BOOTSTRAP PLOT

BOOTSTRAP PLOT

Graphics Command

Generates a bootstrap plot for a given statistic.

The bootstrap is a non-parametric method for calculating a sampling distribution for a statistic. The bootstrap calculates the statistic with N different subsamples. The subsampling is done with replacement. For details on how to calculate bootstrap estimates for unsupported statistics, enter HELP BOOTSTRAP SAMPLE.

For the bootstrap plot, the vertical axis contains the computed value of the statistic and the horizontal axis contains the sample number (for k = 1, 2, ..., N). The number of response variables depends on the number of variables required to compute the statistic (e.g., the MEAN uses one while the LINEAR INTERCEPT uses two). The bootstrap plot is typically followed by some type of distributional plot such as a histogram.

Dataplot also supports BOOTSTRAP computations for the case when there is one group variable. In this case, the horizontal axis is group id and the vertical axis contains the computed values of the statistic for that group. The number of bootstrap samples is applied to each group. For example,if the requested number of bootstrap samples is 100, then each group will have 100 bootstrap samples applied.

For a list of supported statistics in Dataplot, enter

HELP STATISTICS

BOOTSTRAP <stat> PLOT <y1> ... <yk>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> ... <yk> is a list of 1 to 3 response variables (depending on <stat>);
            <stat> is one of Dataplot's supported statistics;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used for the no group-id's case.

BOOTSTRAP <stat> PLOT <y1> ... <yk> <x>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> ... <yk> is a list of 1 to 3 response variables (depending on <stat>);
            <x> is a group id variable;
            <stat> is one of Dataplot's supported statistics;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used for the case when there is one group-id variable.

BOOTSTRAP <stat> PLOT <y1> ... <yk> <x1> <x2>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> ... <yk> is a list of 1 to 3 response variables (depending on <stat>);
            <x1> is the first group id variable;
            <x2> is the second group id variable;
            <stat> is one of Dataplot's supported statistics;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used for the case when there are two group-id variables.

BOOTSTRAP MEAN PLOT Y
BOOTSTRAP LINEAR SLOPE PLOT Y1 X1
BOOTSTRAP MEAN PLOT Y TAG
BOOTSTRAP MEAN PLOT Y TAG SUBSET TAG > 2
BOOTSTRAP DIFFERENCE OF MEAN PLOT Y1 Y2 TAG

The 2010/3 version of Dataplot updated the BOOTSTRAP PLOT command to include reports in addition to the plots.
1. If the BOOTSTRAP PLOT is applied to a statistic (e.g., BOOTSTRAP MEAN PLOT Y), the following tables are generated:
   • An initial summary table.

   • A table containing percent points for the computed statistic.

   • A table containing percentile confidence limits for the statistic for various values of α.

2. If the BOOTSTRAP PLOT is applied to a a distributional fit (e.g., BOOTSTRAP WEIBULL PPCC PLOT Y or BOOTSTRAP WEIBULL MLE PLOT Y), the following tables are generated:
   • An initial summary table.

   • A table containing percentile confidence limits for each of the parameters of the distribution for various levels of α.

   • If the SET MAXIMUM LIKELIHOOD PERCENTILES command was given, a table containing confidence limits for the specified percentiles will be generated.

   Enter HELP DISTRIBUTIONAL BOOTSTRAP for details on this version of the BOOTSTRAP PLOT command.

The SET WRITE DECIMALS command can be used to specify the number of decimals to use in the tables and the CAPTURE HTML, CAPTURE LATEX, and CAPTURE RTF options are supported.

When there are two response variables, the bootstrap is applied differently depending on whether the two response variables are paired (dependent) or unpaired (independent). In the paired case, each bootstrap sample uses the same rows for the two response variables. In the unpaired case, different bootstrap samples are applied to each response variable. If the response variables have different sample sizes, the unpaired case is assumed. If the sample sizes are equal, you can use the following command
SET BOOTSTRAP GROUPS <INDEPENDENT/DEPENDENT>

to specify whether you have paired or unpaired response variables. The default is INDEPENDENT (i.e., unpaired).

