Nonparametric testing

If the normal approximation to the binomial distribution is not valid (that is, more than 20% of expected cell counts are less than 5) for drug therapy and symptom of headache, then you can use Fisher s exact test, which is a nonparametric test, to test for a difference in proportions. To get the p-value using Fisher s exact test, you run the following SAS code ... 

For continuous variables you may be required to provide inferential statistics along with the descriptive statistics that you generate from PROC UNIVARIATE. The inferential statistics discussed here are all focused on two-sided tests of mean values and whether they differ significantly in either direction from a specified value or another population mean. Many of these tests of the mean are parametric tests that assume the variable being tested is normally distributed. Because this is often not the case with clinical trial data, we discuss substitute nonparametric tests of the population means as well. Here are some common continuous variable inferential tests and how to get the inferential statistics you need out of SAS. 

Finally, if the ldl change variable is not normally distributed, then you can run a nonparametric test on the change-from-baseline LDL value, like this ... 

Time-to-event analysis in clinical trials is concerned with comparing the distributions of time to some event for various treatment regimens. The two nonparametric tests used to compare distributions are the log-rank test and the Cox proportional hazards model. The Cox proportional hazards model is more useful when you need to adjust your model for covariates. 

Gehan-Breslow Modification of Kruskal-Wallis Test is a nonparametric test on censored observations. It assigns more weight to early incidences compared to Cox-Tarone test. 

This is a nonparametric test in which the data in each group are first ordered from lowest to highest values, then the entire set (both control and treated values) is ranked, with the average rank being assigned to tied values. The ranks are then summed for each group and U is determined according to... 

Both the Log-Rank and the Generalized Wilcoxon Tests are nonparametric tests, and require no assumptions regarding the distribution of event times. When the event rate is greater early in the trial than toward the end, the Generalized Wilcoxon Test is the more appropriate test since it gives greater weight to the earlier differences. 

Like other statistical methods, the user has to be careful with the requirements of a statistical test. For many statistical tests the data have to follow a normal distribution. If this data requirement is not fulfilled, the outcome of the test can be biased and misleading. A possible solution to this problem are nonparametric tests that are much less restrictive with respect to the data distribution. There is a rich literature on... 

More generally, testing for a trend is a self-defeating exercise, if the observation sets are not connected by causality. This cardinal rule, applying equally to parametric and nonparametric tests, has been amply emphasized in the literature. 

Dependencies may be detected using statistical tests and graphical analysis. Scatter plots may be particularly helpful. Some software for statistical graphics will plot scatter plots for all pairs of variables in a data set in the form of a scatter-plot matrix. For tests of independence, nonparametric tests such as Kendall s x are available, as well as tests based on the normal distribution. However, with limited data, there will be low power for tests of independence, so an assumption of independence should be scientifically plausible. 

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). 

Sensitivity, specificity, odds ratio, and relative risk Types of data and scales of measurement Measures of central tendency and dispersion Inferential statistics Students s t-distribution Comparing means Comparing more than two means Regression and correlation Nonparametric tests The x2-test Clinical trials INTRODUCTION... 

The fractional factorial is designed to reduce the number of qualification trials to a more reasonable number, say, 10, while holding the number of randomly assigned processing variables to a reasonable number as well, say, 9. The technique was developed as a nonparametric test for process evaluation by Box and Hunter [18] and reviewed by Hendrix [19]. Ten is a reasonable number of trials in terms of resource and time commitments and should be considered an upper limit in a practical testing program. This particular design as presented in Table 7 does not include interaction effects. 

Data were expressed as the mean standard error of the mean (SEM). Differences between means were determined using one-way analysis of variance (ANOVA) followed by the Tukey-Kramer post hoc comparison and two-sided t test. For comparing percentages, nonparametric tests were also applied (Mann-Whitney, Kruskal-Wallis). Differences were considered significant when p < 0.05. 

The term parametric test, or analysis, is introduced in Section 6.2.4. In broad terms, statistical analyses can be placed into one of two categories, parametric tests and nonparametric tests. This book almost exclusively discusses parametric tests, but it should be noted here that nonparametric tests are also very valuable analyses in appropriate circumstances. As with the terms experimental design and nonexperimental design (recall Section 5.5), the term nonparametric is not a relative quality judgment compared with parametric. This nomenclature simply differentiates statistical approaches. In circumstances where nonparametric analyses are appropriate, they are powerful tests. 

If normality of the data cannot be accepted, Spearman s correlation coefficient and its corresponding nonparametric test can be used for the null hypothesis Ho =... 

