Students t-Distribution

Construction of an Approximate Confidence Interval. An approxi-mate confidence interval can be constructed for an assumed class of distributions, if one is willing to neglect the bias introduced by the spline approximation. This is accomplished by estimation of the standard deviation in the transformed domain of y-values from the replicates. The degrees of freedom for this procedure is then diminished by one accounting for the empirical search for the proper transformation. If one accepts that the distribution of data can be approximated by a normal distribution the Student t-distribution gives... 

Normally the population standard deviation a is not known, and has to be estimated from a sample standard deviation s. This will add an additional uncertainty and therefore will enlarge the confidence interval. This is reflected by using the Student-t-distribution instead of the normal distribution. The t value in the formula can be found in tables for the required confidence limit and n-1 degrees of freedom. 

Leptokurtic distributions are more outlier-prone. When fitting distributions to data, it may sometimes be difficult to decide whether one should assume a leptokurtic distribution (say, a Student t distribution with relatively few degrees of freedom) or assume the presence of a few outliers. 

There is often a particular concern for the effects of outliers or heavy-tailed distributions when using standard statistical techniques. To address this type of a situation, a parametric approach would be to use ML estimation assuming a heavy-tailed distribution (perhaps a Student t distribution with few degrees of freedom). However, simple ad hoc methods such as trimmed means may be useful. There is a large statistical literature on robust and outlier-resistant methods, (e.g., Hoaglin et al. 1983 Barnett and Lewis 1994). 

Table 2.10 Critical Values of x (from CRC Handbook) for the Student t Distribution...

The next three subsections describe the background and principles of random error treatment, and they introduce two important quantities standard deviation a- and 95 percent confidence limits. The four subsections following these— Uncertainty in Mean Valne, Small Samples, Estimation of Limits of Error, and Presentation of Numerical Results—are essential for the kind of random error analysis most frequently required in the experiments given in this book. The Student t distribution is particularly important and useful. 

Student t Distribution. The way to proceed for small sample sets is to consider the joint probability and integrate it after a different change of variables than that used previously. In the present case, the only variable that we do not integrate over is... 

The distribution represented in Eqs. (25) and (26) and is known as the Student t distribution Student is a pseudonym for W. S. Gosset, who developed this distribution for application to biometrics. Note that a does not appear in the expression. The shape of the Student distribution, unlike that of the normal distribution, is variable, depending on N. When Nis large, it rapidly approaches the normal distribution. The distribution functions for several values of the number of degrees of freedom v v = N— 1 for determining the mean x) are shown in Fig. 5. 

Final Caution. The Student t distribution is to be used when the numerical value to which it attaches is a mean of a definite number of direct observations or is a numerical result calculated from such a mean by a procedure that introduces no uncertainties comparable in magnitude to the random errors of the direct observations. This is notihe case for the specific rotation [a]j discussed above. The uncertainty contribution to the final result that is due to random errors in the raw data on optical rotations Oy is not large compared to the contributions due to the estimated uncertainties in the other variables (particularly Vand Z) that are required for calculating the final result The number of degrees of... 

From a specific PK/PD perspective, where parameters are often assumed to be lognormally distributed, relaxation of this assumption to include a distribution with heavier tails (e.g., a log f-distribution) may be worth considering. This allows for the influence of outliers to be considered explicitly. To accommodate possible outliers, a f-distribution could be used where the degrees of freedom of a student t-distribution can be chosen empirically (29) or estimated during MCMC analysis (30) to provide appropriate weighting to the tails of the distribution. 

Equation (11.73) provides an estimate of the standard deviation of the population of experimental errors. We assume that the errors in y are distributed according to the Student t distribution, so the expected error in y at the 95% confidence level is given by... 

Ardia, D., Hoogerheide, L. F. Dijk, ff. K. v. 2009. Adaptive Mixture of Student-t Distributions as a Flexible Candidate Distribution for Efficient Simulation The R Package KAMii. Journal of Statistical Software, 29(3) 1-32. 

Apart from a simultaneous test for all parameters, tests on the significance of each individual parameter are also at hand. In this case, a stochastic variable following the Student t distribution is constructed from the parameter estimate. This is done by subtracting the value to be tested, c.q., zero, from the estimated value and by dividing this difference by the standard deviation of the parameter estimate ... 

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