Two-sided test

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 most widely used test is that for detecting a deviation of a test object from a standard by comparison of the means, the so-called t-test. Note that before a f-test is decided upon, the confidence level must be declared and a decision made about whether a one- or a two-sided test is to be performed. For details, see shortly. Three levels of complexity, a, b, and c, and subcases are distinguishable. (The necessary equations are assembled in Table 1.10 and are all included in program TTEST.)... 

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. 

This table is used for the two-sided test, that is one simply asks are the two distributions different Approximations to tabulated x -values for different confidence levels can be made using the algorithm and the coefficients given in section 5.1.4. 

For continuous variables you may be required to provide inferential statistics along with the descriptive statistics that you generate from PROC UNIVARIATE. The inferential statistics discussed here are all focused on two-sided tests of mean values and whether they differ significantly in either direction from a specified value or another population mean. Many of these tests of the mean are parametric tests that assume the variable being tested is normally distributed. Because this is often not the case with clinical trial data, we discuss substitute nonparametric tests of the population means as well. Here are some common continuous variable inferential tests and how to get the inferential statistics you need out of SAS. 

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. 

Standardised difference [i.e. Difference between treatments / standard deviation based on a two-sided test at the 0-05 level]... 

The p-value calculation detailed in the previous section gives what we call a two-tailed or a two-sided test since we calculate p by taking into account values of the test statistic equal to, or more extreme, than that observed, in both directions. So... 

It is important to clarify whether one- or two-sided tests of statistical significance will be used, and in particular to justify prospectively the use of one-sided tests... The approach of setting type I errors for one-sided tests at half the conventional type I error used in two-sided tests is preferable in regulatory settings. ... 

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. 

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. 

For a two-sided test, we noted that when Z was less than za/2 or greater than... 

This interval estimate is really based on the two-sided test of the third set of hypotheses previously given. Although it is possible to define one-sided confidence intervals based on the other two sets of hypotheses (1.59) and (1.60), such one-sided intervals are rarely used. By one-sided, we mean an interval estimate that extends from plus or minus infinity to a single random confidence limit. The one-sided confidence interval may be understood as the range one limit of which is the probability level a and the other one °°. 

If we wish to test whether a sample is drawn from a population of a specific known variance, we have a two-sided test ... 

The confidence interval that is equivalent to the two-sided test is obtained from the critical regions ... 

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. 

Show that, in marginal cases, data may be non-significant when assessed by a two-sided test, andyetbesignificantwith the one-sided version of the same test... 

It often perplexes people that they cannot simply consider how an experiment was performed in order to tell whether a one or a two-sided test should be used. The fact is that the decision depends upon what question the experiment was designed to answer. If the purpose was to look for any old change - use a two-sided test. If your only interest lay in checking for a change in a particular, specified direction, a one-sided test should be used. 

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 . 

Data may be Significant with a one-sided test, even though it was non-significant with a two-sided test... 

These are shown diagrammatically in Figure 10.3. For the one-sided test, the confidence limit began life as a part of a 90 per cent confidence interval, which is narrower than the 95 per cent Cl used for the two-sided test. In this case, the difference in width just happens to make a critical difference - the two-sided test is not significant, but the one-sided test is. 

Figure 10.3 A one-sided test (for an increase) and a two-sided test (for any difference) in clearance...

The P value for the one-sided test is 0.038, in line with our previous conclusion that it produced a significant result. The two-sided test would yield a P value exactly twice as large, i.e. 0.076 (non-significant). 

This obviously raises the possibility of abuse. If we initially performed the experiment intending to carry out a two-sided test and obtained a non-significant result, we might then be tempted to change to the one-sided test in order to force it to be significant. 

Data may be non-significant with a two-sided test, and yet significant with a onesided test. This can be abused to convert a non-significant finding to apparent significance. Such abuses raise the risk of false positives from 5 to 10 per cent. 

It is claimed that we planned to test for an increase in fertility and therefore performed a one-sided test . Did they Maybe they were looking for a contraceptive effect but found that there were actually more babies and the initial two-sided test produced a P value of 0.07. Was the pirate box at the end of Chapter 10... 

You did the experiment and analysed the results by your usual two-sided test. The result fell just short of significance (P somewhere between 0.05 and 0.1) There is a simple solution, guaranteed to work every time. Re-run the analysis, but change to a one-sided test, testing for a change in whatever direction you now know the results actually suggest. 

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