Re-expressing coefficients from regression models for inclusion in a meta-analysis

Table 4 Comparison of fitted and re-expressed β coefficients and relative bias in β for three methods of re-expression

First author, year	β_Estimand from analysis of raw data	β_RB^a	Relative Bias^b	β_Dz	Relative Bias^b	β_Alt^c	Relative Bias^b
Abraham, 2020 [28]	-0.0636 log_e/ ng·ml⁻¹	-0.0324	-0.49	-0.0368	-0.42	-0.0384	-0.40
Apelberg, 2007 [29]	-12.9 g/(ng/ml)	-11.7	-0.10	-13.2	0.03	-13.2	0.03
Bulka, 2021 [21]	0.015 log_e(OR)/ ng·ml⁻¹	0.043	1.87	0.0512	2.41	0.0531	2.54
Cheang, 2021 [27]	0.192 mg·dl/pmol·g⁻¹	0.335	0.74	0.352	0.83	0.353	0.84
Chen, 2012 [32]	-11.3 g/(ng/ml)	-16.3	0.44	-17.9	0.58	-17.8	0.58
Darrow, 2013 [33]	-2.3 g/(ng/ml)	-1.95	-0.15	-2.02	-0.12	-2.00	-0.13
Hamm, 2010 [31]	1.5 g/(ng/ml)	3.66	1.44	3.92	1.62	3.85	1.57
Lee, 2020 [22]	1.444 log_e(OR)/μg·dl⁻¹	1.31	-0.09	3.30	1.28	3.46	1.4
Odebeatu, 2019 [23]	6.86 × 10^–4 log_e(OR)/ng·ml⁻¹	0.0114	15.69	0.0122	16.82	0.0116	15.86
Pilkerton, 2018 [26]	-0.0049 log_e/ng·ml⁻¹	-0.0207	3.22	-0.0302	5.17	-0.0306	5.24
Steenland, 2009 [10]	0.00105 log_e/ ng·ml⁻¹	0.00116	0.11	0.00127	0.21	0.00126	0.20
Stein, 2016 [25]	-0.0039 log_e/ng·ml⁻¹	-0.00676	0.73	-0.0069	0.77	-0.00694	0.78
Washino, 2009 [30]	-10.94 g/(ng/ml)	-11.4	0.04	-12.1	0.10	-10.0	-0.08
Xu, 2020a [24]	0.513 log_e(OR)/ ng·ml⁻¹	0.508	-0.01	0.962	0.87	1.01	0.96
Xu, 2020b [24]	29.3 mg·dl/ng·ml⁻¹	25.4	-0.13	48.1	0.64	50.4	0.72
Median			0.11		0.77		0.78
First quartile			-0.095		0.155		0.115
Third quartile			1.09		1.45		1.485
Minimum			-0.49		-0.42		-0.40
Maximum			15.69		16.82		15.86

^a Using the notation of Rodriguez-Barranco et al., k = base of log transformation used; c = 1
^b Proportional difference between β in column to the left compared with the one from the analysis of raw data, calculated using the same method as in previous table ((beta in column to left – beta from analysis of raw data)/beta from analysis of raw data). Note that the β in column to the left was calculated using β with different units in denominator than for the raw data analysis shown in the table
^c Let the log unit increment I in untransformed units = b^(log_b^{(median) +0.5)}-b^(log_b^{(median) – 0.5)}, where b = 2, e, or 10, depending on the base. For observed β_o with units of ∆y/∆log_b(x), to get re-expressed β_r with units ∆y/∆x, calculate β_r = β_o/I. If the units of β_o are ∆y/∆x, to get β_r with units ∆y/∆log_b(x), calculate β_r = β_o· I

ISSN: 1471-2288