In addition to the supported statistics, the following additional statistics are supported
LINEAR CALIBRATION
QUADRATIC CALIBRATION

The BOOTSTRAP PLOT command saves the following internal parameters.

BMEAN	-	the mean of the plotted bootstrap values
BSD	-	standard deviation of the plotted bootstrap values
B001	-	the 0.1% percentile of the plotted bootstrap values
B005	-	the 0.5% percentile of the plotted bootstrap values
B01	-	the 1% percentile of the plotted bootstrap values
B025	-	the 2.5% percentile of the plotted bootstrap values
B05	-	the 5% percentile of the plotted bootstrap values
B10	-	the 10% percentile of the plotted bootstrap values
B20	-	the 20% percentile of the plotted bootstrap values
B80	-	the 80% percentile of the plotted bootstrap values
B90	-	the 90% percentile of the plotted bootstrap values
B95	-	the 95% percentile of the plotted bootstrap values
B975	-	the 97.5% percentile of the plotted bootstrap values
B99	-	the 99% percentile of the plotted bootstrap values
B999	-	the 99.9% percentile of the plotted bootstrap values

The values are typically used in setting confidence intervals.

Note that for the replication case (i.e., you have a group id variable), the above quantities are computed using all the groups. In most cases, you will want these quantities for each group. See the Note section below regarding information written to DPST1F.DAT, DPST2F.DAT, and DPST3F.DAT.

A number of these statistics require one or more parameters to be defined before entering the BOOTSTRAP PLOT command. Specifically,
1. The trimmed and Winsorized statistics, the percentage of data in each tail to be trimmed/Winsorized needs to be specified. For example, to trim/Winsorize 10% of the data in each tail, enter the commands:
   LET P1 = 10%
   LET P2 = 10%
   

2. For the QUANTILE and QUANTILE STANDARD ERROR, the quantile being estimated needs to be specified by entering the command
   LET XQ = <value>

   where <value> is a number in the range 0 to 1.

   Likewise, for the PERCENTILE, the desired percentile needs to be specified by entering the command

   LET P100 = <value>

   where <value> is a number in the range 0 to 100.

3. For the PERCENTAGE BEND MIDVARIANCE and PERCENTAGE BEND CORRELATION, the value of beta needs to be specified by entering the command:
   LET BETA = <value>

   where <value> is a number between 0 and 0.5. A default value of 0.1 is used if beta is not defined.

4. For the LINEAR CALIBRATION and QUADRATIC CALIBRATION statistics, the value of Y for which the calibration is to be computed is specified by entering the command
   LET Y0 = <value>

The 1/2004 version of Dataplot now writes information to the files dpst1f.dat and dpst2f.dat (in the current directory).
1. The file dpst1f.dat will contain the group-id (will be a column of 1's if no group variable was specified) and the estimate of the statistic of interest for each bootstrap sample.

   This can sometimes be useful if you want to some additional processing of the bootstrap estimates (e.g., generate percentiles not automatically computed by Dataplot).

2. The file dpst2f.dat is only written if at least one group-id variable is specified. In that case, the dpst2f.dat file will contain:
   • first group-id
   • the second group-id (if applicable)
   • mean of the bootstrap estimates for the given group
   • median of the bootstrap estimates for the given group
   • the 2.5 percentile of the bootstrap estimates for the given group
   • the 97.5 percentile of the bootstrap estimates for the given group
   • the 5.0 percentile of the bootstrap estimates for the given group
   • the 95.0 percentile of the bootstrap estimates for the given group
   • the 0.5 percentile of the bootstrap estimates for the given group
   • the 99.5 percentile of the bootstrap estimates for the given group

These files are useful if you want to perform further processing on the bootstrap samples. For example, you can generate histograms of the bootstrap samples.

