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Influential observation

Influential observation 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 Influential observation rather than just read about it. In short: In statistics, an influential observation is an observation for a statistical calculation whose deletion from the dataset would noticeably change the result of the calculation. In particular, in regression analysis an influential observation is one whose deletion has a large effect on the parameter estimates.

Influential observation — main illustration
Influential observation — illustration

Key takeaways

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

Reference excerpt

In statistics, an influential observation is an observation for a statistical calculation whose deletion from the dataset would noticeably change the result of the calculation. In particular, in regression analysis an influential observation is one whose deletion has a large effect on the parameter estimates.

Assessment Various methods have been proposed for measuring influence. Assume an estimated regression y = X b + e {\displaystyle \mathbf {y} =\mathbf {X} \mathbf {b} +\mathbf {e} } , where y {\displaystyle \mathbf {y} } is an n×1 column vector for the response variable, X {\displaystyle \mathbf {X} } is the n×k design matrix of explanatory variables (including a constant), e {\displaystyle \mathbf {e} } is the n×1 residual vector, and b {\displaystyle \mathbf {b} } is a k×1 vector of estimates of some population parameter β ∈ R k {\displaystyle \mathbf {\beta } \in \mathbb {R} ^{k}} . Also define H ≡ X ( X T X ) − 1 X T {\displaystyle \mathbf {H} \equiv \mathbf {X} \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}} , the projection matrix of X {\displaystyle \mathbf {X} } . Then we have the following measures of influence:

DFBETA i ≡ b − b ( − i ) = ( X T X ) − 1 x i T e i 1 − h i i {\displaystyle {\text{DFBETA}}_{i}\equiv \mathbf {b} -\mathbf {b} _{(-i)}={\frac {\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {x} _{i}^{\mathsf {T}}e_{i}}{1-h_{ii}}}} , where b ( − i ) {\displaystyle \mathbf {b} _{(-i)}} denotes the coefficients estimated with the i-th row x i {\displaystyle \mathbf {x} _{i}} of X {\displaystyle \mathbf {X} } deleted, h i i = x i ( X T X ) − 1 x i T {\displaystyle h_{ii}=\mathbf {x} _{i}\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {x} _{i}^{\mathsf {T}}} denotes the i-th value of matrix's H {\displaystyle \mathbf {H} } main diagonal. Thus DFBETA measures the difference in each parameter estimate with and without the influential point. There is a DFBETA for each variable and each observation (if there are N observations and k variables there are N·k DFBETAs). Table shows DFBETAs for the third dataset from Anscombe's quartet (bottom left chart in the figure):

Outliers, leverage and influence An outlier may be defined as a data point that differs markedly from other observations. A high-leverage point are observations made at extreme values of independent variables. Both types of atypical observations will force the regression line to be close to the point. In Anscombe's quartet, the bottom right image has a point with high leverage and the bottom left image has an outlying point.

See also Influence function (statistics) Outlier Leverage Partial leverage Regression analysis Cook's distance § Detecting highly influential observations Anomaly detection

References

… excerpt ends here. Continue reading the full article.

Illustrations

Influential observation: In Anscombe's quartet the two datasets on the bottom both contain influential points. All four sets are identical when examined using simple summary statistics, but vary considerably when graphed. If one point is removed, the line would look very different.
In Anscombe's quartet the two datasets on the bottom both contain influential points. All four sets are identical when examined using simple summary statistics, but vary considerably when graphed. If one point is removed, the line would look very different.

Worked examples

Example 1 — a first encounter with Influential observation

Start with the simplest possible case. Write down what Influential observation 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 Influential observation 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 Influential observation 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 Influential observation

In research
Influential observation 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 Influential observation 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
Influential observation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Actuarial science, Regression diagnostics, Robust statistics, so understanding it makes those chapters shorter.
In everyday life
Look for Influential observation 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 Influential observation in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Influential observation means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Influential observation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Influential observation in simple terms?

In statistics, an influential observation is an observation for a statistical calculation whose deletion from the dataset would noticeably change the result of the calculation. In particular, in regression analysis an influential observation is one whose deletion has a large effect on the parameter…

Why does Influential observation 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 Influential observation?

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 Influential observation.

Tags

  • Actuarial science
  • Regression diagnostics
  • Robust statistics

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