Friedman test (stable seasonality test)

Overview

The Friedman test is a non-parametric method for testing that samples are drawn from the same population or from populations with equal medians. The significance of the month (or quarter) effect is tested. The Friedman test requires no distributional assumptions. It uses the rankings of the observations. If the null hypothesis of no stable seasonality is rejected at the 0.10% significance level then the series is considered to be seasonal and the test’s outcome is displayed in green.

The test statistic is constructed as follows. Consider first the matrix of data \(\left\{x_{ij}\right\}_{n \times k}\) with \(n\) rows (the blocks, i.e. number of years in the sample), \(k\) columns (the treatments, i.e. either 12 months or 4 quarters, depending on the frequency of the data).
The data matrix needs to be replaced by a new matrix \(\left\{r_{ij}\right\}_{n \times k}\), where the entry \(r_{ij}\) is the rank of \(x_{ij}\) within block \(i\) .

The test statistic is given by

where \(SS_t=n \sum_{j=1}^{k}(\bar{r}_{.j}-\bar{r})^2\) and \(SS_e=\frac{1}{n(k-1)} \sum_{i=1}^{n}\sum_{j=1}^{k}(r_{ij}-\bar{r})^2\) It represents the variance of the average ranking across treatments j relative to the total.

Under the hypothesis of no seasonality, all months can be equally treated. For the sake of completeness:

• \(\bar{r}_{.j}\) is the average ranks of each treatment (month) j within each block (year)
• The average rank is given by \(\bar{r}= \frac{1}{nk}\sum_{i=1}^{n}\sum_{j=1}^{k}(r_{ij})\)

For large \(n\) or \(k\) , i.e. n > 15 or k > 4, the probability distribution of \(Q\) can be approximated by that of a chi-squared distribution. Thus, the p-value is given by \(P( \chi^2_{k-1}>Q)\) .

Use

The test can be applied directly to any series by selecting the option Statistical Methods » Seasonal Adjustment » Tools » Seasonality Tests. This is an example of how results are displayed for the case of a monthly series:

If the null hypothesis of no stable seasonality is rejected at the 1% significance level, then the series is considered to be seasonal and the outcome of the test is displayed in green.

The test can be applied to the input series before any seasonal adjustment method has been applied. It can also be applied to the seasonally adjusted series or to the irreguar component. In the case of X-13ARIMA-SEATS, the test is applied to the preliminary estimate of the unmodified Seasonal-Irregular component¹ (time series shown in Table B3). In this estimate, the number of observations is lower than in the final estimate of the unmodified Seasonal-Irregular component. Thus, the number of degrees of freedom in the stable seasonality test is lower than the number of degrees of freedom in the test for the presence of seasonality assuming stability. For example, X-13ARIMA-SEATS uses a centred moving average of order 12 to calculate the preliminary estimation of trend. Consequently, the first six and last six points in the series are not computed at this stage of calculation. The preliminary estimation of the trend is then used for the calculation of the preliminary estimation of the unmodified Seasonal-Irregular.

Related tests

• When using this kind of design for a binary response, one instead uses the Cochran’s Q test.
• Kendall’s W is a normalization of the Friedman statistic between 0 and 1.
• The Wilcoxon signed-rank test is a nonparametric test of non-independent data from only two groups.

Implementation

This test is implemented in the class ec.satoolkit.diagnostics.FriedmanTest

References

• Friedman, Milton (December 1937). “The use of ranks to avoid the assumption of normality implicit in the analysis of variance”. Journal of the American Statistical Association (American Statistical Association) 32 (200): 675–701. doi:10.2307/2279372. JSTOR 2279372.

• Friedman, Milton (March 1939). “A correction: The use of ranks to avoid the assumption of normality implicit in the analysis of variance”. Journal of the American Statistical Association (American Statistical Association) 34 (205): 109. doi:10.2307/2279169. JSTOR 2279169.

• Friedman, Milton (March 1940). “A comparison of alternative tests of significance for the problem of m rankings”. The Annals of Mathematical Statistics 11 (1): 86–92. doi:10.1214/aoms/1177731944. JSTOR 2235971.