Confidence limits with time

Fig. X-7. Advancing and receding contact angles of octane on mica coated with a fluo-ropolymer FC 722 (3M) versus the duration of the solid-liquid contact. The solid lines represent the initial advancing and infinite time advancing and receding contact lines and the dashed lines are 95% confidence limits. (From Ref. 75.)...

Acute toxicity studies are often dominated by consideration of lethaUty, including calculation of the median lethal dose. By routes other than inhalation, this is expressed as the LD q with 95% confidence limits. For inhalation experiments, it is convenient to calculate the atmospheric concentration of test material producing a 50% mortaUty over a specified period of time, usually 4 h ie, the 4-h LC q. It is desirable to know the nature, time to onset, dose—related severity, and reversibiUty of sublethal toxic effects. 

Confidence limits are partial integrations over a probability density function. There are two special cases failure with time and failure with demand. 

Loss of offsite power at nuclear power plants is addressed in EPRI NP-2301, 1982 giving data on the frequency of offsite power loss and subsequent recoveiy at nuclear power plants. Data analysis includes point estimate frequency with confidence limits, assuming a constant rate of occurrence. Recovery time is analyzed with a lognormal distribution for the time to recover. 

The Production Department was not amused, because lower values had been expected. Quality Control was blamed for using an insensitive, unse-lective, and imprecise test, and thereby unnecessarily frightening top management. This outcome had been anticipated, and a better method, namely polarography, was already being set up. The same samples were run, this time in duplicate, with much the same results. A relative confidence interval of 25% was assumed. Because of increased specificity, there were now less doubts as to the amounts of this particular heavy metal that were actually present. To rule out artifacts, the four samples were sent to outside laboratories to do repeat tests with different methods X-ray fluorescence (XRFi °) and inductively coupled plasma spectrometry (ICP). The confidence limits were determined to be 10% resp. 3%. Figure 4.23 summarizes the results. Because each method has its own specificity pattern, and is subject to intrinsic artifacts, a direct statistical comparison cannot be performed without first correcting the apparent concentrations in order to obtain presumably true... 

SHELFLIFE.dat The content (% of nominal) of two active components in a dosage form was assayed at various times (0-60 months) during a pharmaceutical stability trial to determine the acceptable shelf-life of the formulation the point at which the lower 90% confidence limit of the finear regression model intersects the 90%-of-nominal line gives the answer. Use with SHELFLIFE or LINREG. 

The variance about the mean, and hence, the confidence limits on the predicted values, is calculated from all previous values. The variance, at any time, is the variance at the most recent time plus the variance at the current time. But these are equal because the best estimate of the current time is the most recent time. Thus, the predicted value of period t+2 will have a confidence interval proportional to twice the variance about the mean and, in general, the confidence interval will increase with the square root of the time into the future. 

An approach for analyzing data of a quantitative attribute that is expected to change with time is to determine the time at which the 95% one-sided confidence limit for the mean curve intersects the acceptance criterion. If analysis shows that the batch-to-batch variability is small, it is advantageous to combine the data into one overall estimate by applying appropriate statistical tests (e.g., p-values for level of significance of rejection of more than 0.25) to the slopes of the regression lines and zero-time intercepts for individual batches. If it is inappropriate to combine data from several batches, the overall shelf life should be based on the minimum time a batch can be expected to remain within the acceptance criteria. 

Reference materials are often real samples that have been carefully prepared and analyzed by many laboratories by many different methods. In this way, their known value and accompanying confidence limits are determined. The regular use of reference materials not only provides for calibration and standardization, but it also can demonstrate an analyst s proficiency with a method. It should be noted that SRMs are expensive, however, and are not often used for routine calibration and standardization work. Usually, primary and secondary standards are used for that. Another important fact about RMs is that they are considered to have a finite shelf life and cannot be confidently used as RMs after a certain period of time. 

Questions 2 and 3 imply a choice between expressing effects in terms of magnitude, frequency, and certainty. In practice, the assessment endpoint may often need to be dehned in terms of 2 or 3 of these dimensions. For example, it may be desirable to estimate the proportion of species (frequency) that will experience different levels of mortality (magnitude), and to provide confidence limits (certainty). Indeed, the risk manager s questions may imply an assessment endpoint with more than 3 dimensions, for example, if it is desired to express frequency in terms of space (e.g., number of hectares) and time (proportion of years). The dimensionality of the assessment endpoint will have major implications for all aspects of the analysis and for communication of results, so it is essential to discuss it carefully with the risk manager at the outset to ensure it meets their needs. 

Performance is defined by the sensitivity threshold, or the minimum concentration of element in solution that will yield an analytical signal with amplitude equal to twice that of the average background signal. This classical definition leads to optimistic values that can vary from element to element. The limit of detection represents the concentration of an element that can be detected with a 95% confidence limit (cf. chapter 21). In general, measurements are made in a concentration domain that corresponds to 50 times the limit of detection. 

For SK-500 the rate at 573°K and 400 sec after the initiation of reactant flow is independent of reactant mole ratio for Ce C2 = 0.7 to 10. Under these conditions the 400-sec point is just beyond the maximum in the rate curve. Similar behavior was observed at one other condition. Initial rate of reaction estimated by extrapolating the decay portion of the rate curves for this data to zero time (see below) indicates a maximum in the rate at C6 C2 == 3.5 (Figure 2). Error bars represent estimated 95% confidence limits. The observed activity for HY is about twice that of SK-500, that for LaY is about two-thirds that of SK-500 (Figure 2). This is consistent with the trend expected (7) since all catalysts were activated to the same temperature. The temperature dependence of the observed rate is large for all systems studied indicating the absence of external mass transfer limitations. 

In this calculation it has been assumed that the value of data points in the table can be used, thus making it unnecessary to repeat the experiments many times. Furthermore, if the distribution is considered to be Gaussian, then the confidence limits for the estimates of the gradient and intercept of the fitted line can be determined. For this type of distribution it can be shown 8 that 95 per cent of the readings will lie within 2ffof the mean. Hence it is possible to say with 95 per cent certainty (or with 95 per cent confidence limits) that the equation of the fitted line is ... 

Again, the minimum and maximum loading configurations should be studied. Thermocouples will be placed both inside and outside the container at the cool spot location(s), in the steam exhaust line, and in constant-temperature baths outside the chamber. The F0 value will be calculated based on the temperature recorded by the thermocouple inside the container at the coolest area of the load. Upon completion of the cycle, the F0 value will indicate whether the cycle is adequate or if alterations must be made. Following the attainment of the desired time-temperature cycle, cycles are repeated until the user is satisfied with the repeatability aspects of the cycle validation process. Statistical analysis of the F0 values achieved at each repeated cycle may be conducted to verify the consistency of the process and the confidence limits for achieving the desired F0 value. 

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