The 1/2004 version of Dataplot added support for the BCA BOOTSTRAP PLOT. The BCA BOOTSTRAP plot is used to generate more accurate confidence intervals.

The regular BOOTSTRAP PLOT command generates confidence intervals based on the percentiles of the bootstrap statistics. This method of computing confidence intervals, referred to as the percentile bootstrap, is first order accurate for confidence intervals.

Section 14.3 of Efron and Tibshirani discuss the BCa method of generating bootstrap confidence intervals that is second order accurate.

BCa is an abbreviation for "acceleration" and "bias-correction". The BCa confidence interval is given by:

\( \mbox{BCa}: (\hat{\theta} \alpha_1, \hat{\theta} \alpha_2) \)

where

\( \hat{\theta} \alpha \)	=	100 x α-th percentile of the bootstrap replications
\( \alpha_{1} \)	=	\( \Phi \left( \hat{z}_0 + \frac{z_{\alpha}} {1 - \hat{\mbox{a}} (\hat{z}_0 + z_{\alpha})} \right) \)
\( \alpha_{2} \)	=	\( \Phi \left( \hat{z}_0 + \frac{z_{1 - \alpha}} {1 - \hat{\mbox{a}} (\hat{z}_0 + z_{1 - \alpha})} \right) \)
\( \Phi \)	=	cumulative distribution function of the standard normal distribution

If \( \hat{\mbox{a}} \) and \( \hat{\mbox{z}}_{0} \) are zero, the BCa confidence interval reduces to the percentile bootstrap. Non-zero values change the percentiles used for the BCa interval.

The BCa confidence interval above depends on \( \hat{\mbox{a}} \) and \( \hat{\mbox{z}}_{0} \). The \( \hat{\mbox{z}}_{0} \) is the bias correction and \( \hat{\mbox{a}} \) is the acceleration.

The value of \( \hat{\mbox{z}}_{0} \). is computed by

\( \hat{\mbox{z}}_{0} = \Phi^{-1}(\mbox{number of } (\hat{\theta}(\mbox{b}) < \hat{\theta}/B)) \)

where

\( \Phi^{-1} \)	=	the normal percent point function
\( \hat{\theta} \)	=	the full sample estimate of the statistic of interest
\( \hat{\theta} (b) \)	=	the estimate of the statistic of interest for the b-th bootstrap replication
B	=	the number of bootstrap replications taken

To compute \( \hat{\mbox{a}} \), let \( \hat{\theta}_{i} \) be the estimate of the statistic of interest with the i-th point deleted (i.e., the jackknife values). Let \( \hat{\theta}_{.} \) equal the mean of the \( \hat{\theta}_{i} \). Then \( \hat{\mbox{a}} \) can be computed by

\( \hat{\mbox{a}} = \frac{\sum_{i=1}^{n} {(\hat{\theta}_{.} - \hat{\theta}_{i})^{3}}} {6 \sum_{i=1}^{n} {(\hat{\theta}_{.} - \hat{\theta}_{i}^2)^{3/2}}} \)

The details of the BCa method are explained in more detail in Efron and Tibshirani.

To have Dataplot generate BCa confidence intervals, enter a command like

BCA BOOTSTRAP MEAN PLOT Y

For the BCA BOOTSTRAP, Dataplot will generate the same plot as if the BCA option was not given. It will write the following values to the file dpst3f.dat (in the current directory):

1. lower confidence interval
2. upper confidence interval
3. zhat(0)
4. ahat
5. alpha1
6. alpha2
7. group-id 1 (if applicable)
8. group-id 2 (if applicable)

These values are written to a single row. If you have the group case, one row is written for each group. The confidence interval printed is for the 95% two-sided case. If you want a different significance level or a one-sided interval, you can read these values into Dataplot to compute the interval (see the relevant definitions above). To read these values back into Dataplot, you can enter the commands

BCA BOOTSTRAP MEAN PLOT Y
SKIP 6
SET READ FORMAT 4E15.7,2F8.4,2F10.0
READ PARAMETER DPST1F.DAT LCL UCL ZOHAT A0HAT ALPHA1 ALPHA2


Note that you can use the BCA option with all of the syntax options given above.

