Confidence interval one-sided

As just seen, when using a two-sided confidence interval, interest lies with both the lower and the upper limit. In contrast, a one-sided confidence interval focuses on the placement of a single interval on one specified side of the treatment effect point estimate. In certain circumstances, interest lies with a drug response in one direction only. In these cases it is legitimate to calculate and present a single limit, which is usually referred to as the lower bound of the confidence interval (when placed below the treatment effect point estimate) or the upper bound when placed above it. 

It must be clarified and emphasized here that the upper bound of a one-sided 95 % Cl will not fall at the same place above the treatment effect point estimate as the upper limit of a two-sided 95 % Cl. The upper bound of a one-sided 95 % Cl will fall at the same place as the upper limit of a two-sided 90 % Cl. It is vital to be clear about precisely what type of confidence interval has been calculated and presented when reporting the results of a clinical trial. 

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One-sided confidence intervals cnf(y) = y + Ay and cnffiy) = y — Ay, respectively, are of importance for the control of limiting values and for test statistics. 

Such a parameter may be, e.g., standard deviation, or a given multiple of it, or a one-sided confidence interval attributed to a fixed level of confidence. In general, uncertainty of measurement comprises many components. These uncertainty components are subdivided into... 

The confidence intervals we have used to date are all two-sided. We will talk later about one-sided confidence intervals. 

For non-inferiority a one-sided confidence interval should be used. ... 

Q, therefore a lower one-sided confidence interval for the population mean could be used as an acceptance limit. The criterion is that the lower bound on the confidence interval must be greater than Q. 

This interval estimate is really based on the two-sided test of the third set of hypotheses previously given. Although it is possible to define one-sided confidence intervals based on the other two sets of hypotheses (1.59) and (1.60), such one-sided intervals are rarely used. By one-sided, we mean an interval estimate that extends from plus or minus infinity to a single random confidence limit. The one-sided confidence interval may be understood as the range one limit of which is the probability level a and the other one °°. 

In Example 2.3, we have calculated that 14 samples are needed to reach the decision with a 95 percent level of confidence. To be on the safe side, we collected and analyzed 20 samples. The collected samples have the concentrations of lead ranging from 5 to 210mg/kg the mean concentration is 86 mg/kg the standard deviation is 63 mg/kg and the standard error is 14mg/kg. From Appendix 1, Table 2 we determine that the t-value for 19 degrees of freedom (the number of samples less one) and a one-sided confidence interval for a — 0.05 is 1.729. Entering these data into Equation 10, Appendix 1, we calculate the 95 percent confidence interval of the mean 86 24 mg/kg. The upper limit of the confidence interval is 110 mg/kg and it exceeds the action level. Therefore, the null hypothesis Hq p > lOOmg/kg, formulated in Example 2.2 is true, as supported by the sample data. Based on this calculation we make a decision not to use the soil as backfill. 

Statistical tests may be one-sided (one-tailed) or two-sided (two-tailed). One-sided confidence intervals are used for testing the data that are compared to action levels to determine whether the mean concentration is greater or lower than the action level. Two-sided confidence limits are used for comparing two sets of data to each other to establish whether they differ, for example, for comparing sample concentrations to background concentrations. One-sided and two-sided confidence intervals are illustrated in Figure 2. 

The term one-sided confidence interval arises since attention only needs to be paid to one limit, the lower end, of the 95% Cl to investigate noninferiority. Therefore, the other end need not be calculated, and can be left unspecified. 

Using statistical packages to obtain one-sided confidence intervals... 

Figure 5.8 shows a useful way to present one-sided confidence intervals. The idea is to emphasize the fact that we have established a definite lower limit, but are making no comment concerning how great the value might be. The figure also shows a normal two-sided 95 per cent Cl for the same data it places limits both above and below the mean. 

The width of the confidence interval will depend upon the SD for the sample, the size of the sample and the degree of confidence required. The width is especially dependent upon sample size - small samples lead to very wide intervals. One-sided confidence intervals can be used to specify a value that the population mean is unlikely to exceed (or be less than). 

Describe the use of one-sided confidence intervals to test such null hypotheses... 

The presently used statistical method is the confidence interval approach. The main concern is to rule out the possibility that the test product is inferior to the comparator pharmaceutical product by more than the specified amount. Hence a one-sided confidence interval (for efficacy and/or safety) may be appropriate. The confidence intervals can be derived from either parametric or nonparametric methods. 

If the interval defines a region below which the population parameter lies, then y, is replaced with -oo. Likewise, y is replaced with °o to determine the minimum value of the population parameter. Both situations are termed one-sided confidence intervals, as the interval is bounded on only one side. Otherwise, the interval is two-sided and the interval has bounds on two sides. Distributions, y, and y are chosen to be centered about y = 0, so that y, = -y. ... 

Exampie. From the ISO statistical methods (1981) we learn that the one sided confidence interval for the population mean is defined as foliows... 

Where fi is Student s-f, based on v degrees of freedom for a one-sided confidence interval of 1—a and ctq is the standard deviation of the true value (expectation). [It also notes that] the correct estimation of LOD must take into account degrees of freedom, a and P, and the distribution of L as influenced by factors such as analyte concentration, matrix effects and interference. 

The information collection taken from the test will be evaluated with respect to a relevant standard recommendation (see lEC 60605-4) as a set of data taken from the test terminated by time and carried out by replacing failed items. The calculation of the mean time to failure of one side confidence interval lower threshold will be performed according to the paragraph 5.1.2.2.1 of the standard using the following formula ... 

It means that lover limit of one side confidence interval for MTTF of the item is approximately... 

Hypothesis testing is conducted to establish assay sensitivity. Assay sensitivity is demonstrated by rejection of the null hypothesis in favor of the research hypothesis. Assay sensitivity is tested by placing the lower bound of a one-sided 95 % confidence interval on the treatment effect point estimate (calculated as moxifloxacin minus placebo) for each time point. In this context, therefore, the null hypothesis is the intersection of several hypotheses, i.e., that the lower bound of the one-sided 95 % confidence interval is <5 ms for all specified time points. Rejection of the null hypothesis occurs if the lower bound of the one-sided confidence interval for any of the measurement times is above 5 ms, i.e., a union of the rejection regions for each of the individual times. Thus, the nature of the two hypotheses leads to the name union-intersection test. This result indicates that there is compelling statistical evidence that the experimental methodology will be able to detect QT/QTc prolongation induced by the drug if it truly exists (based on the 5 ms threshold). 

In the pilot experiment, a confidence interval for m will be helpful because of a nice property of the function -exp Sp(o.as) /m) in Equation 21 it is decreasing in n and m as mentioned before. Let nu be the lower bound for a /3) level one-sided confidence interval for m, that is, Pr m>mi = -p. Then, if /w is greater than trii, the value of the fimction at m will be less than the fimction value at m = mi. The procedure for computing nti at P = 0.05 level is briefly given along with the associated coefficients in Table A4 in the appendix. 

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