Univariate tests

In many engineering applications, however, we can easily reduce the problem to the univariate tests presented in the previous section by assuming that the covariance matrix of the errors can be written as... 

Baxter, M.J. and Gale, N.H. (1998). Testing for multivariate normality via univariate tests a case study using lead isotope ratio data. Journal of Applied Statistics 25 671-683. 

Second, a univariate test for normality is usually conducted. Many software packages have these built-in to their procedures, e.g., Shapiro Wilks test in the Uni-... 

Many tests exist for detecting outliers in univariate data, but most are designed to check for the presence of a single rogue value. Univariate tests for outliers are not designed for multivariate outliers. Consider Figure 1.6, the majority of data exists in the highlighted pattern space with the exception of the two points denoted A and B. Neither of these points may be considered a univariate outlier in terms of variable x or x2, but both are well away from the main cluster of data. It is the combination of the two variables that identifies the presence of these outliers. Outlier detection and treatment is of major concern to analysts, particularly with multivariate data where the presence of outliers may not be immediately obvious from visual inspection of tabulated data. 

Fig. 8.5 Identification of the most discriminative mass spectral features between commensal ( non-CC17 ) and hospital-associated ( CC17 ) strains oiE.faecium. For this purpose, univariate /-tests were earned out on the basis of altogether 266 mass spectra from which peak tables with 50 entries per mass spectrum were extracted. Independent /-tests were systematically carried out for each m/z region (width of 700 ppm). The p values of the /-tests were plotted against the centers of the m/z regions. Small p values cast doubt on the null-hypothesis of equal class means. Note the inverse logarithmie scaling. Reproduced from Lasch et al. (2014), with permission...

In the introduction to Part A we discussed the arch of knowledge [1] (see Fig. 28.1), which represents the cycle of acquiring new knowledge by experimentation and the processing of the data obtained from the experiments. Part A focused mainly on the first step of the arch a proper design of the experiment based on the hypothesis to be tested, evaluation and optimization of the experiments, with the accent on univariate techniques. In Part B we concentrate on the second and third steps of the arch, the transformation of data and results into information and the combination of information into knowledge, with the emphasis on multivariate techniques. 

Fig. 33.14. Density estimate for a test set using normal potential functions (univariate case).

For continuous variables you may be required to provide inferential statistics along with the descriptive statistics that you generate from PROC UNIVARIATE. The inferential statistics discussed here are all focused on two-sided tests of mean values and whether they differ significantly in either direction from a specified value or another population mean. Many of these tests of the mean are parametric tests that assume the variable being tested is normally distributed. Because this is often not the case with clinical trial data, we discuss substitute nonparametric tests of the population means as well. Here are some common continuous variable inferential tests and how to get the inferential statistics you need out of SAS. 

PROC UNIVARIATE can also be used to perform the one-sample f-test as follows ... 

We will not repeat Anscombe s presentation, but we will describe what he did, and strongly recommend that the original paper be obtained and perused (or alternatively, the paper by Fearn [15]). In his classic paper, Anscombe provides four sets of (synthetic, to be sure) univariate data, with obviously different characteristics. The data are arranged so as to permit univariate regression to be applied to each set. The defining characteristic of one of the sets is severe nonlinearity. But when you do the regression calculations, all four sets of data are found to have identical calibration statistics the slope, y-intercept, SEE, R2, F-test and residual sum of squares are the same for all four sets of data. Since the numeric values that are calculated are the same for all data sets, it is clearly impossible to use these numeric values to identify any of the characteristics that make each set unique. In the case that is of interest to us, those statistics provide no clue as to the presence or absence of nonlinearity. 

As for testing other characteristics of a univariate calibration, there are also ways to test for statistical significance of the slope, to see whether unity slope adequately describes the relationship between test results and analyte concentration. These are described in the book Principles and Practice of Spectroscopic Calibration [10]. The Statistics are described there, and are called the Data Significance t test and the Slope Significance t test (or DST and SST tests ). Unless the DST is statistically significant, the SST is meaningless, though. 

To evaluate the quantitation capabilities of the experimental design, univariate calibration curves were constructed at 770 nm for four tested vapors as shown in Fig. 4.13. Upon exposure to the highest tested concentration of DCM and toluene... 

Fig. 4.13 Univariate calibration curves constructed at 770 nm for four tested vapors over 0 0.1 PI P0 concentration range. Reprinted from Ref. 15 with permission. 2008 Institute of Electrical and Electronics Engineers...

Univariate case data from normally distributed populations generally have a higher information value associated with them but the traditional hypothesis testing techniques (which include all the methods described in this section) are generally neither resistant nor robust. All the data analyzed by these methods are also, effectively, continuous that is, at least for practical purposes, the data may be represented by any number and each such data number has a measurable relationship to other data numbers. 

The test assumes that the data are univariate, continuous and normally distributed. 

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