Population methods parametric

Parametric population methods also obtain estimates of the standard error of the coefficients, providing consistent significance tests for all proposed models. A hierarchy of successive joint runs, improving an objective criterion, leads to a final covariate model for the pharmacokinetic parameters. The latter step reduces the unexplained interindividual randomness in the parameters, achieving an extension of the deterministic component of the pharmacokinetic model at the expense of the random effects. Recently used individual empirical Bayes estimations exhibit more success in targeting a specific individual concentration after the same dose. 

In Chapter 10 we saw that there are various methods for the analysis of categorical (and mostly binary) efficacy data. The same is true here. There are different methods that are appropriate for continuous data in certain circumstances, and not every method that we discuss is appropriate for every situation. A careful assessment of the data type, the shape of the distribution (which can be examined through a relative frequency histogram or a stem-and-leaf plot), and the sample size can help justify the most appropriate analysis approach. For example, if the shape of the distribution of the random variable is symmetric or the sample size is large (> 30) the sample mean would be considered a "reasonable" estimate of the population mean. Parametric analysis approaches such as the two-sample t test or an analysis of variance (ANOVA) would then be appropriate. However, when the distribution is severely asymmetric, or skewed, the sample mean is a poor estimate of the population mean. In such cases a nonparametric approach would be more appropriate. 

Software Available for Population Analysis Parametric Maximum Likelihood Method Packages... 

Consideration of the white blood cell (WBC) and differential counts leads to another problem. The total WBC is, typically, a normal population amenable to parametric analysis, but differential counts are normally determined by counting, manually, one or more sets of one hundred cells each. The resulting relative percentages of neutrophils are then reported as either percentages or are multiplied by the total WBC count with the resulting count being reported as the absolute differential WBC. Such data, particularly in the case of eosinophils (where the distribution does not approach normality), should usually be analyzed by nonpara-metric methods. It is widely believed that relative (%) differential data should not be reported because they are likely to be misleading. 

The statistical methods discussed up to now have required certain assumptions about the populations from which the samples were obtained. Among these was that the population could be approximated by a normal distribution and that, when dealing with several populations, these have the same variance. There are many situations where these assumptions cannot be met, and methods have been developed that are not concerned with specific population parameters or the distribution of the population. These are referred to as non-parametric or distribution-free methods. They are the appropriate methods for ordinal data and for interval data where the requirements of normality cannot be assumed. A disadvantage of these methods is that they are less efficient than parametric methods. By less efficient is meant... 

The most commonly employed univariate statistical methods are analysis of variance (ANOVA) and Student s r-test [8]. These methods are parametric, that is, they require that the populations studied be approximately normally distributed. Some non-parametric methods are also popular, as, f r example, Kruskal-Wallis ANOVA and Mann-Whitney s U-test [9]. A key feature of univariate statistical methods is that data are analysed one variable at a rime (OVAT). This means that any information contained in the relation between the variables is not included in the OVAT analysis. Univariate methods are the most commonly used methods, irrespective of the nature of the data. Thus, in a recent issue of the European Journal of Pharmacology (Vol. 137), 20 out of 23 research reports used multivariate measurement. However, all of them were analysed by univariate methods. 

There are often data sets used to estimate distributions of model inputs for which a portion of data are missing because attempts at measurement were below the detection limit of the measurement instrument. These data sets are said to be censored. Commonly used methods for dealing with such data sets are statistically biased. An example includes replacing non-detected values with one half of the detection limit. Such methods cause biased estimates of the mean and do not provide insight regarding the population distribution from which the measured data are a sample. Statistical methods can be used to make inferences regarding both the observed and unobserved (censored) portions of an empirical data set. For example, maximum likelihood estimation can be used to fit parametric distributions to censored data sets, including the portion of the distribution that is below one or more detection limits. Asymptotically unbiased estimates of statistics, such as the mean, can be estimated based upon the fitted distribution. Bootstrap simulation can be used to estimate uncertainty in the statistics of the fitted distribution (e.g. Zhao Frey, 2004). Imputation methods, such as... 

The parametric method for the determination of percentiles and their confidence intervals assumes a certain type of distribution, and it is based on estimates of population parameters, such as the mean and the standard deviation. We are, for example, using a parametric method if we believe that the true distribution is Gaussian and determine the reference limits (percentiles) as the values located 2 standard... 

The sample mean is a poor measure of central tendency when the distribution is heavily skewed. Despite our best efforts at designing well-controlled clinical trials, the data that are generated do not always compare with the (deliberately chosen) tidy examples featured in this book. When we wish to make an inference about the difference in typical values among two or more independent populations, but the distributions of the random variables or outcomes are not reasonably symmetric, nonparametric methods are more appropriate. Unlike parametric methods such as the two-sample t test, nonparametric methods do not require any assumption about the shape of a distribution for them to be used in a valid manner. As the next analysis method illustrates, nonparametric methods do not rely directly on the value of the random variable. Rather, they make use of the rank order of the value of the random variable. 

Parametru/non-parametric techniques This first distinction can be made between techniques that take account of the information on the population distribution. Non parametric techniques such as KNN, ANN, CAIMAN and SVM make no assumption on the population distribution while parametric methods (LDA, SIMCA, UNEQ, PLS-DA) are based on the information of the distribution functions. LDA and UNEQ are based on the assumption that the population distributions are multivariate normally distributed. SIMCA is a parametric method that constructs a PCA model for each class separately and it assumes that the residuals are normally distributed. PLS-DA is also a parametric technique because the prediction of class memberships is performed by means of model that can be formulated as a regression equation of Y matrix (class membership codes) against X matrix (Gonzalez-Arjona et al., 1999). 

The population analysis methods use all the available data to estimate the population. The best estimates for the parameters of an individual study are only obtained after the population distribution has been estimated by Bayesian estimation. Essentially, the various methods estimate the population parameters d in h(fi,6 ). The methods differ primarily in the form that h(fi,6jc) is assumed to have. Despite the fact that all arrive at a quantitative description of h(fi,Ooc), the different forms have been divided into parametric, semiparametric, and nonparametric. Each of these will be described. 

The outlier tests described above assume that the sample comes from a normal population. It is important to realize that a result that seems to be an outlier on the assumption of a normal population distribution may well not be an outlier if the sample actually comes from (for example) a log-normal distribution (Section 2.3). Therefore outlier tests should not be used if there is a suspicion that the population may not have a normal distribution. This difficulty, along with the extra complications arising in cases of multiple outliers, explains the increasing use of the non-parametric and robust statistical methods described in Chapter 6. Such methods are either insensitive to extreme values, or at least give them less weight in calculations, so the problem of whether or not to reject outliers is avoided. 

The statistical methods based on assumption that the lifetime were drawn from known distributed populations such as normal etc., and below there are techniques that do not make such assumptions. The methods are recognized as distribution-free statistics or nonparametric statistics. In situations where the known assumption holds, the nonparametric tests are less efficient than parametric methods. 

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