Sample size confidence intervals

Table 5. Intraclass correlations, confidence intervals, samples sizes, and test utilized for IQ in five studies of monozygotic twins reared apart...

Confidence intervals also can be reported using the mean for a sample of size n, drawn from a population of known O. The standard deviation for the mean value. Ox, which also is known as the standard error of the mean, is... 

In the previous section we noted that the result of an analysis is best expressed as a confidence interval. For example, a 95% confidence interval for the mean of five results gives the range in which we expect to find the mean for 95% of all samples of equal size, drawn from the same population. Alternatively, and in the absence of determinate errors, the 95% confidence interval indicates the range of values in which we expect to find the population s true mean. 

In a case-control study of the relation between occupational exposures to various suspected estrogenic chemicals and the occurrence of breast cancer, the breast cancer odds ratio (OR) was not elevated above unity (OR=0.8 95% 01=0.2-3.2) for occupational exposure to endosulfan compared to unexposed controls (Aschengrau et al. 1998) however, the sample sizes were very small (three exposed seven not exposed), and co-exposure to other unreported chemicals also reportedly occurred. Both of these factors may have contributed to the high degree of uncertainty in the OR indicated by the wide confidence interval. 

The use of confidence intervals is one way to state the required precision. Confidence limits provide a measure of the variability associated with an estimate, such as the average of a characteristic. Table I is an example of using confidence intervals in planning a sampling study. This table shows the interrelationships of variability (coefficient of variation), the distribution of the characteristic (normal or lognormal models), and the sample frequency (sample sizes from 4 to 365) for a monitoring program. 

If a different amount is taken, other than which is specified in the certificate, then this has a significant impact on the confidence interval for the certified value in that particular sample. Extrapolation of uncertainty to different sample sizes, in particular uncertainties due to inhomogeneity at smaller sample size, is not possible without extensive sampling studies. Even so, RM producers should support analysis procedures that require different sample sizes by supplying sampling information such as sampling constants see also Section 4.3. 

The target number of commodity samples to be obtained in the OPMBS was 500, as determined using statistical techniques. A sample size of 500 provided at least 95% confidence that the 99th percentile of the population of residues was less than the maximum residue value observed in the survey. In other words, a sample size of 500 was necessary to estimate the upper limit of the 95% confidence interval around the 99th percentile of the population of residues. 

The command CONFIDENCE, a, n) gives the confidence interval about the mean for a sample size n. To obtain 95 percent confidence limits, use a =. 025 2a = 1 - A. 

If the analysis of a sample for iron content yields a mean result of 35.40% with a standard deviation of 0.30%, the size of the confidence interval will vary inversely with the number of measurements made. For two measurements, the confidence interval (90%) is... 

Additional measurements were made with the 17-)im sizing screen to obtain more information on the variability of our measurement techniques. Eight lint samples from a single source of cotton were analyzed by the procedures outlined previously. The dust levels obtained in this test were 11.7, 12.1, 13.5, 11.8, 10.8, 11.2, 10.9, and 9.7 mg, respectively, per 20 g of lint. The mean and standard deviation of these measurements were 11.5 and 1.1, respectively. The estimated standard error of the mean was 0.42, and the interval from 10.5 to 12.5 represented a 95% confidence interval for the lot mean. 

It is well recognised that the faecal bile acid content of random stool samples is highly variable with marked daily variation.Therefore, studies testing the association between luminal bile acid exposure and the presence of colorectal neoplasia have usually measured serum bile acid levels, which demonstrate less variability and are believed to reflect the total bile acid pool more accurately. Serum DCA levels have been shown to be higher in individuals with a colorectal adenoma compared with individuals without a neoplasm. Only one study has assessed future risk of CRC in a prospective study of serum bile-acid levels. The study was hampered by the small sample size (46 CRC cases). There were no significant differences in the absolute concentrations of primary and secondary bile acids or DCA/CA ratio between cases and controls although there was a trend towards increased CRC risk for those with a DCA/ CA ratio in the top third of values (relative risk 3.9 [95% confidence interval 0.9-17.0 = 0.1]). It will be important to test the possible utility of the DCA/ CA ratio as a CRC risk biomarker in larger, adequately powered studies. A recent study has demonstrated increased levels of allo-DCA and allo-LCA metabolites in the stool of CRC patients compared with healthy controls. ... 

One question that is often asked of statisticians is in what sense can we be 95% confident that the population mean lies within the limits 3.84 and 4.13 To answer the question we can again conduct a sampling experiment as follows. Suppose that the 40 blood glucose measurements in Figure 8.3 comprised the total population of values. For random sample of size 10 from the populations of blood glucose values determine the sample mean, standard error and the corresponding 95% confidence interval. Repeat the process 100 times. The results of such an experiment are shown in Figure 8.6. 

Fig. 8.6 Confidence intervals from 100 samples of size 10 from the population of blood glucose levels.

From the formula for a confidence interval, its width is determined by three parameters the sample size, population variability and the degree of confidence. Plainly, if the sample size is increased then we have seen the standard error will be reduced and hence the width of the interval will also be reduced. If we can reduce the variability of the characteristic being studied then... 

