Intervals, estimation

Confidence-Interval Estimates. Confidence-interval estimates for the expected hfe or rehabihty can be obtained easily in the case of the exponential. Here only the limits for failure-censored (Type II) and time-censored (Type I) life testing are given. It is possible to specify a test as either time- or failure-tmncated, whichever occurs first. The theory for such tests is explained in References 16 and 17. 

Reliability Estimation. Both a point estimate and a confidence interval estimate of product rehabUity can be obtained. Point Estimate. The point estimate of the component rehabUity is given by... 

Watters, R. L., Jr., Carroll, R. J., and Spiegelman, C. H., Error Modeling and Confidence Interval Estimation for Inductively Coupled Plasma Calibration Curves, Anal. Chem. 59, 1987, 1639-1643. 

Once the model functional form has been decided upon and the experimental data have been collected, a value for the model parameters (point estimation) and a confidence region for this value (interval estimation) must be estimated... 

Watts (1994) dealt with the issue of confidence interval estimation when estimating parameters in nonlinear models. He proceeded with the reformulation of Equation 16.19 because the pre-exponential parameter estimates "behaved highly nonlinearly." The rate constants were formulated as follows... 

As a rule, the level of precision of a risk estimate cannot exceed the precision of the exposure and effects data from which it is obtained. In the following we will focus upon carcinogenic risk estimation, for which it will often be possible to achieve at least interval estimates of risk. 

The estimation of means, variances, and covariances of random variables from the sample data is called point estimation, because one value for each parameter is obtained. By contrast, interval estimation establishes confidence intervals from sampling. 

The technique for obtaining interval estimates for X, discussed in this section, is presented in the paper by Lieberman, Miller, and Hamilton ( 2 ) and based on the Bonferroni inequality ( ) described below Other methods are found in the references ( 23,24 ) ... 

Bonferroni Interval Estimates Interval estimates for the unknown X, referred to as unlimited simultaneous discrimination intervals ( 21 ), are based on the estimated regression line of y, on X., and the confidence interval (on the Y-axis) about the response y for an unknown The resulting interval estimates have the property that for at least 100(1- )% of the different calibration sets, at least 100P% of the amount intervals estimated from that calibration will contain true unknown amounts... 

The Bonferroni interval estimate of X, given Y, is found in three moves. First, the Working-Hotelling confidence band for the regression line... 

Bonferroni inequality is invoked to combine the two proceeding confidence statements, each made with the confidence (l-a/2), to yield an interval estimate for X with confidence at least (1-a). The confidence band on the regression line and the confidence interval on U are intersected and the Bonferroni interval estimate of X is found by projecting the intersection onto the x-axis. Figure Ic illustrates the procedure. If is in the interval on the Y-axis and if the hyperbolic confidence band contains the line... 

The frequency interpretation of the interval estimates on the unknown amounts is given by the following ( 27 ) With at least 1- a confidence, based on the sampling characteristics of the observations on the standards, at least P proportion of the interval estimates made from a particular calibration will contain the true amounts. The Bonferroni inequality insures the 1-a confidence since the confidence interval about the regression line and the upper bound on cr are each performed using a 1- a/2 confidence coefficient. Hence, the frequency interpretation states that at least (1-a) proportion of the standard calibrations are such that at least P proportion of the intervals produced by the method cover the true unknown amounts. For the remaining a proportion of standard calibrations the proportion of intervals which cover the true unknown values may be less than P. 

Prediction of Unknowns. Step 5. The point estimate of unknown amounts is obtained from the intersection of the corresponding response value and the regressed line which is then projected down to the amount axis. The interval estimate for an unknown amount... 

Estimated Amount Intervals. Estimated amount intervals, calculated from the example data sets, are shown in Table XIII. 

Calibration Data Extrapolation Caveat. Since extrapolations cannot be done in performing proper regression line calculations, it follows that there will be concern about interval estimates from data that do occur at the extreme ends of the range of standards. To avoid extrapolation we suggest that response values be limited to those values corresponding to the range of amounts of the calibration standards. At the extreme ends, however, one end of the estimated amount interval would then extend into an extrapolated region, the lower end at the minimum amount and the upper end at the maximum amount. 

However, with improper transformation the calculation of confidence bands and amount interval estimates is erroneous because of the non-constant variance ... 

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. 

A somewhat different computational procedure is often used in practice to carry out the test described in the previous section. The procedure involves two questions What is the minimum calculated interval about bg that will include the value zero and, Is this minimum calculated interval greater than the confidence interval estimated using the tabular critical value of t If the calculated interval is larger than the critical confidence interval (see Figure 6.7), a significant difference between Po and zero probably exists and the null hypothesis is disproved. If the calculated interval is smaller than the critical confidence interval (see Figure 6.8), there is insufficient reason to believe that a significant difference exists and the null hypothesis cannot be rejected. 

Estimates of two kinds can be made, point estimate and interval estimate. 

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. 

J. Neuman calls these intervals confidence intervals, for as parameter-interval estimates, ranges with known confidence level are chosen. An interval estimate gives a range of values that can be expected to include the correct value with a certain specified percentage of the time. This provides a measure of the error involved in the estimate. The wider the range of the interval estimate, the poorer the point estimate. 

As it has been mentioned, apart from point estimates there exist the parameter interval estimates. No matter how well the parameter estimate has been chosen, it is only logical to test the estimate deviation from its correct value, as obtained from the sample. For example, if in numerical analysis one obtains that the solution of an equation is approximately 3.24 and that 0.03 is the maximal possible deviation from the unknown correct solution of the equation, then we are absolutely sure that the range (3.24-0.03=3.21 3.24+0.03=3.27) contains the unknown correct solution of the equation. Therefore the problem of determining the interval estimate is formulated in the following way ... 

From the reactor data given earlier, 32, 55, 58, 59, 59, 60, 63, 63, 63, 63, 67 determine an interval estimate of yield, presuming that the population variance is ax=81. 

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 °°. 

Make an interval estimate of the yield from the reactor in Example 1.14 for a 95% confidence level. 

A measure of the scatter or variability of data is the variance, as discussed earlier. We have seen that a large variance produces broad-interval estimates of the mean. Conversely, a small variability, as indicated by a small value of variance, produces narrow interval estimates of the mean. In the limiting case, when no random fluctuations occur in the data, we obtain exact identical measurements of the mean. In this case, there is no scatter of data and the variance is zero, so that the interval estimate reduces to an exact point estimate. 

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. 

Determine point and interval estimates for the population variance of the reactor yield data of earlier Example 1.12 ... 

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