Where does invariance collide?

Whether a measurement instrument, such as a questionnaire scale, records the same underlying attribute in the same way across different groups of people or across repeated occasions: only when it holds can a gap in scores be read as a real difference in the attribute rather than an artefact of how the scale behaves for different people. Meredith's 1993 Measurement invariance, factor analysis and factorial invariance gave the formal factor-analytic definition and its graded levels (configural, metric, and scalar), and Vandenberg and Lance's 2000 review codified how each level is tested. This referent is a property a researcher must establish by testing, and it frequently fails; it is not a fact of nature secured by a symmetry, a fixed property of an abstract object, or a condition asserted to prove a program correct.[1][2]

Unit

A level (configural, metric, or scalar) that a scale does or does not meet across specified groups

Recovery means

A scale that fails at one level can sometimes be salvaged as partial invariance by freeing the few offending items

Measured by

Tested by fitting nested multi-group confirmatory factor models and comparing fit as loadings, then intercepts, are constrained equal

A quantity or a law is invariant when it stays the same under a specified transformation of the frame or coordinates, such as a shift in time, a rotation in space, or a change to a moving observer: these invariances are bound to conservation laws, so that every continuous symmetry yields a conserved quantity. Noether's 1918 theorem, in the English translation Invariant variation problems, established this link between symmetry and conservation, and Lorentz invariance is the requirement that the laws of physics take the same form for every observer in uniform motion. This referent is a discovered fact about how nature behaves under transformation, not a property one must test on a questionnaire, a definition chosen for a mathematical object, or an assertion inserted into a loop.[3]

Unit

A conserved quantity, or a law whose form is unchanged, under a given group of transformations

Recovery means

Not applicable

Measured by

Established by showing the equations of motion or the action are unchanged under the transformation, then confirmed by the conservation it predicts

An invariant is a property of a mathematical object that stays unchanged when the object is put through a whole class of transformations or mappings, so that two objects with different invariants cannot be turned into one another: the Euler characteristic of a surface, unchanged under continuous deformation, and the algebraic quantities studied in invariant theory are examples. Hilbert's 1890 Ueber die Theorie der algebraischen Formen proved that the invariants of a system of forms are finitely generated and founded modern invariant theory. This referent is a definitional property, fixed once the object and the allowed transformations are chosen and settled by proof; it is not a fact of nature, a condition asserted midway through a loop, or a scale property that must be checked in data and can fail.[4]

Unit

A number or algebraic quantity attached to an object, equal for any two objects related by the allowed transformations

Recovery means

Not applicable

Measured by

Derived and proved invariant by deduction: one shows the quantity is unchanged under each transformation in the class

A loop invariant is a condition that is true just before a loop begins and remains true after every single iteration, so that when the loop finally exits the invariant together with the exit condition guarantees the intended result: it is the central device for proving that a loop, and so a program, does what it claims. Hoare's 1969 An axiomatic basis for computer programming introduced the logical framework, now called Hoare logic, in which such invariants are stated and used to reason about correctness. This referent is something a programmer chooses and asserts in order to prove a program correct, not a fact of nature, a fixed property of an abstract object, or a scale property tested across groups of people.[5]

Unit

A logical assertion about program state that holds before the loop and is preserved by each pass through the body

Recovery means

Not applicable

Measured by

Verified by proving the invariant holds initially and that one arbitrary iteration preserves it, then discharging the proof by hand or with a checker