Confidence interval construction

Confidence interval The numerical interval constructed around a point estimate of a population parameter. It is combined with a probability statement linking it to the populations true parameter value, for example, a 90% confidence interval. If the same confidence interval construction technique and assumptions are used to calculate future intervals, they will include the unknown population parameter with the same specified probability. For example a 90% confidence interval around an arithmetic mean implies that 90% of the intervals calculated from repeated sampling of a population will include the unknown (true) arithmetic mean. 

The confidence interval constructed using the D-optimal interval shown in Figure 8.22 is narrower than the confidence interval constructed using the nonoptimal design, giving a reduced range, y, y, for the prediction of y. 

Note that confidence interval construction for the Cmax ratio represents a challenge because of the difficulty of formulating Cmax as a model parameter. Bootstrap (10) allows this construction, though, because in each bootstrap run, the predicted Cmax for the test and reference formulation, and thus their ratio, can be calculated from the population model parameters. The percentile bootstrap method then uses the 5% and 95% percentiles of the bootstrap runs to form the 90% confidence interval. Specifically, in each bootstrap run, a bootstrap data set can be generated where the subjects were resampled with replacement. Parameter estimates can be obtained for the bootstrap data set, and thus a ratio of ACC and Cmax- Results of all bootstrap data sets can be assembled and the 5% and 95% percentiles used to construct the 90% bootstrap confidence intervals. 

We consider the above mentioned study designs together with a few corresponding analysis methods and explore their potential. Several analysis methods may be appealing, depending on the particular study design. For example, measurements of various individuals could be pooled. Alternatively, one could consider mixed effect models, accounting for intersubject variability in certain parameters. We discuss the impact of estimation and confidence interval construction methods separately. 

We also looked at confidence interval construction based on standard errors of parameter estimates. If the asymptotic 90% confidence intervals are truly accurate, then they should cover the true parameter 90% of the time. We examined the confidence interval coverages in simulation scenario I, shown in Figure 16.2. 

We briefly explore later the potential of using likelihood profile and bootstrap as confidence interval construction methods through application examples with broncodUaion and broncoprovocation data. 

We also attempted the likelihood profile method to construct the 90% conhdence interval. However, the likelihood profile of F turned out to be extremely flat in this case. As a result, the 90% confidence interval included (0.1,10). This was considered unreasonable, given the similarity between the two formulations. Thus, the likelihood profile method did not seem suitable for confidence interval construction. 

Donaldson and Schnabel (1987) used Monte Carlo simulation to determine which of the variance estimators was best in constructing approximate confidence intervals. They conclude that Eq. (3.47) is best because it is easy to compute, and it gives results that are never worse and sometimes better than the other two, and is more stable numerically than the other methods. However, their simulations also show that confidence intervals obtained using even the best methods have poor coverage probabilities, as low as 75% for a 95% confidence interval. They go so far as to state confidence intervals constructed using the linearization method can be essentially meaningless (Donaldson and Schnabel, 1987). Based on their results, it is wise not to put much emphasis on confidence intervals constructed from nonlinear models. 

In Section 4D.2 we introduced two probability distributions commonly encountered when studying populations. The construction of confidence intervals for a normally distributed population was the subject of Section 4D.3. We have yet to address, however, how we can identify the probability distribution for a given population. In Examples 4.11-4.14 we assumed that the amount of aspirin in analgesic tablets is normally distributed. We are justified in asking how this can be determined without analyzing every member of the population. When we cannot study the whole population, or when we cannot predict the mathematical form of a population s probability distribution, we must deduce the distribution from a limited sampling of its members. 

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. 

Construct an appropriate standard additions calibration curve, and use a linear regression analysis to determine the concentration of analyte in the original sample and its 95% confidence interval. 

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. 

In constructing confidence intervals, it is essential to use suitable random variables whose values are determined by the sample data as well as by the parameters, but whose distributions do not involve the parameters in question. 

One can also construct 95% confidence intervals using unequal tails (for example, using the upper 2% point and the lower 3% point). We usually want our confidence interval to be as short as possible, however, and with a symmetric distribution such as the normal or t, this is achieved using equal tails. The same procedure very nearly minimizes the confidence interval with other nonsymmetric distributions (for example, chi-square) and has the advantage of avoiding rather tedious computation. 

Vertzoni et al. (30) recently clarified the applicability of the similarity factor, the difference factor, and the Rescigno index in the comparison of cumulative data sets. Although all these indices should be used with caution (because inclusion of too many data points in the plateau region will lead to the outcome that the profiles are more similar and because the cutoff time per percentage dissolved is empirically chosen and not based on theory), all can be useful for comparing two cumulative data sets. When the measurement error is low, i.e., the data have low variability, mean profiles can be used and any one of these indices could be used. Selection depends on the nature of the difference one wishes to estimate and the existence of a reference data set. When data are more variable, index evaluation must be done on a confidence interval basis and selection of the appropriate index, depends on the number of the replications per data set in addition to the type of difference one wishes to estimate. When a large number of replications per data set are available (e.g., 12), construction of nonparametric or bootstrap confidence intervals of the similarity factor appears to be the most reliable of the three methods, provided that the plateau level is 100. With a restricted number of replications per data set (e.g., three), any of the three indices can be used, provided either non-parametric or bootstrap confidence intervals are determined (30). 

