Confidence Intervals and Hypothesis Testing

Near-IR spectroscopic tomography is discussed in a paper by Tosteson et al. [156], They extend basic concepts of statistical hypothesis testing and confidence intervals to images generated by this new procedure as used for breast cancer diagnosis. 

The mean of the difference was calculated by using the statistical hypothesis test and confidence interval estimation on the 20 participants data sample. The normality of the distribution of the sample data was tested by the Ryan—Joiner test at 5% significant level. The significance of the results with over 80% confidence level is reported in the following section. 

An advantage of LR in comparison to LDA is the fact that statistical inference in the form of tests and confidence intervals for the regression parameters can be derived (compare Section 4.3). It is thus possible to test whether the /th regression coefficient bj = 0. If the hypothesis can be rejected, the jth regressor variable xj... 

ML is the approach most commonly used to fit a distribution of a given type (Madgett 1998 Vose 2000). An advantage of ML estimation is that it is part of a broad statistical framework of likelihood-based statistical methodology, which provides statistical hypothesis tests (likelihood-ratio tests) and confidence intervals (Wald and profile likelihood intervals) as well as point estimates (Meeker and Escobar 1995). MLEs are invariant under parameter transformations (the MLE for some 1-to-l function of a parameter is obtained by applying the function to the untransformed parameter). In most situations of interest to risk assessors, MLEs are consistent and sufficient (a distribution for which sufficient statistics fewer than n do not exist, MLEs or otherwise, is the Weibull distribution, which is not an exponential family). When MLEs are biased, the bias ordinarily disappears asymptotically (as data accumulate). ML may or may not require numerical optimization skills (for optimization of the likelihood function), depending on the distributional model. 

In the previous sections we discussed probability distributions for the mean and the variance as well as methods for estimating their confidence intervals. In this section we review the principles of hypothesis testing and how these principles can be used for statistical inference. Hypothesis testing requires the supposition of two hypotheses (1) the null hypothesis, denoted with the symbol //, which designates the hypothesis being tested and (2) the alternative hypothesis denoted by Ha. If the tested null hypothesis is rejected, the alternative hypothesis must be accepted. For example, if... 

In Chapter 6 we described the basic components of hypothesis testing and interval estimation (that is, confidence intervals). One of the basic components of interval estimation is the standard error of the estimator, which quantifies how much the sample estimate would vary from sample to sample if (totally implausibly) we were to conduct the same clinical study over and over again. The larger the sample size in the trial, the smaller the standard error. Another component of an interval estimate is the reliability factor, which acts as a multiplier for the standard error. The more confidence that we require, the larger the reliability factor (multiplier). The reliability factor is determined by the shape of the sampling distribution of the statistic of interest and is the value that defines an area under the curve of (1 - a). In the case of a two-sided interval the reliability factor defines lower and upper tail areas of size a/2. 

The philosophy underlying hypothesis testing is easy to understand. The term that appears in the denominator of Eq. (2.29) is an example of a standard error (in this case, of the average of the differences xa — xg). The estimate i is the deviation of the sample value relative to the population value corresponding to the null h3q)othesis, measured in standard error units. The larger this deviation, the less are the chances that the null hypothesis is true. Confidence intervals can always be transformed into hypothesis tests, for which the numerator is an estimate of the parameter of interest and the denominator is the corresponding standard error. For the difference between two averages, for example, the standard error is. 

When. V/5f is substituted for o, the estimated variance is given by these two equations. Letting. v = 0 in the variance expression for u, t gives the variance for /io, which can be used for hypothesis and confidence interval testing for This has applications for some regression problems where some intrinsic meaning other than an arbitrary fitting... 

There will be incidences when the foregoing assumptions for a two-tailed test will not be true. Perhaps some physical situation prevents p from ever being less than the hypothesized value it can only be equal or greater. No results would ever fall below the low end of the confidence interval only the upper end of the distribution is operative. Now random samples will exceed the upper bound only 2.5% of the time, not the 5% specified in two-tail testing. Thus, where the possible values are restricted, what was supposed to be a hypothesis test at the 95% confidence level is actually being performed at a 97.5% confidence level. Stated in another way, 95% of the population data lie within the interval below p + 1.65cr and 5% lie above. Of course, the opposite situation might also occur and only the lower end of the distribution is operative. 

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. 

In screening studies of standard design, the tendency has been to concentrate mainly on hypothesis testing. However, presentation of the results in the form of estimates with confidence intervals can be a useful adjunct for some analyses and is very important in studies aimed specifically at quantifying the size of an effect. 

A somewhat different computational procedure is often used in practice to carry out the test described in the previous section. The procedure involves two questions What is the minimum calculated interval about bg that will include the value zero and, Is this minimum calculated interval greater than the confidence interval estimated using the tabular critical value of t If the calculated interval is larger than the critical confidence interval (see Figure 6.7), a significant difference between Po and zero probably exists and the null hypothesis is disproved. If the calculated interval is smaller than the critical confidence interval (see Figure 6.8), there is insufficient reason to believe that a significant difference exists and the null hypothesis cannot be rejected. 

Historically, the role of statistics in biomedical research has been largely to test hypotheses. More recently, there has been a move to supplant hypothesis tests from their dominant position by confidence intervals. This move has been endorsed by The International Committee of Medical Journal Editors and climaxed with the publication, under the auspices of the British... 

A test of the null h)rpothesis that the rates of infection are equal - Hq x jii/hnj = 1 gives a p-value of 0.894 using a chi-squared test. There is therefore no statistical evidence of a difference between the treatments and one is unable to reject the null hypothesis. However, the contrary statement is not true that therefore the treatments are the same. As Altman and Bland succinctly put it, absence of evidence is not evidence of absence. The individual estimated infection rates are jTi = 0.250 and = 0.231 that gives an estimated RR of 0.250/0.231 = 1.083 with an associated 95% confidence interval of 0.332-3.532. In other words, inoculation can potentially reduce the infection by a factor of three, or increase it by a factor of three with the implication that we are not justified in claiming that the treatments are equivalent. 

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. 

If the 95 per cent confidence interval for the treatment effect not only lies entirely above - A but abo above zero, then there is evidence of superiority in terms of statistical significance at the 5 per cent level (p < 0.05). In this case, it is acceptable to calculate the exact probability associated with a test of superiority and to evaluate whether this is sufficiently small to reject convincingly the hypothesis of no differenc... Usually this demonstration of a benefit is sufficient for licensing on its own, provided the safety profiles of the new agent and the comparator are similar. ... 

Ford 1, Norrie J and Ahmedi S (1995) Model inconsistency, illustrated by the Cox Proportional Flazards model Statistics in Medicine, 14, 735-746 Gardner MJ and Altman DG (1989) Estimation rather than hypothesis testing confidence intervals rather than p-values In Statistics with Confidence (eds MJ Gardner and DG Altman), Fondon British Medical Journal, 6-19... 

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