NAG Library Routine Document

G07EBF

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 = {(x_{1}, x_{2}, \dots, x_{n})}^{T}

, have distribution

F (x)

and the random sample,

y = {(y_{1}, y_{2}, \dots, y_{m})}^{T}

, have distribution

F (x - θ)

G07EBF finds a point estimate,

\hat{θ}

, of the difference in location

θ

together with an associated confidence interval. The estimates are based on the ordered differences

y_{j} - x_{i}

. The estimate

\hat{θ}

is defined by

\hat{θ} = median \{y_{j} - x_{i}, i = 1, 2, \dots, n; j = 1, 2, \dots, m\} .

Let

d_{k}

, for

k = 1, 2, \dots, n m

, denote the

n m

(ascendingly) ordered differences

y_{j} - x_{i}

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

. Then

if $n m$ is odd, $\hat{θ} = d_{k}$ where $k = (n m - 1) / 2$ ;
if $n m$ is even, $\hat{θ} = (d_{k} + d_{k + 1}) / 2$ where $k = n m / 2$ .

This estimator arises from inverting the two sample Mann–Whitney rank test statistic,

U (θ_{0})

, for testing the hypothesis that

θ = θ_{0}

. Thus

U (θ_{0})

is the value of the Mann–Whitney

U

statistic for the two independent samples

\{(x_{i} + θ_{0}), for ​ i = 1, 2, \dots, n\}

and

\{y_{j}, for ​ j = 1, 2, \dots, m\}

. Effectively

U (θ_{0})

is a monotonically increasing step function of

θ_{0}

with

\begin{matrix} mean ​ (U) = μ = \frac{n m}{2}, \\ var (U) = σ^{2} \frac{n m (n + m + 1)}{12} . \end{matrix}

The estimate

\hat{θ}

is the solution to the equation

U (\hat{θ}) = μ

; two methods are available for solving this equation. These methods avoid the computation of all the ordered differences

d_{k}

; 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

y_{j} - x_{i}

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, 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

U (θ_{0})

which is asymptotically linear as a function of

θ_{0}

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;

\begin{matrix} U_{l} = μ - 0.5 + (σ \times Φ^{- 1} (α / 2)) \\ U_{u} = μ + 0.5 + (σ \times Φ^{- 1} ((1 - α) / 2)) \end{matrix}

where

Φ^{- 1}

is the inverse cumulative Normal distribution function.

U_{l}

and

U_{u}

are rounded to the nearest integer values. These estimates are refined using an exact method, without taking ties into account, if

n + m \leq 40

and

\max (n, m) \leq 30

and a Normal approximation otherwise, to find

U_{l}

and

U_{u}

satisfying

\begin{array}{l} P (U \leq U_{l}) \leq α / 2 \\ P (U \leq U_{l} + 1) > α / 2 \end{array}

and

\begin{array}{l} P (U \geq U_{u}) \leq α / 2 \\ P (U \geq U_{u} - 1) > α / 2 . \end{array}

The function

U (θ_{0})

is a monotonically increasing step function. It is the number of times a score in the second sample,

y_{j}

, precedes a score in the first sample,

x_{i} + θ

, where we only count a half if a score in the second sample actually equals a score in the first.

Let

U_{l} = k

; then

θ_{l} = d_{k + 1}

. This is the largest value

θ_{l}

such that

U (θ_{l}) = U_{l}

Let

U_{u} = n m - k

; then

θ_{u} = d_{n m - k}

. This is the smallest value

θ_{u}

such that

U (θ_{u}) = U_{u}

As in the case of

\hat{θ}

, these equations may be solved using either the exact or iterative methods to find the values

θ_{l}

and

θ_{u}

Then

(θ_{l}, θ_{u})

is the confidence interval for

θ

. The confidence interval is thus defined by those values of

θ_{0}

such that the null hypothesis,

θ = θ_{0}

, is not rejected by the Mann–Whitney two sample rank test at the

(100 \times 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(1)Input

On entry: specifies the method to be used.

$METHOD ='E'$: The exact algorithm is used.
$METHOD ='A'$: The iterative algorithm is used.

Constraint:

METHOD ='E'

'A'

2: N – INTEGERInput

On entry:

n

, the size of the first sample.

Constraint:

N \geq 1

3: X(N) – REAL (KIND=nag_wp) arrayInput

On entry: the observations of the first sample,

x_{i}

, for

i = 1, 2, \dots, n

4: M – INTEGERInput

On entry:

m

, the size of the second sample.

Constraint:

M \geq 1

5: Y(M) – REAL (KIND=nag_wp) arrayInput

On entry: the observations of the second sample,

y_{j}

, for

j = 1, 2, \dots, m

6: CLEVEL – REAL (KIND=nag_wp)Input

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 – REAL (KIND=nag_wp)Output

On exit: the estimate of the difference in the location of the two populations,

\hat{θ}

8: THETAL – REAL (KIND=nag_wp)Output

On exit: the estimate of the lower limit of the confidence interval,

θ_{l}

9: THETAU – REAL (KIND=nag_wp)Output

On exit: the estimate of the upper limit of the confidence interval,

θ_{u}

10: ESTCL – REAL (KIND=nag_wp)Output

On exit: an estimate of the actual percentage confidence of the interval found, as a proportion between

(0.0, 1.0)

11: ULOWER – REAL (KIND=nag_wp)Output

On exit: the value of the Mann–Whitney

U

statistic corresponding to the lower confidence limit,

U_{l}

12: UUPPER – REAL (KIND=nag_wp)Output

On exit: the value of the Mann–Whitney

U

statistic corresponding to the upper confidence limit,

U_{u}

13: WRK( $3 \times (M + N)$ ) – REAL (KIND=nag_wp) arrayWorkspace

14: IWRK( $3 \times N$ ) – 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.

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.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 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,	$METHOD \neq'E'$ or $'A'$ ,
or	$N < 1$ ,
or	$M < 1$ ,
or	$CLEVEL \leq 0.0$ ,
or	$CLEVEL \geq 1.0$ .

$IFAIL = 2$: 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 $\hat{θ}$ , $θ_{l}$ and $θ_{u}$ , 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 \times (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.

NAG Library Routine DocumentG07EBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G07EBF