naginterfaces.library.univar.robust_​2var_​ci

naginterfaces.library.univar.robust_2var_ci(method, x, y, clevel)[source]

robust_2var_ci calculates a rank based (nonparametric) estimate and confidence interval for the difference in location between two independent populations.

For full information please refer to the NAG Library document for g07eb

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/g07/g07ebf.html

Parameters
methodstr, length 1

Specifies the method to be used.

The exact algorithm is used.

The iterative algorithm is used.

xfloat, array-like, shape

The observations of the first sample, , for .

yfloat, array-like, shape

The observations of the second sample, , for .

clevelfloat

The confidence interval required, ; e.g., for a confidence interval set .

Returns
thetafloat

The estimate of the difference in the location of the two populations, .

thetalfloat

The estimate of the lower limit of the confidence interval, .

thetaufloat

The estimate of the upper limit of the confidence interval, .

estclfloat

An estimate of the actual percentage confidence of the interval found, as a proportion between .

ulowerfloat

The value of the Mann–Whitney statistic corresponding to the lower confidence limit, .

uupperfloat

The value of the Mann–Whitney statistic corresponding to the upper confidence limit, .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: or .

Warns
NagAlgorithmicWarning
(errno )

Not enough information to compute an interval estimate since each sample has identical values. The common difference is returned in , and .

(errno )

The iterative procedure used to estimate has not converged.

(errno )

The iterative procedure used to estimate, , the upper confidence limit has not converged.

(errno )

The iterative procedure used to estimate, , the lower confidence limit has not converged.

Notes

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 .

robust_2var_ci 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

Let , for , denote the (ascendingly) ordered differences , for , for . Then

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 . Thus is the value of the Mann–Whitney statistic for the two independent samples and . Effectively is a monotonically increasing step function of with

The estimate 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 and 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 , for , for . 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 and initial estimates of the upper and lower confidence limits for the Mann–Whitney statistic are found;

where is the inverse cumulative Normal distribution function.

and are rounded to the nearest integer values. These estimates are refined using an exact method, without taking ties into account, if and and a Normal approximation otherwise, to find and satisfying

and

The function 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.

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