None

None

JACKNIFE PLOT	= Generate a jacknife plot.
BOOTSTRAP SAMPLE	= Set the sample size for the bootstrap
BOOTSTRAP FIT	= Compute a bootstrap linear/multilinear fit.
HISTOGRAM	= Generates a histogram.
PLOT	= Generates a data/function plot.

Efron and Gong, "A Leisurely Look at the Bootstrap, the Jacknife, and Cross-Validation," The American Statistician, February, 1983.

Efron and Tibshirabi (1993), "An Introduction to the Bootstrap", Springer-Verlang.

Sample Distribution of a Statistic

1989/2: Original implementation
1998/5: added the saving of the parameters (BMEAN, BSD, etc.)
2001/3: added GEOMETRIC MEAN, GEOMETRIC STANDARD DEVIATION, HARMONIC MEAN
2001/9: added IQ RANGE
2001/11: added BIWEIGHT LOCATION, BIWEIGHT SCALE
2002/7: activated CORRELATION, COVARIANCE, RANK CORRELATION, RANK COVARIANCE
2002/7: added LINEAR CALIBRTION, QUADRATIC CALIBRATION, WINSORIZED VARIANCE, WINSORIZED CORRELATION, WINSORIZED COVARIANCE, BIWEIGHT MIDVARIANCE, BIWEIGHT MIDCOVARIANCE, PERCENTAGE BEND MIDVARIANCE, PERCENTAGE BEND CORRELATION, HODGE LEHMAN, TRIMMED MEAN STANDARD ERROR, QUANTILE, QUANTILE STANDARD ERROR
2003/3: added support for the "DIFFERENCE OF" statistics
2003/3: added documentation for the replication (i.e., groups) case
2003/5: Added support for SN SCALE, QN SCALE, DIFFERENCE OF SN, DIFFERENCE OF QN
2004/1: Added support for two group variables
2004/1: Added support for BCA BOOTSTRAP commands
2010/3: Added support for tabular report in addition to the plot

 
LET Y = UNIFORM RANDOM NUMBERS FOR I = 1 1 1000
BOOTSTRAP SAMPLE SIZE 500
MULTIPLOT CORNER COORDINATES 0 0 100 100
MULTIPLOT SCALE FACTOR 2
MULTIPLOT 2 3
TITLE AUTOMATIC
BOOTSTRAP MEAN PLOT Y
LET YMEAN = YPLOT
BOOTSTRAP MEDIAN PLOT Y
LET YMEDIAN = YPLOT
BOOTSTRAP MIDRANGE PLOT Y
LET YMIDR = YPLOT
XLIMITS 0.45 0.55
HISTOGRAM YMEAN
HISTOGRAM YMEDIAN
HISTOGRAM YMIDR
END OF MULTIPLOT
    

 
SKIP 25
READ GEAR.DAT Y X
XLIMITS 1 10
MAJOR XTIC MARK NUMBER 10
MINOR XTIC MARK NUMBER 0
XTIC OFFSET 0.5 0.5
TIC OFFSET UNITS DATA
X1LABEL BATCH
Y1LABEL BOOTSTRAP ESTIMATES OF THE MEAN
CHARACTER CIRCLE ALL
CHARACTER HW 0.5 0.375 ALL
CHARACTER FILL ON ALL
LINE BLANK ALL
SET WRITE DECIMALS 4
BOOTSRAP MEAN PLOT Y X
    



