Single Interval Estimate

Rarely will a researcher want to use a t distribution for evaluating only one confidence interval. The researcher will want, more than likely, all contrasts. 

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Differences in calibration graph results were found in amount and amount interval estimations in the use of three common data sets of the chemical pesticide fenvalerate by the individual methods of three researchers. Differences in the methods included constant variance treatments by weighting or transforming response values. Linear single and multiple curve functions and cubic spline functions were used to fit the data. Amount differences were found between three hand plotted methods and between the hand plotted and three different statistical regression line methods. Significant differences in the calculated amount interval estimates were found with the cubic spline function due to its limited scope of inference. Smaller differences were produced by the use of local versus global variance estimators and a simple Bonferroni adjustment. 

Point estimate uses the sample data to calculate a single best value, which estimates a population parameter. The point estimate is one number, a point on a numeric axis, calculated from the sample and serving as approximation of the unknown population distribution parameter value from which the sample was taken. Such a point estimate alone gives no idea of the error involved in the estimation. If parameter estimates are expressed in ranges then they are called interval estimates. 

This interval estimate is really based on the two-sided test of the third set of hypotheses previously given. Although it is possible to define one-sided confidence intervals based on the other two sets of hypotheses (1.59) and (1.60), such one-sided intervals are rarely used. By one-sided, we mean an interval estimate that extends from plus or minus infinity to a single random confidence limit. The one-sided confidence interval may be understood as the range one limit of which is the probability level a and the other one °°. 

For a sample size of 100 (99 df) the reliability factors for two-sided 90%, 95%, and 99% confidence intervals are 1.66, 1.98, and 2.63. The implication of these three values is that, all other things being equal (that is, x, s, and ri), requiring greater confidence in the interval estimate results in wider interval estimates. The more confidence that is required, the less reliable is the single sample estimate, and therefore greater numerical uncertainty is expressed in the interval estimate. This very important point is illustrated in the following example. 

Although this approach has its advantages, one disadvantage is that no single numerical estimate, either a point estimate ot an interval estimate, can convey the extent to which the populations differ because the test of the location shift is based on relative tank and not the original scale. 

The assessment of bioequivalence is based on 90% confidence intervals for the ratio of the population geometric means (test/reference) for the parameters under consideration. This method is equivalent to two one-sided tests with the null hypothesis of bio-inequivalence at the 5% significance level. Two products are declared bioequivalent if upper and lower limits of the confidence interval of the mean (median) of log-transformed AUC and Cmax each fall within the a priori bioequivalence intervals 0.80-1.25. It is then assumed that both rate (represented by Cmax) and extent (represented by AUC) of absorption are essentially similar. Cmax is less robust than AUC, as it is a single-point estimate. Moreover, Cmax is determined by the elimination as well as the absorption rate (Table 2.1). Because the variability (inter- and intra-animal) of Cmax is commonly greater than that of AUC, some authorities have allowed wider confidence intervals (e.g., 0.70-1.43) for log-transformed Cmax, provided this is specified and justified in the study protocol. 

Comparing two variances. To compare the averages of two independent samples, we combine the two sample variances to form a single pooled estimate. Since the pooled value has more degrees of freedom, the confidence interval becomes narrower and the test more sensitive, that is, it becomes capable of detecting smaller systematic differences. Obviously it only makes sense to combine sample variances if they are estimates of the same population variance. To justify calculating a pooled estimate, we need to test the nuU hypothesis that and s are estimates of identical population variances, = cr - This can be done by performing an F-test, which is based on Eq. (2.23). If the two population variances are equal, Eq. (2.23) becomes... 

One may consider the effect of the variability of EURs and other operational characteristics over a shale play using Monte Carlo. We describe the procedure in detail in our previous work (2). In brief, we conduct N (e.g. 5000) trials in which we select input data from their distributions at random and conduct the LCA. This results in N sets of results (e.g. carbon footprints, water footprints), which constitute distributions for those results in lieu of single point estimates. We report 80% confidence intervals (CIs) for these estimates in Table 2. From the overlap of the CIs, it is evident that there is no statistically significant difference among the carbon footprints, water footprints, and other characteristics of Barnett and Marcellus shale gases. 

