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

Table 5 Comparison of fitted and re-expressed β coefficients and proportional difference in β for three methods of re-expression, adjusted by their OLS describing the relationship between sigma and relative bias

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	-0.0202	-0.68	-0.0214	-0.66	-0.0214	-0.66
Apelberg, 2007 [29]	-12.9 g/(ng/ml)	-10.6	-0.18	-9.41	-0.27	-9.37	-0.27
Bulka, 2021 [21]	0.015 log_e(OR)/ ng·ml⁻¹	0.0339	1.26	0.0303	1.02	0.0301	1.01
Cheang, 2021 [27]	0.192 mg·dl/pmol·g⁻¹	0.224	0.17	0.288	0.50	0.290	0.51
Chen, 2012 [32]	-11.3 g/(ng/ml)	-14.8	0.31	-12.8	0.14	-12.8	0.13
Darrow, 2013 [33]	-2.3 g/(ng/ml)	-1.69	-0.27	-1.56	-0.32	-1.56	-0.32
Hamm, 2010 [31]	1.5 g/(ng/ml)	3.79	1.52	3.24	1.16	3.24	1.16
Lee, 2020 [22]	1.444 log_e(OR)/μg·dl⁻¹	0.957	-0.34	1.79	0.24	1.79	0.24
Odebeatu, 2019 [23]	6.86 × 10^–4 log_e(OR)/ng·ml⁻¹	0.00374	4.45	0.00361	4.26	0.00364	4.30
Pilkerton, 2018 [26]	-0.0049 log_e/ng·ml⁻¹	-0.021	3.26	-0.0232	3.74	-0.0231	3.71
Steenland, 2009 [10]	0.00105 log_e/ ng·ml⁻¹	0.00088	-0.16	0.00090	-0.14	0.00090	-0.14
Stein, 2016 [25]	-0.0039 log_e/ng·ml⁻¹	-0.00527	0.35	-0.00542	0.39	-0.00542	0.39
Washino, 2009 [30]	-10.94 g/(ng/ml)	-11.6	0.06	-9.34	-0.15	-9.40	-0.14
Xu, 2020a [24]	0.513 log_e(OR)/ ng·ml⁻¹	0.380	-0.26	0.535	0.04	0.534	0.04
Xu, 2020b [24]	29.3 mg·dl/ng·ml⁻¹	19.0	-0.35	26.7	-0.09	26.7	-0.09
Median			0.06		0.14		0.13
First quartile			-0.265		-0.145		-0.140
Third quartile			0.805		0.760		0.760
Minimum			-0.68		-0.66		-0.66
Maximum			4.45		4.26		4.30

^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