Statistical test ANOVA

Table 35-1 illustrates the ANOVA results for each individual sample in our hypothetical study. This test indicates whether any of the reported results from the analytical methods or locations is significantly different from the others. From the table it can be observed that statistically significant variation in the reported analytical results is to be expected based on these data. However, there is no apparent pattern in the method or location most often varying from the others. Thus, this statistical test is inconclusive and further investigation is warranted. 

Table 35-3 illustrates the ANOVA results comparing laboratories (i.e., different locations) performing the same METHOD A for analysis. This statistical test indicates that for the mid-level concentration spiked samples (i.e. 4 and 4 at 3.40 and 3.61% levels, respectively) difference in reported average values occurred. However, this trend did not continue for the highest concentration sample (i.e., Sample No. 6) with a concentration of 3.80%. The Lab 1 was slightly lower in reported value for Samples 4 and 5. There is no significant systematic error observed between laboratories using the METHOD A. 

Table 35-4 reports ANOVA comparing the METHOD B procedure to the METHOD A procedure for combined laboratories. Thus the combined METHOD B analyses for each sample were compared to the combined METHOD A analyses for the same sample. This statistical test indicates whether there is a significant bias in the reported results for each method, irrespective of operator or location. An apparent trend is indicated using this statistical analysis, that trend being a positive bias for METHOD B as compared to... 

This set of articles presents the computational details and actual values for each of the statistical methods shown for collaborative tests. These methods include the use of precision and estimated accuracy comparisons, ANOVA tests, Student s t-testing, The Rank Test for Method Comparison, and the Efficient Comparison of Methods tests. From using these statistical tests the following conclusions can be derived ... 

The MathCad worksheets used for this Chemometrics in Spectroscopy collaborative study series are given below in hard copy format. Unless otherwise noted, the worksheets have been written by the authors. The text files for the MathCad v7.0 Worksheets used for the statistical tests in this report are attached as Collabor GM, Collabor TV, ANOVA s4, ANOVA s2, CompareT, and Comp Meth. References [1-11] are excellent sources of information of the details on these statistical methods. 

Figure 9.5 emphasizes the relationships among three other sums of squares in the ANOVA tree - the sum of squares due to lack of fit, SS f , the sum of squares due to purely experimental uncertainty, SS and the sum of squares of residuals, 55,. Two of the resulting variances, and were used in Section 6.5 where a statistical test was developed for estimating the significance of the lack of fit of a model to a set of data. The null hypothesis... 

The practical consequence from this is that in the study type under consideration, always the dam/litter rather than the individual fetus is the basic statistical unit (see Chapters 23, 33, 34 and 35). Six malformed fetuses from six different litters in a treated group of dams is much more likely to constitute a teratogenic effect of the test substance than ten malformed fetuses all from the same litter. It is, therefore, important to report all fetal observations in this context and to select appropriate statistical tests (e.g., Fisher s exact test with Bonferroni correction) based on litter frequency. For continuous data, a procedure to calculate the mean value over the litter means (e.g., ANOVA followed by Dunnet s test) is preferred. An increase in variance (e.g., standard deviation), even without a change in the mean, may indicate that some animals were more susceptible than others, and may indicate the onset of a critical effect. 

If there is replication of the experiment then an independent estimate of the error terms can be calculated and valid statistical tests, such as ANOVA, can be constructed. 

Hence, hypothesis testing (ANOVA analysis followed by multiple comparison analysis) was used to determine NOEC and LOEC values expressed as % v/v of effluent. In order to satisfy statistical analysis requirements enabling NOEC and LOEC determinations, some bioassay protocols were adjusted to make sure that there were at least three replicates per effluent concentration and at least five effluent concentrations tested. TC % effluent values were then determined as follows ... 

Analysis of variance (ANOVA) Anhedonia Statistical test to compare the mean values from two or more groups. Inability to derive pleasure from situations that usually induce pleasure. This is a characteristic... 

Statistical analyses result in a test statistic being calculated. For example, two common tests that will be introduced in this chapter are the f-test and a test called analysis of variance (ANOVA). The /-tests result in a test statistic called /, and ANOVA results in a test statistic called F. When you read the Results sections of regulatory submissions and clinical communications, you will become very familiar with these test statistics. The test statistic obtained determines whether the result of the statistical test attains statistical significance or not. 

As noted in Section 7.5, the test statistic in ANOVAs is called F, and the test is sometimes called the F-test. The name pays respect to Sir Ronald Fisher, the statistician who developed this approach. Similarly to the calculation of the test statistic t in a f-test, F is calculated as a ratio, as follows ... 

