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**.