Population, statistical data points

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. 

The answer is a technique called population kinetics. In this, blood samples are taken on a few occasions, carefully timed in relation to the previous drug dose, in as big a population as can be observed. The blood samples may be obtained at widely different time points after dosing and ah are analyzed for drug concentration. The next step is a statistical treatment of the results which makes the assumption that ah the patients belong to one big, if variable, population. A spread of data points is obtained over the dose interval and one gigantic curve of concentration-time relationships created. If the population is big enough, the mathematics iron out any awkward individuals whose data do not tit the overall pattern and from this derived curve the kinetic parameters we have been discussing can be deduced. 

Quantum mechanical, classical and statistical probabilities agree, on average, reasonably well with the experimental results [133] shown in Fig. 37 (vibrational distributions of NO were also measured by Harrison et al. [310]). In the experiment a high population of the state n o = 1 is found already 100 cm above its threshold. Moreover, the measured probabilities show some indications of fluctuations. Because of the limited number of data points, the inevitable incoherent averaging over several overall rotational states of NO2 and the averaging over the various possible electronic states of the 0 and NO products, these fluctuations are less pronounced than in the quantum mechanical calculations on a single adiabatic PES and for J = 0. 

Statistical laws have been derived for populations often they must be modified substantially when applied to a small sample because a few data points may not represent the entire population. In the discussion that follows, we first describe the Gaussian statistics of populations. Then we show how these relationships can be modified and applied to small samples of data. 

With only a few data points, the resulting sample mean and variance are not equal to the true population parameters. The t distribution [9] then describes the distribution of sample means, where the test statistic is... 

Table 9.1 Diagnostic statistics of the 300 datapoint population, and the two 150 data point sub-populations from Figure 9.1.

Before adopting statistical tests to assess the reliability of data, outliers should be first analyzed carefully to identify any anomaly in instrament fidelity, calibration, procedure, environmental conditions, recording, etc. The first objective is to reject an outlier based on physical evidence that the data point was unrepresentative of the sample population. If this exercise fails to identify probable cause (or if details on the experimental methods are unavailable), Chauvenet s criterion may be applied to assess the reliability of the data point (Holman 2001). Simply put, the criterion reconunends rejecting a data point if the probability of obtaining the deviation is less than the reciprocal of two times the number of data points—l/(2n). For an outlier smaller than the mean, the data point may be rejected when ... 

FIGURE 3.40 Example of data interpretation of IMS-linked PCA. In this study, dots seen in the 2D plot represent the case, that is, the spectrum from distinct data points. If dots from distinct sampie are separated (A), it means that the molecular expression patterns of these two regions were statistically distinct from each other. If not, PCA failed to extract the statistical differences between the two populations (B). 

As has been emphasized in this chapter, many statistical tests assume that the data used are drawn from a normal population. One method of testing this assumption, using the chi-squared test, was mentioned in the previous section. Unfortunately, this method can only be used if there are 50 or more data points. It is common in experimental work to have only a small set of data. A simple visual way of seeing whether a set of data is consistent with the assumption of normality is to plot a cumulative frequency curve on special graph paper known as normal probability paper. This method is most easily explained by means of an example. 

The data points in Fig. 4.3 are obtained from numeric aggregate populations with 100 aggregates per each mass fraction, which leads to some statistic scattering in the curves. Regarding this uncertainty, the ratio xq to Xg can be considered virtually universal for DLCA aggregates (xc/xg 1.66). That means that the geometric size obeys the same fractal law as the radius of gyration N ocR (cf. Eq. (4.9)). 

Alloy Composition and Board Finish Effects. ilie data in Fig. 1(b) is replotted in Fig. 2(a) and 2(b) where, instead of showing component types, results are shown according to alloy composition or board finish. Out of the 27 datasets of interest, five had the nominal solder composition Sn-3.9Ag-0.6Cu (SAC 3906) while all the others had the nominal composition Sn-3.8Ag-0.7Cu (SAC 3807). In Fig. 2(a), the SAC 3906 data points are all below the correlation center line. It might be tempting to conclude that the SAC 3807 alloy leads to a slightly longer life than SAC 3906. However, the SAC 3906 population in the analysis is small (5 data points out of a total of 27), and more data is needed to test the statistical differences between the SAC 3807 and SAC 3906 populations. 

Risk assessment pertains to characterization of the probability of adverse health effects occurring as a result of human exposure. Recent trends in risk assessment have encouraged the use of realistic exposure scenarios, the totality of available data, and the uncertainty in the data, as well as their quality, in arriving at a best estimate of the risk to exposed populations. The use of "worst case" and even other single point values is an extremely conservative approach and does not offer realistic characterization of risk. Even the use of arithmetic mean values obtained under maximum use conditions may be considered to be conservative and not descriptive of the range of exposures experienced by workers. Use of the entirety of data is more scientific and statistically defensible and would provide a distribution of plausible values. 

Obviously, we need tests and estimates on the variability of our experimental data. We can develop procedures that parallel the tests and estimates on the mean as presented in the previous section. We might test to determine whether the sample was drawn from a population of a given variance or we might establish point or interval estimates of the variance. We may wish to compare two variances to determine whether they are equal. Before we proceed with these tests and estimates, we must consider two new probability distributions. Statistical procedures for interval estimates of a variance are based on chi-square and F-distributions. To be more precise, the interval estimate of a a2 variance is based on x -distribution while the estimate and testing of two variances is part of a F-distribution. 

For noncarcinogenic hazardous chemicals, NCRP believes that the threshold for deterministic effects in humans should be estimated using EPA s benchmark dose method, which is increasingly being used to establish allowable doses of noncarcinogens. A benchmark dose is a dose that corresponds to a specified level of effects in a study population (e.g., an increase in the number of effects of 10 percent) it is estimated by statistical fitting of a dose-response model to the dose-response data. A lower confidence limit of the benchmark dose (e.g., the lower 95 percent confidence limit of the dose that corresponds to a 10 percent increase in number of effects) then is used as a point of departure in establishing allowable doses. 

Although dose-response assessments for deterministic and stochastic effects are discussed separately in this Report, it should be appreciated that many of the concepts discussed in Section 3.2.1.2 for substances that cause deterministic effects apply to substances that cause stochastic effects as well. The processes of hazard identification, including identification of the critical response, and development of data on dose-response based on studies in humans or animals are common to both types of substances. Based on the dose-response data, a NOAEL or a LOAEL can be established based on the limited ability of any study to detect statistically significant increases in responses in exposed populations compared with controls, even though the dose-response relationship is assumed not to have a threshold. Because of the assumed form of the dose-response relationship, however, NOAEL or LOAEL is not normally used as a point of departure to establish safe levels of exposure to substances causing stochastic effects. This is in contrast to the common practice for substances causing deterministic effects of establishing safe levels of exposure, such as RfDs, based on NOAEL or LOAEL (or the benchmark dose) and the use of safety and uncertainty factors. 

---