Confidence intervals for cost

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. 

One of the most dependably accurate methods for deriving 95% confidence intervals for cost-effectiveness ratios is the nonparametric bootstrap method. In this method, one resamples from the smdy sample and computes cost-effectiveness ratios in each of the multiple samples. To do so requires one to (1) draw a sample of size n with replacement from the empiric distribution and use it to compute a cost-effectiveness ratio (2) repeat this sampling and calculation of the ratio (by convention, at least 1000 times for confidence intervals) (3) order the repeated estimates of the ratio from lowest (best) to highest (worst) and (4) identify a 95% confidence interval from this rank-ordered distribution. The percentile method is one of the simplest means of identifying a confidence interval, but it may not be as accurate as other methods. When using 1,000... 

Chaudhary MA, Stearns SC. Estimating confidence intervals for cost-effectiveness ratios an example from a randomized trial. Stat Med 1996 15 1447-58. 

Polsky DP, Glick HA, Willke R, Schuknan K. Confidence intervals for cost-effectiveness ratios a comparison of four methods. Health Econ 1997 6 243-52. 

Willan AR, O Brien BJ. Confidence intervals for cost-effectiveness ratios an application of Fieller s theorem. Health Econ 1996 5 297-305. 

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-... 

Suppose your marginal cost is 10. Based on the least squares regression, compute a 95% confidence interval for the expected value of the profit maximizing output. 

Using a well known result, for a linear demand curve, marginal revenue is MR = (-a/ ) + (2/p)q. The profit maximizing output is that at which marginal revenue equals marginal cost, or 10. Equating MR to 10 and solving for q produces q = a/2 + 5p, so we require a confidence interval for this combination of the parameters. 

Frequentist methods are fundamentally predicated upon statistical inference based on the Central Limit Theorem. For example, suppose that one wishes to estimate the mean emission factor for a specific pollutant emitted from a specific source category under specific conditions. Because of the cost of collecting measurements, it is not practical to measure each and every such emission source, which would result in a census of the actual population distribution of emissions. With limited resources, one instead would prefer to randomly select a representative sample of such sources. Suppose 10 sources were selected. The mean emission rate is calculated based upon these 10 sources, and a probability distribution model could be fit to the random sample of data. If this process is repeated many times, with a different set of 10 random samples each time, the results will vary. The variation in results for estimates of a given statistic, such as the mean, based upon random sampling is quantified using a sampling distribution. From sampling distributions, confidence intervals are obtained. Thus, the commonly used 95% confidence interval for the mean is a frequentist inference... 

Table 8 illustrates the effect of CRN on the inventory simulation when the goal is to estimate 0(1) 0(2) tije expected difference between the cost per period of inventory policies 1 and 2. The experiment design is the same one described in Section 9, so the basic data for each inventory policy are k = 20 approximately i.i.d. normal batch means (for the purposes of this illustration they can be thought of as k i.i.d. replications). The table gives the point estimate, estimated standard error, and a 95% confidence interval for the expected difference. 

Figure 4.5. Estimated total analytical cost for one batch of tablets versus the attained confidence interval CI(X). 640 (UV) resp. 336 (HPLC) parameter combinations were investigated (some points overlap on the plot).

Table II displays numerical results for values of ag = 1.0 and 1.5 and =. 25,. 50, 1.00, 1.50 and 2.00. Values of J and K are determined for confidence intervals with upper error bound set at 50 percent and 100 percent and confidence coefficients at 95 percent and 99 percent. In general, the table shows that it is very expensive to go from 95 percent to 99 percent confidence. It is also significantly more expensive to obtain an estimate with a 50 percent error than a 100 percent error. For example, if Og = Ol = 1.00, to be 95 percent certain that the error in the estimated level of airborne asbestos is less than 50 percent requires J = 47 and K = 1 at a cost of 24,440. If a 100 percent error in the estimate is acceptable, J is reduced to 16, K remains at 1 and the cost is 8,320.

What does optimization mean in an analytical chemical laboratory The analyst can optimize responses such as the result of analysis of a standard against its certified value, precision, detection limit, throughput of the analysis, consumption of reagents, time spent by personnel, and overall cost. The factors that influence these potential responses are not always easy to define, and all these factors might not be amenable to the statistical methods described here. However, for precision, the sensitivity of the calibration relation, for example (slope of the calibration curve), would be an obvious candidate, as would the number of replicate measurements needed to achieve a target confidence interval. More examples of factors that have been optimized are given later in this chapter. 

If the confidence interval of the estimate is known, then the contingency charges can also be estimated based on the desired level of certainty that the project will not overrun the projected cost. For example, if the cost estimate is normally distributed, then the estimator has the following confidence levels ... 

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. 

In an inertial microfiuidic device, concentration, efficiency, and purity are the most broadly used parameters to quantitatively characterize device performance. For this, it is necessary to know concentration of separated particles at each outlet. The most common approach is to collect samples fi om each outlet and then to use flow cytometry to count and size particles. An alternative approach is to use a hemocytometer for direct measurement of particle concentration in each outlet. While hemocytometers offer a low-cost option, the increased error rate (as high as 10 %) will require a larger sample size to maintain the confidence interval. Once particle counts in each outlet are known, purity and efficiency of the separation can be calculated. Purity is calculated as the number of target particles over total particles in one outlet. Efficiency is calculated as the number of target particles from one outlet over total number of target particles from all outlets. Next, the modulated aspect-ratio device is used as an example to demonstrate these calculations. 

It is dear that the difference between sample and population may become larger, the smaller a sample is. The minimum sample size of N = 30 as specified in standards represents a compromise between the large cost of specimen preparation and accuracy, although it should be noted that for N = 30 the uncertainty remains relatively large. Figure 12.10 shows the confidence intervals which result from the sampling procedure, for the Weibull modulus and the function (the scatter of the... 

Much more difficult is the situation in the case of a nonlinear objective function of the type (12.2.3). Numerous case studies have shown that the optimum solution according to the objective function (12.2.2), i.e. optimisation as the cost of the measurement concerns, yields a solution that is unacceptable from the point of view of the precision of results. It means that the solution with minimum costs is, at the same time, the solution with some unmeasured quantities that are theoretically observable, but with unacceptably low precision (for example, the confidence interval wider than the value of the quantity). In this case, the following optimisation method can be recommended. The method is analogous to the direct search in graphs that proved efficient for optimisation of measurement designs in single-component mass balancing see Subsection 12.1.2. 

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