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Bootstrap mean differences from two samples.

Available in version 6.4.0 and later.


	function bootstrap_diff (
		x         : numeric,  
		y         : numeric,  
		nBoot [1] : integer,  
		nDim  [*] : integer,  
		opt   [1] : logical   

	return_val [ variable of type 'list' containing multiple estimates] 



A numeric array of up to four dimensions: x(NX), x(NX,:), x(NX,:,:), x(NX,:,:,:). 'NX' represents the original sample size.


A numeric array of up to four dimensions: x(NY), x(NY,:), x(NY,:,:), x(NY,:,:,:). 'NY' represents the original sample size. NOTE: NX and NX may be different.


An integer specifying the number of bootstrap data samples to be generated.


The dimension(s) of x and y on which to calculate the statistic. Most commonly, this is set to (/0,0/) or, if they are both the same, simply, 0.


A logical scalar to which optional attributes may be attached. If opt=False, all default values are used. If opt=True and no optional attributes are present, default values will be used. If opt=True then:

  • opt@sample_size_x and opt@sample_size_y allow the user to specify the sample sizes used to estimate the respective means. The defaults are: opt@sample_size_x=NX and opt@sample_size_y=NY.
  • opt@sample_size_x=nx where (nx.le.NX) and/or opt@sample_size_y=ny where (ny.le.NY). When these options are used, nx and ny are typically, 10-25% the size of NX and NY.

  • opt@rseed1=rseed1: allows user to set the first random seed integer value. Default is to use the system initial random seed. (See: random_setallseed)
  • opt@rseed2=rseed2: allows user to set the second random seed integer value. Default is to use the system initial random seed. (See: random_setallseed)
  • optrseed3="clock": tells NCL to use the 'date' clock to set the two random seeds. (See: random_setallseed)

Return value

A variable of type 'list'. Members of a list can be accessed directly. However, it is clearer if the members are explicity extracted and given meaningful names.

                                    ; typeof(Bootstrap) is 'list'
   BootStrap = bootstrap_diff(x, y, stat, nBoot, 0, opt)
   dBoot     = BootStrap[0]        ; bootstrapped differences in ascending order
   dBootAvg  = BootStrap[1]        ; Average of the bootstrapped differences
   dBootStd  = BootStrap[2]        ; Std. Deviation of the bootstrapped differences
   delete(BootStrap)       ; no longer needed


Bootstrapping is a statistical method that uses data resampling with replacement (see: generate_sample_indices) to estimate the properties of nearly any statistic. It is particularly useful when dealing with small sample sizes. A key feature is that bootstrapping makes no apriori assumption about the distribution of the sample data.


Computer Intensive Methods in Statistics 
   P. Diaconis and B. Efron 
   Scientific American (1983), 248:116-130  
An Introduction to the Bootstrap 
   B. Efron and R.J. Tibshirani, Chapman and Hall (1993) 
Bootstrap Methods and Permutation Tests: Companion Chapter 18 to the Practice of Business Statistics
   Hesterberg, T. et al (2003)

Climate Time Series Analysis: Classical Statistical and Bootstrap Methods
   M. Mudelsee (2014) Second edition. Springer, Cham Heidelberg New York Dordrecht London
   ISBN: 978-3-319-04449-1, e-ISBN: 978-3-319-04450-7
   doi: 10.1007/978-3-319-04450-7
   xxxii + 454 pp; Atmospheric and Oceanographic Sciences Library, Vol. 51

See Also

bootstrap_stat, bootstrap_correl, bootstrap_regcoef, bootstrap_estimate, generate_sample_indices, ListIndexFromName


Please see the Bootstrap and Resampling application page.

Example 1: Let x(NX); y(NY)

   nBoot       = 1000                ; user set
   nDim        = 0                   ; (/0,0/) since they refer to the same dimension
   opt         = False

   BootStrap   = bootstrap_diff(x, y, nBoot, nDim, opt)
   diffBoot    = BootStrap[0] ; All the bootstrapped differences
   diffBootAvg = BootStrap[1] ; Average of the bootstrapped differences
   diffBootStd = BootStrap[2] ; Std. Dev. of the boot strapped samples
   delete(BootStrap)         ; no longer needed

   diffBootLow = bootstrap_estimate(diffBoot, 0.025, False)   ;  2.5% lower confidence bound 
   diffBootMed = bootstrap_estimate(diffBoot, 0.500, False)   ; 50.0% median of bootstrapped estimates
   diffBootHi  = bootstrap_estimate(diffBoot, 0.975, False)   ; 97.5% upper confidence bound

   printVarSummary(diffBoot)   ; information only

Example 2: Let x(NX,:,:); y(NY,: :) where NX=100 and NY=50. Use subsampling:


   nBoot       = 2000                ; user set
   nDim        = 0                   
   opt         = True
   opt@sampling_size_x = 30
   opt@sampling_size_y = 10

   BootStrap   = bootstrap_diff(x, y, nBoot, nDim, opt)