Two-sided hypothesis tests

The significance of any calculated correlation coefficient is evaluated by adopting the usual two-sided hypothesis test. 

The null hypothesis is true only at the single point = do so it is called a point null hypothesis. Its negation, the alternative hypothesis is two-sided. This would be called a two-sided hypothesis test. Under the Bayesian approach with a continuous prior, the posterior will also be continuous, so the posterior probability of the point null hypothesis will always be 0. So we can t test a two-sided hypothesis by calculating the posterior probability of the null hypothesis. Clearly we can t test the "truth" of the null hypothesis. Our choice of a continuous prior means we do not believe the null hypothesis can be literally true. Instead we calculate a (1 — a) x 100% credible interval for the parameter 0. If the null value do lies in the credible interval, then we cannot reject the null hypothesis and do remains a credible value. Note we are not testing the "truth" of the null hypothesis, but rather its "credibility."... 

There is (1 - a) posterior probability that 9 lies in the interval 9i,0u)-We can test the two sided hypothesis Hq 0 = 0o Hi 0 0q zt the a level of significance by observing whether or not the null value lies inside the (1 - q) x 100% credible interval for 0. If it does not we reject the null hypothesis at the level a. If it does, we cannot reject the null hypothesis and conclude remains a credible value. 

Equal tail (1 - a) x 100% credible interval for 0 is the interval (0i,0u) where the proportion of the posterior sample less than 9i is f and the proportion of the posterior sample is greater than is. We can test a two-sided hypothesis at the a level of significance by seeing whether the null value 9q lies inside or outside the (1 — a) x 100% credible interval for6 . 

The macro CredIntNum.mac calculates a lower, upper, or two-sided credible interval from the numerical posterior CDF. The Minitab commands for invoking this macro are shown in Table A.2. We can test a two-sided hypothesis... 

Finding a credibie intervai. We want to use our random sample from the posterior to do an inference such as finding a credible interval for the parameter, or calculating the posterior probability of a one-sided null hypothesis about the parameter. The macro CredIntSamp.mac calculates a lower, upper, or two-sided credible interval from a random sample from the posterior. The Minitab commands for running this macro are given in Table A.4. We can test a two-sided hypothesis... 

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. 

Interpretation If the alternate hypothesis had been stated as //i Xmean is different from /r, a two-sided test is applied with 2.5% probability being provided for each possibility Xmean smaller than p" resp. Xmean larger than /i . Because 1.92 is smaller than 2.45, the test criterion is not exceeded, so Hi is rejected. On the other hand, if it was known beforehand that Xmean can only be smaller than p, the one-sided test is conducted under the alternate hypothesis H Xn,ean smaller than p in this case the result is elose, with 1.92 almost exceeding 1.94. 

Negative values of A indicate disordered regions are evolving faster than ordered. d / Values for a two-sided test of the null hypothesis. 

For each site and parameter we utilized the Mann-Kendall test to detect temporal trends [21]. The two-sided test for the null hypothesis that no trend is present was rejected for p-values below 0.05. In addition we quantified trends with the method of [22]. Results are shown in Table 3. 

For a two-sided test, the null hypothesis, Hq, is that the variance of the population from which the data giving is drawn is equal to and the alternative hypothesis is that it is not equal. Hq is rejected at the 95% level if r> xEi.025.1,-1 or T < x o.975.11-1 In Excel the probability of a particular value of chi-square is given by =CHIDIST, df), and the critical value of chi-square is =CHIINV( q, df) for probability ct and df degrees of freedom. 

Because of the relationship k( - a) < k( 1 - a/2) one-sided tests are sharper , i.e. they will indicate significance earlier. Therefore a null hypothesis is rejected earlier than it would be by use of a two-sided test. 

If we are going to test a one-sided question, we need to modify our null and alternative hypotheses. For two-sided testing, the null hypothesis would be that there is no difference in clearance and the alternative would be that there is. For a one-sided test (looking for a greater clearance) we want our alternative hypothesis to be there is an increase in clearance . The null hypothesis then has to cover all other possibilities - clearance is either unchanged or reduced . 

