Power of the test

Figure 1.34. Alternative hypothesis and the power of a t-test. Alpha (a) is the probability of rejecting an event that belongs to the population associated with it is normally in the range 0.05. .. 0.01. Beta (/3) is the probability that an event that is effectively to be associated with H is accepted as belonging to the population associated with Hq. Note that the power of the test to discriminate between hypotheses increases with the distance between ha and hb- >-a is fixed either by theory or by previous measurements, while hb can be adjusted (shifted along the x-axis), for examples see H - H4, Section 4.1. Compare with program HYPOTHESIS.

The existence of the [3 probability provides us with the tool for determining what is called the power of the test, which is just 1 - j8, the probability of coming to the correct conclusion when in fact the data did not come from the hypothesized population P0. This is the answer to our earlier question once we have defined the alternate population Pa, we can determine the /3 probability of a sample having come from Pa, just as we can determine the a probability of that sample having come from P0. 

The wrong acceptance of a false hypothesis leads to committing a type II error. The probability of a type II error is designated ft and (1 — j8) is called the power of the test. For a certain Hi, it is not possible to make both a and ff arbitrarily small. Decreasing the probability of one type of error increases the probability of the other and vice versa. The balance between both types of errors depends on the purpose of the test. 

Minimizes both type I (false positive) and type II (false negative) error rates, thereby increasing power of the test statistic to be employed while decreasing inconsistent significant effects. 

The first precise or calculable aspect of experimental design encountered is determining sufficient test and control group sizes to allow one to have an adequate level of confidence in the results of a study (that is, in the ability of the study design with the statistical tests used to detect a true difference, or effect, when it is present). The statistical test contributes a level of power to such a detection. Remember that the power of a statistical test is the probability that a test results in rejection of a hypothesis, H0 say, when some other hypothesis, H, say, is valid. This is termed the power of the test with respect to the (alternative) hypothesis H. ... 

There are a multiplicity of methods for testing significance in fife tables, with (as is often the case) the power of the tests increasing as does the difficulty of computation (Salsburg, 1980 Cox, 1972 Haseman, 1977 and Tarone, 1975). 

Is the probability of calling a polluted panel polluted. Is called the power of the test, and Is shown as dotted area to the right of CV In Figure 1. 

Human data and their relevance have to be assessed carefully on a case-by-case basis due to limitations of the techniques available. In particular, attention should be paid to the adequacy of the exposure information, confounding factors, and to sources of bias in the smdy design. The statistical power of the test may also be considered. 

Determine the power of the test in Example 1.10 for the alternate hypothesis that the mean is really 50, using the sample size 1000. 

Odhiambo and Manene presented a performance analysis of stepwise screening that assumes a2 > 0, where statistical tests are fallible even if all assumptions are correct. They derived expected values of the number of runs required, the number of factors mistakenly classified as active, and the number of factors mistakenly classified as not active, in terms of p, f, k, and the significance level and power of the tests used. These expressions are fairly complicated and are not repeated here, but Odhiambo and Manene also provide simpler approximations that are appropriate for small values of p. 

Be aware that the decision of a statistical test does not supply 100% certainty. A differentiation between the test decision and reality must always be made. In Table 3.4 the two kinds of errors that may occur are shown a Type I error is to reject the null hypothesis when it is true, and a Type II error is to accept the null hypothesis when it is not true. The probability a for the Type I error is called the level of the test. The probability (i for the Type II error depends on this a-level, the sample size, the expression change to be detected and the variance of the measured values. The probability of rejecting the null hypothesis is called the power of the test. It should be very small when the null hypothesis is true and very big when... 

Figures 2 and 3 show coral data in plots of 5234u versus °Th/ U. These plots illustrate both the power of the test for diagenesis and...

The propulsion system is powered in a hybrid configuration by using lead batteries and the 20 kW PEM stack described in the above paragraphs. The experimental tests in dynamic operation are carried out on a laboratory test bench utilizing the European R40 driving cycle, varying both dynamics and maximum power of the test cycle for different hybridization levels between FCS and batteries. 

USE OF HIGH CONTAMINATION LEVEL CERTIFIED REFERENCE MATERIALS IN MICROBIOLOGY. THEORY OF TESTING THE METHODS TYPE II ERROR AND POWER OF THE TEST. 

Fig. 3.3. Relationship between the power of the test, the geometric mean count and the number of capsules (1= I V I = 2A I = 3 0 I=4 ) examined, using two replicates per capsule for the Bacillus cereus CRM 528 enumerated on MEYP agar after 24 h incubation at 30 C (certified value 53.4).

Statistical analyses of results from the clinical trials, showing statistical power of the test... 

The unpaired t-test is an example of a parametric method, which means that it is based on the assumption that the two samples are taken from normal, or approximately normal distributions. Generally, parametric tests should be used where possible because they are more powerful (effectively, more sensitive) than the alternative non-parametric methods [32]. However, significance levels obtained from parametric tests may be inaccurate, and the true power of the test may decrease, if the assumption of normality is poor. The non-parametric alternative to the unpaired t-test is the Mann-Whitney test [32]. In this test, a rank is assigned to each observation (1 = smallest, 2 = next smallest, etc.), and the test statistic is computed from these ranks. Obviously, the test is less sensitive to departures from normality, such as the presence of outliers, since, for example, the rank assigned to the smallest observation will always be 1, no matter how small that observation is. 

Pavelka and Kov f studied 24 derivatives of protoberberine for their interaction with liver alcohol dehydrogenase. The inhibitory power of the tested compounds was correlated with their structures and with some properties following from those structures. The most effective inhibitor of liver alcohol dehydrogenase is 13-ethylberberine, which is bound more firmly to the enzyme at pH 10 than NAD and NADH (670). 

It is possible to show, with an application of Bayes theorem, that if we allow a certain prior probability that a product is effective, then the posterior probability of the effectiveness of the product, given a significant result, is an increasing function of the power of the test. Suppose, for example, that the prior odds for a given alternative hypothesis against the null hypothesis are pr /pr. Let be the likelihood of observing a given piece of evidence under and Lq be the likelihood under H. Let po jpo be the posterior odds. Then Bayes theorem implies (see Chapter 4) that... 

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