Three-way analysis

Multiway and particularly three-way analysis of data has become an important subject in chemometrics. This is the result of the development of hyphenated detection methods (such as in combined chromatography-spectrometry) and yields three-way data structures the ways of which are defined by samples, retention times and wavelengths. In multivariate process analysis, three-way data are obtained from various batches, quality measures and times of observation [55]. In image analysis, the three modes are formed by the horizontal and vertical coordinates of the pixels within a frame and the successive frames that have been recorded. In this rapidly developing field one already finds an extensive body of literature and only a brief outline can be given here. For a more comprehensive reading and a discussion of practical applications we refer to the reviews by Geladi [56], Smilde [57] and Henrion [58]. 

A.K. Smilde, Three-way analysis. Problems and prospects. Chemom. Intell. Lab. Syst., 15 (1992) 143-157. 

In the case that interactions prove to be insignificant, it should be gone over to the ab model the estimations of which for the various variance components is more reliable than that of the 2ab model. A similar scheme can be used for three-way ANOVA when the factor c is varied at two levels. In the general, three-way analysis bases on block-designed experiments as shown in Fig. 5.1. 

The wheat bran used in these studies was milled for us from a single lot of Waldron hard red spring wheat. Other foods and diet ingredients were purchased from local food suppliers. Data from HS-I was analyzed statistically by Student s paired t test, each subject acting as his own control. A three-way analysis of variance (ANOVA) was performed to test for significant differences betwen diet treatments, periods and individuals in HS-II and HS-III. 

Finally, we might add a third independent variable, catalyst concentration, to type of catalyst and temperature of setting. We then would use a three-way analysis of variance to determine whether differences in means exist. Of course, as additional independent variables are added, the calculations become much more complex, so that they are better carried out on a digital computer. 

Based on models and assumptions of one-way and two-way analyses of variance with or without replications of design points, it is possible to generalize for multiple-way analysis of variance. It is of interest to present the three-way analysis of variance for it is used quite often. In the case of a three-way analysis of variance the total number of observations is N=IxJxKxL, where I, J and K are numbers of levels or columns, rows and layers. L is the number of design-point replications or the number of observations in cells. Fig. 1.18 shows the tridimensional arrangement of columns, rows and layers. 

In calculation for the three-way analysis without replicating L=l, the same conventions in the notations were followed as for one and two-way analysis of variance. 

Presentation of definitions and procedures for the three-way analysis of variance, no replications of the design points L=l, is given in Table 1.26. For this kind of analysis with replicating L>1, the calculation is given in Table 1.27. The convention on notations were followed here ... 

Table 1.26 Three-way analysis of variance without replications... 

This is an example of three-way analysis of variance with no design-point replication. As we have only one value for each set of factors, the variance or the mean square within the cell as an estimate of system variance cannot be calculated. In the lack of error variance, or rather Reproducibility variance. Interaction of a higher order can be used as error estimate for the F-test. Although all statisticians do not agree with this approach, the three-way interaction variance C x R x L was taken as the error estimate for F-test. The tabular results show that only the effects of columns and layers, or temperature and catalyst, are significant. Pressure and interaction are not important at the 95% confidence level. The other approach in estimating repro-... 

In developing a procedure for bacteriological testing of milk, samples were tested in an apparatus that includes two components bottles and kivets. All six combinations of two bottle types and three kivet types were tested ten times for each sample. The table contains data on the number of positive tests in each of ten testings. If we remember section 1.1.1 then the obtained values of positive tests are a random variable with the binomial distribution. For a correct application of the analysis of variance procedure, the results should be normally distributed. It is therefore possible to transform the obtained results by means of arcsine mathematical transformation for the purpose of example of three-way analysis of variance with no replications, no such transformations are necessary. The experiment results are given in the table ... 

As the values with an asterix in Problem 1.41 mean that they were obtained for one thickness of the sample and those with no asterix for another one, apply the three-way analysis of variance for all the data in Problem 1.41 taking into account the new sample thickness factor. 

There are two competing and equivalent nomenclature systems encountered in the chemical literature. The description of data in terms of ways is derived from the statistical literature. Here a way is constituted by each independent, nontrivial factor that is manipulated with the data collection system. To continue with the example of excitation-emission matrix fluorescence spectra, the three-way data is constructed by manipulating the excitation-way, emission-way, and the sample-way for multiple samples. Implicit in this definition is a fully blocked experimental design where the collected data forms a cube with no missing values. Equivalently, hyphenated data is often referred to in terms of orders as derived from the mathematical literature. In tensor notation, a scalar is a zeroth-order tensor, a vector is first order, a matrix is second order, a cube is third order, etc. Hence, the collection of excitation-emission data discussed previously would form a third-order tensor. However, it should be mentioned that the way-based and order-based nomenclature are not directly interchangeable. By convention, order notation is based on the structure of the data collected from each sample. Analysis of collected excitation-emission fluorescence, forming a second-order tensor of data per sample, is referred to as second-order analysis, as compared with the three-way analysis just described. In this chapter, the way-based notation will be arbitrarily adopted to be consistent with previous work. 

As with univariate and multivariate calibration, three-way calibration assumes linear additivity of signals. When the sample matrix influences the spectral profiles or sensitivities, either care must be taken to match the standard matrix to those of the unknown samples, or the method of standard additions must be employed for calibration. Employing the standard addition method with three-way analysis is straightforward only standard additions of known analyte quantity are needed [42], When the standard addition method is applied to nonbilinear data, the lowest predicted analyte concentration that is stable with respect to the leave-one-out cross-validation method is unique to the analyte. 

There are numerous other considerations not covered in this chapter that a thorough treatment of three-way analysis would demand. Perhaps the most important of these is the choosing of the optimal number of factors, N, to include in the three-way... 

The analysis of results from a 23 factorial experiment requires a three-way analysis of variance. That is, in addition to having row and column effects, there are also layer effects and interactions among the three. 

The second part deals with multivariate pseudo three-way analysis, i.e. two-way analysis of two-way and unfolded three-way NMR data. These methods are included, since they may offer an alternative choice for analysing three-way NMR data matrices and, furthermore, they are an important supplement to the real three-way methods described in the third part of the section. Here, application of three-way methods to three-way data is described and exemplified. The fourth part covers a short description of some related three-way methods primarily to inform about their existence and possible usage. 

In order to explain the basics of three-way analysis, it is easiest to start with two-way data and then extend the concepts to three-way. Two-way data are represented in a two-way data matrix typically with columns as variables and rows as objects (Figure 1.1). Three-way... 

One of the typical purposes of using three-way analysis on a block of data is exploring the interrelations in those data. An example is a three-way environmental data set consisting of measured concentrations of different chemical compounds on several locations in a geographical area at several points in time. Three-way analysis of such a data set can help in distinguishing patterns, e.g., temporal and spatial behavior of the different chemical... 

Finally, two important decomposition methods for two-way analysis (PCA) and three-way analysis (PARAFAC) are introduced briefly, because these methods are needed in the following chapter. 

The third product which is useful in three-way analysis is the Khatri-Rao product [McDonald 1980, Rao Mitra 1971] which is defined as follows ... 

---