For the present illustrative example, 11 pairs of observations constitute the sample/population. From these 11 pairs of data, sampling with replacement is used to create a new sample of size 11 (a bootstrap sample). In this new sample, the data pair (.18, .20) might appear only once or could appear multiple times in the sample of 11 data pairs. A Pearson's correlation coefficient is calculated for each of these bootstrap samples. The standard deviation of these bootstrap correlations is calculated thus giving an estimate of the standard error of the correlation coefficient. The calculation of the 2.5th and 97.5th percentiles of the distribution of the bootstrap correlations will give an estimate of the 95th confidence intervals for the correlation coefficient (percentile method). A bias adjustment for the percentile method (bias corrected percentile method) is discussed in Efron and Tibshirani (1993); here, we discuss only the unadjusted percentile method.
Thompson (1993) discusses using the bootstrap methodology in conjunction with traditional statistical significance testing to explore result replicability. Thompson's data set (page 370)is used as our example:
The SPSS syntax included here uses the SPSS INPUT PROGRAM to generate 1000 samples (n=11 per sample)of randomly sampled case id's (sampling with replacement). The MATCH FILES procedure is used to copy data from the original file (the x,y pairs) into the working data file.
**** Bootstrap Confidence Intervals for **** the Correlation Coefficient **** create 1000 bootstrap samples of size **** n=11, use sampling with replacement input program. loop samp=1 to 1000. + loop #i=1 to 11. + compute id=trunc(uniform(11))+1. + end case. + end loop. + leave samp. end loop. end file. end input program. execute. sort cases by id. match files file=* /table='a:\thompson.sav' /by id. sort cases by samp. split file by samp. execute. **** calculate a correlation coefficient for each bootstrap sample CORRELATIONS /VARIABLES=y x /PRINT=TWOTAIL SIG /MISSING=PAIRWISE .Once the correlation output has been saved to an output text file, one removes the sample ids, correlation values, and pvalues from the output file of the 1000 bootstrap samples:
SET WIDTH=80. FILE TYPE NESTED FILE='a:\corr.out' RECORD=1-80 (A). RECORD TYPE SAMP:'. DATA LIST / sample 9-16. RECORD TYPE X'. DATA LIST RECORDS=3 / corr 13-18 // pvalue 16-19 . END FILE TYPE. FORMATS corr (F8.2) pvalue (F8.2) sample (F8.2) . execute.Next, the lower 2.5th and upper 97.5th percentiles of the empirical distribution of correlation coefficients are calculated:
FREQUENCIES VARIABLES=corr /FORMAT=NOTABLE /PERCENTILES= 2.5 97.5 /STATISTICS=STDDEV MEAN. CORR Mean .573 Std dev .162 Percentile Value Percentile Value 2.50 .179 97.50 .828 Valid cases 1000 Missing cases 0Our bootstrap 95th percentiles are (.179, .828). Since these intervals do ot include 0, this is taken to be a rejection of the null hypothesis, that the correlation coefficient is zero in the population Power estimation (Cohen, 1988) with the bootstrap is accomplished by counting the proportion of redrawn samples that lead to a statistically significant estimator (for a given alpha level):
**** Probability to reject an assumed false **** null hypothesis (simulated power). do if pvalue<<=.05). compute count=1. else if pvalue>>(.05). compute count=0. end if. execute. FREQUENCIES VARIABLES=count. COUNT Valid Cum Value Label Value Frequency Percent Percent Percent .00 523 52.3 52.3 52.3 1.00 477 47.7 47.7 100.0 - - - Total 1000 100.0 100.0 Valid cases 1000 Missing cases 0Power estimate based on distributional assumptions (Using Cohen's power tables) = .460
Efron, B & Tibshirani, R.J. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York.
Fox, J & Long, J.S. (1990). Modern Methods of Data Analysis. Sage Publications, Newbury, Park, CA.
Thompson, B. (1993). The Use of Statistical Significance Tests in Research: Bootstrap and Other Alternatives. Journal of Experimental Education, 61(4), 361-377.
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