The r-statistic is (207.6 - 199.6)/3.78 = 2.12. Entering Table 5 with 60 degrees of freedom, we obtain a critical t-value for a one-sided test at the 0.025 level of significance equal to 2.00. As the alternative hypothesis is that the two QC specimens differ, without regard to which has the higher and which has the lower value, the test is a two-sided test. The critical table value at 0.025, one-sided, is the critical value to use for a two-sided test at 0.05. The calculated statistic exceeds this critical value, and we reject the hypothesis that the two sets of QC samples were prepared identically. If we have the stock solutions used to prepare the two QC specimens, we would probably analyze them to see whether they have identical concentrations. 

When we have two (or more) treatments , we will usually wish to carry out a hypothesis test. This is making a decision about whether it is plausible that the difference (if any) between the treatments is real . In Table 7.2, we might wish to consider whether treatment B really does give a higher rate of side-effects than treatment A. To address this question, the alternative non-causal explanations for the apparent difference must be considered. They are (a) bias in allocation or group membership, (b) assessment or measurement bias, and (c) chance. 

Similarly to the case of the standard normal distribution, the critical values can be obtained from a series of tabulated values or from statistical software. A number of percentiles of various t distributions are provided in Appendix 2. It is important to note that there is not just one t distribution there are many of them, and their shapes are determined by the number of degrees of freedom. As either low or high values of the test statistic could lead to rejection, the hypothesis test is considered a two-sided test. The probability of committing a type I error is O.OS, but, because the critical region is evenly split between low values and high values, the probability of committing a type I error in favor of one direction (for example, large values of t) is a/2. 

The one-sample t test will be used to test the null hypothesis. As there are 10 observations and assuming the change scores (the random variable of interest) are normally distributed, the test statistic will follow a t distribution with 9 df. A table of critical values for the t distribution (Appendix 2) will inform us that the two-sided critical region is defined as t < -2.26 and t > 2.26 - that is, under the null hypothesis, the probability of observing a t value < -2.26 is 0.025 and the probability of observing a t value > 2.26 is 0.025. 

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. 

Regulatory agencies have traditionally accepted only two-sided hypotheses because, theoretically, one could not rule out harm (as opposed to simply no effect) associated with the test treatment. If the value of a test statistic (for example, the Z-tesl statistic) is in the critical region at the extreme left or extreme right of the distribution (that is, < -1.96 or > 1.96), the probability of such an outcome by chance alone under the null hypothesis of no difference is 0.05. However, the probability of such an outcome in the direction indicative of a treatment benefit is half of 0.05, that is, 0.025. This led to a common statistical definition of "firm" or "substantial" evidence as the effect was unlikely to have occurred by chance alone, and it could therefore be attributed to the test treatment. Assuming that two studies of the test treatment had two-sided p values < 0.05 with the direction of the treatment effect in favor of a benefit, the probability of the two results occurring by chance alone would be 0.025 X 0.025, that is, 0.000625 (which can also be expressed as 1/1600). 

The equality of two proportions is being tested with the null hypothesis, Hq. p p - Ppy cEBO = 0- Given that this is a two-sided test, what is the p value that corresponds to the following values of the / approximation test statistic ... 

Under the null hypothesis of equal population means, the test statistic follows a t distribution with Kj 4- 2 - 2 degrees of freedom (df), assuming that the sample size in each group is large (that is, > 30) or the underlying distribution is at least mound shaped and somewhat symmetric. As the sample size in each group approaches 200, the shape of the t distribution becomes more like a standard normal distribution. Values of the test statistic that ate fat away from zero would contradict the null hypothesis and lead to its rejection. In particular, for a two-sided test of size a, the critical region (that is, those values of the test statistic that would lead to rejection of the null hypothesis) is defined by t[Pg.148]

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