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With the exception of special cases (see below) for the t distribution, the distribution of the t statistic is the N(0,1), which has considerable variance and, therefore, considerable imprecision. The magnitude of the expected value of the t statistic is not defined for data when the distribution is non-normal, whereas the expected value of the z statistic is defined for all N(0,1) distributions. Therefore, the z statistic is very useful when we can only make nonparametric comparisons. The difference between parametric and nonparametric testing is illustrated in Figure 2. The t distribution is symmetric about the vertical line t = 0, so the percentiles are defined by the same equation to the right and left of the line. This has an important consequence: we can use the t statistic without loss of generality to represent the data. This follows from the fact that any symmetric line is the t value of a simple linear model with slope zero. 27,28 In parametric tests, this type of linear relationship between the independent and dependent variables is considered unnatural, because it requires that the relationship between the dependent and independent variables be linear. Principles Of Biostatistics Pagano Free Download 128 Thus, in this context, a data point along a line would mean the sample mean of all samples, and a data point that is not a sample mean is an outlier. Note that the outlier could be the best measurement of the mean or it could be a measurement that occurred in a high-velocity, high-acceleration or high-inclination environment. These outlying measurements are called extreme outliers, because they are as far from the mean of all the samples as is possible, i.e. they form the extreme end of the distribution. High outliers are more likely to occur in high-inclination orbits as they would be the outermost planet in the Solar System. Principles Of Biostatistics Pagano Free Download 128 Not only are outliers problematic for parametric tests (the number of outliers is not even known) but because the t distribution is a skewed distribution (due to the low values of the normalizing constant) with a long tail, the theory of parametric tests such as the t test is not applicable to the analysis of the possibility of outliers, especially extreme outliers, due to the danger of treating them as equally distributed. In contrast, nonparametric tests allow for the analysis of unusual values such as outliers and we will discuss these tests in more detail. Other reasons that high outlying values are to be rejected include the presence of measurement error, which may be especially problematic when the error is similar to or larger than the actual signal. The other problem is multicollinearity, which occurs when the error term has a linear relationship with a fixed set of underlying variables. In simulations such as ours, we tested the robustness of our results to various errors. A fifth reason to discard outliers is that they may be related to a different model for the system.
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