If the null hypothesis is true, we expect to see similar numbers of lower and higher ranks that are both positive and negative i. If the research hypothesis is true we expect to see more higher and positive ranks in this example, more children with substantial improvement in repetitive behavior after treatment as compared to before, i.
Next we must determine whether the observed test statistic W supports the null or research hypothesis. This is done following the same approach used in parametric testing. Specifically, we determine a critical value of W such that if the observed value of W is less than or equal to the critical value, we reject H 0 in favor of H 1 , and if the observed value of W exceeds the critical value, we do not reject H 0.
Note that when we analyzed the data previously using the Sign Test, we failed to find statistical significance. The discrepant results are due to the fact that the Sign Test uses very little information in the data and is a less powerful test. A study is run to evaluate the effectiveness of an exercise program in reducing systolic blood pressure in patients with pre-hypertension defined as a systolic blood pressure between mmHg or a diastolic blood pressure between mmHg.
A total of 15 patients with pre-hypertension enroll in the study, and their systolic blood pressures are measured. Each patient then participates in an exercise training program where they learn proper techniques and execution of a series of exercises. Patients are instructed to do the exercise program 3 times per week for 6 weeks. After 6 weeks, systolic blood pressures are again measured. The most important statistical tests are listed in the Table. The group comparison for two categorical endpoints is illustrated here with the simplest case of a 2 x 2 table four field table Figure 1.
However, the procedure is similar for the group comparison of categorical endpoints with multiple values Table. In such a case, one has to perform the McNemar test 7. Figure 2 shows a decision algorithm for test selection. Normally distributed variables—parametric tests: So-called parametric tests can be used if the endpoint is normally distributed. Where subjects in both groups are independent of each other persons in first group are different from those in second group , and the parameters are normally distributed and continuous, the unpaired t-test is used.
If a comparison is to be made of a normally distributed continuous parameter in more than two independent unpaired groups, analysis of variance ANOVA can be used. One example would be a study with three or more treatment arms.
ANOVA is a generalization of the unpaired t-test. ANOVA only informs you whether the groups differ, but does not say which groups. This requires methods of multiple testing The paired t-test is used for normally distributed continuous parameters in two paired groups. If a normally distributed continuous parameter is compared in more than two paired groups, methods based on analysis of variance are also suitable. The factor describes the paired groups—for example, more than two points of measurement in the use of a therapy.
Non-normally distributed variables—non-parametric tests: If the parameter of interest is not normally distributed, but at least ordinally scaled, non-parametric statistical tests are used. This necessitates putting the values in order of size and giving them a running number. The test variable is then calculated from these rank numbers. If the necessary preconditions are fulfilled, parametric tests are more powerful than non-parametric tests. However, the power of parametric tests may sink drastically if the conditions are not fulfilled.
The Mann-Whitney U test also known as the Wilcoxon rank sum test can be used for the comparison of a non-normally distributed, but at least ordinally scaled, parameter in two unpaired samples 5. If more than two unpaired samples are to be compared, the Kruskal-Wallis test can be used as a generalization of the Mann-Whitney U test The Wilcoxon signed rank test can be used for the comparison of two paired samples of non-normally distributed, but at least ordinally scaled, parameters Alternatively, the sign test should be used when the two values are only distinguished on a binary scale—for example, improvement versus deterioration 7.
If more than two paired samples are being compared, the Friedman test can be used as a generalization of the sign test. If the point of interest is not the endpoint itself, but the time till it is reached, survival time analysis is the most suitable procedure.
This compares two or more groups with respect to the time when an endpoint is reached within the period of observation One example is the comparison of the survival time of two groups of cancer patients given different therapies. The endpoint here is death, although it could just as well be the occurrence of metastases.
In contrast to the previous tests, it almost never happens that all subjects reach the endpoint in survival time analysis, as the period of observation is limited. For this reason, the data are also described as right censored, as it is still unclear when all subjects will reach the endpoint when the study ends. The log rank test is the usual statistical test for the comparison of the survival functions between two groups. A formula is used to calculate the test variable from the observed and the expected numbers of events.
This value can be compared with the known distribution which would have been expected if the null hypothesis were correct—the chi-square distribution in this case. A p-value can thus be calculated.
A rule can then be given for deciding for or against the null hypothesis. Correlation analysis examines the strength of the correlation between two test variables, for example, the strength of the correlation between the body weight of a neonate and its body length. The selection of a suitable measure of association depends on the scale of measurement and the distribution of the two parameters.
The parametric variant Pearson correlation coefficient exclusively tests for a linear correlation between continuous parameters. On the other hand, the non-parametric variant—the Spearman correlation coefficient—solely tests for monotonous relationships for at least ordinally scaled parameters.
