Parametric test

Metrology - The goal of most metrology machine efforts is to keep the process under control, whether it involves making measurements of physical size of individual features and film thickness, or making electriccd measurements of parametric test structures. Defects are also measured and estimated, including excess particles and misplaced features in the composite. 

Here you can still use the Pearson chi-square test as shown in the 2x2 table example as long as your response variable is nominal and merely descriptive. If your response variable is ordinal, meaning that it is an ordered sequence, and you can use a parametric test, then you should use the Mantel-Haenszel test statistic for parametric tests of association. For instance, if in our previous example the variable called headache was coded as a 2 when the patient experienced extreme headache, a 1 if mild headache, and a 0 if no headache, then headache would be an ordinal variable. You can get the Mantel-Haenszel /pvalue by running 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. 

Group 1 can be seen to approximate a normal distribution (bell-shaped curve) we can proceed to perform the appropriate parametric tests with such data. But group 2 clearly does not appear to be normally distributed. In this case, the appropriate nonparamctric technique must be used. 

It should be clearly understood that for data that do not fulfill the necessary assumptions for parametric analysis, these nonparametric methods are either as powerful or in fact, more powerful than the equivalent parametric test. 

This parametric test assumes that event probabilities are constant over time. That is, the chance that a patient becomes event-positive at time t given that he is event-negative up to time t does not depend on t. A plot of the negative log of the event times distribution showing a linear trend through the origin is consistent with exponential event times. 

The range of values that will contain the true population mean with a stated percentage confidence. Used in parametric tests. 

The range of values that lie between the first and third quartiles and, therefore, represent 50°/o of the data points. Used in non-parametric tests. 

A parametric test for comparison of sample means where... 

The data obtained was analyzed using SPSS-9.0 software. Any differences with a significance level p<0.05 were considered valid. To assess the validity of the differences and correlation, non-parametric tests were used as indicated vide infra). The data are displayed as mean SE. 

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). 

Non-parametric tests, as seen in the two procedures outlined earlier in Section 11.5, are based on some form of ranking of the data. Once the data are ranked then the test is based entirely on those ranks the original data play no further part. It is therefore the behaviour of ranks that determines the properties of these tests and it is this element that gives them their robustness. Whatever the original data looks like, once the rank transformation is performed then the data become well-behaved. 

In terms of summary statistics, means are less relevant because of the inevitable skewness of the original data (otherwise we would not be using non-parametric tests). This skewness frequently produces extremes, which then tend to dominate the calculation of the mean. Medians are usually a better, more stable, description of the average . 

Extending non-parametric tests to more complex settings, such as regression, ANOVA and ANCOVA is not straightforward and this is one aspect of these methods that limits their usefulness. 

Non-parametric methods reduce power. Therefore if the data are normally distributed, either on the original scale or following a transformation, the non-parametric test will be less able to detect differences should they exist. 

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