Statistical tests significance test

A detailed treatment of linearity evaluation is beyond the scope of this present book but a few general points are made below. It is important to establish the homogeneity of the variance ( homoscedasticity ) of the method across the working range. This can be done by carrying out ten replicate measurements at the extreme ends of the range. The variance of each set is calculated and a statistical test (F test) carried out to check if these two variances are statistically significantly different [9]. 

The next step is to arrange the seven differences, Aa to AG, in numerical order (ignoring the sign). To calculate if any of the differences are statistically significant, a statistical test (t-test) is applied. Equation (4.17) is used to compare the difference A with the expected precision of the method, s. The value of t used corresponds to the value obtained from statistical tables for the degrees of freedom appropriate for the estimation of s and the level of confidence used. For example, if the method standard deviation was obtained from ten results, i.e. nine degrees of freedom, t(95%) = 2.262. 

Robust statistical method Significance tests Standard uncertainty True value Type 1 error Type II error Uncertainty j -Residuals... 

Pearson s product-moment correlation coefficient (r) is the most commonly used correlation coefficient. If both variables are normally distributed, then r can be used in statistical tests to test whether the degree of correlation is significant. If one or both variables are not normally distributed you can use Kendall s coefficient of rank correlation (t) or Spearman s coefficient of rank correlation (rs). They require that data are ranked separately and calculation can be complex if there are tied ranks. Spearman s coefficient is said to be better if there is uncertainty about the reliability of closely ranked data values. 

Having carried out as many experiments as there are coefficients in the model equation and not having any independent measurements or estimation of the experimental variance we cannot do any of the standard statistical tests for testing the significance of the coefficients. However all factorial matrices, complete or fractional, have the fundamental property that all coefficients are estimated with equal precision and, like the screening matrices, all the coefficients have the same unit, that of the response variable. This is because they are calculated as contrasts of the experimental response data, and they are coefficients of the dimensionless coded variables X. They can therefore be compared directly with one another. 

Fig. 26.7 Number of C. volutator males that crawled upstream into the arrival tank E or O (males could also stay in the start tank S). The tank O contained 50 females but E was empty. Males were exposed for 24 h (0.01 or 0.1 pg mL-1) to the antifouling compound medetomidine or to control seawater. Statistical tests (G-test) revealed highly significant differences in distribution depending on the treatment. Fewer exposed males left the starting tank than the controls and after exposure to the higher concentration no significant discrimination between scented and unscented water was observed (figure from Krang and Dahlstrom 2006 photo by D. Lackschewitz). Figure reprinted with permission from Elsevier...

The two-sample t-test (or Student s t-tesi) is the most widely used parametric statistical test. This test compares the means of two populations that should be normally distributed when a sample size is small. The test statistic is formed as the mean difference divided by its standard error, that is, the difference of measured expressions normalized by the magnitude of noises. If the difference of the measured expressions is very large relative to its noise, it is claimed as being significant. Formally, suppose we want to test null hypotheses, H/. pji = pj2, against alternative hypotheses, Hj pji pp, foij= 1, 2,..., m. The test statistic for each j is... 

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

Analytical chemists make a distinction between error and uncertainty Error is the difference between a single measurement or result and its true value. In other words, error is a measure of bias. As discussed earlier, error can be divided into determinate and indeterminate sources. Although we can correct for determinate error, the indeterminate portion of the error remains. Statistical significance testing, which is discussed later in this chapter, provides a way to determine whether a bias resulting from determinate error might be present. 

Next, an equation for a test statistic is written, and the test statistic s critical value is found from an appropriate table. This critical value defines the breakpoint between values of the test statistic for which the null hypothesis will be retained or rejected. The test statistic is calculated from the data, compared with the critical value, and the null hypothesis is either rejected or retained. Finally, the result of the significance test is used to answer the original question. 

A statistical test to determine if the difference between two values is significant. 

The variance for the sample of ten tablets is 4.3. A two-tailed significance test is used since the measurement process is considered out of statistical control if the sample s variance is either too good or too poor. The null hypothesis and alternative hypotheses are... 

In this experiment students measure the length of a pestle using a wooden meter stick, a stainless-steel ruler, and a vernier caliper. The data collected in this experiment provide an opportunity to discuss significant figures and sources of error. Statistical analysis includes the Q-test, f-test, and F-test. 

Using an appropriate statistical test, determine whether there is any significant difference between the standard and new methods at a = 0.05. 

Representativeness can be examined from two aspects statistical and deterministic. Any statistical test of representativeness is lacking becau.se many histories are needed for statistical significance. In the absence of this, PSAs use statistical methods to synthesize data to represent the equipment, operation, and maintenance. How well this represents the plant being modeled is not known. Deterministic representativeness can be answered by full-scale tests on like equipment. Such is the responsibility of the NSSS vendor, but for economic reasons, recourse to simplillcd and scaled models is often necessary. System success criteria for a PSA may be taken from the FSAR which may have a conservative bias for licensing. Realism is more expensive than conservatism. 

Outliers, observations that are very inconsistent with the main sample of data, i.e., apparently significantly different from the rest of the data. While there are statistical methods to test whether these values may be aberrant and thus should be removed, caution should be exercised in this practice as these data may also be the most interesting and indicative of a rare but important occurrence. 

The significance level relates to the risk of designating a chance occurrence as statistically significant. Usually a 5% level is utilized for testing treatment effects. If a p-value of 0.04 is reported for a treatment effect, this means that there is only a 4% chance that the difference in response between the active and control treatments occurred due to chance. Keep in mind, however, that if many tests are run in a trial, it is entirely possible that one or two might be significant due to chance. As an extreme example, consider a study in which 100 statistical tests are run. We would expect five of those tests to show significance with a p-value of 0.05 or less due to chance. Therefore, it is essential to specify the main tests to be run in the protocol. Any tests that are conducted after the trial has been completed should be clearly labeled as post hoc exploratory analyses. 

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

A statistical test of these data gave a correlation coefficient of 0.968 (19), with only 0.834 required for 1% significance. Thus the elimination rates are functions dependent... 

In these cases it is not necessary to determine the absolute bioavailability or the absorption rate constant for the product under study. It is only necessary to prove that the plasma concentration versus time curve is not significantly different from the reference product s curve. This is done by comparing the means and standard deviations of the plasma concentrations for the two products at each sampling time using an appropriate statistical test. 

In Table 8.1 three different analytical results are listed, the uncertainties of which are estimated in several ways (A) measurement uncertainty only, as sometimes can be done in analytical practice, (B) additionally uncertainty of calibration considered, and (C) uncertainty of sample preparation included (partially nonstatistically estimated). Whereas in cases (A) and (B) the results are judged to be significantly false, in case (C) the difference is statistically not significant. The situation is illustrated in Fig. 8.4a when a comparison is carried out on the basis of the f-test (Eq. 8.6). 

Statistical interval, e.g., of a mean, y, cnfCy ) = y ycnfwhich express the uncertainty of measured values. CIs are applied for significance tests and to establish quantities for limit values (CV). 

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. 

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