Converting Confidence Intervals

With a ratio it is not possible to obtain a standard error formula directly however it is possible to obtain standard errors for log ratios. (Taking logs converts a ratio into a difference with log A/B = log A — log B.) So we first of all calculate confidence intervals on the log scale. It does, in fact, not make any difference what base we use for the logs but by convention we usually use natural logarithms, denoted fn . 

Finally we convert this back onto the OR scale by taking anti-logs of the ends of this interval to give a 95 per cent confidence interval for the OR as (1.13, 2.01). We can be 95 per cent confident that the OR lies within this range. 

Previously when we had calculated a confidence interval, for example for a difference in rates or for a difference in means, then the confidence interval was symmetric around the estimated difference in other words the estimated difference sat squarely in the middle of the interval and the endpoints were obtained by adding and subtracting the same amount (2 x standard error). When we calculate a confidence interval for the odds ratio, that interval is symmetric only on the log scale. Once we convert back to the odds ratio scale by taking anti-logs that symmetry is lost. This is not a problem, but it is something that you will notice. Also, it is a property of all standard confidence intervals calculated for ratios. 

Confidence intervals for the hazard ratio are straightforward to calculate. Like the odds ratio (see Section 4.5.5), this confidence interval is firstly calculated on the log scale and then converted back to the hazard ratio scale by taking anti-logs of the ends of that confidence interval. 

If this confidence interval is on the log scale, for example with both the odds ratio and the hazard ratio, then both the lower and upper confidence limits should be converted by using the anti-log to give a confidence interval on the original odds ratio or hazard ratio scale. 

For a measurement result to be metrologically traceable, the measurement uncertainty at each level of the calibration hierarchy must be known. Therefore, a calibration standard must have a known uncertainty concerning the quantity value. For a CRM this is included in the certificate. The uncertainty is usually in the form of a confidence interval (expanded uncertainty see chapter 6), which is a range about the certified value that contains the value of the measurand witha particular degree of confidence (usually 95%). There should be sufficient information to convert this confidence interval to a standard uncertainty. Usually the coverage factor ( see chapter 6) is 2, corresponding to infinite degrees of freedom in the calculation of measurement uncertainty, and so the confidence interval can be divided by 2 to obtain uc, the combined standard uncertainty. Suppose this CRM is used to calibrate... 

Blrt Standard addition. Selenium from 0.108 g of Brazil nuts was converted into the fluorescent products in Reaction 18-15. which was extracted into 10.0 mL of cyclohexane. Then 2.00 mL of the cyclohexane solution was placed in a cuvet for fluorescence measurement. Standard additions of fluorescent product containing 1.40 rg Se/mL are given in the table below. Construct a standard addition graph like Figure 5-6 to find the concentration of Se in the 2.00-mL unknown solution. Find the wt% of Se in the nuts and its uncertainty and 95% confidence interval. 

Cl, confidence interval Ml, myocardial infarction PVD, peripheral vascular disease TIA, transient ischemic attack. Hazard ratios derived from the model are used for the scoring system. The score for the five-year risk of stroke is the product the individual scores for each of the risk factors present. The score is converted into a risk with a graph. Source. Rothwell et at. (2005). 

For regression analysis a nonUnear relation often can be transformed into a linear one by plotting a simple function such as the logarithm, square root, or reciprocal of one or both of the variables. Nonlinear transformations should be used with caution because the transformation will convert a distribution from gaussian to nongaussian. Calculations of confidence intervals usually are based on data having a gaussian distribution. 

Based on eqs (4) and (5), the time axis (5) of Fig. 8.20D can be converted to depth (nm) for the entire sampling depth (150 nm). Therefore, the final form of the data now reveals concentrations as a function of depth. Figure 8.20E shows the quantified result of the Cu layer on a steel substrate shown previously in Fig. 8.20C as a time-intensity plot. The experimental results obtained using the quantitative method predict a depth of 54 2.1 nm (1 s at the 95% confidence interval), consistent with the nominal depth of Cu reported for the sample. In addition, this plot provides more valuable information than do the time-intensity plots as it allows the concentration of the multiple layers to be monitored as the sample is being sputtered. 

