Unravelling the Fundamental Data Structure of Experimental Evaluation

Experimental evaluation often involves analyzing groups containing various numbers of elements; for instance, a special variety of units for every treatment task inside each stratum. We subsequently encounter objects which can be like matrices, except they usually are not perfect rectangular blocks; i.e., they usually are not at all times “filled.”

On this note, we define a recent structure, called a tableau, which will be considered a partially filled matrix, and seek to formalize the operations on tableaus which can be utilized in the evaluation of experiment. We then show how tableau notation will be used to specific the important thing equations in a wide range of statistical contexts, including stratification, clustering, and the sum-of-squares decomposition. Furthermore, we express these equations in each an invariant and index form:

1. invariant notation (coordinate-free form) — defined when it comes to objects and operators, very like the matrix-vector product A⋅x, and
2. index notation (coordinate form) — defined explicitly when it comes to indexed arrays and summation of multiple indices, very like expressing the matrix-vector product as ∑ⱼAᵢⱼ xⱼ.

Outline

This post consists of 4 essential sections:

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

In experimental evaluation, there are three essential kinds of notation which can be commonly used:

1. classic notation — treatment task is explicitly enumerated: unit (ijk) describes the kth unit within the jth stratum of the ith treatment group (see [1], [2], and [5]);
2. task notation — the task mechanism is treated as an independent variable, and we consider sums over quantities like ZᵢYᵢ or Zᵢⱼ Yᵢⱼ (see [2], [3], and [4]); and