The assumption of normality

This visual approach based on inspecting the normal probability plot may seem fairly crude. However, most of the test procedures, such as the unpaired t-test, are what we call robust against departures from normality. In other words, the 

Secondly we have a statistical test, the Shapiro-Wilks test, which gives a p-value for the following setting  

A significant p-value is indicating that the data is not normally distributed and leads to the rejection of Hg a non-significant p-value tells us that there is no evidence for non-normality and in practice it will then be safe to assume that the data is at least approximately normally distributed. 

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Given that the assumption of normally distributed data (see Section 1.2.1) is valid, several useful and uncomplicated methods are available for finding the most probable value and its confidence interval, and for comparing such results. 

If the assumption of normality is grossly violated, ML estimates of the parameters can only be obtained using the error-in-variables" method where besides the parameters, we also estimate the true (error-free) value of the measured variables. In particular, assuming that Ey i is known, the parameters are obtained by minimizing the following objective function... 

The main difference between the Z-test and the /-test is that the Z-statistic is based on a known standard deviation, a, while the /-statistic uses the sample standard deviation, s, as an estimate of a. With the assumption of normally distributed data, the variance sample variance, v2 as n gets large. It can be shown that the /-test is equivalent to the Z-test for infinite degrees-of-freedom. In practice, a large sample is usually considered n > 30. 

The principle of using least squares may still be applicable in fitting the best curve, if the assumptions of normality, independence, and reasonably error-free measurement of response are valid. 

Both assumptions are mainly needed for constructing confidence intervals and tests for the regression parameters, as well as for prediction intervals for new observations in x. The assumption of normal distribution additionally helps avoid skewness and outliers, mean 0 guarantees a linear relationship. The constant variance, also called homoscedasticity, is also needed for inference (confidence intervals and tests). This assumption would be violated if the variance of y (which is equal to the residual variance a2, see below) is dependent on the value of x, a situation called heteroscedasticity, see Figure 4.8. 

The remaining diagnostic plots shown in Figure 4.17 are the QQ-plot for checking the assumption of normal distribution of the residuals (upper right), the values of the y-variable (response) versus the fitted y values (lower left), and the residuals versus the fitted y values (lower right). The symbols + for outliers were used for the same objects as in the upper left plot. Thus it can be seen that the... 

Nonparametric statistics (NPS) differs primarily from its traditional, distribution-based counterpart by dealing with data of unknown probability distributions. Its principal attractiveness lies, in fact, in not requiring the knowledge of a probability distribution. NPS is especially inviting when the assumption of normal distribution of small data sets is hazardous (if at all admissible), even if NPS-based calculations are more time consuming than in traditional statistics. The steadily growing importance of NPS has been amply demonstrated by numerous textbooks and monographs published within the last few decades, e.g. [1-7],... 

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). 

B observations in each of several centres) and also with more complex structures which form the basis of ANCOVA and regression. For example, in regression the assumption of normality applies to the vertical differences between each patient s observation y and the value of y on the underlying straight line that describes the relationship between x andy. We therefore look for the normality of the residuals the vertical differences between each observation and the corresponding value on the fitted line. 

If data are normally distributed, the mean and standard deviation are the best description possible of the data. Modern analytical chemistry is often automated to the extent that data are not individually scrutinized, and parameters of the data are simply calculated with a hope that the assumption of normality is valid. Unfortunately, the odd bad apple, or outlier, can spoil the calculations. Data, even without errors, may be more or less normal but with more extreme values than would be expected. These are known has heavy-tailed distributions, and the values at the extremes are called outliers. In interlaboratory studies designed to assess proficiency, the data often have outliers, which cannot be rejected out of hand. It would be a misrepresentation for a proficiency testing body to announce that all its laboratories give results within 2 standard deviations (except the ones that were excluded from the calculations). 

We note from Table 1.19 that the sums of squares between rows and between columns do not add up to the defined total sum of squares. The difference is called the sum of squares for error, since it arises from the experimental error present in each observation. Statistical theory shows that this error term is an unbiased estimate of the population variance, regardless of whether the hypotheses are true or not. Therefore, we construct an F-ratio using the between-rows mean square divided by the mean square for error. Similarly, to test the column effects, the F-ratio is the be-tween-columns mean square divided by the mean square for error. We will reject the hypothesis of no difference in means when these F-ratios become too much greater than 1. The ratios would be 1 if all the means were identical and the assumptions of normality and random sampling hold. Now let us try the following example that illustrates two-way analysis of variance. 

