Calculation of confidence intervals

The use of these equations allow the calculation of confidence intervals to assess, for example, the significance of negative parameter estimates or the additional experimentation required to estimate the parameters with a particular precision. 

This chapter introduces basic concepts in statistical analysis that are of relevance to describing and analyzing the data that are collected in clinical trials, the hallmark of new drug development. (Statistical analysis in nonclinical studies was addressed earlier in Chapter 4.) This chapter therefore sets the scene for more detailed discussion of the determination of statistical significance via the process of hypothesis testing in Chapter 7, evaluation of clinical significance via the calculation of confidence intervals in Chapter 8, and discussions of adaptive designs and of noninferiority/equivalence trials in Chapter 11. 

In order to compensate for the uncertainty incurred by taking small samples of size n, the / probability distribution shown in Figure 3.2 is used in the calculation of confidence intervals, replacing the normal probability distribution based on z values shown in Figure 3.1. When n > 30, the /-distribution approaches the standard normal probability distribution. For small samples of size n, the confidence interval of the mean is inflated and can be estimated using Equation 3.9... 

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. 

The key to the ethics of such studies is informed consent from patients, efficient scientific design and review by an independent research ethics committee. The key interpretative factors in the analysis of trial results are calculations of confidence intervals and statistical significance.The potential clinical significance needs to be considered within the confines of controlled clinical crials.This is best expressed by stating not only the percentage differences, but also the absolute difference or its reciprocal, the number of patients who have to be treated to obtain one desired outcome.The outcome might include both efficacy and safety... 

Another important characteristic of the normal distribution is that 95% of the data values he within 2s, and 99.7% in the range 3s, as shown in Figure 16.2. This distribution of error in a normal distribution allows the calculation of confidence intervals for x. The confidence interval is the range of concentration within which the real sample concentration is expected to occur, for a given degree of confidence. Therefore, the interval size depends of the degree of confidence (for greater... 

To perform unbiased analyses, a collaboration-wide policy of blindness was established, where cut selections are optimized on a fraction of data or on time-scrambled data set. We present upper confidence limits for null results following the treatment described in (Feldman and Cousins, 1998) and incorporate systematic uncertainties into the calculation of confidence intervals according to (Conrad et al., 2003). The contributions to systematic uncertainties is predominantly due to variations of the optical properties of the ice, the absolute sensitivity of the OM, the neutrino cross section and the muon propagation. The combined systematic uncertainty is typically 30%, although the value varies slightly with the analysis method. 

It follows from the fact that the sampling distribution is normally distributed that 95% of the sample statistics will be within 1-96 SDs of the mean (the population parameter). The SD of the sampling distribution (as mentioned above) is referred to as the SE of the estimate. Because only 5% of sample statistics will be more than 1-96 SEs from the population parameter, for any sample statistic taken at random it is 95% likely that the population parameter is within 1-96 SEs of the sample statistic. This is the rationale for the calculation of confidence intervals (Cis) in estimation. So, a 95% CI is found by the expression ... 

Another useful interpretation of confidence intervals is that the values that are enclosed within the confidence interval are those that are considered the most plausible values of the unknown population parameter. Values outside the interval are considered less plausible. All other things being equal, the need for greater confidence in the estimate results in wider confidence intervals, and confidence intervals become narrower (that is, more precise) as the sample size increases. This last fact is explored in greater detail in Chapter 12 because it is directly relevant to the estimation of the required sample size for a clinical trial. The methods to use for the calculation of confidence intervals for other population parameters of interest are provided in subsequent chapters. 

Whereas the mean is a measure of the tendency of a variable within its confidence intervals, the width of the confidence interval itself depends on the sample size and on the variation of data values. On one hand, the larger the sample size, the more reliable is the mean. On the other hand, the larger the variation, the less reliable is the mean. The calculation of confidence intervals is based on the assumption that the variable is normally — or at least equally — distributed in the population. The estimate may not be valid if this assumption is not met, unless the sample size is large. 

In the formula, which leads to the calculation of confidence intervals, the standard deviation of the mean or the uncertainty of the standard deviation may appear, using the following relation that gives the standard deviation of the mean (22.17) ... 

In statistical analysis involving normal distributions some other types of distributions are encountered frequently. The t-distribution is encountered e.g. in the calculation of confidence intervals in various situations. Its limiting distribution is the standard-normal distribution. The ( -distribution Is the sum of squares of several standard-normal distributed variables. It may be encountered in tests on normality of data. 