            Bootstrap Analysis for the MEAN
 
Response Variable One: Y
Group ID Variable One (X       ):                1.0000
 
Number of Bootstrap Samples:                        100
Number of Observations:                              10
Mean of Bootstrap Samples:                       0.9979
Standard Deviation of Bootstrap Samples:         0.0011
Median of Bootstrap Samples:                     0.9978
MAD of Bootstrap Samples:                        0.0008
Minimum of Bootstrap Samples:                    0.9954
Maximum of Bootstrap Samples:                    1.0008
 
 
 
Percent Points of the Bootstrap Samples
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.1    =         0.9954
            0.5    =         0.9954
            1.0    =         0.9954
            2.5    =         0.9955
            5.0    =         0.9958
           10.0    =         0.9961
           20.0    =         0.9970
           50.0    =         0.9978
           80.0    =         0.9991
           90.0    =         0.9993
           95.0    =         0.9996
           97.5    =         1.0001
           99.0    =         1.0008
           99.5    =         1.0008
           99.9    =         1.0008
 
 
            Percentile Confidence Interval for Statistic
 
------------------------------------------
  Confidence          Lower          Upper
 Coefficient          Limit          Limit
------------------------------------------
       50.00         0.9972         0.9988
       75.00         0.9962         0.9993
       90.00         0.9958         0.9996
       95.00         0.9955         1.0001
       99.00         0.9954         1.0008
       99.90         0.9954         1.0008
------------------------------------------
 
 
Response Variable One: Y
Group ID Variable One (X       ):                2.0000
 
Number of Bootstrap Samples:                        100
Number of Observations:                              10
Mean of Bootstrap Samples:                       0.9993
Standard Deviation of Bootstrap Samples:         0.0016
Median of Bootstrap Samples:                     0.9994
MAD of Bootstrap Samples:                        0.0011
Minimum of Bootstrap Samples:                    0.9957
Maximum of Bootstrap Samples:                    1.0031
 
 
 
Percent Points of the Bootstrap Samples
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.1    =         0.9957
            0.5    =         0.9957
            1.0    =         0.9958
            2.5    =         0.9963
            5.0    =         0.9966
           10.0    =         0.9972
           20.0    =         0.9978
           50.0    =         0.9994
           80.0    =         1.0005
           90.0    =         1.0016
           95.0    =         1.0021
           97.5    =         1.0024
           99.0    =         1.0030
           99.5    =         1.0031
           99.9    =         1.0031
 
 
            Percentile Confidence Interval for Statistic
 
------------------------------------------
  Confidence          Lower          Upper
 Coefficient          Limit          Limit
------------------------------------------
       50.00         0.9982         1.0003
       75.00         0.9974         1.0012
       90.00         0.9966         1.0021
       95.00         0.9963         1.0024
       99.00         0.9957         1.0031
       99.90         0.9957         1.0031
------------------------------------------
 
 
Response Variable One: Y
Group ID Variable One (X       ):                3.0000
 
Number of Bootstrap Samples:                        100
Number of Observations:                              10
Mean of Bootstrap Samples:                       0.9955
Standard Deviation of Bootstrap Samples:         0.0010
Median of Bootstrap Samples:                     0.9955
MAD of Bootstrap Samples:                        0.0007
Minimum of Bootstrap Samples:                    0.9925
Maximum of Bootstrap Samples:                    0.9978
 
 
 
Percent Points of the Bootstrap Samples
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.1    =         0.9925
            0.5    =         0.9925
            1.0    =         0.9925
            2.5    =         0.9927
            5.0    =         0.9937
           10.0    =         0.9942
           20.0    =         0.9946
           50.0    =         0.9955
           80.0    =         0.9964
           90.0    =         0.9968
           95.0    =         0.9973
           97.5    =         0.9975
           99.0    =         0.9978
           99.5    =         0.9978
           99.9    =         0.9978
 
