Analysts ANOVA

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. 

Let s use a simple example to develop the rationale behind a one-way ANOVA calculation. The data in Table 14.7 show the results obtained by several analysts in determining the purity of a single pharmaceutical preparation of sulfanilamide. Each column in this table lists the results obtained by an individual analyst. For convenience, entries in the table are represented by the symbol where i identifies the analyst and j indicates the replicate number thus 3 5 is the fifth replicate for the third analyst (and is equal to 94.24%). The variability in the results shown in Table 14.7 arises from two sources indeterminate errors associated with the analytical procedure that are experienced equally by all analysts, and systematic or determinate errors introduced by the analysts. 

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. 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

It should be noted that in this example the performance of only one variable, the three analysts, is investigated and thus this technique is called a one-way ANOVA. If two variables, e.g. the three analysts with four different titration methods, were to be studied, this would require the use of a two-way ANOVA. Details of suitable texts that provide a solution for this type of problem and methods for multivariate analysis are to be found in the Bibliography, page 156. 

Analysis of Variance (ANOVA) is a useful tool to compare the difference between sets of analytical results to determine if there is a statistically meaningful difference between a sample analyzed by different methods or performed at different locations by different analysts. The reader is referred to reference [1] and other basic books on statistical methods for discussions of the theory and applications of ANOVA examples of such texts are [2, 3],... 

Within-laboratory reproducibility studies should cover a period of three or more months and these data may need to be collected during the routine use of the method. It is possible, however, to estimate the intermediate precision more rapidly by deliberately changing the analyst, instrument, etc. and carrying out an analysis of variance (ANOVA) [9]. Different operators using different instruments, where these variations occur during the routine use of the method, should generate the data. 

A basic assumption underlying r-tests and ANOVA (which are parametric tests) is that cost data are normally distributed. Given that the distribution of these data often violates this assumption, a number of analysts have begun using nonparametric tests, such as the Wilcoxon rank-sum test (a test of median costs) and the Kolmogorov-Smirnov test (a test for differences in cost distributions), which make no assumptions about the underlying distribution of costs. The principal problem with these nonparametric approaches is that statistical conclusions about the mean need not translate into statistical conclusions about the median (e.g., the means could differ yet the medians could be identical), nor do conclusions about the median necessarily translate into conclusions about the mean. Similar difficulties arise when - to avoid the problems of nonnormal distribution - one analyzes cost data that have been transformed to be more normal in their distribution (e.g., the log transformation of the square root of costs). The sample mean remains the estimator of choice for the analysis of cost data in economic evaluation. If one is concerned about nonnormal distribution, one should use statistical procedures that do not depend on the assumption of normal distribution of costs (e.g., nonparametric tests of means). 

Figure 3.5 Design for a nested ANOVA in which the factors laboratories, analysts, instruments and days are examined...

Let us use an example to illustrate how the ANOVA calculations are performed on some test data. A chemist wishes to evaluate four different extraction procedures that can be used to determine an organic compound in river water (the quantitative determination is obtained using ultraviolet [UV] absorbance spectroscopy). To achieve this goal, the analyst will prepare a test solution of the organic compound in river water and will perform each of the four different extraction procedures in replicate. In this case, there are three replicates for each extraction procedure. The quantitative data is shown below. 

ANOVA calculations are straightforward in this example and are easily expanded to situations in which there are higher numbers of categories. The first ANOVA quantity we calculate is C, the correction factor. C is simply the square of the sum of all the individual values, divided by N, the total number of values (Lxijkif /N, where i = 1 - 2 (analysts), j = 1 - 2 (days). A = 1 — 3 (specimens), and 1=1—2 values for each specimen for each analyst on each day. Table 8 demonstrates the calculation of this and the other calculated ANOVA quantities. The quantity A is calculated as the sum of the squared individual values The ANOVA quantity for each classi-... 

In ANOVA for the results from the two analysts, MS rror = 1.2953 with 19 degrees of freedom. 

X 5.44/55.12) lower than those of analyst 2 using the assay method. Because the value 0 is not included in the 95%i confidence interval, we can conclude at the 5% level that there is a statistically significant difference between the two analysts, the same conclusion we had with ANOVA. However, a difference of 9.88%i, the maximum confidence interval limit, might not be large enough for us to reject that the two analysts are comparable in their performance of the assay method. This is a decision that was not possible based solely on ANOVA results. 

The factor can be considered the independent variable, whereas the response is the dependent variable. Figure 7-4 illustrates how to visualize ANOVA data for the five analysts determining Ca in triplicate. 

Figure 7-4 Pictorial of the results from the ANOVA study of the determination of calcium by five analysts. Each analyst does the determination in triplicate. Analyst is considered a factor, whereas analyst 1, analyst 2, analyst 3, analyst 4, and analyst 5 are levels of the factor.

Figure 7-5 Pictorial representation of the ANOVA principle. The results of each analyst are considered a group. The triangles (A) represent individual results, and the circles ( ) represent the means. Here the variation between the group means is compared with that within groups.

Example 7-9 shows an application of ANOVA to the determination of calcium by five analysts. The data are those used to construct Figures 7-4 and 7-5. 

From the F table on page 159, the critical value of F at the 95% confidence level for 4 and 10 degrees of freedom is 3.48. Since F exceeds 3.48, we reject //q at the 95% confidence level and conclude that there is a significant difference among the analysts. The ANOVA table is shown here. 

Spreadsheet Summary In Chapter 3 of Applications of Microsoft Excel in Analytical Chemistry, the use of Excel to perform ANOVA procedures is described. There are several ways to do ANOVA with Excel. First, the equations from this section are entered manually into a worksheet, and Excel is invoked to do the calculations. Second, the Analysis ToolPak is used to carry out the entire ANOVA procedure automatically. The results of the five analysts from Example 7-9 are analyzed by both these methods. 

A factor is whatever we are testing in ANOVA, for example an analytical method, sampling position in a silo, the gender of an analyst. Instances of the factor are the particular examples of that factor chosen for study, for example a spectrophotometric method and an electrochemical method, measures at the top, middle, and bottom of a silo, and male analysts and female analysts. (Section 4.3)... 

The row that assesses the interaction gives important information. It was quite obvious from the data that the significant difference between the means for the different pipetting techniques was independent of the analyst. However, if the data were as in spreadsheet 4.9, then you would suspect that a significant difference in means between the two pipetting methods would be dependent on whether Quinn was the analyst. The output of the Two Factor with Replicates ANOVA for this data is shown in spreadsheet 4.10. 

The primary performance measures of a ligand-binding assay are bias/trueness and precision. These measures along with the total error are then used to derive and evaluate several other performance characteristics such as sensitivity (LLOQ), dynamic range, and dilutional linearity. Estimation of the primary performance measures (bias, precision, and total error) requires relevant data to be generated from a number of independent runs (also termed as experiments or assay s). Within each run, a number of concentration levels of the analyte of interest are tested with two or more replicates at each level. The primary performance measures are estimated independently at each level of the analyte concentration. This is carried out within the framework of the analysis of variance (ANOVA) model with the experimental runs included as a random effect [23]. Additional terms such as analyst, instmment, etc., may be included in this model depending on the design of the experiment. This ANOVA model allows us to estimate the overall mean of the calculated concentrations and the relevant variance components such as the within-run variance and the between-run variance. 

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