The verification of transferred reference values or intervals is both important and problematic. The comparison of a locally produced, small subset of values with the large set produced elsewhere using traditional statistical tests often is not appropriate because the underlying statistical assumptions are not fulfilled and because of the unbalanced sample sizes. Alternative methods using nonparametric tests or Monte Carlo sampling have been described. 

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. 

Crawford, C.G., Slack, J.R. and Hirsch, R.M. (1983) Nonparametric Tests for Trends in Water Quality Data using the Statistical Analysis System. United States Geological Survey Report, 83-550. 

Thas, O., Van Vooren, L. and Ottoy, J.P. (1998) Nonparametric test performance for trends in water quality with sampling design applications. Journal of the American Water Resources Association, 34, 347-357. 

There are times when the required assumptions for ANOVA, a parametric test, are not met. One example would be if the underlying distributions are non-normal. In these cases, nonparametric tests are very useful and informative. For example, we saw in Section 11.3 that a nonparametric analog to the two-sample ttest, Wilcoxon s rank sum test, makes use of the ranks of observations rather than the scores themselves. When a one-factor ANOVA is not appropriate in a particular case a corresponding nonparametric approach called the Kruskal-Wallis test can be used. This test is a hypothesis test of the location of (more than) two distributions. 

The daily iodine intake in Ukrainians living in three contaminated areas are compared among sampling years from 1997 to 2001 in Table 122.2. The median ranged from 20.5 to 87.7pg d per person. The intake of 1997, which was the total diet for children, was higher than that in other years. The reason was unclear however, there was a statistically significant difference (p < 0.01) in a nonparametric test (Mann-Whitney test) except in the 1999 intakes. 

For the pre-intervention surveys logarithmic transformation of the UIC produced symmetric and approximately normal distributions and where possible analysis was performed using the transformed iodine concentration. For many of the results, however, medians and results of nonparametric tests are reported to enable comparison with the post-intervention surveys. In examining the differences between years some subjects had been sampled on both occasions. To remove the effect of the assumed correlation between subjects sampled on multiple occasions, a linear mixed model analysis including subjects as random effects was used to test the difference between log iodine in the pre-intervention years. 

For dose-finding, such approaches will exploit the dose ordering, at least if we are prepared to assume that the dose-response is monotonic. (Our two chapter quotations give examples where the response, conversation in the one case and intoxication in the other, is not a monotonic function of the dose, but dose-finding is often performed in the hope that this sort of response is rare.) An early example of a nonparametric test... 

The AUC can be estimated making assumptions about the distribution of test results. Therefore, a nonparametric test is based on the fact that the AUC is related to the test statistic U of the Mann-Whitney rank sum test ... 

Thus, the nonparametric test and parametric estimates of the AUC for T3 data are 0.9935 and 0.9745, respectively. Figures 2 and 3 show examples of ROC plots. 

We often assume that data are drawn from a normal distribution. When the normality assumption is not met, alternatively, a nonparametric test is used. Nonparametric tests do not rely on assumptions that the data are drawn from a given probability distribution. However, it is less efficient if it is used when the distribution assumption is met. 

The Wilcoxon rank-sum test is a nonparametric test for assessing whether two samples of measurements come from the same distribution. That is, as an alternative to the two-sample f-test, this test can be used to discover differentially expressed candidates under two conditions. For example, again consider the measurements of the probe set used for the two-sample t-test. The gene expression values are 12.79, 12.53, and 12.46 for the naive condition and 11.12, 10.77, and 11.38 for the 48-h activated condition. Measurement 12.79 has rank 6, measurement 12.53 has 5, and measurement 12.46 has rank 4. The rank sum of the naive condition is 6 -I- 5 -I-4=15. Then after the sum is subtracted by ni(ni-I-l)/2 = 3 x 4/2 = 6, the Wilcoxon rank-sum test statistic becomes 9. Considering all of the combinations of the three measurements, we can compute the probability that the rank sum happens more extremely than 9. The probability becomes its p-value. This is the most extreme among the 20 combinations thus the p-value is 2 x Pr( W > 9) = 2 x = 0.1 for the two-sided test. It is hard to say that the probe set is differentially expressed since the p-value 0.1 > 0.05. This test is also called the Mann - Whitney- Wilcoxon test because this test was proposed initially by Wilcoxon for equal sample sizes and extended to arbitrary sample sizes by Mann and Whitney. As a nonparametric alternative to the paired t-test for the two related samples, the Wilcoxon signed-rank test can be used. The statistic is computed by ordering absolute values of differences of paired samples. For example, consider a peptide in the platelet study data. Their differences for each... 

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