Statistics hypothesis testing

Suter GW, II. 1996. Abuse of hypothesis testing statistics in ecological risk assessment. Human Ecol Risk Assess 2 331-347. 

In tables where the dispersion of each data set is shown by an appropriate statistical parameter, you must state whether this is the (sample) standard deviation, the standard error (of the mean) or the 95% confidence limits and you must give the value of n (the number of repUcates). Other descriptive statistics should be quoted with similar detail, and hypothesis-testing statistics should be quoted along with the value of P (the probabiUty). Details of any test used should be given in the legend, or in a footnote. 

This chapter outlines the philosophy of hypothesis-testing statistics, indicates the steps to be taken when choosing a test, and discusses features and assumptions of some important tests. For details of the mechanics of tests, consult appropriate texts (e.g. Miller and Miller, 2000). Most tests are now available in statistical packages for computers (see p. 315). 

The hypothesis-testing statistical functions may be reasonably powerful (e.g. t-test, ANOVA, regressions) and they often return the probability P of obtaining the test statistic (where 0 < F < 1), so there may be no need to refer to statistical tables. Again, check on the effects of including empty cells. 

Hypothesis-testing statistics are used to compare the properties of samples either with other samples or with some theory about them. For instance, you may be interested in whether two samples can be regarded as having different means, whether the concentration of a pesticide in a soil sample can be regarded as randomly distributed, or whether soil organic matter is linearly related to pesticide recovery. 

In the domain of proarrhy thmic cardiac safety, QT prolongation can be regarded as an adverse event of special interest, and therefore an inferential (hypothesis-testing) statistical approach is taken. Three statistical methodologies are applicable here the first two, the intersection-union test and the union-intersection test, are discussed in this section concentration- esponse modeling is then discussed in the following section. 

Table 1. Sample information, null hypothesis, test. statistic, and test interval for common tests on parameters of one or two measurement series...

Hypothesis tests Statistical tests that compare two quantities, one calculated and one tabulated, to determine the acceptance or rejection of a hypothesis. 

Next, an equation for a test statistic is written, and the test statistic s critical value is found from an appropriate table. This critical value defines the breakpoint between values of the test statistic for which the null hypothesis will be retained or rejected. The test statistic is calculated from the data, compared with the critical value, and the null hypothesis is either rejected or retained. Finally, the result of the significance test is used to answer the original question. 

The test statistic for evaluating the null hypothesis is called an f-test, and is given as either... 

Jui y trials represent a form of decision making. In statistics, an analogous procedure for making decisions falls into an area of statistical inference called hypothesis testing. 

If the null hypothesis is assumed to be true, say, in the case of a two-sided test, form 1, then the distribution of the test statistic t is known. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero (the midvalue of t) by chance alone. If the sample value oft does, in fact, deviate too far from zero, then this is defined to be sufficient evidence to refute the assumption of the null hypothesis. It is consequently rejected, and the converse or alternative hypothesis is accepted. 

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

There are statistical procedures available to choose models (hypothesis testing), assess outliers (or weight them), and deal with partial curves. 

In passing we remark that there are well-known statistical methods of hypothesis testing and parameter estimation used in decisionmaking. Sequential analysis is a method of sampling used to decide whether to accept or reject a lot with defective items, or whether to continue sampling. Also, there are various statistical methods used in quality control of a manufacturing process, to decide on how much the quality should be improved to be acceptable. 

The results of such multiple paired comparison tests are usually analyzed with Friedman s rank sum test [4] or with more sophisticated methods, e.g. the one using the Bradley-Terry model [5]. A good introduction to the theory and applications of paired comparison tests is David [6]. Since Friedman s rank sum test is based on less restrictive, ordering assumptions it is a robust alternative to two-way analysis of variance which rests upon the normality assumption. For each panellist (and presentation) the three products are scored, i.e. a product gets a score 1,2 or 3, when it is preferred twice, once or not at all, respectively. The rank scores are summed for each product i. One then tests the hypothesis that this result could be obtained under the null hypothesis that there is no difference between the three products and that the ranks were assigned randomly. Friedman s test statistic for this reads... 

This homogeneity value will become positive when the null hypothesis is not rejected by Fisher s E-test (F < Ei a)Vl)V2) and will be closer to 1 the more homogeneous the material is. If inhomogeneity is statistically proved by the test statistic F > Fi a>VuV2, the homogeneity value becomes negative. In the limiting case F = hom(A) becomes zero. 

Statistical tests make it possible, objectively to compare and interpret experimental data. They base on a test statistic to verify a statistical hypothesis about a ... 

Hi = alternative hypothesis a = significance level, usually set at. 10,. 05, or. 01 t = tabled t value corresponding to the significance level a. For a two-tailed test, each corresponding tail would have an area of a/2, and for a one-tailed test, one tail area would be equal to a. If a2 3 4 is known, then z would be used rather than the t. t = (x- Po)/(s/Vn) = sample value of the test statistic. 

We might also note here, almost parenthetically, that if the hypothesis test gives a statistically significant result, it would be valid to calculate the sensitivity of the result to the difference between the two groups (i.e., divide the difference in the means of the two groups by the difference in the values of the variable that correspond to the experimental and control groups). 

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. 

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