The advantages of the latter are its robustness to outliers and skew distributions. The closer they are to 1, the stronger is the association. A test variable and a statistical test can be constructed from the correlation coefficient. The null hypothesis to be tested is then that there is no linear or monotonous correlation. The null hypotheses for these statistical tests described in this article are that the groups are equal. There are however other types of test. A non-inferiority test might examine whether a cheaper new medicine is not much worse than a conventional medicine.
The acceptable level of activity is specified before the start of the study on the basis of expert medical knowledge. An equivalence test is intended to show that a medication has approximately the same activity as a conventional standard medication.
The advantages of the new medication might be simpler administration, fewer side effects, or a lower price. The methods of regression analysis and the related statistical tests will be discussed in more detail in the course of this series on the evaluation of scientific publications. The present selection of statistical tests is obviously incomplete. Our intention has been to make it clear that the selection of a suitable test procedure is based on criteria such as the scale of measurement of the endpoint and its underlying distribution.
Bortz et al. The selection of the statistical test before the study begins ensures that the study results do not influence the test selection. Moreover, the necessary sample size depends on the test procedure selected. Problems in planning sample size will be discussed in more detail later in this series.
Finally, the point must be made that a statistical test is not necessary for every study. Statistical testing can be dispensed with in purely descriptive studies 12 or when the interrelationships are based on scientific plausibility or logical arguments. Statistical tests are also usually not helpful when investigating the quality of a diagnostic test procedure or rater agreement for example, in the form of a Bland-Altman diagram Conflict of interest statement.
The authors declare that no conflict of interest exists according to the guidelines of the International Committee of Medical Journal Editors. National Center for Biotechnology Information , U. Journal List Dtsch Arztebl Int v. Dtsch Arztebl Int. Published online May Jean-Baptist du Prel , Dr.
Specifically, we produce a test statistic based on the ranks. First, we sum the ranks in each group. In the placebo group, the sum of the ranks is 37; in the new drug group, the sum of the ranks is For the test, we call the placebo group 1 and the new drug group 2 assignment of groups 1 and 2 is arbitrary. We let R 1 denote the sum of the ranks in group 1 i.
If the null hypothesis is true i. In this example, the lower values lower ranks are clustered in the new drug group group 2 , while the higher values higher ranks are clustered in the placebo group group 1. This is suggestive, but is the observed difference in the sums of the ranks simply due to chance? To answer this we will compute a test statistic to summarize the sample information and look up the corresponding value in a probability distribution. Is this evidence in support of the null or research hypothesis?
Before we address this question, we consider the range of the test statistic U in two different situations. Consider the situation where there is complete separation of the groups, supporting the research hypothesis that the two populations are not equal. If all of the higher numbers of episodes of shortness of breath and thus all of the higher ranks are in the placebo group, and all of the lower numbers of episodes and ranks are in the new drug group and that there are no ties, then:.
Consider a second situation where l ow and high scores are approximately evenly distributed in the two groups , supporting the null hypothesis that the groups are equal. If ranks of 2, 4, 6, 8 and 10 are assigned to the numbers of episodes of shortness of breath reported in the placebo group and ranks of 1, 3, 5, 7 and 9 are assigned to the numbers of episodes of shortness of breath reported in the new drug group, then:.
Thus, smaller values of U support the research hypothesis, and larger values of U support the null hypothesis. In the example above, U can range from 0 to 25 and smaller values of U support the research hypothesis i.
The procedure for determining exactly when to reject H 0 is described below. In every test, we must determine whether the observed U supports the null or research hypothesis.
This is done following the same approach used in parametric testing. Specifically, we determine a critical value of U such that if the observed value of U is less than or equal to the critical value, we reject H 0 in favor of H 1 and if the observed value of U exceeds the critical value we do not reject H 0. The critical value of U can be found in the table below. However, in this example, the failure to reach statistical significance may be due to low power. The sample data suggest a difference, but the sample sizes are too small to conclude that there is a statistically significant difference.
A new approach to prenatal care is proposed for pregnant women living in a rural community. The new program involves in-home visits during the course of pregnancy in addition to the usual or regularly scheduled visits. A pilot randomized trial with 15 pregnant women is designed to evaluate whether women who participate in the program deliver healthier babies than women receiving usual care.
Each of the 5 criteria is rated as 0 very unhealthy , 1 or 2 healthy based on specific clinical criteria. Infants with scores of 7 or higher are considered normal, low and 0 to 3 critically low. Sometimes the APGAR scores are repeated, for example at 1 minute after birth, at 5 and at 10 minutes after birth and analyzed.
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