MEANCIBOOT
This macro is designed to calculate bootstrap confidence intervals for a population mean.
RUNNING THE MACRO
Calling statement
meanciboot c1 ;
siglev k1 (95) ;
nboot k1 (2000);
means c1 ;
quantiles c1-c3 ;
tvalues c1.
Input
Input to the macro must be a single column, containing only numerical values.
Subcommands
siglev
The significance level of the confidence interval, expressed as a percentage.
The default is 95 (corresponding to 95% significance); other standard choices are
90, 98 or 99.
nboot
The number of bootstrap samples used. The default is 2000. It is not recommend to use
less than 1000 for the construction of confidence intervals.
means
Specify a column in which to store bootstrap sample means.
quantiles Specify three columns in which to store ranks corresponding to the lower and upper
confidence interval limits, for the standard percentile method (column 1), the BC method
(column 2) and the BCa method (column 3).
tvalues
Specify a column in which to bootstrap sample t-statistics.
Output
 Basic information (number of data points, significance level, number of bootstrap samples)
 Sample mean, with associated standard error
 Sample standard deviation
 Bootstrap standard deviation about the estimated mean
 Overall bootstrap mean
 Estimated bias correction (for BC and BCa methods)
 Estimated acceleration (for BCa method)
 Standard bootstrap confidence interval
 Bootstrap confidence intervals using : Estimate -/+ 1.96*bootstrap standard deviation, Bootstrap-t
method, Efron percentile method, Hall percentile method, BC method, BCa method.
Speed of macro : FAST
Missing data : Allowed
ALTERNATIVE PROCEDURES
Standard procedures
tinterval c1
This produces a confidence interval about a mean value, in the situation in which variance is unknown.
zinterval k1 c1
This produces a confidence interval about a mean value, where the variance is known to be k1.
REFERENCES
MANLY, F.J. (1997) Randomization, bootstrap and Monte Carlo methods in biology,
Chapman and Hall, London (Chapter 3).
EFRON, B. & TIBSHIRANI, J. (1993) An introduction to the Bootstrap, Chapman and Hall, London
(Chapters 12-14).
WORKED EXAMPLE FOR MEANCIBOOT
Name of dataset
EXPONENTIAL
Description
The data are 20 realisations from an Exponential distribution with rate parameter 1.
Source
MANLY, F.J. (1997) Randomization, bootstrap and Monte Carlo methods in biology,
Chapman and Hall, London.
Data
Number of observations = 20
Number of variables = 1
3.56 0.69 0.10 1.84 3.93 1.25 0.18 1.13 0.27 0.50 0.67 0.01 0.61
0.82 1.70 0.39 0.11 1.20 1.21 0.72
Worksheet
C1
Data
Aims of analysis
To create confidence intervals for the population mean.
Standard procedure
MTB > Retrieve "N:\resampling\Examples\Exponential.MTW".
Retrieving worksheet from file: N:\resampling\Examples\Exponential.MTW
# Worksheet was saved on 23/08/01 12:16:52
Results for: Exponential.MTW
MTB > OneT c1.
One-Sample T: C1
Variable
C1
N
20
Mean
1.044
StDev SE Mean
1.060 0.237
95.0% CI
(0.549, 1.540)
Resampling procedure
MTB > % N:\resampling\library\meanciboot c1 ;
SUBC> siglev 95 ;
SUBC> nboot 1000 ;
2
SUBC> means c3 ;
SUBC> quantiles c5-c7 ;
SUBC> tvalues c9.
Executing from file: N:\resampling\library\meanciboot.MAC
Data Display
STANDARD CONFIDENCE INTERVALS
Data Display (WRITE)
Number of data values 20
Mean of data values
1.0445
Standard deviation of data values
1.0597
Standard error of the mean 0.23695
Significance level for confidence intervals
95
Estimated confidence interval, lower bound (Standard t method)
Estimated confidence interval, upper bound (Standard t method)
0.54855
1.5404
BOOTSTRAP CONFIDENCE INTERVALS
Data Display (WRITE)
Number of bootstrap samples 1000
Overall mean for bootstrap samples
1.042
Standard deviation of bootstrap means
0.2407
Estimated bias-correction (for BC, BCa)
0.0652
Estimated acceleration (for BCa)
0.0612
Confidence limits
Data Display (WRITE)
Estimate -/+ 1.96*boot sd
Bootstrap-t method
0.5727
0.6405
1.516
1.932
Efron percentile method
Hall percentile method
0.6255
0.5355
1.553
1.463
BC percentile method
BCa percentile method
0.6400
0.6650
1.580
1.718
Modified worksheet
C3
A column containing 1000 sample means, one for each bootstrap resample
C5
Upper and lower rank positions for percentile confidence limits using the Efron method
C6
Upper and lower rank positions for percentile confidence limits using the Efron method
C7
Upper and lower rank positions for percentile confidence limits using the Efron method
C9
A column containing 1000 t-statistics for sample means, one for each bootstrap resample
Columns c5 - c7 each contain 2 values.
3
Discussion
There is a fair amount of variation between the different confidence intervals. All 7 intervals include the
true population mean of one. The Efron percentile method produces the shortest interval in this case, the
bootstrap t-interval the longest. The bootstrap-t and BCa intervals generally imply larger values for the
mean, whilst the Hall, standard t and estimate -/+ 1.96 * bootstrap standard deviation intervals generally
imply smaller values for the mean. These are not general properties of the different methods, however.
Manly (1997) performs a simulation study to investigate the coverage of the different bootstrap intervals
for a sample of 20 observations from an exponential distribution with rate parameter one. He finds that
the bootstrap-t method (with 95.2% coverage) has the closest coverage to the nominal 95% level, closely
followed by the BCa method (with 92.4% coverage).
4