mathematics

Funnel plot

Funnel plot 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 Funnel plot rather than just read about it. In short: A funnel plot is a graph designed to check for the existence of publication bias; funnel plots are commonly used in systematic reviews and meta-analyses. In the absence of publication bias, it assumes that studies with high precision will be plotted near the average, and studies with low precision will be spread evenly on both sides of the average, creating a roughly funnel-shaped distribution.

Key takeaways

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

Reference excerpt

A funnel plot is a graph designed to check for the existence of publication bias; funnel plots are commonly used in systematic reviews and meta-analyses. In the absence of publication bias, it assumes that studies with high precision will be plotted near the average, and studies with low precision will be spread evenly on both sides of the average, creating a roughly funnel-shaped distribution. Deviation from this shape can indicate publication bias.

Quotation Funnel plots, introduced by Light and Pillemer in 1984 and discussed in detail by Matthias Egger and colleagues, are useful adjuncts to meta-analyses. A funnel plot is a scatterplot of treatment effect against a measure of study precision. It is used primarily as a visual aid for detecting bias or systematic heterogeneity. A symmetric inverted funnel shape arises from a 'well-behaved' data set, in which publication bias is unlikely. An asymmetric funnel indicates a relationship between treatment effect estimate and study precision. This suggests the possibility of either publication bias or a systematic difference between studies of higher and lower precision (typically 'small study effects'). Asymmetry can also arise from use of an inappropriate effect measure. Whatever the cause, an asymmetric funnel plot leads to doubts over the appropriateness of a simple meta-analysis and suggests that there needs to be investigation of possible causes. A variety of choices of measures of 'study precision' is available, including total sample size, standard error of the treatment effect, and inverse variance of the treatment effect (weight). Sterne and Egger have compared these with others, and conclude that the standard error is to be recommended. When the standard error is used, straight lines may be drawn to define a region within which 95% of points might lie in the absence of both heterogeneity and publication bias. In common with confidence interval plots, funnel plots are conventionally drawn with the treatment effect measure on the horizontal axis, so that study precision appears on the vertical axis, breaking with the general rule. Since funnel plots are principally visual aids for detecting asymmetry along the treatment effect axis, this makes them considerably easier to interpret.

Criticism The funnel plot is not without problems. If high-precision studies are different from low-precision studies with respect to effect size (e.g., due to different populations examined) a funnel plot may give a wrong impression of publication bias. The appearance of the funnel plot can change quite dramatically depending on the scale on the y-axis — whether it is the inverse square error or the trial size. Researchers have a poor ability to visually discern publication bias from funnel plots.

See also Galbraith plot Forest plot Systematic review

References

Further reading Sterne, J. A. C.; Sutton, A. J.; Ioannidis, J. P. A.; et al. (2011), "Recommendations for examining and interpreting funnel plot asymmetry in meta-analyses of randomised controlled trials", BMJ, 343 d4002, doi:10.1136/bmj.d4002, PMID 21784880 Higgins, J.P.T.; Thomas, J.; Chandler, J.; Cumpston, M.; Li, T.; Page, M.J.; Welch, V.A. (2019), Cochrane handbook for systematic reviews of interventions (2nd ed.), Wiley Blackwell, ISBN 978-1-119-53661-1

Illustrations

Funnel plot: An example funnel plot showing no publication bias. Each dot represents a study (e.g. measuring the effect of a certain drug); the y-axis represents study precision (e.g. the inverse standard error or number of experimental subjects) and the x-axis shows the study's result (e.g. the drug's measured average effect). — An example funnel plot showing no publication bias. Each dot represents a study (e.g. measuring the effect of a certain drug); the y-axis represents study precision (e.g. the inverse standard error or number of experimental subjects) and the x-axis shows the study's result (e.g. the drug's measured average effect).

Worked examples

Example 1 — a first encounter with Funnel plot

Start with the simplest possible case. Write down what Funnel plot 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 Funnel plot 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 Funnel plot 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 Funnel plot

In research: Funnel plot 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 Funnel plot 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: Funnel plot is common in secondary-school and first-year university syllabi. It links to neighbouring topics Meta-analysis, Statistical charts and diagrams, Systematic review, so understanding it makes those chapters shorter.
In everyday life: Look for Funnel plot 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 Funnel plot in 20 minutes

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

Frequently asked questions

What is Funnel plot in simple terms?

Why does Funnel plot 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 Funnel plot?

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 Funnel plot.