One can conclude that ANOVA can be a very useful test for evaluating both systematic and random errors in data, and is a useful addition to the basic statistical tests mentioned previously in this chapter. It is important to note, however, there are other factors that can greatly influence the outcome of any statistical test, as any result obtained is directly affected by the quality of the data used. It is therefore important to assess the quality of the input data, to ensure that it is free from errors. One of the most commonly encountered errors is that of outliers. 

Lack-of-Fit Test The best-known statistical test to evaluate the appropriateness of the chosen regression model is the lack-of-fit test [6]. A prerequisite for this test is the availability of replicate measurements. An analysis of variance (ANOVA) approach is used in this test. The total sum of squares (SS V) can be written as follows ... 

These effects can be analyzed statistically by comparing performance between the treated groups and vehicle control at each test session, or pooled over the 3 sessions, using Student s t tests. ANOVA with repeated measures (sessions) provides a more sensitive assessment by including within the same analysis all the scores obtained per animal. 

Table IV shows the overall analysis of variance (ANOVA) and lists some miscellaneous statistics. The ANOVA table breaks down the total sum of squares for the response variable into the portion attributable to the model, Equation 3, and the portion the model does not account for, which is attributed to error. The mean square for error is an estimate of the variance of the residuals — differences between observed values of suspensibility and those predicted by the empirical equation. The F-value provides a method for testing how well the model as a whole — after adjusting for the mean — accounts for the variation in suspensibility. A small value for the significance probability, labelled PR> F and 0.0006 in this case, indicates that the correlation is significant. The R2 (correlation coefficient) value of 0.90S5 indicates that Equation 3 accounts for 91% of the experimental variation in suspensibility. The coefficient of variation (C.V.) is a measure of the amount variation in suspensibility. It is equal to the standard deviation of the response variable (STD DEV) expressed as a percentage of the mean of the response response variable (SUSP MEAN). Since the coefficient of variation is unitless, it is often preferred for estimating the goodness of fit.

The hypothesis-testing statistical functions may be reasonably powerful (e.g. t-test, ANOVA, regressions) and they often return the probability P of obtaining the test statistic (where 0 < F < 1), so there may be no need to refer to statistical tables. Again, check on the effects of including empty cells. 

The F-distribution has great utility in a statistical test referred to as analysis of variance (ANOVA). ANOVA is a powerful tool for testing the equivalence of means from samples obtained from normally distributed, or approximately normally distributed, populations. As an example, suppose that the following are the content uniformity values on 20 tablets from each of four different lots lot A mean = 99.5%, standard deviation = 2.6% lot B mean = 100.2%, standard deviation = 2.8% lot C mean = 90.5%, standard deviation = 2.1% and lot D mean = 100.3%, standard deviation = 2.7%. 

After outliers have been purged from the data and a model has been evaluated visually and/or by, e.g. residual plots, the model fit should also be tested by appropriate statistical methods [2, 6, 9, 10, 14], The fit of unweighted regression models (homoscedastic data) can be tested by the ANOVA lack-of-fit test [6, 9]. A detailed discussion of alternative statistical tests for both unweighted and weighted calibration models can be found in Ref. [16]. The widespread practice to evaluate a calibration model via its coefficients of correlation or determination is not acceptable from a statistical point of view [9]. 

Normal Distribution is a continuous probability distribution that is useful in characterizing a large variety of types of data. It is a symmetric, bell-shaped distribution, completely defined by its mean and standard deviation and is commonly used to calculate probabilities of events that tend to occur around a mean value and trail off with decreasing likelihood. Different statistical tests are used and compared the y 2 test, the W Shapiro-Wilks test and the Z-score for asymmetry. If one of the p-values is smaller than 5%, the hypothesis (Ho) (normal distribution of the population of the sample) is rejected. If the p-value is greater than 5% then we prefer to accept the normality of the distribution. The normality of distribution allows us to analyse data through statistical procedures like ANOVA. In the absence of normality it is necessary to use nonparametric tests that compare medians rather than means. 

The basic statistical test used for ANOVA is the F test, described in Section 7B-4. Here, a large value of F compared with the critical value from the tables may give us reason to reject Hq in favor of the alternative hypothesis. 

Hyphenated methods Methods involving the combination of two or more types of instrumentation the product is an instrument with greater capabilities than any one instrument alone. Hypothesis testing The process of testing a tentative assertion with various statistical tests. See t-test, F-test, Q-test, onA ANOVA. 

The significant variables will be identified as those for which the corresponding coefficient in the model is significantly larger than the experimental error. The significance can be assessed by statistical tests, such as r-tests, F-tests on ANOVA tables, or from cumulative normal probability plots. 

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