```
# Hollander-Wolfe-Chicken Example
data all ;$ value ;
input group
cards ;1.83
A 0.50
A 1.62
A 2.48
A 1.68
A 1.88
A 1.55
A 3.06
A 1.30
A 0.878
B 0.647
B 0.598
B 2.050
B 1.060
B 1.290
B 1.060
B 3.140
B 1.290
B
; run ;
```

# Non-parametric point estimation in SAS

# Introduction

The Hodges-Lehman estimator (Hodges and Lehmann 1962) provides a point estimate which is associated with the Wilcoxon rank sum statistics based on location shift. This is typically used for the 2-sample comparison with small sample size. Note: The Hodges-Lehman estimates the median of the difference and not the difference of the medians. The corresponding distribution-free confidence interval (CI) is also based on the Wilcoxon rank sum statistics (Moses). In addition, exact CIs can be constructed.

PROC NPAR1WAY provides these estimates in a flexible manner.

*Hodges, J. L. and Lehmann, E. L. (1962) Rank methods for combination of independent experiments in analysis of variance. Annals of Mathematical Statistics, 33, 482-4.*

# Case study

# Hodges-Lehmann estimate and confidence interval

Hodges-Lehmann estimate and Moses confidence interval for the 2-sample case will be generated when putting HL as an option. The direction of the comparison can be controlled via refclass. If the exact confidence interval is required additionally then the exact statement together with the option HL needs to be defined. The Hodges-Lehmann point estimate and confidence interval can be addressed via the HodgesLehmann option under the ODS statement.

```
hl (refclass = "B") data = all ;
proc npar1way
class group ;
var value ;
exact hl ;
ods select HodgesLehmann ; run ;
```

# Results

The NPAR1WAY Procedure

```
Hodges-Lehmann Estimation
Location Shift (A - B) 0.5600
Interval Asymptotic
Type 95% Confidence Limits Midpoint Standard Error
Asymptotic (Moses) -0.3700 1.1830 0.4065 0.3962
Exact -0.2200 1.0820 0.4310
```