Sample statistics, as estimators of population parameters

As before, these sample statistics are estimates of the unknown population parameters, the population means, and the population variances. If the population variances are assumed to be equal, each sample statistic is a different estimate of the same population variance. It is then reasonable to average or "pool" these estimates to obtain the following  

A data set is often considered as a sample from a population and the sample parameters calculated from the data set as estimates of the population parameters (-> statistical indices). Moreover, it is usually used to calculate statistical models such as quantitative -> structure/response correlations. In this case the data set is organized into a data matrix X with n rows and p columns, where each row corresponds to an object of the data set and each column to a variable therefore each element represent the value of the yth variable for the ith object (/ = 1,. .., n j = 1,. .., p). 

On many occasions, sample statistics are used to provide an estimate of the population parameters. It is extremely useful to indicate the reliability of such estimates. This can be done by putting a confidence limit on the sample statistic. The most common application is to place confidence limits on the mean of a sample from a normally distributed population. This is done by working out the limits as F— ( />[ i] x SE) and F-I- (rr>[ - ij x SE) where //>[ ij is the tabulated critical value of Student s t statistic for a two-tailed test with n — 1 degrees of freedom and SE is the standard error of the mean (p. 268). A 95% confidence limit (i.e. P = 0.05) tells you that on average, 95 times out of 100, this limit will contain the population 

Such a data set has a mean and a SD. The mean of the data set of sample statistics will be 11, the population parameter on average the sample statistics will be g and hence p from any sample is an unbiased estimator of g. The variation of the sample statistics, p, can be described in the same way as for any data set the SD of the distribution of sample statistics is known as the standard error (SE) (of the estimate) - here it would be SEp. 

If the value of a sample statistic 6 is used to estimate a parameter 0 of the population, this statistic is called an estimator and its value for the sample the estimate. Sample mean x and variance s2 are the usual estimators of the population mean g and 

To summarize, the computational aspects of confidence intervals involve a point estimate of the population parameter, some error attributed to sampling, and the amount of confidence (or reliability) required for interpretation. We have illustrated the general framework of the computation of confidence intervals using the case of the population mean. It is important to emphasize that interval estimates for other parameters of interest will require different reliability factors because these depend on the sampling distribution of the estimator itself and different calculations of standard errors. The calculated confidence interval has a statistical interpretation based on a probability statement. 

It is not possible to know whether any single sample estimate, like the sample mean, is a good estimate of the population parameter that it is intended to estimate. However, it is possible to use the fact that most estimates of the sample statistic (for example, sample mean) are not too far removed from the population parameter, as specified by the shape of the sampling distribution, to define a range of values of the population parameter (for example, population mean) that are best supported by the sample data. 

In addition, it is very useful to estimate the difference between the population parameters using the difference between the sample statistics. So, the actual difference seen in the data gives a point estimate of the difference. In the data shown in Table 7.12 in which P = 0-017, the difference in percentage success between the two samples is 16-0% (82-7% - 66-7%), which gives a point estimate of the true difference. 

So basic is the notion of a statistical estimate of a physical parameter that statisticians use Greek letters for the parameters and Latin letters for the estimates. For many purposes, one uses the variance, which for the sample is s and for the entire populations is cr. The variance s of a finite sample is an unbiased estimate of cr, whereas the standard deviation 5- is not an unbiased estimate of cr. 

Fundamental to statistical measurement are two basic parameters the population mean, /r, and the population standard deviation, cr. The population parameters are generally unknown and are estimated by the sample mean, x, and sample standard deviation, s. The sample mean is simply the central tendency of a sample set of data that is an unbiased estimate of the population mean, /r. The central tendency is the sum of values in a set, or population, of numbers divided by the number of values in that set or population. For example, for the sample set of values 10, 13, 19, 9, 11, and 17, the sum is 79. When 79 is divided by the number of values in the set, 6, the average is 79 6 = 13.17. The statistical formula for average is 

It follows from the fact that the sampling distribution is normally distributed that 95% of the sample statistics will be within 1-96 SDs of the mean (the population parameter). The SD of the sampling distribution (as mentioned above) is referred to as the SE of the estimate. Because only 5% of sample statistics will be more than 1-96 SEs from the population parameter, for any sample statistic taken at random it is 95% likely that the population parameter is within 1-96 SEs of the sample statistic. This is the rationale for the calculation of confidence intervals (Cis) in estimation. So, a 95% CI is found by the expression  

Ideally, the group of reference individuals should be a random sample of all the individuals fulfilling the defined inclusion criteria in the parent population. Statistical estimation of distribution parameters (and their confidence intervals) and statistical hypothesis testing require this assumption. 

