Exact confidence interval

In view of these considerations, a reasonable goal is to seek methods of analysis for screening experiments that include powerful tests of specified size or level and exact confidence intervals that are tight. An exact confidence interval is analogous to a test of specified size— exact means that the confidence level is at least as high as the level claimed and the level claimed is the greatest lower bound on the confidence level. 

When screening experiments are used, it is generally anticipated that several effects may be nonzero. Hence, one ought to use statistical procedures that are known to provide strong control of error rates. It is not enough to control error rates only under the complete null distribution. This section discusses exact confidence intervals. Size-a tests are considered in Section 5. 

Kinateder, K. J., Voss, D. T., and Wang, W. (1999). Exact confidence intervals in the analysis of nonorthogonal saturated designs. American Journal of Mathematical and Management Sciences, 20, 71-84. 

Next we covered analysis of data. We used probability and random variables to model the irreproducibie part of the experiment. For models that are linear in the parameters, we can perform parameter estimation and construct exact confidence intervals analytically. For models that are nonlinear in the parameters, we compute para ter estimates and construct approximate confidence intervals using nonlinear optimization methods. 

Locke CS (1984) An Exact Confidence-Interval from Untransformed Data for the Ratio of 2 Formulation Means. Journal of Pharmacokinetics and Biopharmaceutics 12 649-655... 

Whether or not to include trials with no cardiovascular events in a meta-analysis has been a highly debated topic. If a trial had a sufficiently long period of follow-up and no cardiovascular events were reported, these data would appear to support the hypothesis of noninferiority however, no formal statistical method exists for estimating the cardiovascular risk from such data, and they are typically excluded from the analysis. Tian and colleagues (2009) discussed a method of obtaining an exact confidence interval for the difference in event rates at a fixed time point that permits combining data from trials that have zero events with data from other trials. This method could be utilized as a supportive sensitivity analysis. 

With nonlinear regression, exact confidence intervals can be found for the parameters. 

Confidence Interval for a Mean For the daily sample tensile-strength data with 4 df it is known that P[—2.132 samples exactly 16 do fall witmn the specified hmits (note that the binomial with n = 20 and p =. 90 would describe the likelihood of exactly none through 20 falling within the prescribed hmits—the sample of 20 is only a sample). 

The performance curve presents graphically the relationship between the probability of obtaining positive results PPRy i.e. x > xLSp on the one hand and the content x within a region around the limit of discrimination xDIS on the other. For its construction there must be carried out a larger number of tests (n > 30) with samples of well-known content (as a rule realized by doped blank samples). As a result, curves such as shown in Fig. 4.10 will be obtained, where Fig. 4.10a shows the ideal shape that can only be imagined theoretically if infinitely exact decisions, corresponding to measured values characterized by an infinitely small confidence interval, exist. 

The coefficients a,- are estimated from the results of experiments carried out according to a design matrix such as Table 5.9 which shows a 23 plan matrix. The significance of the several factors are tested by comparing the coefficients with the experimental error, to be exact, by testing whether the confidence intervals Aai include 0 or not. The experimental error can be estimated by repeated measurements of each experiment or - as it is done frequently in a more effective way - by replications at the centre of the plan (so-called zero replications ), see Fig. 5.2. 

First, amount error estimations in Wegscheider s work were the result of only the response uncertainty with no regression (confidence band) uncertainty about the spline. His spline function knots were found from the means of the individual values at each level. Hence the spline exactly followed the points and there was no lack of fit in this method. Confidence intervals around spline functions have not been calculated in the past but are currently being explored ( 5 ). 

What was missing in the previous section was a definition of what is meant by equivalence. Since it is imlikely that two treatments wiU have exactly the same effect we will need to consider how big a difference between the treatments would force us to choose one in preference to the other. In the t)q)hoid example there was a difference in rates of 1.9% and we may well believe that such a small difference would justify us in claiming that the treatment effects were the same. But had the difference been 5% would we still have thought them to be the same Or 10 There will be a difference, say S %, for which we are no longer prepared to accept the equivalence of the treatments. This is the so-called equivalence boimdary. If we want then to have a high degree of confidence that two treatments are equivalent it is logical to require that an appropriately chosen confidence interval (say 95%) for the treatment differences should have its extremes within the boundaries of equivalence. 

The classical or frequentist approach to probability is the one most taught in university conrses. That may change, however, becanse the Bayesian approach is the more easily nnderstood statistical philosophy, both conceptually as well as numerically. Many scientists have difficnlty in articnlating correctly the meaning of a confidence interval within the classical frequentist framework. The common misinterpretation the probability that a parameter lies between certain limits is exactly the correct one from the Bayesian standpoint. 

We have seen in the previous chapter that it is not possible to make a precise statement about the exact value of a population parameter, based on sample data, and that this is a consequence of the inherent sampling variation in the sampling process. The confidence interval provides us with a compromise rather than trying to pin down precisely the value of the mean p or the difference between two means — p2> for example, we give a range of values, within which we are fairly certain that the true value lies. 

Most of the 95 per cent confidence intervals do contain the true mean of 80 mmHg, but not all. Sample number 4 gave a mean value 3c = 81.58 mmHg with a 95 per cent confidence interval (80.33, 82.83), which has missed the true mean at the lower end. Similarly samples 35, 46, 66, 98 and 99 have given confidence intervals that do not contain p = 80 mmHg. So we have a method that seems to work most of the time, but not all of the time. For this simulation we have a 94 per cent (94/100) success rate. If we were to extend the simulation and take many thousands of samples from this population, constructing 95 per cent confidence intervals each time, we would in fact see a success rate of 95 per cent exactly 95 per cent of those intervals would contain the true (population) mean value. This provides us with the interpretation of a 95 per cent confidence interval in... 