 
            Percentile Confidence Interval for Statistic
 
------------------------------------------
  Confidence          Lower          Upper
 Coefficient          Limit          Limit
------------------------------------------
       50.00         0.9948         0.9962
       75.00         0.9942         0.9967
       90.00         0.9937         0.9973
       95.00         0.9927         0.9975
       99.00         0.9925         0.9978
       99.90         0.9925         0.9978
------------------------------------------
 
 
Response Variable One: Y
Group ID Variable One (X       ):                4.0000
 
Number of Bootstrap Samples:                        100
Number of Observations:                              10
Mean of Bootstrap Samples:                       0.9981
Standard Deviation of Bootstrap Samples:         0.0011
Median of Bootstrap Samples:                     0.9980
MAD of Bootstrap Samples:                        0.0007
Minimum of Bootstrap Samples:                    0.9949
Maximum of Bootstrap Samples:                    1.0016
 
 
 
Percent Points of the Bootstrap Samples
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.1    =         0.9949
            0.5    =         0.9949
            1.0    =         0.9949
            2.5    =         0.9957
            5.0    =         0.9962
           10.0    =         0.9967
           20.0    =         0.9972
           50.0    =         0.9980
           80.0    =         0.9991
           90.0    =         0.9996
           95.0    =         1.0001
           97.5    =         1.0005
           99.0    =         1.0015
           99.5    =         1.0016
           99.9    =         1.0016
 
 
            Percentile Confidence Interval for Statistic
 
------------------------------------------
  Confidence          Lower          Upper
 Coefficient          Limit          Limit
------------------------------------------
       50.00         0.9973         0.9990
       75.00         0.9969         0.9995
       90.00         0.9962         1.0001
       95.00         0.9957         1.0005
       99.00         0.9949         1.0016
       99.90         0.9949         1.0016
------------------------------------------
 
 
Response Variable One: Y
Group ID Variable One (X       ):                5.0000
 
Number of Bootstrap Samples:                        100
Number of Observations:                              10
Mean of Bootstrap Samples:                       0.9919
Standard Deviation of Bootstrap Samples:         0.0022
Median of Bootstrap Samples:                     0.9921
MAD of Bootstrap Samples:                        0.0015
Minimum of Bootstrap Samples:                    0.9860
Maximum of Bootstrap Samples:                    0.9976
 
 
 
Percent Points of the Bootstrap Samples
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.1    =         0.9860
            0.5    =         0.9860
            1.0    =         0.9860
            2.5    =         0.9866
            5.0    =         0.9878
           10.0    =         0.9887
           20.0    =         0.9902
           50.0    =         0.9921
           80.0    =         0.9937
           90.0    =         0.9949
           95.0    =         0.9954
           97.5    =         0.9959
           99.0    =         0.9975
           99.5    =         0.9976
           99.9    =         0.9976
 
 
            Percentile Confidence Interval for Statistic
 
------------------------------------------
  Confidence          Lower          Upper
 Coefficient          Limit          Limit
------------------------------------------
       50.00         0.9904         0.9932
       75.00         0.9892         0.9947
       90.00         0.9878         0.9954
       95.00         0.9866         0.9959
       99.00         0.9860         0.9976
       99.90         0.9860         0.9976
------------------------------------------
 
 
Response Variable One: Y
Group ID Variable One (X       ):                6.0000
 
Number of Bootstrap Samples:                        100
Number of Observations:                              10
Mean of Bootstrap Samples:                       0.9989
Standard Deviation of Bootstrap Samples:         0.0028
Median of Bootstrap Samples:                     0.9989
MAD of Bootstrap Samples:                        0.0019
Minimum of Bootstrap Samples:                    0.9911
Maximum of Bootstrap Samples:                    1.0059
 
 
 
