Table 4. Estimated benchmark dose (BMD) and benchmark dose lower limit (BMDL) for five data sets used in the Benchmark dose report from EFSA (Hardy et al., 2017).

Data were analyzed using PROAST, BMDS and bmd. PROAST and BMDS uses profile likelihood intervals for estimating BMDL while the R package bmd uses the delta method, inverse regression or bootstrap.

					BMD					BMDL
Data set	Data type	Definition	BMR	Model	PROAST	BMDS	bmd		PROAST profile	BMDS profile	bmd delta	bmd inverse	bmd bootstrap
1	Continuous	Relative	5%	Exponential	235.1	233.7	235.5		170	170.4	201.5	203.3	201.1
2	Binomial	Excess	10%	Log–logistic	399	398.6	398.7		171	171.0	204.2	291.4	61.9
3	Binary	Excess	10%	One-stage	173	172.7	172.7a		92.4	92.3	35.4	95.2	75.2
4	Continuous	Relative	5%	Exponential	0.297	0.302	0.304		0.198	0.229	0.112	0.162	0.277
				Hillb,c	0.297	–	0.309		0.198	–	0.159	0.189	0.287
5	Binomial	Excess	10%	Log–logistic	3.2	3.2	3.2		1.84	1.84	1.63	2.1	1.42
				Log-normal	3.31	3.31	3.31		1.98	1.98	1.78	2.23	1.58

Notes.

^aStarting values required to get the reported results.

^bPotential problem with PROAST reporting the same values for the Hill and the exponential model. In the EFSA report results for Hill were: BMD=0.302 and BMDL=0.205.

^cNot possible to fit a three-parameter Hill model for these data in BMDS. In bmd a three-parameter log-logistic model was used instead.