NAG Library Routine Document
G07EBF
1 Purpose
G07EBF calculates a rank based (nonparametric) estimate and confidence interval for the difference in location between two independent populations.
2 Specification
| SUBROUTINE G07EBF ( | METHOD, N, X, M, Y, CLEVEL, THETA, THETAL, THETAU, ESTCL, ULOWER, UUPPER, WRK, IWRK, IFAIL) |
| INTEGER | N, M, IWRK(3*N), IFAIL |
| double precision | X(N), Y(M), CLEVEL, THETA, THETAL, THETAU, ESTCL, ULOWER, UUPPER, WRK(3*(M+N)) |
| CHARACTER*1 | METHOD |
3 Description
Consider two random samples from two populations which have the same continuous distribution except for a shift in the location. Let the random sample, x=x1,x2,…,xnT, have distribution Fx and the random sample, y=y1,y2,…,ymT, have distribution Fx-θ.
G07EBF finds a point estimate,
θ^, of the difference in location
θ together with an associated confidence interval. The estimates are based on the ordered differences
yj-xi. The estimate
θ^ is defined by
Let
dk for
k=1,2,…,nm denote the
nm (ascendingly) ordered differences
yj-xi for
i=1,2,…,n and
j=1,2,…,m. Then
- if nm is odd, θ^=dk where k=nm-1/2;
- if nm is even, θ^=dk+dk+1/2 where k=nm/2.
This estimator arises from inverting the two sample Mann–Whitney rank test statistic,
Uθ0, for testing the hypothesis that
θ=θ0. Thus
Uθ0 is the value of the Mann–Whitney
U statistic for the two independent samples
xi
+
θ0
, for
i=1,2,…,n
and
yj
, for
j=1,2,…,m
. Effectively
Uθ0 is a monotonically increasing step function of
θ0 with
The estimate
θ^ is the solution to the equation
Uθ^=μ; two methods are available for solving this equation. These methods avoid the computation of all the ordered differences
dk; this is because for large
n and
m 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
yj-xi for
i=1,2,…,n and
j=1,2,…,m. 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
Uθ0 which is asymptotically linear as a function of
θ0.
The confidence interval limits are also based on the inversion of the Mann–Whitney test statistic.
Given a desired percentage for the confidence interval,
1-α, expressed as a proportion between
0.0 and
1.0 initial estimates of the upper and lower confidence limits for the Mann–Whitney
U statistic are found;
where
Φ-1 is the inverse cumulative Normal distribution function.
Ul and
Uu are rounded to the nearest integer values. These estimates are refined using an exact method, without taking ties into account, if
n+m≤40 and
maxn,m≤30 and a Normal approximation otherwise, to find
Ul and
Uu satisfying
and
The function
Uθ0 is a monotonically increasing step function. It is the number of times a score in the second sample,
yj, precedes a score in the first sample,
xi+θ, where we only count a half if a score in the second sample actually equals a score in the first.
Let Ul=k; then θl=dk+1. This is the largest value θl such that Uθl=Ul.
Let Uu=nm-k; then θu=dnm-k. This is the smallest value θu such that Uθu=Uu.
As in the case of θ^, these equations may be solved using either the exact or iterative methods to find the values θl and θu.
Then θl,θu is the confidence interval for θ. The confidence interval is thus defined by those values of θ0 such that the null hypothesis, θ=θ0, is not rejected by the Mann–Whitney two sample rank test at the 100×α% level.
4 References
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
5 Parameters
- 1: METHOD – CHARACTER*1Input
On entry: specifies the method to be used.
- If METHOD='E', the exact algorithm is used.
- If METHOD='A', the iterative algorithm is used.
Constraint:
METHOD='E' or 'A'.
- 2: N – INTEGERInput
On entry:
n, the size of the first sample.
Constraint:
N≥1.
- 3: X(N) – double precision arrayInput
On entry: the observations of the first sample, xi for i=1,2,…,n.
- 4: M – INTEGERInput
On entry:
m, the size of the second sample.
Constraint:
M≥1.
- 5: Y(M) – double precision arrayInput
On entry: the observations of the second sample, yj for j=1,2,…,m.
- 6: CLEVEL – double precisionInput
On entry: the confidence interval required, 1-α; e.g., for a 95% confidence interval set CLEVEL=0.95.
Constraint:
0.0<CLEVEL<1.0.
- 7: THETA – double precisionOutput
On exit: the estimate of the difference in the location of the two populations, θ^.
- 8: THETAL – double precisionOutput
-
On exit: the estimate of the lower limit of the confidence interval, θl.
- 9: THETAU – double precisionOutput
-
On exit: the estimate of the upper limit of the confidence interval, θu.
- 10: ESTCL – double precisionOutput
-
On exit: an estimate of the actual percentage confidence of the interval found, as a proportion between 0.0,1.0.
- 11: ULOWER – double precisionOutput
On exit: the value of the Mann–Whitney U statistic corresponding to the lower confidence limit, Ul.
- 12: UUPPER – double precisionOutput
On exit: the value of the Mann–Whitney U statistic corresponding to the upper confidence limit, Uu.
- 13: WRK(3×M+N) – double precision arrayWorkspace
- 14: IWRK(3×N) – INTEGER arrayWorkspace
- 15: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
6 Error Indicators and Warnings
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
- IFAIL=1
-
| On entry, | METHOD≠'E' or 'A', |
| or | N<1, |
| or | M<1, |
| or | CLEVEL≤0.0, |
| or | CLEVEL≥1.0. |
- IFAIL=2
-
Each sample consists of identical values. All estimates are set to the common difference between the samples.
- IFAIL=3
-
For at least one of the estimates θ^, θl and θu, the underlying iterative algorithm (when METHOD='A') failed to converge. This is an unlikely exit but the estimate should still be a reasonable approximation.
7 Accuracy
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 0.00001×THETAU-THETAL.
8 Further Comments
The time taken increases with the sample sizes n and m.
9 Example
The following program calculates a 95% confidence interval for the difference in location between the two populations from which the two samples of sizes 50 and 100 are drawn respectively.
9.1 Program Text
Program Text (g07ebfe.f)
9.2 Program Data
Program Data (g07ebfe.d)
9.3 Program Results
Program Results (g07ebfe.r)