Parametric and nonparametric tests and use of nonparametric tests

CFA level I / Quantitative Methods: Application / Hypothesis Testing / Parametric and nonparametric tests and use of nonparametric tests

A parametric test is based on some measurement of parameters like mean or variance. This test also requires the validity of a set of assumptions like assumptions regarding the distribution of the population producing the sample.

A nonparametric test is not concerned with a parameter or a test that do not require strict assumption regarding the population from where the sample is drawn. It is primarily used in three scenarios: when the data do not meet the distributional assumptions, when the data are given in ranks, or when the hypothesis does not concern a parameter.

We often convert the original data into ranks to use nonparametric tests. Sometimes, the original data are already ranked. Then we also use nonparametric tests as parametric tests require a stronger measurement scales than ranks. If we have to rank investment managers, then a nonparametric test will be used.

The nonparametric tests are also used when the test does not involve a parameter. For example, testing whether a sample is random or not or whether a sample came from a population following a particular probability distribution. Some nonparametric alternatives to parametric tests are given below in the table.

	Parametric Tests	Nonparametric Tests
Tests concerning a single mean	t-test, z-test	Wilcoxon signed-rank test
Tests concerning differences between means	t-test, approximate t-test	Mann-Whitney U test
Tests concerning mean differences (paired comparison tests)	t-test	Wilcoxon signed-rank test, Sign test

Tests concerning correlation: The Spearman Rank Correlation Coefficient: Correlation measures the strength of the linear relationship between two variables. The t-test for the hypothesis testing relies on fairly stringent assumptions. When the assumptions are violated, then we use the test based on the Spearman rank correlation coefficient. The Spearman rank correlation coefficient, r_s, is equivalent to the correlation coefficient calculate don the ranks of the two variables within their respective samples. Its understanding is similar to the actual correlation coefficient i.e. a coefficient of zero means the absence of any straight-line relationships between the variables.

Steps in the calculation of the Spearman rank correlation coefficient:

(1) Rank the observation on X and Y separately from largest to smallest. Assign the number 1 to the largest value and 2 to the second largest value and so on. In the case of ties, assign the average of the ranks to the tied observations. For example, if 4th and 5th observations are tied, then we will assign them a rank of 4.5.

(2) Calculate the difference (d_i) between the ranks of each pair of observations on X and Y.

(3) Calculate the Spearman coefficient, r_s = 1 - [6∑d_i² /n(n² - 1)]

The rejection points for the null hypothesis calculated similarly to that of t-test for a particular level of significance.

If the sample size is large (n>30), then we can conduct a t-test using the below formula.

t = (n-2)^1/2r_s/(1-r_s²)^1/2 with n-2 degrees of freedom

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