Experimental analysis often involves analyzing groups containing varying numbers of elements; for example, a different number of units for each treatment assignment within each stratum. We therefore encounter objects that are *like matrices*, except they are not perfect rectangular blocks; i.e., they are not always “filled.”

In this note, we define a new structure, called a *tableau*, which can be regarded as a partially filled matrix, and seek to formalize the operations on tableaus that are used in the analysis of experiment. We then show how *tableau notation *can be used to express the key equations in a variety of statistical contexts, including stratification, clustering, and the sum-of-squares decomposition. Moreover, we express these equations in both an *invariant* and *index* form:

*invariant notation (coordinate-free form)*— defined in terms of*objects*and*operators*, much like the matrix-vector product A⋅x, and*index notation (coordinate form)*— defined explicitly in terms of indexed arrays and summation of multiple indices, much like expressing the matrix-vector product as*∑ⱼAᵢⱼ xⱼ.*

## Outline

This post consists of four main sections:

- Review of classic notation, the pros and cons;
- Theoretical development of the Tableau Calculus;
- Application to Experiments (completely randomized, block-randomized, adjustment formula, cluster-randomized, block-cluster, and ANOVA sum of squares decomposition);
- Python implementation

In experimental analysis, there are three main styles of notation that are commonly used:

*classic notation —*treatment assignment is explicitly enumerated: unit*(ijk)*describes the*k*th unit in the*j*th stratum of the*i*th treatment group (see [1], [2], and [5]);*assignment notation*— the assignment mechanism is treated as an independent variable, and we consider sums over quantities like*ZᵢYᵢ*or*Zᵢⱼ Yᵢⱼ*(see [2], [3], and [4]); and

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