Hierarchical generalized linear model

Key takeaways

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

Reference excerpt

In statistics, hierarchical generalized linear models extend generalized linear models by relaxing the assumption that error components are independent. This allows models to be built in situations where more than one error term is necessary and also allows for dependencies between error terms. The error components can be correlated and not necessarily follow a normal distribution. When there are different clusters, that is, groups of observations, the observations in the same cluster are correlated. In fact, they are positively correlated because observations in the same cluster share some common features. In this situation, using generalized linear models and ignoring the correlations may cause problems.

Overview and model

Model In a hierarchical model, observations are grouped into clusters, and the distribution of an observation is determined not only by common structure among all clusters but also by the specific structure of the cluster where this observation belongs. So a random effect component, different for different clusters, is introduced into the model. Let y {\displaystyle y} be the response, u {\displaystyle u} be the random effect, g {\displaystyle g} be the link function, η = X β {\displaystyle \eta =X\beta } , and v = v ( u ) {\displaystyle v=v(u)} is some strictly monotone function of u {\displaystyle u} . In a hierarchical generalized linear model, the assumption on y | u {\displaystyle y|u} and u {\displaystyle u} need to be made: y ∣ u ∼ f ( θ , ϕ ) {\displaystyle y\mid u\sim \ f(\theta ,\,\phi )} and u ∼ f u ( α ) . {\displaystyle u\sim \ f_{u}(\alpha ).}

The linear predictor is in the form:

g ( E ( y ) ) = g ( μ ) = η = X β + v {\displaystyle g(E(y))=g(\mu )=\eta =X\beta +v\,}

where g {\displaystyle g} is the link function, μ = E ( y ) {\displaystyle \mu =E(y)} , η = X β + v {\displaystyle \eta =X\beta +v} , and v = v ( u ) {\displaystyle v=v(u)} is a monotone function of u {\displaystyle u} . In this hierarchical generalized linear model, the fixed effect is described by β {\displaystyle \beta } , which is the same for all observations. The random component u {\displaystyle u} is unobserved and varies among clusters randomly. So v {\displaystyle v} takes the same value for observations in the same cluster and different values for observations in different clusters.

Identifiability Identifiability is a concept in statistics. In order to perform parameter inference, it is necessary to make sure that the identifiability property holds. In the model stated above, the location of v is not identifiable, since

X β + v = ( X β + a ) + ( v − a ) {\displaystyle X\beta +v=(X\beta +a)+(v-a)\,}

for constant a {\displaystyle a} . In order to make the model identifiable, we need to impose constraints on parameters. The constraint is usually imposed on random effects, such as E ( v ) = 0 {\displaystyle E(v)=0} .

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hierarchical generalized linear model

Start with the simplest possible case. Write down what Hierarchical generalized linear model claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In biology, 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 Hierarchical generalized linear model 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 Hierarchical generalized linear model 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 Hierarchical generalized linear model

In research: Hierarchical generalized linear model appears in biology 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 Hierarchical generalized linear model 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: Hierarchical generalized linear model is common in secondary-school and first-year university syllabi. It links to neighbouring topics Generalized linear models, Regression models, so understanding it makes those chapters shorter.
In everyday life: Look for Hierarchical generalized linear model 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.

How to study Hierarchical generalized linear model in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hierarchical generalized linear model 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 Hierarchical generalized linear model out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hierarchical generalized linear model in simple terms?

In statistics, hierarchical generalized linear models extend generalized linear models by relaxing the assumption that error components are independent. This allows models to be built in situations where more than one error term is necessary and also allows for dependencies between error terms.

Why does Hierarchical generalized linear model matter?

Because it connects several biology 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 Hierarchical generalized linear model?

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 Hierarchical generalized linear model.