Consistency Criteria for General Datasets ========================================= For every subject whose choices are in the dataset Prest can compute, view and export the total number of violations for each of the axioms/criteria of choice consistency that are listed below. .. note:: Much of the terminology and notation that follows is introduced and explained in the :ref:Datasets  and :ref:Revealed Preference Relations  sections. Weak Axiom of Revealed Preference - WARP ---------------------------------------- For any two distinct alternatives x,y in X .. math:: x\succ^R y\;\; \Longrightarrow\;\; y\not\succsim^R x .. note:: Prest reports two WARP counts for general datasets: **WARP (pairs)** and **WARP (all)**. **WARP (pairs)** is the number of *pairs of menus* that are implicated in a WARP violation. **WARP (all)** is the total number of WARP violations. For example, the data C(\{x,y,z\})=\{x,y\} and C(\{x,z\})=\{z\} is associated with a WARP (pairs) count of 1 and a WARP (all) count of 2, the latter involving alternatives x,z and y,z, respectively. Congruence ---------- For any two distinct alternatives x,y in X .. math:: x\succ^R y\;\; \Longrightarrow\;\; y\not\succsim^{\widehat{R}} x .. note:: In Prest, Congruence violations of length 2 coincide with the **WARP (all)** count. Strict Choice Consistency ------------------------- For any two distinct alternatives x,y in X .. math:: x \succ^{\widehat{R}} y\;\; \Longrightarrow\;\; y\not\succsim^R x Strict Binary Choice Consistency -------------------------------- For any two distinct alternatives x,y in X .. math:: x\succ^{\widehat{B}} y\;\; \Longrightarrow\;\; y\not\succsim^B x Binary Choice Consistency ------------------------- For any two distinct alternatives x,y in X .. math:: x\succsim^{\widehat{B}} y\;\; \Longrightarrow\;\; y\not\succ^B x .. _general-consistency-tip: .. tip:: **To use the consistency-analysis feature:** right-click on the dataset of interest [e.g. "DatasetX.csv"] in the workspace and select *"Analysis -> Consistency analysis"*. **To view the consistency-analysis output:** right-click on the Prest-generated dataset ["DatasetX.csv (consistency)"] in the workspace and then click on "View". **To export the consistency-analysis output (in .xslx or .csv format):** right-click on the Prest-generated dataset ["DatasetX.csv (consistency)"] in the workspace, click on "Export", and then select one of the following options: * **Summary**: lists the total number of violations of each axiom (per subject). * **Congruence violations (wide)**: lists the number of Congruence violations, decomposed by cycle length. * **Strict general cycles (wide)**: lists the number of Strict Choice Consistency violations, decomposed by cycle length. * **Strict binary cycles (wide)**: lists the number of Strict Binary Choice Consistency violations, decomposed by cycle length. * **Binary cycles (wide)**: lists the number of Binary Choice Consistency violations, decomposed by cycle length. Additional Features: Inconsistent Tuples ---------------------------------------- .. _menu-tuples: Inconsistent tuples of menus ............................ By right-clicking on the dataset and then selecting *"Analysis -> Inconsistent tuples of menus"*, Prest computes and enumerates all distinct pairs, triples, quadruples, ..., n-tuples of menus that have led to a Congruence violation, and groups them according to the size of n. .. note:: The number of inconsistent *pairs* of menus coincides with the **WARP (pairs)** count. Following the same steps as above, this output can be viewed within Prest or exported to a .csv or .xslx file. .. _alternative-tuples: Inconsistent tuples of alternatives ................................... By right-clicking on the dataset and then selecting *"Analysis -> Inconsistent tuples of alternatives"*, Prest computes and enumerates all distinct pairs, triples, quadruples, ..., n-tuples of alternatives that have led to a Congruence violation, and groups them according to the size of n. Following the same steps as above, this output can be viewed within Prest or exported to a .csv or .xslx file. .. _merging-tip: .. tip:: If the same menu A appears more than once for the same subject in \mathcal{D}, Prest allows for **merging the choices** made at this menu in the different observations. For example, if the dataset \mathcal{D} is such that A_1=A_5=\{w,x,y\} and C(A_1)=\{x\}, C(A_5)=\{y\} for the same subject, then \mathcal{D} would be altered after the merging operation so that the menu A_1=A_5:=A appears only once, and with C(A)=\{x,y\} being the subject's new choice at this menu. **To use this feature:** right-click on the dataset of interest [e.g. "DatasetX.csv"] in the workspace and select *"Analysis -> Merge options at the same menu"*. The resulting merged dataset appears in the workspace ["DatasetX.csv (merged)"] and can then be analysed separately for consistency analysis or model estimation after the potential "noisiness" of choice data has been accounted for in this way through multi-valued choice. **Remark:** *If the merging operation is applied on a non-forced-choice dataset where a subject has chosen an alternative from menu* A *in one or more instances and has deferred choice/opted for the outside option in at least another, then the merged dataset will feature menu* A *appearing twice: one where* C(A) *comprises all alternatives in* A *that were chosen at least once; and one where* C(A)=\emptyset. **An example of a dataset that may help as an illustration for these merging features is available** here _. .. note:: **We provide an** example general dataset with default alternatives _ **and** an example general dataset without default alternatives _, **that can be analysed for consistency as described above**.