO.Comparison of two-factor analysis via Friedman s statistic and conventional randomized block experiment in (parametric) ANOVA... 

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,... 

Anova Two-Factor With Replication Anova Two-Factor Without Replication Correlation... 

Spreadsheet 2.7. (a) Input data and (b) ANOVA table for two factors with replication. 

It has become uncommon for ANOVA with more than two factors to be analyzed on a computer, owing to considerations of time, ease, and accuracy. It will presume that established computer programs will be used to perform the necessary mathematical manipulation of ANOVA. 

TABLE 6.3. ANOVA Table for a Two-Factor Balanced Design with Replicates... 

The two-way analysis of variance is used where two factors are being varied and all combinations of both factors have been studied. This ANOVA will test whether certain levels of each factor are consistently associated with high or low values for the endpoint. It will also test whether the effect of changing from one level to another within a factor is a constant increase/decrease or whether the effect seen depends upon the level of the other factor ( interaction ). Where interaction is present, a graphical method can be used to clarify what form the interaction takes. 

The functional analysis of variance in (14) is then computed from the estimated corrected effects. Here, the 41 main effects and 820 two-factor-interaction effects together account for about 89% of the total variance of the predictor. Hence, about 11 % of the predictor s total variability is due to higher-order effects. Table 1 shows the estimated main effects and interaction effects that contribute at least 1% to the functional ANOVA These 12 effects together account for about 74% of the total variation. Only six variables appear in these 12 effects they are described in Table 2. 

Tables 13.5 to 13.9 show the results of applying two-way ANOVA to the 1-propanol content in wines resulting from fermentation of the same must with or without skins and in the presence or absence of sulfur dioxide, that corresponds to a 7 factorial design (Herraiz et al. 1990 Martln-Alvarez et al. 2006), obtained with the STATISTICA program (Factorial ANOVA procedure, in the ANOVA module). They include the two-way ANOVA table (only the skins factor has a significant influence at the 95% confidence level, P < 0.05), the mean and the standard error values of 1-propanol content for each combination of levels of factors (greater values in wines elaborated with the addition of skins), and the results of the S-N-K test for comparison of means of the skins factor. Figure 13.2 shows plots of the means of 1-propanol content for each level of factors.

Multifactorial ANOVA can be used to test the effect of more than two factors (Pozo-Bay6n 2003a Martm-Alvarez et al. 2006). 

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. 

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. 

Method of moments estimates (also known as ANOVA estimates) can be calculated directly from the raw data as long as the design is balanced. The reader is referred to Searle et al. (11) for a thorough but rather technical presentation of variance components analysis. The equations that follow show the ANOVA estimates for the validation example. First, a two-factor with interaction ANOVA table is computed (Table 7). Then the observed mean squares are equated to the expected mean squares and solved for the variance components (Table 8 and the equations that follow). 

A first pass analysis was conducted on the primary responses. Note that because we have replicated the center point we now have an estimate (although crude) of the underlying variation allowing for the use of the ANOVA approach to the analysis. An ANOVA model that included all main effects, two-factor interactions, and a curvature term was fit to each of the five key responses. The results are summarized in Table 10. Tabled is the estimated effect for each of the main effects along with a flag indicating the magnitude of the... 

There are also situations where two of the modes of the three-way array can be expressed as qualitative variables in a two-way ANOVA and where the response measured is some continuous (spectral, chromatogram, histogram or otherwise) variable. Calling the ANOVA factors treatment A and treatment B a (Treatment x Treatment x Variable) array is made, see Figure 10.5. An example for particle size distribution in peat slurries was explained in Chapters 7 and 8. The two-way ANOVA can become a three-way ANOVA in treatments A, B and C, leading to a four-way array etc. Having three factors and only one response per experiment will lead to a three-way structure and examples will be given on how these can be modeled efficiently even when the factors are continuous. 

Sometimes two of the modes of a three-way array are formed by a two-way ANOVA layout in qualitative variables and the three-way structure comes from measuring a continuous variable (e.g. a spectrum) in each cell of the ANOVA layout. In such cases each cell of the ANOVA has a multitude of responses that not even MANOVA can handle. When quantitative factors are used, one or two modes of the three-way array may result from an experimental design where the responses are noisy spectra that behave nonlinearly. Such data are treated in this section. 

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