Taking Serial Correlation into Account in Tests of the Mean

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  • 1 Canadian Centre for Climate Modelling and Analysis, Victoria, British Columbia, Canada
  • | 2 Max Planck Institute for Meteorology, Hamburg, Germany
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Abstract

The comparison of means derived from samples of noisy data is a standard pan of climatology. When the data are not serially correlated the appropriate statistical tool for this task is usually the conventional Student's t-test. However, frequently data are serially correlated in climatological applications with the result that the t test in its standard form is not applicable. The usual solution to this problem is to scale the t statistic by a factor that depends upon the equivalent sample size ne.

It is shown, by means of simulations, that the revised t tea is often conservative (the actual significance level is smaller than the specified significance level) when the equivalent sample size is known. However, in most practical cases the equivalent sample size is not known. Then the test becomes liberal (the actual significance level is greater than the specified significance level). This systematic error becomes small when the true equivalent sample size is large (greater than approximately 30).

The difficulties inherent in difference of means tests when there is serial dependence are reexamined. Guidelines for the application of the “usual” t test are provided and two alternative tests are proposed that substantially improve upon the “usual” t test when samples are small.

Abstract

The comparison of means derived from samples of noisy data is a standard pan of climatology. When the data are not serially correlated the appropriate statistical tool for this task is usually the conventional Student's t-test. However, frequently data are serially correlated in climatological applications with the result that the t test in its standard form is not applicable. The usual solution to this problem is to scale the t statistic by a factor that depends upon the equivalent sample size ne.

It is shown, by means of simulations, that the revised t tea is often conservative (the actual significance level is smaller than the specified significance level) when the equivalent sample size is known. However, in most practical cases the equivalent sample size is not known. Then the test becomes liberal (the actual significance level is greater than the specified significance level). This systematic error becomes small when the true equivalent sample size is large (greater than approximately 30).

The difficulties inherent in difference of means tests when there is serial dependence are reexamined. Guidelines for the application of the “usual” t test are provided and two alternative tests are proposed that substantially improve upon the “usual” t test when samples are small.

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