Figure 3 shows the confidence limits of the predicted bottom-hole flowing pressures using single-porosity estimate at 0.20 percent measurement error. The confidence interval is about 186 psi which is practically acceptable. The true pressures are all contained within the confidence interval also. As can be seen in Figure 2, the confidence regions for joint estimation of porosity and permeability at 0.20 percent measurement error indicate that even at the lowest confidence level of 95 percent, the confidence interval for porosity is very wide. The orientation and shape of the ellipses show that porosity is much less well determined than permeability. It seems, therefore, that porosity estimation is very sensitive to measurement error. Also, porosity estimates are not reliable when joint estimation of porosity and other parameter(s) is made or when there is a significant error in the matched performance data. 

As just seen, when using a two-sided confidence interval, interest lies with both the lower and the upper limit. In contrast, a one-sided confidence interval focuses on the placement of a single interval on one specified side of the treatment effect point estimate. In certain circumstances, interest lies with a drug response in one direction only. In these cases it is legitimate to calculate and present a single limit, which is usually referred to as the lower bound of the confidence interval (when placed below the treatment effect point estimate) or the upper bound when placed above it. 

The standard deviation of the mean is a point estimate of//. However, a point estimate does not indicate the confidence that can be placed in such an estimate. When an objective measure of reliability is required, we report a range of values rather than a single value. These interval estimates are called confidence intervals (Cl) or confidence limits. 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

The precision limit P. The interval about a nominal result (single or average) is the region, with 95% confidence, within which the mean of many such results would fall, if the experiment were repeated under the same conditions using the same equipment. Thus, the precision limit is an estimate of the lack of repeatability caused by random errors and unsteadiness. 

An answer to the previous problems is provided by the conditional distribution approach, whereby at each node x of a grid the whole likelihood function of the unknown value p(x) is produced instead of a single estimated value p (x). This likelihood function allows derivation of different estimates corresponding to different estimation criteria (loss functions), and provides data values-dependent confidence intervals. Also this likelihood function can be used to assess the risks a and p associated with the decisions to clean or not. 

In the previous discussion of the one- and two-compartment models we have loaded the system with a single-dose D at time zero, and subsequently we observed its transient response until a steady state was reached. It has been shown that an analysis of the response in the central plasma compartment allows to estimate the transfer constants of the system. Once the transfer constants have been established, it is possible to study the behaviour of the model with different types of input functions. The case when the input is delivered at a constant rate during a certain time interval is of special importance. It applies when a drug is delivered by continuous intravenous infusion. We assume that an amount Z) of a drug is delivered during the time of infusion x at a constant rate (Fig. 39.10). The first part of the mass balance differential equation for this one-compartment open system, for times t between 0 and x, is given by ... 

Usually, one has obtained an estimate for the elimination constant and the distribution volume Vp from a single intravenous injection. These pharmacokinetic parameters, together with the interval between administrations 0 and the single-dose D, then allow us to compute the steady-state peak and trough values. The criterion for an optimal dose regimen depends on the minimum therapeutic concentration (which must be exceeded by and on the maximum safe... 

Accuracy is often used to describe the overall doubt about a measurement result. It is made up of contributions from both bias and precision. There are a number of definitions in the Standards dealing with quality of measurements [3-5]. They are only different in the detail. The definition of accuracy in ISO 5725-1 1994, is The closeness of agreement between a test result and the accepted reference value . This means it is only appropriate to use this term when discussing a single result. The term accuracy , when applied to a set of observed values, describes the consequence of a combination of random variations and a common systematic error or bias component. It is preferable to express the quality of a result as its uncertainty, which is an estimate of the range of values within which, with a specified degree of confidence, the true value is estimated to lie. For example, the concentration of cadmium in river water is quoted as 83.2 2.2 nmol l-1 this indicates the interval bracketing the best estimate of the true value. Measurement uncertainty is discussed in detail in Chapter 6. 

If a single reaction order must be selected, an examination of the 95 % confidence intervals (not shown) indicates that the two-thirds order is a reasonable choice. For this order, however, estimates of the forward rate constants deviate somewhat from an Arrhenius relationship. Finally, some trend of the residuals (Section IV) of the transformed dependent variable with time exists for this reaction order. 

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