Table 1 Simulation scenarios for the true cluster model and alternative hypothesis

Single cluster

(A) One circular-shaped cluster (8%)

(B) One elliptic-shaped cluster (8%)

(C) One irregular-shaped cluster (15%)

(1) p⁽¹⁾ = (0.05, 0.15, 0.35, 0.45)

(2) p⁽¹⁾ = (0.05, 0.25, 0.25, 0.45)

(3) p⁽¹⁾ = (0.10, 0.10, 0.40, 0.40)

(4) p⁽¹⁾ = (0.15, 0.15, 0.15, 0.55)

Two homogeneous clusters

(D) Two circular-shaped clusters (8% each)

(E) Two elliptic-shaped clusters (8% each)

(1) p⁽¹⁾ = p⁽²⁾ = (0.05, 0.15, 0.35, 0.45)

(2) p⁽¹⁾ = p⁽²⁾ = (0.05, 0.25, 0.25, 0.45)

(3) p⁽¹⁾ = p⁽²⁾ = (0.10, 0.10, 0.40, 0.40)

(4) p⁽¹⁾ = p⁽²⁾ = (0.15, 0.15, 0.15, 0.55)

Two heterogeneous clusters

(D) Two circular-shaped clusters (8% each)

(E) Two elliptic-shaped clusters (8% each)

(5) p⁽¹⁾ = (0.05, 0.15, 0.35, 0.45), p⁽²⁾ = (0.05, 0.25, 0.25, 0.45)

(6) p⁽¹⁾ = (0.05, 0.15, 0.35, 0.45), p⁽²⁾ = (0.10, 0.10, 0.40, 0.40)

(7) p⁽¹⁾= (0.05, 0.15, 0.35, 0.45), p⁽²⁾ = (0.15, 0.15, 0.15, 0.55)