
BOOTSTRAP PLOTName:
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
<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 groupid's case.
<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 groupid variable.
<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 groupid variables.
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
to specify whether you have paired or unpaired response variables. The default is INDEPENDENT (i.e., unpaired).
QUADRATIC CALIBRATION
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.
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 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 "biascorrection". The BCa confidence interval is given by:
where
If \( \hat{\mbox{a}} \) and \( \hat{\mbox{z}}_{0} \) are zero, the BCa confidence interval reduces to the percentile bootstrap. Nonzero 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
where
To compute \( \hat{\mbox{a}} \), let \( \hat{\theta}_{i} \) be the estimate of the statistic of interest with the ith 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
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
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):
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% twosided case. If you want a different significance level or a onesided 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
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.
Efron and Tibshirabi (1993), "An Introduction to the Bootstrap", SpringerVerlang.
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 MULTIPLOTProgram 2: 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   
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