In a Latin Square Design (LSD) Split-Plot, one treatment factor is assigned such that each level appears once in each row and column, forming the Latin Square.  Rows and columns are blocking factors, are causing the experimental units (cells in the diagrams) to differ.  This example has four treatments, A1-A4, which requires there be 4 rows and 4 columns.  Then each whole plot (color coded) is split to form sub-plot experimental units for the B treatment factor. 

LSD Split-Plot None - This example shows a 4 by 4 Latin Square with 4 A treatments. Exactly 4 rows and columns are expected, anything else will create unbalance and not be optimal.  Then the row-column whole plot experimental unit is split into sub-plot experimental units (cells) for the B treatment. There is one measurement per experimental unit (eg. A1B1) on each response variable.

Latin Square Split-Plot with Replication - Replication can added for both whole and sub-plots.  This diagram shows replication in the sub-plot, which requires multiple experimental units (cells) for each level of the B treatment.

Latin Square Splitplot with Sampling - This example shows a 4 by 4 Latin Square with 4 A treatment levels assigned to color coded whole plots.  These are split into sub-plots (cells) for the two levels of B treatment, and there are two sampling measurements (for example, the two A1B1 A1B1) per experimental unit (cell) on each response variable.  This is sampling in the sub-plot, but sampling in the whole plot can also be done.

Latin Square Split-Plot with Replication and Sampling - This illustrates replication and sampling in the sub-plot (whole plot can also have replication and/or sampling).  Each color-coded whole plot has replicated experimental units (cells) for each level of B treatment.  Then each experimental unit has multiple measurements (for example, the two A1B1 A1B1) on each response variable.

Latin Square Split-Plot with Covariate - Add a covariate in the sub-plot of a Latin Square Design Split-plot by measuring the covariate (X) on each sub-plot experimental unit (cell).  Then treatments are applied and one measurement per response variable taken (eg. A1B1).  Note covariates can also be measured on the whole plot experimental unit.

Latin Square Split-Plot with Sampling and Covariate - 4 levels of A treatment are assigned to whole plots (color coded) in the Latin Square pattern.  Whole plots are split into sub-plot experimental units (cells), one cell per level of the B treatment.  Then each experimental unit is measured twice, producing sampling.  The covariate measurements (X) are either taken on each sampling unit, OR one measurement on the experimental unit is taken and the value copied to each sample.

Latin Square Split-Plot with Replication and Covariate - The covariate (X) is measured on each sub-plot experimental unit (cell).  Then 4 levels of A treatment are assigned to whole plots (color coded) in the Latin Square pattern.  Whole plots are split into sub-plot experimental units (cells) with enough units to replicate both levels of the B treatment twice.  Note covariates can also be measured on the whole plot experimental unit.

latinsq_splitplot_replication_covariate_sketch

Latin Square Split-Plot with Replication, Sampling and Covariate - 4 levels of A treatment are assigned to whole plots (color coded) in the Latin Square pattern.  Whole plots are split into sub-plot experimental units (cells) with enough units to replicate both levels of the B treatment twice.  Then each experimental unit is measured twice, producing sampling. The covariate measurements (X) are either taken on each sampling unit, OR one measurement on the experimental unit is taken and the value copied to each sample.

latinsq_splitplot_replication_sampling_covariate_sketch