Central tendency

Commonly used descriptive statistics include measures that describe where the middle of the data is. These measures are sometimes called measures of central tendency and include the mean, median, and mode. Another category of measures describes how spread out the data is. These measures are sometimes called measures of variability and include the range, variance, and standard deviation. Additional descriptive measures can include percentages, percentiles, and frequencies. In safety performance measurement, the safety professional must determine the format of the data (i.e., ratio, interval, ordinal, or categorical) that will be collected and match the data format to the appropriate statistic. As will be discussed in the following sections, certain descriptive statistics are appropriate for certain data formats. 

The first group of descriptive statistics are the measures of central tendency. These statistics describe where the middle of the data falls. There are three measures of central tendency. They are the mean, median, and mode (Hays 1998,155-56). 

The median is the point that 50 percent of the values lie above and 50 percent lie below. Using the numbers of lost workdays from the example above, the median would be determined by first arranging the observations in order from lowest to highest. Thus, the observations would be arranged as 2,5,7,8, and 15. Because there is an odd number of observations in this example, the median would be the number in the middle of the distribution. The median would be 7 in this example. 

The mode of a distribution is the most frequently occurring number in that distribution. Using the carbon monoxide data in case above, the value 15 ppm occurs twice while all other values appear only once in the distribution. In this example, the mode of the distribution is 15 ppm. 

It is possible for a distribution to have more than one mode. In the distribution of numbers 2, 3, 4, 5, 5, 6, 7, 8, 8, 10, there are two modes, 5 and 8, since these values occur more frequently than the others do. This would be considered a bimodal distribution (2 modes). 

Once we have assembled individual observations in a sample from a clinical study, our ability to understand the nature of those observations as a whole is limited by our ability to synthesize several disparate pieces of observation into an overall impression. Imagine that you have observed the following 10 observations of age of study participants in an early exploratory therapeutic clinical trial 45, 62, 32, 38, 77, 28, 25, 62, 41, and 50. 

Regulatory authorities are concerned about how well study participants match those in the general population of patients with the condition. How might such a question be answered There are several strategies here. 

One fundamental idea in the development of new pharmaceutical products is that pharmaceutical companies (sponsors) would like to demonstrate that participants who receive a test treatment tend to fare better than those who receive some alternate therapy. This alternate therapy could be an inactive control (a placebo) or some other approved therapy (an active control). We said tend to fare better because participants will not all respond in the same way to the same test treatment. It is also true that, if and when the drug is approved for marketing and prescribed for patients, some patients will do better on the drug than others, but it is still very useful to clinicians to know how patients will tend to respond. 

When we flip a fair coin ten times, we do not always expect to observe five heads and five tails. If we do several series of ten flips, we know that, by chance, we will observe six heads and four tails sometimes, and even more lopsided results would not be all that surprising. The same phenomenon happens with the response to test treatments in clinical studies. When doctors prescribe a new medicine to a patient it would be helpful to know what kind of response could be expected. Although we might expect that a fair coin flipped ten times will result in five heads, we also would expect that four or three heads could be observed. The determination of values that might be expected is the next topic in this chapter, that is, measures of central tendency. 

One possible way to answer this question is to report that the most common value of age is 62. There are two such observations with this value of age. This measure of central tendency is known as the mode. The mode is most commonly used with non-numeric data (for example, most of the study participants were female), but it may also be useful for numeric data if there are only a few unique values. Unfortunately, the choice of the mode as the typical value in this case is a little misleading. Although there are two 62-year-olds in the study, most study participants (seven of them) are younger than that. 

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One way to characterize the data in Table 4.1 is to assume that the masses of individual pennies are scattered around a central value that provides the best estimate of a penny s true mass. Two common ways to report this estimate of central tendency are the mean and the median. 

The mean is the most common estimator of central tendency. It is not considered a robust estimator, however, because extreme measurements, those much larger or smaller than the remainder of the data, strongly influence the mean s value. For example, mistakenly recording the mass of the fourth penny as 31.07 g instead of 3.107 g, changes the mean from 3.117 g to 7.112 g ... 

As shown by Examples 4.1 and 4.2, the mean and median provide similar estimates of central tendency when all data are similar in magnitude. The median, however, provides a more robust estimate of central tendency since it is less sensitive to measurements with extreme values. For example, introducing the transcription error discussed earlier for the mean only changes the median s value from 3.107eto3.112e. 

If the mean or median provides an estimate of a penny s true mass, then the spread of the individual measurements must provide an estimate of the variability in the masses of individual pennies. Although spread is often defined relative to a specific measure of central tendency, its magnitude is independent of the central value. Changing all... 

Realizing that our data for the mass of a penny can be characterized by a measure of central tendency and a measure of spread suggests two questions. Eirst, does our measure of central tendency agree with the true, or expected value Second, why are our data scattered around the central value Errors associated with central tendency reflect the accuracy of the analysis, but the precision of the analysis is determined by those errors associated with the spread. 

Accuracy is a measure of how close a measure of central tendency is to the true, or expected value, Accuracy is usually expressed as either an absolute error... 

Although the mean is used as the measure of central tendency in equations 4.2 and 4.3, the median could also be used. 

Consider, for example, the data in Table 4.1 for the mass of a penny. Reporting only the mean is insufficient because it fails to indicate the uncertainty in measuring a penny s mass. Including the standard deviation, or other measure of spread, provides the necessary information about the uncertainty in measuring mass. Nevertheless, the central tendency and spread together do not provide a definitive statement about a penny s true mass. If you are not convinced that this is true, ask yourself how obtaining the mass of an additional penny will change the mean and standard deviation. 

A binomial distribution has well-defined measures of central tendency and spread. The true mean value, for example, is given as... 