The analysis of variance technique for testing equality of means is a rather robust procedure. That is, when the assumption of normality and homogeneity of variances is slightly violated the F-test remains a good procedure to use. In the one-way model, for example, with an equal number of observations per column it has been exhibited that the F-test is not significantly effected. However, if the sample size varies across columns, then the validity of the F-test can be greatly affected. There are various techniques for testing the equality of k variances Oi, 02,..., crj,. We discuss... 

SED-TOX scores were determined for each sediment sample investigated in the two studies and Pearson s correlations were estimated with benthic community metrics and levels of contamination (SAS, 1988). Contamination levels were expressed as the mean ratios of individual contaminant concentrations in a sample relative to their respective SQG values. Logarithmic transformations were applied to mean SQG quotient values to respect the assumption of normality. 

A third and often neglected reason for the need for care fill application of chemometric methods is the problem of the type of distribution of environmental data. Most basic and advanced statistical methods are based on the assumption of normally distributed data. But in the case of environmental data, this assumption is often not valid. Figs. 1-7 and 1-8 demonstrate two different types of experimentally found empirical data distribution. Particularly for trace amounts in the environment, a log-normal distribution, as demonstrated for the frequency distribution of N02 in ambient air (Fig. 1-7), is typical. 

The assumption of normality should be verified with each variable. 

The great majority of statistical procedures are based on the assumption of normality of variables, and it is well known that the central limit theorem protects against failures of normality of the univariate algorithms. Univariate normality does not guarantee multivariate normality, though the latter is increased if all the variables have normal distributions in any case, it avoids the deleterious consequences of skewness and outliers upon the robustness of many statistical procedures. Numerous transformations are also able to reduce skewness or the influence of outlying objects. 

To check the assumptions of the model, Bartlett s or Levene s tests can be used to assess the assumption of equality of variance, and the normal probability plot of the residuals (etj = Xij - Xj) to assess the assumption of normality. If either equality or normality are inappropriate, we can transform the data, or we can use the nonparametric Kruskal-Wallis test to compare the k groups. In any case, the ANOVA procedure is insensitive to moderate departures from the assumptions (Massart et al. 1990). 

The calculations are less lengthy than those based on equation (17) if the temperatures can be selected such that (Tj—Ti) is virtually constant, and lead to almost the same values for the activation parameters but the estimates of the standard errors can be a little larger when the simpler method is employed (see p. 136). However, the errors of E j and depend essentially on (Tj—Ti) so that different statistical weights (gry) must be assigned to each Efj (or in the subsequent calculations if Tj-T() varies appreciably. The assumption of normal (Gaussian) distribution suggests that... 

Further statistical analyses can be used to determine the relative influence that any factor or set of factors has on the total variation (global uncertainty). One of these methods is the analysis of variance (ANOVA). This is an important technique for analyzing the effects of categorical factors on a response. However, the assumption of normality of the data has to be checked prior to the use of ANOVA to decompose the variability in the response variable between the different factors. Depending upon the type of analysis, it may be important to determine (a) which factors have a significant effect on the response, and/or (b) how much of the variability in the response variable is attributable to each factor (as described in the statistical software STATGRAPHICS, Vs 5.0). 

Prediction of the log reduction of an inoculated organism as a function of acid concentration, time, and temperature can also be done by a mathematical model developed for this purpose, using the second-order polynomial equation to fit the data. The following tests justified the reliability of the model the analysis of variance for the response variable indicated that the model was significant (P < 0.05 and R2 = 0.9493) and had no significant lack of fit (P > 0.05). Assumptions underlying the ANOVA test were also investigated and it was demonstrated that with the normal probability plot of residuals, plot of residuals versus estimated values for the responses, and plot of residuals versus random order of runs, that the residuals satisfied the assumptions of normality, independence, and randomness (Jimenez et al., 2005). 

Applying Lagrange s equation to Eqs. (40) and (41) leads to an infinite number of second-order differential equations in r" ", with an infinite number of normal-mode frequencies. However, the assumption of normal modes for the helix is equivalent to assuming that all r" vary with the same frequency and with a phase factor that depends only on m, viz., that... 

Sometimes, plots of individual PC scores can be used for preliminary analysis of variables that contribute to an out-of-control signal. The control limits for new t scores under the assumption of Normality at significance level a at any time interval k is given by [100]... 

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