Analysis of variances

Analysis of variance (ANOVA) is used to analyze observations that depend on the operation of one or more effects. These effects are caused by factors the levels of which are also called groups, for example, different laboratories. 

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. 

To test for systematic differences between the groups, it is assumed that each measurement, can be described by the sum of the total mean, Jtotai, the group mean, Jj, and the residual random error, e, according to the following  

The total variance, expressed as the sum of squares of deviations from the grand mean, is partitioned into the variances within the different groups and between the groups. This means that the sum of squares corrected for the mean, is obtained from 

For deciding on the acceptance of the null hypothesis - that the groups belong to the same population and differ only randomly - an f test is performed (see Eq. (2.37))  

Analysis of variance is, perhaps, the most powerful statistical technique available to the research worker in view of its wide applicability and ease of application. It will therefore be fitting to spend considerable time discussing not only the mechanics of this technique but also the type of problem to which it has legitimately been applied. 

Arithmetically, analysis of variance is nothing more than a device for subdividing the total variation in a set of observations (as measured by the sum of the squares of the deviations of each observation from the 

The estimation of the true average effect of the various factors. 

The estimation of the amount of variation to be expected among repeated observations of a similar type and, eventually, the estimation of the proportion of the total variability that may be ascribed to each factor. 

The estimation (or discovery) of relationships among the different factors involved. 

On a pilot-plant for producing composite rocket propellant four batches of the same composition were made under the same process conditions with the two different burning-rate catalysts. From the obtained propellant experimental rocket motors were static fired. The following burning rates at 70 [bar] pressure were determined from the calculated burning rate laws  

Are there significant differences in catalyst effects on burning rates  

The technique known as analysis of variance (ANOVA)2) uses tests based on variance ratios to determine whether or not significant differences exist among the means of several groups of observations, where each group follows a normal distribution. The analysis of variance technique extends the t-test used to determine whether or not two means differ to the case where there are three or more means. 

The analysis of variance is used very widely in the biological, social and physical sciences. The technique was first developed by R. A. Fisher and his colleagues in England in the 1920s. Fisher has said that the analysis of variance is merely a convenient way of arranging the arithmetic . This statement points out that the statistical principles underlying the analysis of variance are quite simple but the calculations can become quite involved, so that they require careful and systematic arrangement. 

We might extend our first example using four different catalysts to study the effect of temperature as well. We could pick three different temperatures and determine the setting rate for each of the four catalysts. This would require a two-way analysis of variance to determine significant differences among the 12 setting times that we would obtain. 

Finally keep in mind that analysis of variance (ANOYA) is the most powerful statistical technique for evaluating the results of factorial designs with replications if the significance of factors is of interest, rather than the models of their relationship. 

To look at the experimental design from the point of view of ANOVA we invert the scheme used so far. For the two-factorial case we can build Tab. 3-2. 

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]). 

In addition it is now time to think about the two assumption models, or types of analysis of variance. ANOVA type 1 assumes that all levels of the factors are included in the analysis and are fixed (fixed effect model). Then the analysis is essentially interested in comparing mean values, i.e. to test the significance of an effect. ANOVA type 2 assumes that the included levels of the factors are selected at random from the distribution of levels (random effect model). Here the final aim is to estimate the variance components, i.e. the variance fractions with respect to total variance caused by the samples taken or the measurements made. In that case one is well advised to ensure balanced designs, i.e. equally occupied cells in the above scheme, because only then is the estimation process straightforward. 

Assume that we have sampled soil four times at each field location given by the factor combinations, with the following result  

In Section 3.3 a method was described for comparing two means to test whether they differ significantly. In analytical work there are often more than two means to be compared. Some possible situations are comparing the mean concentration of protein in solution for samples stored under different conditions comparing the mean results obtained for the concentration of an analyte by several different methods and comparing the mean titration results obtained by several different experimentalists using the same apparatus. In all these examples there are two possible sources of variation. The first, which is always present, is due to the random error in measurement. This was discussed in detail in the previous chapter it is this error which causes a different result to be obtained each time a measurement is repeated under the same conditions. The second possible source of variation is due to what is known as a controlled or fixed-effect factor. For the examples above. 

The tests and examples discussed above have concentrated on the statistics associated with a single variable and comparing two samples. When more samples are involved a new set of techniques is used, the principal methods being concerned with the analysis of variance. Analysis of variance plays a major role in statistical data analysis and many texts are devoted to the subject. Here, we will only discuss the topic briefly and illustrate its use in a simple example. 

Consider an agricultural trial site sampled to provide six soil samples which are subsequently analysed colorimetrically for phosphate concentration. The task is to decide whetW the phosphate content is the same in each sample. 

Source of variation Sum of squares Degrees of freedom Mean squares F-Test 

The total variation in the data can be partitioned between the variation amongst the sub-samples and the variation within the sub-samples. The computation proceeds by determining the sum of squares for each source of variation and then the variances. 

The total variance for all replicates of all samples analysed is given, from Equation (3), by 

Calculate the mean change score for each treatment group. 