Note that ifp is exactly equal to 0.05 then one end of the confidence interval will be equal to zero, this is the boundary between the two conditions above. 

If the 95 per cent confidence interval for the treatment effect not only lies entirely above - A but abo above zero, then there is evidence of superiority in terms of statistical significance at the 5 per cent level (p < 0.05). In this case, it is acceptable to calculate the exact probability associated with a test of superiority and to evaluate whether this is sufficiently small to reject convincingly the hypothesis of no differenc... Usually this demonstration of a benefit is sufficient for licensing on its own, provided the safety profiles of the new agent and the comparator are similar. ... 

Following calculation of the exact p-value for superiority the 95 per cent confidence interval allows the clinical relevance of the finding to be evaluated. Presumably, however, any level of benefit would be of value given that at the outset we were looking only to demonstrate non-inferiority. 

A particular use of the t-statistic is calculating confidence intervals (Cl). When we calculate the mean of a sample we do not expect that it will be exactly equal to the mean of the population from which the sample was drawn. Nonetheless, we can expect that it will be reasonably close to the population mean. A confidence interval provides an estimate as to how close. The 95% confidence interval is a random interval such that, in 95% of hypothetical replications of the sampling process, the confidence intervals obtained will include the true value of p The confidence interval for p is of the form x multiples of s.e.m. The multiple used is tl-a/2 (n-1), which is the 100(l-o/2) percentage point of the t-distribution with n-1 degrees of freedom. Thus, the 95% Cl (o=0.05) is given by ... 

Using the same estimates for S and A as before, we find that Ax 0.32. Hence, when A equals 1, a total of four data points (x = 0, 0.33, 0.67 and 1) is sufficient to describe the capacity factor within 2.5% (an error in In k of 5= 0.025 corresponds to an error of about 2.5% in fc). When more than two data points are available, a better estimate for A may of course be obtained from the data. For instance, when the verification of the first predicted optimum yields exactly the same capacity factors as were predicted, then apparently all A values are equal to zero and the confidence intervals extend over the entire parameter space. 

In the example quoted earlier, we found that 42 out of a sample of 50 patients (84 per cent) showed a successful response to treatment, but, what would happen if we were to adopt this treatment and record the outcomes for thousands of patients over the next few years The proportion of successful outcomes would (hopefully) settle down to a figure in the region of 84 per cent, but it would be most surprising if our original sample provided an exact match to the long-term figure. To deal with this, we quote 95 per cent confidence intervals for the proportion in the population based upon a sample proportion. 

Figure 15.3 refers back to our trial where 42 out of 50 patients showed a successful outcome. Notice that the interval is asymmetrical. The asymmetry arises because possible values are more tightly constrained on one side than the other. The upper limit of the interval could not logically be greater than 100 per cent, so the upper limit cannot be far above the point estimate of 84 per cent. However, the lower limit could be anything down to 0 per cent. Confidence intervals for proportions are always asymmetrical, unless the point estimate happens to be exactly 50 per cent (as in Figure 15.2). 

Critical values for individual tests and confidence intervals are based on the null distribution of i /aL, that is, on the distribution of this statistic when all effects Pi are zero. Lenth proposed a /-distribution approximation to the null distribution, whereas Ye and Hamada (2000) obtained exact critical values by simulation of Pi /ai under the null distribution. From their tables of exact critical values, the upper 0.05 quantile of the null distribution of Pi /aL is CL = 2.156. On applying Lenth s method for the plasma etching experiment and using a = 0.05 for individual inferences, the minimum significant difference for each estimate is calculated to be cl x l = 60.24. Hence, the effects A, AB, and E are declared to be nonzero, based on individual 95% confidence intervals. 

An application of this lemma shows that an exact 100 (1 — a)% confidence interval for /3, is given by... 

A non-parametric test is the Reverse Arrangements Test, in which a statistic, called 2I, is calculated in order to assess the trend of a time series. The exact procedure of calculation as well as tables containing confidence intervals is described in Bendat Piersol (2000). If A is too big or too small compared to these standard values could mean there is a significant trend in the data, therefore the process should not be considered in steady state. The test is applied sequentially to data windows of a given... 

Method validation is the process of demonstrating the ability of a method to produce reliable results [8], An analytical result should be expresed along with a confidence interval and a confidence level. The confidence interval can be described by a mean value and a interval width. Therefore the validation depends on the reliability of the confidence interval width estimation. The accuracy test can be performed exactly only, after that step. 

Figure 21-45 illustrates how the size of the confidence interval normalized with the sample variance decreases as the number of random samples n increases. The confidence interval depicts the accuracy of the analysis. The smaller the interval, the more exactly the mix quality can be estimated from the measured sample variance. If there are few samples, the mix quality s confidence interval is very large. An evaluation of the mix quality with a high degree of accuracy (a small confidence interval) requires that a large number of samples be taken and analyzed, which can be expensive and can require great effort. Accuracy and cost of analysis must therefore be balanced for the process at hand. 

There is nothing definitive about the selected number of 20. Quite generally, the estimate of the imprecision improves the more observations that are available. Exact confidence limits for the standard deviation can be derived from the distribution. Estimates of the variance, SD, are distributed according to the distribution (tabulated in most statistics textbooks) (N-l)SDVa X(v-i)j where (N-1) is the degrees of freedom. Then the two-sided 95% confidence interval (Cl) (95% Cl) is derived from the relation ... 

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