Confidence interval and

The distribution of the /-statistic (x — /ji)s is symmetrical about zero and is a function of the degrees of freedom. Limits assigned to the distance on either side of /x are called confidence limits. The percentage probability that /x lies within this interval is called the confidence level. The level of significance or error probability (100 — confidence level or 100 — a) is the percent probability that /X will lie outside the confidence interval, and represents the chances of being incorrect in stating that /X lies within the confidence interval. Values of t are in Table 2.27 for any desired degrees of freedom and various confidence levels. 

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

Given that the assumption of normally distributed data (see Section 1.2.1) is valid, several useful and uncomplicated methods are available for finding the most probable value and its confidence interval, and for comparing such results. 

A form of this approach has long been followed by RT Corporation in the USA. In their certification of soils, sediments and waste materials they give a certified value, a normal confidence interval and a prediction interval . A rigorous statistical process is employed, based on that first described by Kadafar (1982,), to produce the two intervals the prediction interval (PI) and the confidence interval (Cl). The prediction interval is a wider range than the confidence interval. The analyst should expect results to fall 19 times out of 20 into the prediction interval. In real-world QC procedures, the PI value is of value where Shewhart (1931) charts are used and batch, daily, or weekly QC values are recorded see Section 4.1. Provided the recorded value falls inside the PI 95 % of the time, the method can be considered to be in control. So occasional abnormal results, where the accumulated uncertainty of the analytical procedure cause an outher value, need no longer cause concern. 

Mention has already been made of the EPA recommended use of both confidence interval and prediction interval. However, many users and their customers may be satisfied by some simplistic comparisons. Two methods for comparing experimental results with certified values are presented here. The users are referred to statistical handbooks for comparing results of sets of determinations such as using/test, etc. 

Using the initial rate data given above do the following (a) Determine the parameters, kR, kH and KA for model-A and model-B and their 95% confidence intervals and (b) Using the parameter estimates calculate the initial rate and compare it with the data. Shah (1965) reported the parameter estimates given in Table 16.14. 

Fig. 6.3. Three-dimensional model of calibration, analytical evaluation and recovery spatial model (A) the three relevant planes are given separately in (B) as the calibration function with confidence interval, in (C) as the recovery function with confidence interval, and in (C) as the evaluation function with prediction interval (D)... 

Fig. 7.8. Schematic three-dimensional representation of a calibration straight line of the form y = a + bx with the limits of its two-sided confidence interval and three probability density function (pdf) p(y) of measured values y belonging to the analytical values (contents, concentrations) X(A) = 0 (A), x = x(B) (B) and X(q = ld (C) yc is the critical value of the measurement quantity a the intercept of the calibration function yBL the blank x(B) the analytical value belonging to the critical value yc (which corresponds approximately to Kaiser s a3cr-limit ) xLD limit of detection... 

Both assumptions are mainly needed for constructing confidence intervals and tests for the regression parameters, as well as for prediction intervals for new observations in x. The assumption of normal distribution additionally helps avoid skewness and outliers, mean 0 guarantees a linear relationship. The constant variance, also called homoscedasticity, is also needed for inference (confidence intervals and tests). This assumption would be violated if the variance of y (which is equal to the residual variance a2, see below) is dependent on the value of x, a situation called heteroscedasticity, see Figure 4.8. 

The denominator n 2 is used here because two parameters are necessary for a fitted straight line, and this makes s2 an unbiased estimator for a2. The estimated residual variance is necessary for constructing confidence intervals and tests. Here the above model assumptions are required, and confidence intervals for intercept, b0, and slope, b, can be derived as follows ... 

Toxicity Assays. The computed LC50 and 95% confidence interval and the synergistic ratio (SR) for each insecticide are contained in Table I. Control mortality was 5% or less in all experiments. 

If / ijc > /jri, (see Figure 6.7), the minimum confidence interval is greater than the critical confidence interval and there is strong reason to believe that Po is significantly different firom zero. 

The width of the confidence interval depends on both F(j .p) and s. But F(, is a function of n, p, and the level of confidence the experimenter chooses to set for the particular confidence interval. And s] depends on both the lack of fit of the model to the data and the repeatability of experimentation. Because the values of these quantities depend on the experimenter, the model, and the system, F(j and can be removed to give a normalized confidence interval half width that depends on the design only ... 

Any inferences about the difference between the effects of the two treatments that may be made upon such data are the observed rates, or proportions of deteriorations by the intrathecal route. In this example, amongst those treated by the intrathecal route 22/58 = 0.379 of patients deteriorated, and the corresponding control rate is 37/60 = 0.617. The observed rates are estimates of the population incidence rates, jtt for the test treatment and Jtc for the controls. Any representation of differences between the treatments will be based upon these population rates and the estimated measure of the treatment effect will be reported with an associated 95% confidence interval and/or p-value. 

In Figure 8.9, we illustrate various cases that can arise from studies intended to show equivalence and the relationship between significance in the traditional sense and clinical significance as determined by the confidence interval and the boundaries of equivalence. In case (A), the 95% confidence interval includes both the null hypothesis of no difference and is within the boundaries of equivalence and from both a statistical and clinical perspective there is no evidence of a difference between the treatments. In case (B), in contrast, the confidence interval is still within the boundaries, but does include the null hypothesis, so from a statistical perspective there is a difference between the treatments but it is not clinically relevant. Case (C) shows both statistical and clinical significance, as the confidence interval lies outside the equivalence boundaries and therefore cannot include the null hypothesis. In the final case, (D), the confidence interval includes... 

As discussed in the previous section the standard error simply provides indirect information about reliability, it is not something we can use in any specific way, as yet, to tell us where the truth lies. We also have no way of saying what is large and what is small in standard error terms. We will, however, in the next chapter cover the concept of the confidence interval and we will see how this provides a methodology for making use of the standard error to enable us to make statements about where we think the true (population) value lies. 

For the first sample in the computer simulation the 99 per cent confidence interval is (78.64, 81.80). This is a wider interval than the 95 per cent interval the more confidence we require the more we have to hedge our bets. It is fairly standard to use 95 per cent confidence intervals and this links with the conventional use of 0.05 (or 5 per cent) for the cut-off for statistical significance. Under some circumstances we also use 90 per cent confidence intervals and we will mention one such situation later. In multiple testing it is also sometimes the case that we use confidence coefficients larger than 95 per cent, again we will discuss the circumstances where this might happen in a later chapter. 

The formula for the 95 per cent confidence interval (and also for the 99 per cent confidence interval) given above is in fact not quite correct. It is correct up to a... 

The regulators are not only interested in statistical significance but also in clinical importance. This allows them, and others, to appropriately balance benefit and risk. It is good practice therefore to present both p-values and confidence intervals and indeed this is a requirement within a submission. Most journals nowadays also require results to be presented in the form of confidence intervals in addition to p-values. 

The studies with the highest precision are those with the narrowest confidence intervals and usually these aspects of the different trials are emphasised by having squares at the estimated values, whose size is related to the precision within that trial. These plots, as seen in Figure 15.1, are so-called Forest... 

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