Calculate the difference between the mean change score for the drug treatment group and the mean change score for the placebo group, i.e., the effect size. 

Divide the effect size by the error variance to give the test statistic f. 

Calculate the degrees of freedom associated with the f-value. 

---

Hirsch, R. F. Analysis of Variance in Analytical Chemistry, Anal. Chem., 49 691A (1977). Jaffe, A. J., and H. F. Spirer, Misused Statistics—Straight Talk for Twisted Numbers, Marcel Dekker, New York, 1987. 

A variety of statistical methods may be used to compare three or more sets of data. The most commonly used method is an analysis of variance (ANOVA). In its simplest form, a one-way ANOVA allows the importance of a single variable, such as the identity of the analyst, to be determined. The importance of this variable is evaluated by comparing its variance with the variance explained by indeterminate sources of error inherent to the analytical method. 

Variance was introduced in Chapter 4 as one measure of a data set s spread around its central tendency. In the context of an analysis of variance, it is useful to see that variance is simply a ratio of the sum of squares for the differences between individual values and their mean, to the degrees of freedom. For example, the variance, s, of a data set consisting of n measurements is given as... 

Once a significant difference has been demonstrated by an analysis of variance, a modified version of the f-test, known as Fisher s least significant difference, can be used to determine which analyst or analysts are responsible for the difference. The test statistic for comparing the mean values Xj and X2 is the f-test described in Chapter 4, except that Spool is replaced by the square root of the within-sample variance obtained from an analysis of variance. 

An analysis of variance can be extended to systems involving more than a single variable. For example, a two-way ANOVA can be used in a collaborative study to determine the importance to an analytical method of both the analyst and the instrumentation used. The treatment of multivariable ANOVA is beyond the scope of this text, but is covered in several of the texts listed as suggested readings at the end of the chapter. 

Collaborative testing provides a means for estimating the variability (or reproducibility) among analysts in different labs. If the variability is significant, we can determine that portion due to random errors traceable to the method (Orand) and that due to systematic differences between the analysts (Osys). In the previous two sections we saw how a two-sample collaborative test, or an analysis of variance can be used to estimate Grand and Osys (or oJand and Osys). We have not considered, however, what is a reasonable value for a method s reproducibility. 

The following texts provide additional information about ANOVA calculations, including discussions of two-way analysis of variance. Graham, R. C. Data Analysis for the Chemical Sciences. VCH Publishers New York, 1993. 

This experiment demonstrates how control charts and an analysis of variance can be used to evaluate the quality of results in a quantitative analysis for chlorophyll a and b in plant material. 

Lambs received saline, oST at 40 )-lg/kg BW, or the indicated dose of hGRF per kg BW four times per day for 42 or 56 days. Half of the lambs were withdrawn from treatment after 42 days. Carcass data shown are for lambs treated 56 days. Carcass composition data were analy2ed by analysis of variance using carcass weight as the covariate. Data are summarized in Ref. 85. 

W. Mendenhall, Introduction to EinearMode/s and the Design andAna/ysis of Experiments, Duxbury Press, Belmont, Calif., 1968. This book provides an introduction to basic concepts and the most popular experimental designs without going into extensive detail. In contrast to most other books, the emphasis in the development of many of the underlying models and analysis methods is on a regression, rather than an analysis-of-variance, viewpoint. 

The resultant analysis-of-variance tables will remain exactly the same. However,... 

Suppose we have two methods of preparing some product and we wish to see which treatment is best. When there are only two treatments, then the sampling analysis discussed in the section Two-Population Test of Hypothesis for Means can be used to deduce if the means of the two treatments differ significantly. When there are more treatments, the analysis is more detailed. Suppose the experimental results are arranged as shown in the table several measurements for each treatment. The goal is to see if the treatments differ significantly from each other that is, whether their means are different when the samples have the same variance. The hypothesis is that the treatments are all the same, and the null hypothesis is that they are different. The statistical validity of the hypothesis is determined by an analysis of variance. 

The following quantities are needed for the analysis of variance table. 

One often wishes to determine if, in a least squares treatment, addition of a new parameter will improve significantly the fit of the data. This is readily achieved by analysis of variance. Since this technique is little known, it will be briefly outlined here. 

Figure 61.7 Cumulative cash flow Analysis of variances ...

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). 

FIGURE 11.4 Two-way analysis of variance. Arrangement of data in rows and columns such that each row of the cell culture plate (shown at the top of the figure) defines a single dose-response curve to the agonist. Also, data is arranged by plate in that each plate defines eight dose-response curves and the total data set is comprised of 32 dose-response curves. The possible effect of location with respect to row on the plate and/or which plate (order of plate analysis) can be tested with the two-way analysis of variance. 

Results of the two-way analysis of variance for data shown in Figure 11.4 and Table 11.5. 

Possible significant differences between samples can be estimated by one-way and two-way analysis of variance. 

Analysis of variance (ANOVA), a statistical procedure that quantifies differences between means of samples and the extent of variances within and between those means to determine the probability of there being a difference in the samples. 

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... 

---