Table 3.

Simulation study results

		PM			DL			REML
${\tau }_{\beta }^2$	${\tau }_{\omega }^2$	${\widehat{\tau }}_{\beta }^2$	${\widehat{\tau }}_{\omega }^2$	CP	${\widehat{\tau }}_{\beta }^2$	${\widehat{\tau }}_{\omega }^2$	CP	${\widehat{\tau }}_{\beta }^2$	${\widehat{\tau }}_{\omega }^2$	CP	ρβ	ρω
0	0	0.03 (0.05)	0.01 (0.06)	0.97	0.01 (0.01)	0.01 (0.01)	0.97	0.00 (0.00)	0.00 (0.01)	0.96	(0.94, 0.11, 0.24)	(0.16, 0.27, 0.10)
0	0.25	0.03 (0.05)	0.20 (0.18)	0.88	0.01 (0.01)	0.26 (0.29)	0.89	0.00 (0.02)	0.26 (0.17)	0.91	(0.92, 0.50, 0.57)	(0.25, 0.77, 0.52)
0	0.50	0.03 (0.05)	0.40 (0.28)	0.88	0.01 (0.01)	0.51 (0.55)	0.87	0.00 (0.02)	0.50 (0.29)	0.91	(0.93, 0.47, 0.54)	(0.26, 0.82, 0.47)
0.25	0	0.26 (0.15)	0.08 (0.14)	0.96	0.26 (0.26)	0.11 (0.21)	0.94	0.23 (0.12)	0.05 (0.10)	0.95	(0.30, 0.84, 0.45)	(0.23, 0.77, 0.37)
0.25	0.25	0.27 (0.15)	0.27 (0.27)	0.92	0.25 (0.25)	0.30 (0.44)	0.88	0.25 (0.13)	0.25 (0.24)	0.92	(0.29, 0.85, 0.44)	(0.28, 0.88, 0.39)
0.25	0.50	0.26 (0.16)	0.52 (0.41)	0.91	0.25 (0.25)	0.57 (0.77)	0.87	0.25 (0.14)	0.51 (0.38)	0.91	(0.30, 0.88, 0.45)	(0.28, 0.93, 0.37)
0.50	0	0.51 (0.24)	0.12 (0.20)	0.96	0.52 (0.52)	0.22 (0.43)	0.95	0.46 (0.19)	0.08 (0.15)	0.96	(0.30, 0.87, 0.40)	(0.20, 0.82, 0.25)
0.50	0.25	0.51 (0.23)	0.32 (0.35)	0.93	0.50 (0.49)	0.44 (0.73)	0.90	0.48 (0.20)	0.28 (0.31)	0.92	(0.33, 0.90, 0.41)	(0.27, 0.89, 0.34)
0.50	0.50	0.52 (0.24)	0.56 (0.48)	0.92	0.50 (0.51)	0.64 (1.00)	0.88	0.50 (0.21)	0.54 (0.45)	0.92	(0.31, 0.92, 0.38)	(0.27, 0.94, 0.32)
0.75	0	0.75 (0.30)	0.17 (0.27)	0.96	0.72 (0.72)	0.33 (0.66)	0.94	0.69 (0.24)	0.13 (0.23)	0.95	(0.33, 0.90, 0.39)	(0.33, 0.85, 0.39)
0.75	0.25	0.76 (0.31)	0.35 (0.43)	0.93	0.72 (0.77)	0.53 (0.89)	0.90	0.72 (0.27)	0.33 (0.39)	0.92	(0.31, 0.92, 0.37)	(0.26, 0.91, 0.30)
0.75	0.50	0.75 (0.31)	0.59 (0.55)	0.93	0.72 (0.76)	0.83 (1.28)	0.89	0.73 (0.27)	0.56 (0.52)	0.93	(0.28, 0.93, 0.34)	(0.26, 0.93, 0.30)
1.00	0	1.02 (0.38)	0.20 (0.34)	0.95	1.00 (0.98)	0.45 (0.89)	0.94	0.95 (0.31)	0.15 (0.27)	0.95	(0.29, 0.90, 0.35)	(0.18, 0.85, 0.19)
1.00	0.25	1.01 (0.39)	0.38 (0.48)	0.94	1.01 (1.01)	0.63 (1.13)	0.91	0.96 (0.34)	0.33 (0.43)	0.93	(0.24, 0.93, 0.31)	(0.33, 0.90, 0.34)
1.00	0.50	1.01 (0.41)	0.57 (0.61)	0.92	1.03 (1.01)	0.85 (1.39)	0.89	0.98 (0.36)	0.53 (0.58)	0.92	(0.31, 0.94, 0.38)	(0.32, 0.93, 0.33)

One thousand simulated datasets were produced for each run so that 15 000 simulated datasets were produced in total. We show the mean ${\widehat{\tau }}_{\beta }^2$ and ${\widehat{\tau }}_{\omega }^2$ for all 3 estimation methods Empirical standard deviations are shown in parentheses. We also show the proportion of nominal 95% confidence intervals for the basic parameters that contain the true values (the estimated coverage probability, “CP”). Finally, we show the correlations between the 3 sets of estimates of ${\tau }_{\beta }^2$ and ${\tau }_{\omega }^2$ as the ordered triples ρ_β and ρ_ω.

Abbreviations: PM, the proposed Paule and Mandel estimation method; DL, a generalisation of DerSimonian and Laird's univariate method; REML: restricted maximum likelihood.