Simultaneous Inference on All Linear Combinations of Means with Heteroscedastic Errors

Table 2

Comparison of interval widths between Scheffé’s and generalized Scheffé’s methods. Their interval widths differ in quantities: 𝑄₁=√𝐼⋅𝐹_{𝛼,𝐼,𝑁−𝐼}⋅MSE in Scheffé’s method and 𝑄₂=𝐹_𝛼,𝜈̂_1,𝜈̂₂⋅∑𝑆^2𝑖 in the generalized Scheffé’s method (𝛼=0.05).

Sample size	Equal variances				Unequal variances
	(0.1, 0.1, 0.1, 0.1)		(1, 1, 1, 1)		(0.3, 0.3, 0.1, 0.1)		(3, 3, 1, 1)
Balanced	𝑄1	𝑄2	𝑄1	𝑄2	𝑄1	𝑄2	𝑄1	𝑄2
(5, 5, 5, 5)	0.343	0.370	3.422	3.680	0.754	0.909	7.598	9.182
(10, 10, 10, 10)	0.322	0.331	3.229	3.323	0.718	0.813	7.166	8.105
(20, 20, 20, 20)	0.315	0.319	3.153	3.194	0.703	0.778	7.032	7.780
(50, 50, 50, 50)	0.310	0.312	3.105	3.120	0.694	0.759	6.945	7.595
Unbalanced
(5, 5, 10, 10)	0.329	0.350	3.284	3.490	0.602	0.905	6.052	9.125
(5, 5, 20, 20)	0.318	0.340	3.196	3.423	0.489	0.905	4.894	9.093
(10, 10, 20, 20)	0.318	0.326	3.173	3.250	0.597	0.812	5.951	8.092
(10, 10, 50, 50)	0.312	0.321	3.128	3.218	0.466	0.810	4.669	8.138