A Distribution-Free Approach to Stochastic Efficiency Measurement with Inclusion of Expert Knowledge

Table 13

Hypothesis tests for mean differences of efficiency scores. Sample 1 is denoted as the “Control group” and sample 2 is denoted as the “Virtual group”.

	Simulation 1		Simulation 2		Simulation 3		Simulation 4
	Control group	Virtual group	Control group	Virtual group	Control group	Virtual group	Control group	Virtual group
DEA
Mean	0.999	0.943	0.996	0.95	0.973	0.986	0.951	0.978
Variance	0.00001	0.00187	0.00011	0.00153	0.0007	0.0009	0.0015	0.0019
Observations	15	15	15	15	15	15	15	15
Pearson correlation	0.7117		0.1166		−0.5253		−0.1409
Hypothesized mean difference	0		0		0		0
Df	14		14		14		14
Rank-sum test	−3.09		−3.2146		1.3688		1.7213
𝑡 stat	5.2614		4.5917		−1.0167		−1.6501
P (T ≤ t)  two tail	0.00012		0.00042		0.3266		0.1212
𝑡 critical two tail	2.145		2.145		2.145		2.145
CCP efficiency evaluation
Mean	0.972	0.944	0.994	0.955	0.955	0.992	0.926	0.964
Variance	0.0016	0.0019	0.00028	0.0011	0.00176	0.00018	0.0025	0.00231
Observations	15	15	15	15	15	15	15	15
Pearson correlation	0.35661		−0.5373		−0.14		−0.5035
Hypothesized mean difference	0		0		0		0
Df	14		14		14		14
Rank-sum test	−1.8873		−3.0072		2.136		2.0117
𝑡 stat	2.2373		3.334		−3.1453		−1.7383
P (T ≤ t)   two tail	0.042		0.005		0.0072		0.1041
𝑡 critical two tail	2.145		2.145		2.145		2.145
DCF
Mean	0.971	0.944	0.993	0.956	0.961	0.987	0.934	0.962
Variance	0.00168	0.0019	0.0003	0.0011	0.0013	0.0008	0.00304	0.00217
Observations	15	15	15	15	15	15	15	15
Pearson correlation	0.3296		−0.5296		−0.4235		−0.0966
Hypothesized mean difference	0		0		0		0
Df	14		14		14		14
Rank-sum test	−1.8873		−2.9657		1.8665		1.2236
𝑡 stat	2.1038		3.2401		−1.8448		−1.4533
P (T ≤ t)   two tail	0.05396		0.0059		0.08633		0.1682
𝑡 critical two tail	2.145		2.145		2.145		2.145

The Rank-sum test shown previously is used to determine if the two samples being tested are of the same population. If they are of the same population, then we can conclude that the two frontiers for both the samples respectively, are one, and the same or that they consistently overlap one another, thus they can be assumed to be of the same surface.