NAG Library Routine Document
G07EBF calculates a rank based (nonparametric) estimate and confidence interval for the difference in location between two independent populations.
|SUBROUTINE G07EBF (
||METHOD, N, X, M, Y, CLEVEL, THETA, THETAL, THETAU, ESTCL, ULOWER, UUPPER, WRK, IWRK, IFAIL)
||N, M, IWRK(3*N), IFAIL
||X(N), Y(M), CLEVEL, THETA, THETAL, THETAU, ESTCL, ULOWER, UUPPER, WRK(3*(M+N))
Consider two random samples from two populations which have the same continuous distribution except for a shift in the location. Let the random sample, , have distribution and the random sample, , have distribution .
G07EBF finds a point estimate,
, of the difference in location
together with an associated confidence interval. The estimates are based on the ordered differences
. The estimate
is defined by
, denote the
(ascendingly) ordered differences
- if is odd, where ;
- if is even, where .
This estimator arises from inverting the two sample Mann–Whitney rank test statistic,
, for testing the hypothesis that
is the value of the Mann–Whitney
statistic for the two independent samples
is a monotonically increasing step function of
is the solution to the equation
; two methods are available for solving this equation. These methods avoid the computation of all the ordered differences
; this is because for large
both the storage requirements and the computation time would be high.
The first is an exact method based on a set partitioning procedure on the set of all differences
. This is adapted from the algorithm proposed by Monahan (1984)
for the computation of the Hodges–Lehmann estimator for a single population.
The second is an iterative algorithm, based on the Illinois method which is a modification of the regula falsi
method, see McKean and Ryan (1977)
. This algorithm has proved suitable for the function
which is asymptotically linear as a function of
The confidence interval limits are also based on the inversion of the Mann–Whitney test statistic.
Given a desired percentage for the confidence interval,
, expressed as a proportion between
initial estimates of the upper and lower confidence limits for the Mann–Whitney
statistic are found;
is the inverse cumulative Normal distribution function.
are rounded to the nearest integer values. These estimates are refined using an exact method, without taking ties into account, if
and a Normal approximation otherwise, to find
is a monotonically increasing step function. It is the number of times a score in the second sample,
, precedes a score in the first sample,
, where we only count a half if a score in the second sample actually equals a score in the first.
Let ; then . This is the largest value such that .
Let ; then . This is the smallest value such that .
As in the case of , these equations may be solved using either the exact or iterative methods to find the values and .
Then is the confidence interval for . The confidence interval is thus defined by those values of such that the null hypothesis, , is not rejected by the Mann–Whitney two sample rank test at the level.
Lehmann E L (1975) Nonparametrics: Statistical Methods Based on Ranks Holden–Day
McKean J W and Ryan T A (1977) Algorithm 516: An algorithm for obtaining confidence intervals and point estimates based on ranks in the two-sample location problem ACM Trans. Math. Software 10 183–185
Monahan J F (1984) Algorithm 616: Fast computation of the Hodges–Lehman location estimator ACM Trans. Math. Software 10 265–270
- 1: METHOD – CHARACTER(1)Input
: specifies the method to be used.
- The exact algorithm is used.
- The iterative algorithm is used.
- 2: N – INTEGERInput
On entry: , the size of the first sample.
- 3: X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the observations of the first sample,
, for .
- 4: M – INTEGERInput
On entry: , the size of the second sample.
- 5: Y(M) – REAL (KIND=nag_wp) arrayInput
On entry: the observations of the second sample,
, for .
- 6: CLEVEL – REAL (KIND=nag_wp)Input
On entry: the confidence interval required, ; e.g., for a confidence interval set .
- 7: THETA – REAL (KIND=nag_wp)Output
On exit: the estimate of the difference in the location of the two populations, .
- 8: THETAL – REAL (KIND=nag_wp)Output
On exit: the estimate of the lower limit of the confidence interval, .
- 9: THETAU – REAL (KIND=nag_wp)Output
On exit: the estimate of the upper limit of the confidence interval, .
- 10: ESTCL – REAL (KIND=nag_wp)Output
On exit: an estimate of the actual percentage confidence of the interval found, as a proportion between .
- 11: ULOWER – REAL (KIND=nag_wp)Output
On exit: the value of the Mann–Whitney statistic corresponding to the lower confidence limit, .
- 12: UUPPER – REAL (KIND=nag_wp)Output
On exit: the value of the Mann–Whitney statistic corresponding to the upper confidence limit, .
- 13: WRK() – REAL (KIND=nag_wp) arrayWorkspace
- 14: IWRK() – INTEGER arrayWorkspace
- 15: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
|On entry,|| or ,|
Each sample consists of identical values. All estimates are set to the common difference between the samples.
For at least one of the estimates , and , the underlying iterative algorithm (when ) failed to converge. This is an unlikely exit but the estimate should still be a reasonable approximation.
G07EBF should return results accurate to five significant figures in the width of the confidence interval, that is the error for any one of the three estimates should be less than .
The time taken increases with the sample sizes and .
The following program calculates a 95% confidence interval for the difference in location between the two populations from which the two samples of sizes and are drawn respectively.
9.1 Program Text
Program Text (g07ebfe.f90)
9.2 Program Data
Program Data (g07ebfe.d)
9.3 Program Results
Program Results (g07ebfe.r)