Testing for normality

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- 

Under the null hypothesis that the sample comes from a normal distribution, Zj is approximately normally distributed. Eqs. (4.39)-(4.43) to are based on the skewness of the samples. The other half of the omnibus test is based on the kurtosis. Compute 

Under the null hypothesis that the residuals come from a normal distribution, Z2 is approximately normally distributed. The omnibus test statistic is then calculated as 

Under the null hypothesis, K2 has a chi-squared distribution with 2 degrees of freedom. If K2 is greater than the critical value, the null hypothesis of a normal distribution is rejected. While these are a lot of equations, they can easily be calculated within a spreadsheet program. 

Sometimes it is useful to transform a nonlinear model into a linear one when the distribution of error terms is approximately normal and homoscedastic. Such a case might be when a suitable nonlinear function cannot be found to model the data. One might then try to change the relationship between x and Y so that a model can be found. One way to do this is to change the model 

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Since we do not know the proper values for X and t, we need a way of Judging plausible values of X and t from the data. We do this by testing the transformed background measurements for normality. Our choice of a test for normality is the probability plot correlation coefficient r (12). The coefficient r is the correlation between the ordered measurements and predicted values for an ordered set of normal random observations. We denote the ordered background measure-ments by yB(l). where yB(l) < yB(2) < yBCnn) denote the... 

Routine Lab Testing for Normal Pregnancies (First Trimester Unless Otherwise Indicated)... 

Rapid test for normal distribution Range test of David et al. [1954]... 

In a well-behaved calibration model, residuals will have a Normal (i.e., Gaussian) distribution. In fact, as we have previously discussed, least-squares regression analysis is also a Maximum Likelihood method, but only when the errors are Normally distributed. If the data does not follow the straight line model, then there will be an excessive number of residuals with too-large values, and the residuals will then not follow the Normal distribution. It follows, then, that a test for Normality of residuals will also detect nonlinearity. 

Bartlett s test does not test for normality, but rather homogeneity of variance (also called equality of variances or homoscedasticity). 

Shapiro Wilks W-test for normal data Shapiro Wilks W-test for exponential data Maximum studentlzed residual Median of deviations from sample median Andrew s rho for robust regression Classical methods of multiple comparisons Multivariate methods... 

If there are more suspect outliers, perhaps some scrutiny should be directed at the data set as a whole and the data tested for normality. 

US military requirements and tests for Normal LSt are given after the spec requirements and tests for Basic LSt Refs 1) Beil 6, 830, (405), [825] 4354 ... 

US Military requirements and test for Normal LSt for use in priming compositions are described in US Military Specification MIL-L-757A(1967) Amendment 1(1968)... 

Semivariograms shall be applied for the characterization of the spatial structure of the data. First, the data have to be tested for normal distribution. The results of the test hypothesis are acceptable for potassium only, in all other cases the hypothesis has to be rejected. After logarithmic transformation of the original data, however, a normal distribution can be obtained. Thus, the following calculations must be performed on the logarithms of the data - only for potassium can the untransformed values be used. 

Shapiro SS, Wilk MB (1965) An analysis of variance test for normality (Complete samples). Biometrika 52 591-611 Sidak Z (1967) Rectangular confidence regions for the means of multivariate normal distributions. J Am Statist Assoc 62 626-631... 

Data of each set are first tested for normal distribution using the Kolmogoroff/Smimow test. The normal distribution hypothesis is eliminated if the data having a significance level of 5 % are not normally distributed. 

T.A. Ryan, Jr. and B.L. Joiner (1976). "Normal Prohahility Plots and Tests for Normality," Technical Report, Statistics Department, The Pennsylvania State University. (Available from Minitab Inc.)... 

The wide availability of tests based on the normal distribution and their relative simplicity means you may wish to transform your data to make them more like a normal distribution. Table 41.1 provides transformations that can be applied. The transformed data should be tested for normality as described above before proceeding - don t forget that you may need to check that transformed variances are homogeneous for certain tests (see below). 

In terms of the statistical methods of the partial life cycle whole-effluent tests, survival, growth, and reproduction data from the 7 day cladoceran or fish exposure are often analyzed using hypothesis testing to determine acceptable concentrations. In order to determine the appropriateness of using parametric statistical methods, the data are first tested for normality of distribution and homogeneity of variance, for which the US EPA recommends the use of Shapiro-Wilk s test and Bartlett s test, respectively. Kolmogorov test for normality and Levine s test for homogeneity can be also used for these purposes. Dunnett s anova test is typically used for a... 

Total range of xl measurements Total range of x2 measurements Total range of differences (x2 - xl) Test for normality of differences (Anderson-DarUng test) Statistical analysis of differences Median Mean SD... 

Figure 14-15 Cumulative frequency distribution of relative differences for the comparison of drug assays example.The lighter curve indicates the Gaussian cumulative frequency distribution curve. In accordance with the test for normality, a good agreement is observed.

A choice between a t test and the comparable test for ordinal data often presents diflGculty. As well as requiring that the outcome measure be quantitative, a t test requires that the samples should be from a parent population in which the measurement is normally distributed. Testing for normality is difficult if the study is very small. If two groups are being compared, the unpaired t test assumes that the two samples come from... 

Approximate (1 — a)100% confidence intervals can be developed using any of the methods presented in the bootstrapping section of the book appendix. Using the previous example, CL was simulated 1,000 times from a normal distribution with mean 50 L/h and variance 55 (L/h)2 while V was simulated 10,000 times with a mean of 150 L and variance 225 L2. The correlation between V and CL was fixed at 0.18 given the covariance matrix in Eq. (3.70). The simulated mean and variance of CL was 49.9 L/h and 55.5 (L/h)2, while the simulated mean and variance of V was 149.8 L with variance 227 L2. The simulated correlation between CL and V was 0.174. The mean estimated half life was 2.12 h with a variance of 0.137 h2, which was very close to the Taylor series approximation to the variance. The Sha-piro Wilk test for normality indicated that the distribution of half life was not normally distributed (p < 0.01). Hence, even though CL and V were normally distributed the resulting distribution for half life was not. Based on the 5 and 95% percentiles of the simulated half life... 

However, a test for normality on the residuals indicated that the residuals were not normally distributed (p < 0.01). Of what impact was this violation on the model inferences Probably little. First, it is well known... 

It is assumed that the residuals are independent, normally distributed, with mean zero and constant variance. These are standard assumptions for maximum likelihood estimation and can be tested using standard methods examination of histograms, autocorrelation plots (ith residual versus lag-1 residual), univariate analysis with a test for normality, etc. 

Figure 10 Bootstrap distribution for the linear regression (slope) of 5-FU clearance versus dihydrouracil to uracil ratio as reported in Table 1. The solid line is the theoretical probability assuming a normal distribution. The bootstrap distribution was normally distributed using Kolmogorov—Smirnoff s test for normality. The arrow shows the observed slope.

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