It sounds like you're tackling a very interesting and important issue in agroecology, where accurately measuring competition and facilitation between crop species is essential. As you correctly mentioned, many of the commonly used indices for plant interactions (including the Relative Index of Interaction (RII)) are indeed closely related to the Land Equivalent Ratio (LER), which complicates their independent use in your study.

Here are a few other indices for competition/facilitation that might be more suitable for your needs, specifically designed to avoid overlap with LER:

1. Competition Index (CI)

• Competition Index (CI) can be used to quantify the relative intensity of competition between crop species in polyculture compared to monoculture. It is often derived from per capita growth (or yield) data from the different systems.
• A typical formula for CI is: CI=YSC−YMCYSCCI = \frac{Y_{\text{SC}} - Y_{\text{MC}}}{Y_{\text{SC}}}
• Where:YSCY_{\text{SC}} is the yield of the crop in monoculture, YMCY_{\text{MC}} is the yield of the crop in a polyculture system.
• The CI index focuses on comparing the yields within the monoculture and polyculture setups but does not rely on LER for its calculation. The result can indicate the relative competition or facilitation experienced by the species.

2. Relative Yield Total (RYT)

• While RYT is sometimes confused with LER, it can be used as a comparison tool for the total relative yield between multiple crops in polyculture systems. However, there are some specific modifications of RYT that can break it away from being a direct function of LER. The RYT can be calculated by: RYT=∑YMCiYSCiRYT = \sum \frac{Y_{\text{MC}_i}}{Y_{\text{SC}_i}}
• This is just a basic sum of relative yields of the crops in polyculture versus monoculture. The modification here could focus on measuring the yield interaction without referencing the system-wide total (LER), making it potentially more sensitive to the dynamics of individual species interactions.

3. Interaction Intensity Index (III)

• The Interaction Intensity Index (III) is designed to quantify the intensity of interactions (both competition and facilitation) based on species yield difference.
• This index compares the yield of species in monoculture versus polyculture for each species individually: III=YSC−YMCYSCIII = \frac{Y_{\text{SC}} - Y_{\text{MC}}}{Y_{\text{SC}}}
• It provides insight into the relative interaction strength between the species in a polyculture setting, independent of LER. The larger the value, the greater the interaction intensity.

4. Net Interaction Index (NII)

• The Net Interaction Index (NII) quantifies the net effect (whether positive or negative) of species interactions in polyculture relative to monoculture.
• It is calculated as: NII=YMC−YSCYSCNII = \frac{Y_{\text{MC}} - Y_{\text{SC}}}{Y_{\text{SC}}}
• This index reflects whether species benefit from each other (facilitation) or whether one negatively impacts the other (competition).

5. Resource Overlap Index (ROI)

• The Resource Overlap Index (ROI) is based on the concept that competition intensity increases as species overlap in resource use. This method could be useful in polyculture setups where spatial or temporal resource overlap occurs. The ROI quantifies the degree of resource sharing and can indicate competition when overlap is high.
• This index could involve assessing soil nutrients, water usage, and light to assess whether crops compete more intensely in polyculture compared to monoculture, potentially focusing on the relationship between these ecological factors and species interactions.

6. Normalized Difference Index (NDI)

• The NDI focuses on the difference in yield of individual crops between monoculture and polyculture but normalizes the data to eliminate bias from absolute yield levels. It can be defined as: NDI=(YMC−YSC)(YMC+YSC)NDI = \frac{(Y_{\text{MC}} - Y_{\text{SC}})}{(Y_{\text{MC}} + Y_{\text{SC}})}
• This will give you an index that reflects the relative change in yield due to competition or facilitation without needing to reference LER.

Summary:

The RII, CI, and III are some of the most prominent indices that measure competition/facilitation independently of LER. Of these, the Competition Index (CI) or Net Interaction Index (NII) might offer the most direct insight into the competitive dynamics you are interested in, as they don't depend on the overall yield ratio (which LER often uses).

In your case, where the focus is on behavioral interactions within polycropping systems and analyzing competition vs facilitation, I would recommend trying CI or III, as both offer measures of interaction intensity without being tied to LER. These indices can be used independently to assess individual crop species’ performance in polycultures versus monocultures.