The confidence intervals discussed in this subsection are shown in Table 3.11 [2] in two-sided form. These intervals can be converted to one-sided form by removing the appropriate inequality and replacing the remaining (1 - a/2) or ot/2 term with (1 - a) or a. For example, the generic two-sided confidence interval y /2 < y < yi a/2 is replaced with y < yi to define a one-sided interval with an upper bound. 

Convert a quoted uncertainty that defines a confidence interval having a stated level of confidence, such as 95% or 99%, to a standard uncertainty by treating the quoted uncertainty as if a normal distribution had been used to calculate it (unless otherwise indicated) and dividing it by the appropriate factor for such a distribution. These factors are 1.960 and 2.576 for the two levels of confidence given. 

While the calculation of confidence intervals for a correlation is straightforward, it is rarely used in the cheminformatics literature. As such, we will provide a brief review of the method for calculating a confidence interval on a Pearson r. Since values of Pearson s r cannot exceed 1, its distribution is not normal. The distribution is closer to normal for lower values of r and becomes more skewed as r approaches 1. In order to calculate a confidence interval, values of r must be converted to Fisher s i distribution using Equation 1.10.1. 

Strictly speaking this equation estimates for the concentrations of the highest and lowest monitoring reference materials, so the estimate is a little pessimistic for concentrations between those extremes (see Figure 5.6). As usual the value can be converted to a confidence interval by multiplying by t, which has 2/ degrees of freedom in this case. 

In the next step, standard uncertainties are defined for each source of uncertainty. GUM defines two different methods for estimating uncertainty. Type A is a method of evaluation by the statistical analysis of series of observations. Standard deviations can be calculated through repeated observations. Type B is a method of evaluation of uncertainty by means other than the statistical analysis of series of observations. Calibration results or tolerances given in manuals can be used here. They are usually expressed in the form of limits or confidence intervals. Typical rules for converting such information to an estimated standard uncertainty u are introduced in [5] (p. 164). 

If none of the above holds, then the following method can be used to obtain an estimate for the converted confidence interval for a small initial confidence intervals. This method can also be used if the function depends on more than one of the parameters. The general formula is given as... 

Finally, though not less important, is the fact that SAM is an empirical tool that analytical chemists developed to take account of a serious practical problem. But it will not appear in any statistical textbook. The reason is that somehow chemists create a situation where an artificial signal (jo=0) gives rise to a theoretical concentration (which even in most papers and textbooks is negative and it is converted subjectively to a more convenient positive value ). Hence, serious problems arise from a statistical point of view when attempting accurately to define the variance associated with such a prediction. There will not be exact mathematical solutions, and different approaches (all of them approximately, but not totally, correct) can be considered. This will have important consequences in the calculation of the confidence intervals, as will be shown next. 

A multicenter, multinational study of 2225 patients with ischaemic and nonischaemic heart failure (HF) randomised to receive amiodarone, placebo or single lead implantable converter-defibrillator was followed for 5 years to determine the relation of thyroid status to mortality [4 ]. Abnormal thyroid fxmction in moderately symptomatic HF patients whether evident at base line (12% with TSH >5 nU/ml 1% < 0.3 gU/ml) or occurring during tiie study was associated with a higher mortality risk (hazard ratio (FIR) of 1.58 95% confidence interval (Cl) of 1.29-1.94, p < 0.0001) [4C],... 

The confidence limit selected was 0.95. The resultant calculated interval about xg was 0.078 Ib/ft, which converted to a percentage accuracy is 0.11. The end result is that there are only 5 chances in 100 that 5 differs from the curve presented by more than 0.078 Ib/ft. ... 

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