Percent Points of the Bootstrap Samples
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.1    =         0.9911
            0.5    =         0.9911
            1.0    =         0.9912
            2.5    =         0.9924
            5.0    =         0.9945
           10.0    =         0.9955
           20.0    =         0.9965
           50.0    =         0.9989
           80.0    =         1.0013
           90.0    =         1.0027
           95.0    =         1.0040
           97.5    =         1.0047
           99.0    =         1.0058
           99.5    =         1.0059
           99.9    =         1.0059
 
 
            Percentile Confidence Interval for Statistic
 
------------------------------------------
  Confidence          Lower          Upper
 Coefficient          Limit          Limit
------------------------------------------
       50.00         0.9971         1.0009
       75.00         0.9957         1.0021
       90.00         0.9945         1.0040
       95.00         0.9924         1.0047
       99.00         0.9911         1.0059
       99.90         0.9911         1.0059
------------------------------------------
 
 
Response Variable One: Y
Group ID Variable One (X       ):                7.0000
 
Number of Bootstrap Samples:                        100
Number of Observations:                              10
Mean of Bootstrap Samples:                       1.0016
Standard Deviation of Bootstrap Samples:         0.0025
Median of Bootstrap Samples:                     1.0016
MAD of Bootstrap Samples:                        0.0018
Minimum of Bootstrap Samples:                    0.9946
Maximum of Bootstrap Samples:                    1.0072
 
 
 
Percent Points of the Bootstrap Samples
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.1    =         0.9946
            0.5    =         0.9946
            1.0    =         0.9946
            2.5    =         0.9966
            5.0    =         0.9970
           10.0    =         0.9984
           20.0    =         0.9993
           50.0    =         1.0016
           80.0    =         1.0040
           90.0    =         1.0048
           95.0    =         1.0056
           97.5    =         1.0060
           99.0    =         1.0072
           99.5    =         1.0072
           99.9    =         1.0072
 
 
            Percentile Confidence Interval for Statistic
 
------------------------------------------
  Confidence          Lower          Upper
 Coefficient          Limit          Limit
------------------------------------------
       50.00         1.0000         1.0036
       75.00         0.9986         1.0046
       90.00         0.9970         1.0056
       95.00         0.9966         1.0060
       99.00         0.9946         1.0072
       99.90         0.9946         1.0072
------------------------------------------
 
 
Response Variable One: Y
Group ID Variable One (X       ):                8.0000
 
Number of Bootstrap Samples:                        100
Number of Observations:                              10
Mean of Bootstrap Samples:                       1.0003
Standard Deviation of Bootstrap Samples:         0.0010
Median of Bootstrap Samples:                     1.0001
MAD of Bootstrap Samples:                        0.0005
Minimum of Bootstrap Samples:                    0.9974
Maximum of Bootstrap Samples:                    1.0028
 
 
 
Percent Points of the Bootstrap Samples
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.1    =         0.9974
            0.5    =         0.9974
            1.0    =         0.9974
            2.5    =         0.9979
            5.0    =         0.9982
           10.0    =         0.9990
           20.0    =         0.9996
           50.0    =         1.0001
           80.0    =         1.0010
           90.0    =         1.0018
           95.0    =         1.0021
           97.5    =         1.0026
           99.0    =         1.0027
           99.5    =         1.0028
           99.9    =         1.0028
 
 
            Percentile Confidence Interval for Statistic
 
------------------------------------------
  Confidence          Lower          Upper
 Coefficient          Limit          Limit
------------------------------------------
       50.00         0.9998         1.0009
       75.00         0.9991         1.0016
       90.00         0.9982         1.0021
       95.00         0.9979         1.0026
       99.00         0.9974         1.0028
       99.90         0.9974         1.0028
------------------------------------------
 
 
Response Variable One: Y
Group ID Variable One (X       ):                9.0000
 
Number of Bootstrap Samples:                        100
Number of Observations:                              10
Mean of Bootstrap Samples:                       0.9982
Standard Deviation of Bootstrap Samples:         0.0012
Median of Bootstrap Samples:                     0.9982
MAD of Bootstrap Samples:                        0.0009
Minimum of Bootstrap Samples:                    0.9953
Maximum of Bootstrap Samples:                    1.0011
 
