In physics education research (PER), concept inventories (CIs) have become standard instruments for assessing students' learning throughout instruction. To promote widespread use of concept inventories, previous studies have developed an approach to split a full length CI into short versions of CIs. This research extends the existing method to fully utilize the item response theory framework in equating and linking between the short CIs and the full length CIs. Three quantitative studies have been conducted: First, the extended algorithm is applied to divide the Brief Electricity and Magnetism Assessment (BEMA) into two half-length BEMAs (HBEMAs). Through a series of test-equating and validation analysis, the HBEMAs are confirmed to measure the same latent constructs of student understanding of electricity and magnetism to that of the original BEMA at a similar level of reliability. The second study establishes equivalent score conversions among the three versions of BEMA using the Stocking-Lord method, which has the best performance on equating error reduction among several methods explored. It is also confirmed that the equivalent statistical characteristics of the three versions of BEMA are equity and population invariant. In the third study, the extended algorithm is applied to link and compare the BEMA and the Conceptual Survey of Electricity and Magnetism (CSEM). After linking the BEMA and CSEM assessment scales, it becomes possible to directly convert and compare students' performances on the two CIs. It is found that the scales of BEMA and CSEM are almost identical after scale transformation. Based on these studies, it can be suggested that all short and long versions of BEMA and CSEM can be used interchangeably after scale transformation.