Statistics, the art and science of learning from data by A. Agresti (fourth edition) – Chapter 15 summary

COMPARE TWO GROUPS BY RANKING
Nonparametric statistical methods are inferential methods that do not assume a particular form of distribution (e.g: the assumption of a normal distribution) for the population distribution. The Wilcoxon test is the best known nonparametric method. Nonparametric methods are useful when the data are ranked and when the assumption of normality is inappropriate.

The Wilcoxon test sets up a distribution using the probability of each difference of the mean rank. This test has five steps:

1. Assumptions
   Independent random samples from two groups.
2. Hypotheses
   
   
   
3. Test statistic
   This is the difference between the sample mean ranks for the two groups.
4. P-value
   This is a one-tail or two-tail probability, depending on the alternative hypothesis.
5. Conclusion
   The null hypothesis is either rejected in favour of the alternative hypothesis or not.

The sum of the ranks can also be used, instead of the mean of the ranks. When conducting the Wilcoxon test, a z-test can also be conducted if the sample is large enough. This z-test has the following formula:

A Wilcoxon test can also be conducted by converting quantitative observations to ranks. The Wilcoxon test is not affected by outliers (e.g: an extreme outlier gets the lowest/highest rank, no matter if it’s a bit higher or lower than the number before that). The difference between the population medians can also be used if the distribution is highly skewed, but this requires the extra assumption that the population distribution of the two groups have the same shape. The point estimate of the difference between two medians equals the median of the differences between the two groups. A sample proportion can also be used, by checking what the proportion is of observations in group one that’s better than group two. If there is a proportion of 0.50, then there is no effect. The closer the proportion gets to 0 or 1, the greater the difference between the two groups.

NONPARAMETRIC METHODS FOR SEVERAL GROUPS AND FOR MATCHED PAIRS
The test for comparing mean ranks of more than two groups is called the Kruskal-Wallis test. This test has five steps:

1. Assumptions
   Independent random samples.
2. Hypotheses
   
   
3. Test statistic
   The test statistic is based on the between-groups variability in the sample mean ranks. The test statistic uses the following formula:
   The test statistic has an approximate chi-squared distribution with g-1 degrees of freedom.
4. P-value
   The right-tail probability above observed test statistic value from chi-squared distribution.
5. Conclusion
   The null hypothesis is either rejected in favour of the alternative hypothesis or not.

It is also possible to compare matched pairs, which are dependent samples. This test uses proportions of two groups. The sign test for matched pairs has five steps:

1. Assumptions
   Random sample of matched pairs
2. Hypotheses
   
   
3. Test statistic
   The test statistic uses the following formula:

   
   The standard error has the following formula:
   

4. P-value
   This uses the tail probabilities from standard normal distribution for samples where p is equal to or larger than 30.
5. Conclusion
   The null hypothesis is either rejected in favour of the alternative hypothesis or not.

The Wilcoxon signed-ranks test is a nonparametric test designed for cases in which the comparisons of the paired observations can themselves be ranked. For each matched pair of responses, it measures the difference between the responses. This test has five steps:

1. Assumptions
   Random sample of matched pairs for which the difference of observations have a symmetric population distribution and can be ranked.
2. Hypotheses
   
   
3. Test statistic
   Rank the absolute values of the difference scores for the matched pairs and then find the sum of ranks of the differences that were positive.
4. P-value
   It uses an approximate normal distribution and a P-value can be found using software.
5. Conclusion
   The null hypothesis is either rejected in favour of the alternative hypothesis or not.

Advantages of nonparametric methods are that they make weaker assumptions than the parametric groups. This is useful for small samples. Disadvantages of the nonparametric methods are that the confidence intervals are not as thoroughly developed as significance tests.