Significance Testing#

Significance testing is important for assessing whether a given initialized prediction system is skillful. Some questions that significance testing can answer are:

  • Is the correlation coefficient of a lead time series significantly different from zero?

  • What is the probability that the retrospective forecast is more valuable than a historical/uninitialized simulation?

  • Are correlation coefficients statistically significant despite temporal and spatial autocorrelation?

All of these questions deal with statistical significance. See below on how to use climpred to address these questions. Please also have a look at the significance testing example.

p value for temporal correlations#

For the correlation metrics, like _pearson_r() and _spearman_r(), climpred also hosts the associated p-value, like _pearson_r_p_value(), that this correlation is significantly different from zero. _pearson_r_eff_p_value() also incorporates the reduced degrees of freedom due to temporal autocorrelation. See example.

Bootstrapping with replacement#

Testing statistical significance through bootstrapping is commonly used in the field of climate prediction. Bootstrapping relies on resampling the underlying data with replacement for a large number of iterations, as proposed by the decadal prediction framework [Boer et al., 2016, Goddard et al., 2013]. This means that the initialized ensemble is resampled with replacement along a dimension (init or member) and then that resampled ensemble is verified against the observations. This leads to a distribution of initialized skill. Further, a reference forecast uses the resampled initialized ensemble, which creates a reference skill distribution. Lastly, an uninitialized skill distribution is created from the underlying historical members or the control simulation.

The probability or p value is the fraction of these resampled initialized metrics beaten by the uninitialized or resampled reference metrics calculated from their respective distributions. Confidence intervals using these distributions are also calculated.

This behavior is incorporated by HindcastEnsemble.bootstrap() and PerfectModelEnsemble.bootstrap(), see example.

Field significance#

Please use esmtools.testing.multipletests() to control the false discovery rate (FDR) in geospatial data from the above obtained p-values [Wilks, 2016]. See the FDR example.

Sign test#

Use DelSole’s sign test relying on the statistics of a random walk to decide whether one forecast is significantly better than another forecast [Benjamini and Hochberg, 1994, DelSole and Tippett, 2016], see xskillscore.sign_test() and sign test example.



Yoav Benjamini and Yosef Hochberg. Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing. Journal of the Royal Statistical Society: Series B (Methodological), 57(1):13, 1994. doi:10.1111/j.2517-6161.1995.tb02031.x.


G. J. Boer, D. M. Smith, C. Cassou, F. Doblas-Reyes, G. Danabasoglu, B. Kirtman, Y. Kushnir, M. Kimoto, G. A. Meehl, R. Msadek, W. A. Mueller, K. E. Taylor, F. Zwiers, M. Rixen, Y. Ruprich-Robert, and R. Eade. The Decadal Climate Prediction Project (DCPP) contribution to CMIP6. Geosci. Model Dev., 9(10):3751–3777, October 2016. doi:10/f89qdf.


Timothy DelSole and Michael K. Tippett. Forecast Comparison Based on Random Walks. Monthly Weather Review, 144(2):615–626, February 2016. doi:10/f782pf.


L. Goddard, A. Kumar, A. Solomon, D. Smith, G. Boer, P. Gonzalez, V. Kharin, W. Merryfield, C. Deser, S. J. Mason, B. P. Kirtman, R. Msadek, R. Sutton, E. Hawkins, T. Fricker, G. Hegerl, C. a. T. Ferro, D. B. Stephenson, G. A. Meehl, T. Stockdale, R. Burgman, A. M. Greene, Y. Kushnir, M. Newman, J. Carton, I. Fukumori, and T. Delworth. A verification framework for interannual-to-decadal predictions experiments. Climate Dynamics, 40(1-2):245–272, January 2013. doi:10/f4jjvf.


D. S. Wilks. “The Stippling Shows Statistically Significant Grid Points”: How Research Results are Routinely Overstated and Overinterpreted, and What to Do about It. Bulletin of the American Meteorological Society, 97(12):2263–2273, March 2016. doi:10/f9mvth.