 
 
Percent Points of the Bootstrap Samples
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.1    =         0.9953
            0.5    =         0.9953
            1.0    =         0.9953
            2.5    =         0.9957
            5.0    =         0.9963
           10.0    =         0.9966
           20.0    =         0.9971
           50.0    =         0.9982
           80.0    =         0.9995
           90.0    =         0.9998
           95.0    =         1.0000
           97.5    =         1.0005
           99.0    =         1.0010
           99.5    =         1.0011
           99.9    =         1.0011
 
 
            Percentile Confidence Interval for Statistic
 
------------------------------------------
  Confidence          Lower          Upper
 Coefficient          Limit          Limit
------------------------------------------
       50.00         0.9972         0.9992
       75.00         0.9967         0.9998
       90.00         0.9963         1.0000
       95.00         0.9957         1.0005
       99.00         0.9953         1.0011
       99.90         0.9953         1.0011
------------------------------------------
 
 
Response Variable One: Y
Group ID Variable One (X       ):               10.0000
 
Number of Bootstrap Samples:                        100
Number of Observations:                              10
Mean of Bootstrap Samples:                       0.9946
Standard Deviation of Bootstrap Samples:         0.0015
Median of Bootstrap Samples:                     0.9947
MAD of Bootstrap Samples:                        0.0010
Minimum of Bootstrap Samples:                    0.9911
Maximum of Bootstrap Samples:                    0.9982
 
 
 
Percent Points of the Bootstrap Samples
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.1    =         0.9911
            0.5    =         0.9911
            1.0    =         0.9911
            2.5    =         0.9912
            5.0    =         0.9920
           10.0    =         0.9925
           20.0    =         0.9933
           50.0    =         0.9947
           80.0    =         0.9959
           90.0    =         0.9967
           95.0    =         0.9975
           97.5    =         0.9978
           99.0    =         0.9982
           99.5    =         0.9982
           99.9    =         0.9982
 
 
            Percentile Confidence Interval for Statistic
 
------------------------------------------
  Confidence          Lower          Upper
 Coefficient          Limit          Limit
------------------------------------------
       50.00         0.9936         0.9957
       75.00         0.9927         0.9965
       90.00         0.9920         0.9975
       95.00         0.9912         0.9978
       99.00         0.9911         0.9982
       99.90         0.9911         0.9982
------------------------------------------
 
 
Response Variable One: Y
Group ID Variable One (All Data):
 
Number of Bootstrap Samples:                        100
Number of Observations:                             100
Mean of Bootstrap Samples:                       0.9976
Standard Deviation of Bootstrap Samples:         0.0032
Median of Bootstrap Samples:                     0.9978
MAD of Bootstrap Samples:                        0.0020
Minimum of Bootstrap Samples:                    0.9860
Maximum of Bootstrap Samples:                    1.0072
 
 
 
Percent Points of the Bootstrap Samples
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.1    =         0.9860
            0.5    =         0.9878
            1.0    =         0.9887
            2.5    =         0.9904
            5.0    =         0.9918
           10.0    =         0.9932
           20.0    =         0.9952
           50.0    =         0.9978
           80.0    =         1.0001
           90.0    =         1.0013
           95.0    =         1.0026
           97.5    =         1.0040
           99.0    =         1.0049
           99.5    =         1.0058
           99.9    =         1.0072
 
 
            Percentile Confidence Interval for Statistic
 
------------------------------------------
  Confidence          Lower          Upper
 Coefficient          Limit          Limit
------------------------------------------
       50.00         0.9957         0.9998
       75.00         0.9938         1.0009
       90.00         0.9918         1.0026
       95.00         0.9904         1.0040
       99.00         0.9878         1.0058
       99.90         0.9860         1.0072
------------------------------------------