Counting confidence limits

If there is no mathematical fit, find the level below which 95% of the counts fall (for 95% confidence limit). 

Accuracy. The more accurate the sampling method the better. Given the very large environmental variability, however, sampling and analytical imprecision is rardy a significant contribution to overall error, or width of confidence limits, of the final result. Even highly imprecise methods, such as dust count methods, do not add much to overall variability when the variability between workers and overtime is considered. An undetected bias, however, is more serious because such bias is not considered by the statistical analysis and can, therefore, result in gross unknown error. 

FIGURE 13.29 Correlation between scratch counts and LPC determined for particles with diameter >0.469 pm (SO3 + SO5) with mixtures of slurry A (0 wt% of slurry B) and slurry B. Weight percentage of slurry B for the mixtures is labeled at the corresponding data point. The weighted linear regression fit to the data set and 95% confidence limits are represented by the solid line and dashed lines, respectively. Error bars correspond to 1 standard deviation (from Ref. 40). 

MPN and 95% Confidence Limits for Conforms Counts for Various Combinations of Positive and Negative Results when Five 10-mL, Five 1 -mL, and Five 0.1-mL Portions are Used... 

Fig. 12 Antitussive effect of compound 21 (TRK-851) on capsaicin-induced coughing in rats (p.o. administration). The number of coughs over 3 min were counted in the same rats before and 60 min after p.o. administration. The ED50 value (the dose that reduced the number of coughs to 50% vs control) for the inhibition of coughing was calculated to be 35.5 pg/kg (95% confidence limits 16.8-72.9 pg/kg). Reprinted from [6] with permission from Elsevier. Copyright (2008)...

Confidence limits are 2a-values based solely on counting statistics. Ra data provided by the Kansas Geological Survey. 

A sample was counted for 5 min and gave 2250 counts the background, also recorded for 5 min, gave 2050 counts. Is this sample radioactive Assume confidence limits of both 95% and 90%. 

The combined low background, high detection efficiency, and 5.5-ml sample volume lead to a detection limit of 10 Bq (600 dpm) per liter for a 100-min count (95% confidence limit). These results are achieved routinely in field tests of the FDTAS at monitoring wells, surface streams, ground water remediation facilities, sewage treatment plants, and the Savannah River. Results have been confirmed by parallel sampling and laboratory analyses. 

The phase contact ratios obtained should be accurate to about 0.5% at 95% confidence limits. The technique described here cancels out most of the common minor sources of error in liquid scintillation counting, for example ... 

FIGURE 1. Plasma Fe clearance curves for a normal subject determined by gamma counting and by liquid scintillation counting dots represent experimental data and full lines are model prediction. The 0.955 confidence limits of each net count rate are also indicated. 

The detection limits are determined by the number of fibers in suitably prepared blank filters and by the random nature of the filter and of the search. In some electron microscopy preparations, where no fibers are found in the blank controls, the detection limit has been taken as 3.69 times the sensitivity (the upper 95% confidence limit for zero in a Poisson distribution). Otherwise, as in PCOM, where a nonzero blank control is common, a higher detection limit prevails. Thus a series of blank filters might produce an average fiber count of five fibers in a standard analysis, which for routine air samples ( 0.5m ) could translate into a detection limit of 0.01 fibers per milliliter. 

These relationships are valid whatever the distribution of our measured values. When we make a radioactive count, our ultimate intention is to estimate the sample activity and a degree of confidence in that estimate of activity. Statistically we can achieve the former aim by identifying the measured count, C, as the expected number of decays, E n), and relating the confidence limit to the variance var(w). Thus, in principle ... 

Suppose that we have made two measurements of the same parameter and have calculated the uncertainty associated with them. For example, we might have taken measurements on two separate sub-samples of the same radioactive sample and calculated the activity, and 2> in Becquerels per gram with confidence limits of Sj and S2, which for simplicity we will take as the 68.3 % confidence limit (one standard uncertainty). We will assume that these confidence limits include all sources of uncertainty, not only those due to counting uncertainty ... 

If a count, N, is found to be valid (i.e. greater than Lq), then the result may be quoted as a value with an appropriate confidence limit represented by k standard deviations of A, as explained in Section 5.3.1 ... 

Table 5.5 95 % confidence limits for low numbers of gross counts... 

For small numbers of counts, the Poisson distribution is not symmetrical. This implies that confidence limits associated with a small count will not be symmetrical either. Again, appropriate limits can be tabulated and examples. 

The most obvious source of uncertainty is, of course, that due to the statistical nature of radioactivity counting. In many cases, it will be the major component of the total uncertainty and it is not uncommon for that alone to be the basis of the quoted confidence limits. However, that alone must underestimate the true uncertainty, especially when the counting uncertainties themselves are small and other sources of uncertainty become dominant. In setting up the budget, the uncertainty of all parameters which contribute to the final result must be assessed and quantified. Some may be justifiably found to be negligible. That conclusion in itself is part of the budget and should be documented. 

Providing the coincidence correction is less than 5% see Table XVII), the precision of counts made with the Coulter Counter is a function of the number of particles counted. The precision decreases with the total number of particles counted but is independent of the size of the aperture tube employed. The following data will serve as a guide to the precision of count data. (Confidence limits given are two standard deviations.)... 

Fig. 4 shows the calibration curve for the standard potassium iodide solutions. The counts in the Kiy peak show a linear relationship with the iodine concentration in the standard solution. Using a 99% confidence limit, the minimum detectable concentration of iodine in the cuvette was O.Ol mg/ml corresponding to 0.02 mg/ml of thyroid tissue. 

If background (B in c.p.m.) is counted for t minutes, its standard deviation will be y/ Bt. The minimum limit of detection at la confidence level and one-minute counting time requires ... 

Example Suppose you are attempting to observe a process that should produce 1 decay per week, and you observe no counts (with a 100% efficient detector) in 4 weeks. At the 95% confidence level, the upper limit on the decay rate is... 

X is not usually available because most experimental samples are counted only once. As a result of this limitation it has become customary to indicate a proportional error of a given count. At a confidence level of 95% (i.e., there is a 95% chance the observed value will fall within the interval x - l-96o-) the proportional error may be calculated from the relationship... 

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