Single-factor ANOVA

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

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 ... 

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. 

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... 

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... 

Select ANOVA Single Factor from the list of options to bring up the dialog box. 

Figure 2.2. ANOVA (single factor) with Excel. Output of ANOVA data analysis of NAD(H) assays of 15 samples from each of two tissues using Excel is shown. The difference in NAD(H) content of the two tissues are indicated by a small P value and F > Fom (reject H0).

The ANOVA calculation can be performed in Excel only if the Data Analysis Toolbox has been installed. To perform the calculation, you can select the ANOVA tool under ToolsVData Analysis ANOVA Single Factor from the Excel toolbar. Using the following example of arsenic content of coal taken from different parts of a ship s hold, where there are five sampling points and four aliquots or specimens taken at each point, we have the data as shown below ... 

To perform the analysis, enter the data into an Excel spreadsheet (start at the top left-hand comer cell Al), then select the ANOVA Single Factor option from the Tool Menu. Select all the data by entering B 2 E 6 in the input range box (or select the data using the mouse). Now ensure that you select the Grouped By Rows Radio Button, as the default is to assume the data are grouped in columns (remember we want to... 

The one-way analysis of variance is used where there is a single factor that will be set to three or more levels. It is not appropriate to analyse such data by repeated /-tests as this will raise the risk of false positives above the acceptable level of 5 per cent. If the ANOVA produces a significant result, this only tells us that at least one level produces a different result from one of the others. It does not tell us which level differs from... 

The type of ANOVA shown in Figure 7-4 is known as a single-factor, or oneway, ANOVA. Often, several factors may be involved, such as in an experiment to... 

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. 

Let us start with an ANOVA in case of a single factor, termed a one-way analysis of variance. Table 2.12 demonstrates the general scheme of the measurements of this type of ANOVA. 

This result can also be derived from the ANOVA Table 6.3. There the p value is given with 0.0391. This value is lower than the significance level of 0.05 (1 - 0.95) considered here, and therefore, the test is significant. In other words, the risk that all of the factors are different from zero is 3.91%. Since in our model, we only have a single factor, x, we can assume with (100 - 3.91) = 96.09% probabihty that the effect oix is realistic. 

The Cl is defined as the displacement along the direction of the gradient divided by the total migration distance and is used to quantify the cell motility toward the chemokine gradient. The results were evaluated by the Student s f-test and single factor analysis of variance (ANOVA). The number (JV) of cells is indicated in Fig. 5. 

One of the assumptions of one-way (and other) ANOVA calculations is that the uncontrolled variation is truly random. However, in measurements made over a period of time, variation in an uncontrolled factor such as pressure, temperature, deterioration of apparatus, etc., may produce a trend in the results. As a result the errors due to uncontrolled variation are no longer random since the errors in successive measurements are correlated. This can lead to a systematic error in the results. Fortunately this problem is simply overcome by using the technique of randomization. Suppose we wish to compare the effect of a single factor, the concentration of perchloric acid in aqueous solution, at three different levels or treatments (0.1 M, 0.5 M, and 1.0 M) on the fluorescence intensity of quinine (which is widely used as a primary standard in fluorescence spectrometry). Let us suppose that four replicate intensity measurements are made for each treatment, i.e. in each perchloric acid solution. Instead of making the four measurements in 0.1 M acid, followed by the four in 0.5 M acid, then the four in 1 M acid, we make the 12 measurements in a random order, decided by using a table of random numbers. Each treatment is assigned a number for each replication as follows ... 

As we may remember from Sections 2.3 and 2.4.10, the ANOVA technique is useful in cases where the number of results in each cell is different (but see below ). This may happen sometimes when single experiments fail or, in environmental analysis, when some samples are exhausted more quickly than others or when sampling fails. We also recognize ANOVA to be a valuable technique for the evaluation of data from planned (designed) environmental analysis. In this context the principle of ANOVA is to subdivide the total variation of the data of all cells, or factor combinations, into meaningful component parts associated with specific sources of variation for the purpose of testing some hypothesis on the parameters of the model or estimating variance components (ISO 3534/3 in [ISO STANDARDS HANDBOOK, 1989]). 

A Factor is an aspect of an experiment that we can alter to see if this changes the endpoint we are measuring. The various different possibilities for each factor are then referred to as levels . While f-tests are used with the simplest experimental designs - a single experimental factor that has just two levels - for more complex designs, analyses of variance (ANOVAs) are called for. 

In this first section, we will consider the statistical methods to process data, originating in the observation of a single continuous random variable. We will distinguish three possible situations, with one, two or more than two data sets of the observed variable. In the last case, we will present the Analysis of Variance (ANOVA) for one or more factors. 

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. 

Each single combination of the factors has been replicated twiee to evaluate the significance of the factor modification on every dependent variable. Furthermore, the factorial analysis of variance (ANOVA) and the response surfaee method (RSM) have been applied, and the first-order regression models linking the dependent variable to the two control factors have been found and analyzed with an ANOVA. 

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