The data we collect are characterized by their central tendency (where the values are clustered), and their spread (the variation of individual values around the central value). Central tendency is reported by stating the mean or median. The range, standard deviation, or variance may be used to report the data s spread. Data also are characterized by their errors, which include determinate errors... 

Variance was introduced in Chapter 4 as one measure of a data set s spread around its central tendency. In the context of an analysis of variance, it is useful to see that variance is simply a ratio of the sum of squares for the differences between individual values and their mean, to the degrees of freedom. For example, the variance, s, of a data set consisting of n measurements is given as... 

The median particle diameter is the diameter which divides half of the measured quantity (mass, surface area, number), or divides the area under a frequency curve ia half The median for any distribution takes a different value depending on the measured quantity. The median, a useful measure of central tendency, can be easily estimated, especially when the data are presented ia cumulative form. In this case the median is the diameter corresponding to the fiftieth percentile of the distribution. 

Many distribution functions can be apphed to strength data of ceramics but the function that has been most widely apphed is the WeibuU function, which is based on the concept of failure at the weakest link in a body under simple tension. A normal distribution is inappropriate for ceramic strengths because extreme values of the flaw distribution, not the central tendency of the flaw distribution, determine the strength. One implication of WeibuU statistics is that large bodies are weaker than small bodies because the number of flaws a body contains is proportional to its volume. 

One further point might be made here. Although the example illustrates the difference between the two types of molecular weight average, the weight average molecular weight in this example cannot be said to be truly representative, an essential requirement of any measure of central tendency. In such circumstances where there is a bimodal, i.e. two-peaked, distribution additional data should be provided such as the modal values (100 and 100000 in this case) of the two peaks. 

Distributions are characterized by measures of central tendency The median is the value of X (e.g., crap scores) that divides the distribution into equal areas. The value of x at the peak... 

Mean The measure of central tendency of a distribution, often referred to as its arithmetic average. 

Variability arises from true heterogeneity in characteristics such as dose-response differences within a population, or differences in contaminant levels in tlie enviromiient The values of some variables used in an assessment change witli time and space, or across tlie population whose exposure is being estimated. Assessments should address tlie resulting variability in doses received by members of the target population. Individual exposure, dose, and risk can vary widely in a large population. The central tendency and high end individual risk descriptors are intended to capture tlie variability in exposure, lifestyles, and other factors tliat lead to a distribution of risk across a population. 

A location parameter is the abscissa of a location point and may be a measure of central tendency, such as a mean. 

Populations are very large collections of values. In practice, experimental pharmacology deals with samples (much smaller collections) from a population. The statistical tools used to deal with samples differ somewhat from those used to deal with populations. When an experimental sample is obtained, the investigator often wants to know about two features of the sample central tendency and variability. The central tendency refers to the most representative estimate of the value, while the variability defines the confidence that the estimate is a true reflection of that value. Central tendency estimates can be the median (value that divides the sample into two equal halves) or the... 

Descriptive statistics quantify central tendency and variance of data sets. The probability of occurrence of a value in a given population can be described in terms of the Gaussian distribution. 

Central tendency, 226-227 Chemical genomics, 178 Chemical tools, in target-based drug discovery, 178-179 Chemokine C receptor type 1, 133 Chemokine receptors, 6, 44, 53, 129, 132, 177... 

Mean, arithmetic More simply called the mean, it is the sum of the values in a distribution divided by the number of values. It is the most common measure of central tendency. The three different techniques commonly used are the raw material or ungrouped, grouped data with a calculator, and grouped data with pencil and paper. 

Population pharmacokinetics is the application of pharmacokinetic and statistical methods to sparse data to derive a pharmacokinetic profile of central tendency. 

Classic parameter estimation techniques involve using experimental data to estimate all parameters at once. This allows an estimate of central tendency and a confidence interval for each parameter, but it also allows determination of a matrix of covariances between parameters. To determine parameters and confidence intervals at some level, the requirements for data increase more than proportionally with the number of parameters in the model. Above some number of parameters, simultaneous estimation becomes impractical, and the experiments required to generate the data become impossible or unethical. For models at this level of complexity parameters and covariances can be estimated for each subsection of the model. This assumes that the covariance between parameters in different subsections is zero. This is unsatisfactory to some practitioners, and this (and the complexity of such models and the difficulty and cost of building them) has been a criticism of highly parameterized PBPK and PBPD models. An alternate view assumes that decisions will be made that should be informed by as much information about the system as possible, that the assumption of zero covariance between parameters in differ-... 

Figure 22.3 The drug dose-response model was augmented by nsing data for the comparator drug. Because the mechanism of the drugs was the same, this comprised additional data for the model. This enhanced the predictive power of the model, in a better estimate for central tendency (solid line compared with dotted line) bnt also in smaller confidence intervals. This is especially prononnced at the higher doses— precisely where data on the drug were sparse. See color plate.

In particle size analysis it is important to define three terms. The three important measures of central tendency or averages, the mean, the median, and the mode are depicted in Figure 2.4. The mode, it may be pointed out, is the most common value of the frequency distribution, i.e., it corresponds to the highest point of the frequency curve. The distribution shown in Figure 2.4 (A) is a normal or Gaussian distribution. In this case, the mean, the median and the mode are found to fie in exactly the same position. The distribution shown in Figure 2.4 (B) is bimodal. In this case, the mean diameter is almost exactly halfway between the two distributions as shown. It may be noted that there are no particles which are of this mean size The median diameter lies 1% into the higher of the two distri-... 

The statistical parameters of central tendency obtained for the four groups of areas which reflect the geochemistry of different periods of the mine abandon are presented in Tables 1Aand 1B. 

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