A Spatial Lag Design Model Only in Treatments Using Perpendicular Projection Operators

Spatial modeling literature typically presents models that incorporate spatial lags within classical spatial regression frameworks. This research proposes a design model that integrates both treatment and block effects, selectively incorporating the spatial lag only in the submatrix associated with treatments. This choice is based on the practical reasoning that the application of a treatment is more likely to cause interference or spillover effects among neighboring experimental units, rather than the environmental conditions motivating blocking. Through simulation, a response is generated using a Gaussian variogram with varying intensities of spatial dependence, to study the effect of including such a spatial lag in a simple factorial design arranged in generalized complete blocks. Orthogonal projection matrices incorporating the spatial lag are constructed, and a formal analogy is established with covariance analysis. From these, the corresponding sums of squares are extracted from the different sources of variation, enabling analysis of how the effect of the treatment‐induced spatial lag changes as the magnitude of spatial dependence in the response varies. This methodology is not only straightforward to implement but also offers a key structural advantage: it preserves the block component of the design matrix unaltered, maintaining its original interpretation. This feature facilitates its adoption within experimental design models that explicitly incorporate spatial weight matrices, extending its applicability toward what could be termed a spatial variance analysis corrected for treatment‐induced lag.