Using the data with built-in error generated in the previous section (six replications per data set), for every c value, two test data sets (b = 0.5 and b = 1.5) were separately compared with a reference data set (b = 1). The estimated total amount dissolved (W0) of the test and the reference data sets were compared by constructing confidence intervals at the 0.05 level for their mean differences. Estimated shape parameter, c, and scale parameter, b, of the test and the reference... 

In a series of papers (23-26), Polli and colleagues proposed alternative direct curve comparison metrics on this level. In their papers, attention was focused on two aspects (i) are means or medians more suitable for comparison and (ii) how can symmetric confidence intervals be constructed that are invariant when exchanging reference and test In addition, this work was devoted to bioavailability and bioequivalence, i.e., time profiles in vivo, but the conclusions apply likewise to in vitro-release profiles. 

Both assumptions are mainly needed for constructing confidence intervals and tests for the regression parameters, as well as for prediction intervals for new observations in x. The assumption of normal distribution additionally helps avoid skewness and outliers, mean 0 guarantees a linear relationship. The constant variance, also called homoscedasticity, is also needed for inference (confidence intervals and tests). This assumption would be violated if the variance of y (which is equal to the residual variance a2, see below) is dependent on the value of x, a situation called heteroscedasticity, see Figure 4.8. 

The denominator n 2 is used here because two parameters are necessary for a fitted straight line, and this makes s2 an unbiased estimator for a2. The estimated residual variance is necessary for constructing confidence intervals and tests. Here the above model assumptions are required, and confidence intervals for intercept, b0, and slope, b, can be derived as follows ... 

We will describe an accurate statistical method that includes a full assessment of error in the overall calibration process, that is, (I) the confidence interval around the graph, (2) an error band around unknown responses, and finally (3) the estimated amount intervals. To properly use the method, data will be adjusted by using general data transformations to achieve constant variance and linearity. It utilizes a six-step process to calculate amounts or concentration values of unknown samples and their estimated intervals from chromatographic response values using calibration graphs that are constructed by regression. 

Move 2. Next, the confidence interval for the true mean of Y is constructed such that, for a given Y, the 100P% confidence interval for is... 

Construction of an Approximate Confidence Interval. An approxi-mate confidence interval can be constructed for an assumed class of distributions, if one is willing to neglect the bias introduced by the spline approximation. This is accomplished by estimation of the standard deviation in the transformed domain of y-values from the replicates. The degrees of freedom for this procedure is then diminished by one accounting for the empirical search for the proper transformation. If one accepts that the distribution of data can be approximated by a normal distribution the Student t-distribution gives... 

Suppose we could use bg and s from only one set of experiments to construct a confidence interval about bg such that there is a given probability that the interval... 

Each of these confidence intervals (the calculated interval and the critical interval) can be expressed in terms of b, s, and some value of t (see Equation 6.5). Because the same values of bg and are used for the construction of these intervals, the... 

We can test the significance of the parameter estimate bp by calculating the value of t required to construct a confidence interval extending from and including the value zero (see Section 6.3). 

The 80% confidence bands for a given concentration level are constructed such that all the blocks within the band are those whose 80% confidence interval contains the given concentration level. That is, if we want to estimate the 80% confidence band for 250 ppm lead, all those blocks whose lower limits are greater than 250 ppm lead are classified as blocks whose concentrations are greater than 250 ppm. Those blocks whose upper limits are less than 250 ppm are classified as blocks whose concentrations are less than 250 ppm. The blocks which are left over, those containing 250 ppm in the 80% confidence interval, constitute the confidence band about the 250 ppm concentration level. Figures 15 through 22 show the 80% confidence bands for 2500 ppm, 1000 ppm, and 500 ppm concentration levels for the RSR and DMC and 500 ppm and 250 ppm concentration levels for REF, respectively. 

There are a number of sources of uncertainty surrounding the results of economic assessments. One source relates to sampling error (stochastic uncertainty). The point estimates are the result of a single sample from a population. If we ran the experiment many times, we would expect the point estimates to vary. One approach to addressing this uncertainty is to construct confidence intervals both for the separate estimates of costs and effects as well as for the resulting cost-effectiveness ratio. A substantial literature has developed related to construction of confidence intervals for cost-effectiveness ratios. 

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... 

The previous sections in this chapter are applicable when we are dealing with means. As noted earlier these parameters are relevant when we have continuous, count or score data. With binary data we will be looking to construct confidence intervals for rates or proportions plus differences between those rates. 

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