The covariance matrices of Eq. (6.21) or (6.22) can be used analogously for calculations of confidence intervals, if more than two parameters are to be determined. 

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. 

Given the manipulations carried out and the diverse data bases employed, calculation of confidence intervals for the various toxicity constants presented is a difficult undertaking. At this point the accuracy of these values is of the scale of an order of magnitude caution must be exercised as these first approximations will be subject to further refinement. 

From Equation 4, x is 77.11 and from Equation 5, is 0.24 for 4 degrees of freedom. Because cr is not known, the Student 975 (2.78 for 4 degrees of freedom) is used to calculate the confidence interval at the 95% probability level. 

In the introduction to this section, two differences between "classical" and Bayes statistics were mentioned. One of these was the Bayes treatment of failure rate and demand probttbility as random variables. This subsection provides a simple illustration of a Bayes treatment for calculating the confidence interval for demand probability. The direct approach taken here uses the binomial distribution (equation 2.4-7) for the probability density function (pdf). If p is the probability of failure on demand, then the confidence nr that p is less than p is given by equation 2.6-30. 

When a small number of observations is made, the value of the standard deviation s, does not by itself give a measure of how close the sample mean x might be to the true mean. It is, however, possible to calculate a confidence interval to estimate the range within which the true mean may be found. The limits of this confidence interval, known as the confidence limits, are given by the expression ... 

Calculate the confidence interval (90%) about the mean of the data in question 3. 

Once the reliability of a replicate set of measurements has been established the mean of the set may be computed as a measure of the true mean. Unless an infinite number of measurements is made this true mean will always remain unknown. However, the t-factor may be used to calculate a confidence interval about the experimental mean, within which there is a known (90%) confidence of finding the true mean. The limits of this confidence interval are given by ... 

With what confidence can the mean of a set of experimental results be quoted Calculate the confidence interval (equation as a measure of the true mean (2.7))... 

For circumstances where wide variability is observed, or a statistical evaluation of f2 metric is desired, a bootstrap approach to calculate a confidence interval can be performed (8). 

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

Perform some analyses (e.g. at least five ), then calculate the confidence interval of the arithmetic mean and check if the reference value is within this interval... 

Fig. 5. Interpretation of the pignals from a multidetector SEC direct calculations 95% confidence interval included. Low conversion, low styrene content SAN copolymer 0 styrene content, molecular weight.

We will first look at the way we calculate the confidence interval for a single mean p and then talk about its interpretation. Later in this chapter we will extend the methodology to deal with pj — p2 and other parameters of interest. 

So if we were calculating a confidence interval for a mean p. from a sample of size 16 then we would look in row 15 for the multiplying constant and use 2.13 in place of 1.960 in the calculation of the 95 per cent confidence interval and 2.95 in place of 2.576 for the 99 per cent confidence interval. 

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. 

If the treatment effect in each of the individual trials is the difference in the mean responses, then d represents the overall, adjusted mean difference. If the treatment effect in the individual trials is the log odds ratio, then d is the overall, adjusted log odds ratio and so on. In the case of overall estimates on the log scale we generally anti-log this final result to give us a measure back on the original scale, for example as an odds ratio. This is similar to the approach we saw in Section 4.4 when we looked at calculating a confidence interval for an odds ratio. 

Calculate a confidence interval from the standard deviation of replicate measurements. 

Figure 4-5 illustrates the meaning of confidence intervals. A computer chose numbers at random from a Gaussian population with a population mean (p.) of 10 000 and a population standard deviation (o) of 1 000 in Equation 4-3. In trial 1, four numbers were chosen, and their mean and standard deviation were calculated with Equations 4-1 and 4-2. The 50% confidence interval was then calculated with Equation 4-6, using t = 0.765 from Table 4-2 (50% confidence, 3 degrees of freedom). This trial is plotted as the first point at the left in Figure 4-5a the square is centered at the mean value of 9 526, and the error bar extends from the lower limit to the upper limit of the 50% confidence interval ( 290). The experiment was repeated 100 times to produce the points in Figure 4-5a. 

Quality of objects separation into groups was checked statistically. If we suggest objects be distributed normally inside zoning groups it s possible to calculate the confidence interval with which the chosen groups can be considered as different ... 

---