mathematics

Partial least squares path modeling

Partial least squares path modeling is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Partial least squares path modeling rather than just read about it. In short: The partial least squares path modeling or partial least squares structural equation modeling (PLS-PM, PLS-SEM) is a method for structural equation modeling that allows estimation of complex cause-effect relationships in path models with latent variables. Overview PLS-PM is a component-based estimation approach that differs from the covariance-based structural equation modeling.

Key takeaways

Partial least squares path modeling belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Partial least squares path modeling to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Partial least squares path modeling from memory before moving on to harder problems.

Reference excerpt

The partial least squares path modeling or partial least squares structural equation modeling (PLS-PM, PLS-SEM) is a method for structural equation modeling that allows estimation of complex cause-effect relationships in path models with latent variables.

Overview PLS-PM is a component-based estimation approach that differs from the covariance-based structural equation modeling. Unlike covariance-based approaches to structural equation modeling, PLS-PM does not fit a common factor model to the data, it rather fits a composite model. In doing so, it maximizes the amount of variance explained (though what this means from a statistical point of view is unclear and PLS-PM users do not agree on how this goal might be achieved). In addition, by an adjustment PLS-PM is capable of consistently estimating certain parameters of common factor models as well, through an approach called consistent PLS-PM (PLSc-PM). A further related development is factor-based PLS-PM (PLSF), a variation of which employs PLSc-PM as a basis for the estimation of the factors in common factor models; this method significantly increases the number of common factor model parameters that can be estimated, effectively bridging the gap between classic PLS-PM and covariance‐based structural equation modeling. The PLS-PM structural equation model is composed of two sub-models: the measurement models and the structural model. The measurement models represent the relationships between the observed data and the latent variables. The structural model represents the relationships between the latent variables. An iterative algorithm solves the structural equation model by estimating the latent variables by using the measurement and structural model in alternating steps, hence the procedure's name, partial. The measurement model estimates the latent variables as a weighted sum of its manifest variables. The structural model estimates the latent variables by means of simple or multiple linear regression between the latent variables estimated by the measurement model. This algorithm repeats itself until convergence is achieved. PLS is viewed critically by several methodological researchers. A major point of contention has been the claim that PLS-PM can always be used with very small sample sizes. A recent study suggests that this claim is generally unjustified, and proposes two methods for minimum sample size estimation in PLS-PM. Another point of contention is the ad hoc way in which PLS-PM has been developed and the lack of analytic proofs to support its main feature: the sampling distribution of PLS-PM weights. However, PLS-PM is still considered preferable (over covariance‐based structural equation modeling) when it is unknown whether the data's nature is common factor- or composite-based.

See also Partial least squares regression Principal component analysis Structural equation modeling

References

Worked examples

Example 1 — a first encounter with Partial least squares path modeling

Start with the simplest possible case. Write down what Partial least squares path modeling claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Partial least squares path modeling before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Partial least squares path modeling ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Partial least squares path modeling

In research: Partial least squares path modeling appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Partial least squares path modeling in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Partial least squares path modeling is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graphical models, Least squares, Structural equation models, so understanding it makes those chapters shorter.
In everyday life: Look for Partial least squares path modeling outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Partial least squares path modeling in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Partial least squares path modeling means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Partial least squares path modeling out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Partial least squares path modeling in simple terms?

Why does Partial least squares path modeling matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Partial least squares path modeling?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Partial least squares path modeling.