One-sided interval

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 °°. 

Let us have another look at the table of Appendix B and understand why we may need two -values with the same confidence level P and the same total risk a. If we compare columns 2 and 3 we note that one can use a fractile of the order q = 0.975, which means that a risk oe/2 remains that both sides of the symmetrical region around the estimated parameter do not include the true value. On the other hand the fractile of the order g = 0.95 may be useful for constructing a one-sided interval hence concentrating the risk a on one side. 

Another example may be even more interesting one-sided intervals are especially use fill in comparing heavy metal contamination found in an environmental compartment with a legally fixed threshold value. Because of the experimental random error it is not recom-... 

Most packages will produce one-sided intervals. There are two possible approaches ... 

Find an option that allows you to select a one-sided interval in place of the default two-sided version. In this case you still request 95 per cent confidence. 

Figure 5.8 also shows that, when a one-sided interval is calculated, the one limit that is calculated is closer to the mean than the corresponding limit for a two-sided interval. This is related to the amount of risk we are taking. With the one-sided interval, we are prepared to take a 5 per cent risk that the true mean might be below the indicated figure, but with the two-sided interval, we can only allow a 2.5 per cent chance of error. 

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. 

One of the options was to carry out a one-side interval calculation of an item reliability measure at a required confidence level. This intention was easy to be fidfiUed since the data on the item operation was carefidly and systematically collected. The aim of this procedure has been completed and calculated also with the statistical comparison of the evaluated sets. 

The aim of the analysis was to calculate the one-side item reliability interval. The item made in previous manner was assessed first, and the item being produced currently was assessed as the second. The calculation of a reliability one-side interval determined for each set separately was the outcome of the analysis. 

The next step was to compare both items sets and decide whether the slight technology change can/cannot affect the item s reliability. A one-side interval was determined at a required confidence level and it specifies a minimal reliability level of an item set obtained by a calculation. 

Following the standard (lEC 60605-4) recommendation a lower limit of mean time to failure at the required confidence level was calculated. In order to estimate one-side interval of a lower level of mean time to failure we used the following equation (see also Lipson Sheth 1973, Neson 1982) ... 

The water-vapor transmission rate (WVTR) is another descriptor of barrier polymers. Strictly, it is not a permeabihty coefficient. The dimensions are quantity times thickness in the numerator and area times a time interval in the denominator. These dimensions do not have a pressure dimension in the denominator as does the permeabihty. Common commercial units for WVTR are (gmil)/(100 in. d). Table 2 contains conversion factors for several common units for WVTR. This text uses the preferred nmol/(m-s). The WVTR describes the rate that water molecules move through a film when one side has a humid environment and the other side is dry. The WVTR is a strong function of temperature because both the water content of the air and the permeabihty are direcdy related to temperature. Eor the WVTR to be useful, the water-vapor pressure difference for the value must be reported. Both these facts are recognized by specifying the relative humidity and temperature for the WVTR value. This enables the user to calculate the water-vapor pressure difference. Eor example, the common conditions are 90% relative humidity (rh) at 37.8°C, which means the pressure difference is 5.89 kPa (44 mm Hg). 

Extrapolation is required if f(x) is known on the interval [a,b], but values of f(x) are needed for x values not in the interval. In addition to the uncertainties of interpolation, extrapolation is further complicated since the function is fixed only on one side. Gregory-Newton and Lagrange formulas may be used for extrapolation (depending on the spacing of the data points), but all results should be viewed with extreme skepticism. 

The true standard deviation Ox is expected inside the confidence interval CI(5 , ) = /Vi. .. /V with a total error probability 2 p (in connection with F and x P taken to be one-sided). 

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

The relation between systematic and random deviations as well as the character of outliers is shown in Fig. 4.1. The scattering of the measured values is manifested by the range of random deviations (confidence interval or uncertainty interval, respectively). Measurement errors outside this range are described as outliers. Systematic deviations are characterized by the relation of the true value p and the mean y of the measurements, and, in general, can only be recognized if they are situated beyond the range of random variables on one side. 

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... 

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A second possibility consists of experimentally determining the SST limits from measurements at the worst-case conditions (n measurements with standard deviation 9,12,13 gg-p limit is defined as the lower or upper limit of the one-sided 95% confidence interval around the worst-case average result. For example, for resolution, the lower limit will be considered, while for migration time it would be the upper. The confidence intervals are defined as in Equations (16) and (17), when considering the lower or the upper limit, respectively. 

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

In order to demonstrate non-inferiority, it is only one end of the confidence interval that matters in our example it is simply the lower end that needs to be above —2 mmHg. It is therefore not really necessary to calculate the upper end of the interval and sometimes we leave this unspecified. The resulting confidence interval with just the lower end is called a one-sided 97.5per cent confidence interval the two-sided 95 per cent confidence interval cuts off 2.5 per cent at each of the lower and upper ends, having the upper end undefined leaves just 2.5 per cent cut off at the lower end. The whole of this confidence interval must be entirely to the right of the non-inferiority margin for non-inferiority to be established. 

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

The non-inferiority margin has been set at —15 per cent. Figure 12.4 displays the non-inferiority region and we need the (two-sided) 95 per cent confidence interval, or the one-sided 97.5 per cent confidence interval, to be entirely within this non-inferiority region for non-inferiority to be established. 

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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