Similar to the single refinery planning example in Section 7.5.1, the problem was solved for different sample sizes N and N to illustrate the variation of optimality gap confidence intervals, as shown in Table 7.5 and Figure 7.5. The results illustrate the trade-off between model solution accuracy and computational effort. Furthermore, the increase in the sample size N has a more pronounced effect on reducing the optimality gap, however, due to computa-... 

The problem was solved for different sample sizes N and N to illustrate the variation of optimality gap confidence intervals, while fixing the number of replications I to 30. The replication number R need not be very large to get an insight into Vn variability. Table 9.3 shows different confidence interval values of the optimality gap when the sample size of N assumes values of 1000, 2000, and 3000 while varying N from 5000, 10 000, to 20 000 samples. The sample sizes N and N were limited to these values due to increasing computational effort. In our case study, we ran into memory limitations when N and N values exceeded 3000 and 20 000, respectively. The solution of the three refineries network and the PVC complex using the SAA scheme with N = 3000 and N = 20000 required 1114 CPU s to converge to the optimal solution. 

Also, the analysis plan should identify the statistical methods that will be used and how hypotheses will be tested (e.g., a p value cutoff or a confidence interval for the difference that excludes 0). And the plan should prespecify whether interim analyses are planned, indicate how issues of multiple testing will be addressed, and predefine any subgroup analyses that will be conducted. Finally, the analysis plan should include the results of power and sample size calculations. 

One of the most dependably accurate methods for deriving 95% confidence intervals for cost-effectiveness ratios is the nonparametric bootstrap method. In this method, one resamples from the smdy sample and computes cost-effectiveness ratios in each of the multiple samples. To do so requires one to (1) draw a sample of size n with replacement from the empiric distribution and use it to compute a cost-effectiveness ratio (2) repeat this sampling and calculation of the ratio (by convention, at least 1000 times for confidence intervals) (3) order the repeated estimates of the ratio from lowest (best) to highest (worst) and (4) identify a 95% confidence interval from this rank-ordered distribution. The percentile method is one of the simplest means of identifying a confidence interval, but it may not be as accurate as other methods. When using 1,000... 

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. 

For sample sizes beyond about 30 the multiplying constant for the 95 per cent confidence interval is approximately equal to two. Sometimes for reasonably large sample sizes we may not agonise over the value of the multiplying constant and simply use the value two as a good approximation. This gives us an approximate formula for the 95 per cent confidence interval as (3c — 2se, x + 2se). 

Finally, returning again to the formula for the standard error, sl.y/n, we can, at least in principle, see how we could make the standard error smaller increase the sample size n and reduce the patient-to-patient variability. These actions will translate into narrower confidence intervals. 

Note that for binary data and proportions the multiplying constant is 1.96, the value used previously when we first introduced the confidence interval idea. Again this provides an approximation, but in this case the approximation works well except in the case of very small sample sizes. 

There are invariably rules for how to obtain the multiplying constant for a specific confidence coefficient, but as a good approximation and providing the sample sizes are not too small, using the value 2 for the 95 per cent confidence interval and 2.6 for the 99 per cent confidence interval would get you very close. 

Clearly the main advantage of a non-parametric method is that it makes essentially no assumptions about the underlying distribution of the data. In contrast, the corresponding parametric method makes specific assumptions, for example, that the data are normally distributed. Does this matter Well, as mentioned earlier, the t-tests, even though in a strict sense they assume normality, are quite robust against departures from normality. In other words you have to be some way off normality for the p-values and associated confidence intervals to be become invalid, especially with the kinds of moderate to large sample sizes that we see in our trials. Most of the time in clinical studies, we are within those boundaries, particularly when we are also able to transform data to conform more closely to normality. 

Although conventional p-values have no role to play in equivalence or noninferiority trials there is a p-value counterpart to the confidence intervals approach. The confidence interval methodology was developed by Westlake (1981) in the context of bioequivalence and Schuirmann (1987) developed a p-value approach that was mathematically connected to these confidence intervals, although much more difficult to understand It nonetheless provides a useful way of thinking, particularly when we come later to consider type I and type II errors in this context and also the sample size calculation. We will start by looking at equivalence and use A to denote the equivalence margins. 

We will focus our attention to the situation of non-inferiority. Within the testing framework the type I error in this case is as before, the false positive (rejecting the null hypothesis when it is true), which now translates into concluding noninferiority when the new treatment is in fact inferior. The type II error is the false negative (failing to reject the null hypothesis when it is false) and this translates into failing to conclude non-inferiority when the new treatment truly is non-inferior. The sample size calculations below relate to the evaluation of noninferiority when using either the confidence interval method or the alternative p-value approach recall these are mathematically the same. 

In an anti-infective non-inferiority study it is expected that the true cure rates for both the test treatment and the active control will be 75 per cent. A has been chosen to be equal to 15 per cent. Using the usual approach with a one-sided 97.5 per cent confidence interval for the difference in cure rates a total of 176 patients per group will give 90 per cent power to demonstrate non-inferiority. Table 12.1 gives values for the sample size per group for 90 per cent power and for various departures from the assumptions. 

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