Power of a statistical test

Why First consider which factors influence the power of a statistical test. Gad (1988) established the basic factors that influence the statistical performance of any... 

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. ... 

Why First consider which factors influence the power of a statistical test. Gad [11] established the basic factors that influence the statistical performance of any bioassay in terms of its sensitivity and error rates. Recently, Healy [21] presented a review of the factors that influence the power of a study (the ability to detect a dose-related effect when it actually exists). In brief, the power of a study depends on seven aspects of study design ... 

Our previous two chapters based on references [1,2] describe how the use of the power concept for a hypothesis test allows us to determine a value for n at which we can state with both a- and / -% certainty that the given data either is or is not consistent with the stated null hypothesis H0. To recap those results briefly, as a lead-in for returning to our main topic [3], we showed that the concept of the power of a statistical hypothesis test allowed us to determine both the a and the j8 probabilities, and that these two known values allowed us to then determine, for every n, what was otherwise a floating quantity, D. 

Statistical methods are based on specific assumptions. Parametric statistics, those most familiar to the majority of scientists, have more stringent underlying assumptions than do nonparametric statistics. Among the underlying assumptions for many parametric statistical methods (such as the analysis of variance) is that the data are continuous. The nature of the data associated with a variable (as described previously) imparts a value to that data, the value being the power of the statistical tests which can be employed. 

When an observable value was missing, no collocated difference was obtained for that observable. However, to calculate a mean value if one of the observables was missing, the valid collocated measurement was taken as the mean. For a week which contained one or more missing daily samples the derived weekly and the corresponding measured weekly samples were discarded from the comparison only if the precipitation of the missing daily sample(s) accounted for more than 20% of the week s total. This was done to maximize the number of data points and the power of the statistical tests. 

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... 

The power of the statistical test is a quantitative measure of the ability to differentiate accurately differences in populations. The usual case in toxicity testing is the comparison of a treatment group to control group. Depending on the expected variability of the data and the confidence level chosen, an enormous sample size or number of replicates may be required to achieve the necessary discrimination. If the sample size or replication is too large, then the experimental design may have to be altered. 

However, by definition, these univariate methods of hypothesis testing are inappropriate for multispecies toxicity tests. As such, these methods are an attempt to understand a multivariate system by looking at one univariate projection after another, attempting to find statistically significant differences. Often the power of the statistical tests is quite low due to the few replicates and the high inherent variance of many of the biotic variables. 

The probability of committing a type I error is the probability of rejecting the null hypothesis when it is true (for example, claiming that the new treatment is superior to placebo when they are equivalent in terms of the outcome). The probability of committing a type I error is called a, which is sometimes referred to as the size of the test. The probability of committing a type II error is the probability of failing to reject the null hypothesis when it is false. This probability is also called beta (P). The quantity (1 - P) is referred to as the power of the statistical test. It is the probability of rejecting the null hypothesis (in favor of the alternate) when the alternate is true. As stated earlier it is desirable to have low error probabilities associated with a test. As we would like a and p to be as low as possible the quanti-... 

Power relates to the probability of a statistical test rejecting the null hypothesis when it is false. 

Random variability of the experimental data plays a major role in the power and detection levels of a statistic. The smaller the variance s, the greater the power of any statistical test. The lesser the variability, the smaller the value and the greater the detection level of the statistic. An effective way to determine if the power of a specific statistic is adequate for the researcher is to compute the detection limit S. The detectimi limit simply informs the researcher how sensitive the test is by stating what the difference needs to be between test groups to state that a significant difference exists. 

The po-wer and Type II error rate (beta) Power and its related Type II error rate are probably the most neglected aspects of a statistical test. Power refers to the probability that a relationship in the population will be detected when one in fact exists. Therefore, power might reflect what R. A. Fisher called the "sensitivity" of an experiment (Fisher, 1942). Importantly, the power of an obtained statistical test reflects the probability that such a result can be replicated (Goodman, 1992). The effects of low statistical power on the reproducibility of research findings has been well documented (Goodman, 1992 Harris, 1997). The Type II error (beta) occurs when one fails to reject a felse null hypothesis. 

To demonstrate that this was not the case, the shaded trays shown in Figure 4A were sampled and for each tray 3x2 vials were assayed for protein content. The protein content results were analyzed with a two-cell analysis-of-variance model including a factor, the left/right positioning, and a covariate, the shelf number. In order to increase the power of the statistical testing, the shelf number was handled as a covariate and not as a factor, based on the assumption that the filling was progressing at a constant rate. 

Power The probability of rejecting the null hypothesis in a statistical test when a particular alternative hypothesis happens to be true. 

The following examples illustrate the procedure for a statistical test [7]. In the first, we consider a very simple test on a single observation. The second applies the seven-step procedure to a test on the mean of a binomial population using a normal approximation. Here, and in the third example, we introduce the idea of one-sided and two-sided tests, while in the fourth example we illustrate the calculation of Type II error, and the power function of a test. 

The QSARs obtained were then tested by cross validation and visual examination of plots of fitted values against residuals. Cross-validation was performed by leave-one-out (LOO) testing. Each data point was omitted in turn from a regression, and the actual value of the omitted point compared to the value predicted by the revised model. The difference was referred to as a deletion residual. Q2 values (an analog of the summary statistic R2) were then calculated from the sum of squares of the deletion residuals. The Q2 statistic provides a measure of the predictive power of a regression, and is therefore more relevant for QSAR modeling than the R2 statistic (Damborsky and Schultz, 1997). 

Sensory evaluation (sensory science) is a scientific discipline that concerns the presentation of a stimulus (in this case a flavor compound, a flavor, or flavored product) to a subject and then evaluation of the subject s response. The response is expressed as, or translated into, a numerical form so that the data can be statistically analyzed. The sensory scientist then collaborates with the research or product development team to interpret the results and to reach decisions. Sensory scientists stress that decisions, such as product formulation, are made by people, not by the results of a sensory test, although such results may provide powerful guidance in the decision-making process. 

Coverage factor Expanded uncertainty Method of least squares Multivariate statistics Null hypothesis Power of a test Probability levels Regression of j on jc... 

The use of these two measures is very scarce in secondary structure prediction literature, despite their obvious superiority over Q3. In one of the few publications that utilize Mathews correlation coefficients, Holley and Karplus (1989) reported values of Ca=41%, Cp=32% and Cc=36%. In the new analysis those values were sensibly decreased (Ca=32%, Cp=25% and Cc=31%), clearly indicating there is a poor generalization power of the method to a larger set of proteins. It also strengthens an important fact for secondary structure evaluation, already noted by Rost and Sander (1994), which is the need of a representative testing set in terms of size and structural composition that permits gathering reliable statistical information from the results. 

The F-distribution has great utility in a statistical test referred to as analysis of variance (ANOVA). ANOVA is a powerful tool for testing the equivalence of means from samples obtained from normally distributed, or approximately normally distributed, populations. As an example, suppose that the following are the content uniformity values on 20 tablets from each of four different lots lot A mean = 99.5%, standard deviation = 2.6% lot B mean = 100.2%, standard deviation = 2.8% lot C mean = 90.5%, standard deviation = 2.1% and lot D mean = 100.3%, standard deviation = 2.7%. 

In the design of a toxicity test there is often a compromise between the statistical power of the toxicity test and the practical considerations of personnel and logistics. In order to make these choices in an efficient and informed manner, several parameters are considered ... 

---