Then, given a model for data from a specific drug in a sample from a population, mixed-effect modeling produces estimates for the complete statistical distribution of the pharmacokinetic-dynamic parameters in the population. Especially, the variance in the pharmacokinetic-dynamic parameter distributions is a measure of the extent of inherent interindividual variability for the particular drug in that population (adults, neonates, etc.). The distribution of residual errors in the observations, with respect to the mean pharmacokinetic or pharmacodynamic model, reflects measurement or assay error, model misspecification, and, more rarely, temporal dependence of the parameters. 

One of the primary goals of Statistics is to use data from a sample to estimate an unknown quantity from an underlying population, called a population parameter. In general, we typically use the arithmetic mean as the measure of central tendency of choice because the sample mean is an unbiased estimator of the population mean, typically represented by the symbol p. The main conceptual point about unbiased estimators is that they come closer to estimating the true population parameter, in this case the population mean, than biased estimators. When extreme observations influence the value of the mean such that it really is not representative of a typical value, use of the median is recommended as a measure of central tendency. 

Statistical evaluations of data are warranted by the fact that the true mean concentration // (the population mean) will never be known and that we can only estimate it with a sample mean x. As a reflection of this fact, there are two parallel systems of symbols. The attributes of the theoretical distribution of mean concentrations are called parameters (true mean p, variance a2, and standard deviation from sample results are called statistic (sample mean x, variance s2, and standard deviation 5). 

Statistical estimation uses sample data to obtain the best possible estimate of population parameters. The p value of the Binomial distribution, the p value in Poison s distribution, or the p and a values in the normal distribution are called parameters. Accordingly, to stress it once again, the part of mathematical statistics dealing with parameter distribution estimate of the probabilities of population, based on sample statistics, is called estimation theory. In addition, estimation furnishes a quantitative measure of the probable error involved in the estimate. As a result, the engineer not only has made the best use of this data, but he has a numerical estimate of the accuracy of these results. 

The sample standard deviation, s, provides an estimate of the population standard deviation, a. The (n — 1) term in equations (6.4) and (6.6) is often described as the number of degrees of freedom (frequently represented in statistical tables by the parameter v (Greek letter, pronounced nu ). It is important for judging the reliability of estimates of statistics, such as the standard deviation. In general, the number of degrees of freedom is the number of data points (n) less the number of parameters already estimated from the data. In the case of the sample standard deviation, for example, v = n — 1 since the mean (which is used in the calculation of s) has already been estimated from the same data. 

Linear mixed effects models are primarily used in pharmacodynamic analysis or in the statistical analysis of pharmacokinetic parameters. Linear mixed effects models could also be used to analyze concentrationtime data from a 1-compartment model with bolus administration after Ln-transformation. The advantages to using mixed effects in an analysis are that observations within a subject may be correlated and that in addition to estimation of the model parameters, between- and within-subject variability may be estimated. Also, the structural model is based on the population, not on data from any one particular subject, thus allowing for sparse sampling. Most statistical packages now include linear mixed effects models as part of their analysis options, as do some pharmacokinetic software (Win-Nonlin). While linear mixed effects models are not cov- 

Figure 6-4a shows two Gaussian curves in which we plot the relative frequency y of various deviations from the mean versus the deviation from the mean. As shown in the margin, curves such as these can be described by an equation that contains just two parameters, the population mean p. and the population standard deviation a. The term parameter refers to quantities such as pu and a that define a population or distribution. This is in contrast to quantities such as the data values x that are variables. The term statistic refers to an estimate of a parameter that is made from a sample of data, as discussed below. The sample mean and the sample standard deviation are examples of statistics that estimate parameters p. and a, respectively. 

The unknown quantities of interest described in the previous section are examples of parameters. A parameter is a numerical property of a population. One may be interested in measures of central tendency or dispersion in populations. Two parameters of interest for our purposes are the mean and standard deviation. The population mean and standard deviation are represented by p and cr, respectively. The population mean, p, could represent the average treatment effect in the population of individuals with a particular condition. The standard deviation, cr, could represent the typical variability of treatment responses about the population mean. The corresponding properties of a sample, the sample mean and the sample standard deviation, are typically represented by x and s, which were introduced in Chapter 5. Recall that the term "parameter" was encountered in Section 6.5 when describing the two quantities that define the normal distribution. In statistical applications, the values of the parameters of the normal distribution cannot be known, but are estimated by sample statistics. In this sense, the use of the word "parameter" is consistent between the earlier context and the present one. We have adhered to convention by using the term "parameter" in these two slightly different contexts. 

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