Species Accumulation Curves and Rarefaction: How to Compare Biodiversity Samples Fairly

Why Unequal Sampling Creates Misleading Comparisons

Observed species richness is the number of species detected in a sample. It is not necessarily the total number of species present in the community.

As researchers collect more individuals, examine more quadrats, operate more traps or make more site visits, they usually encounter additional species. Common species tend to be detected early, while uncommon, seasonal or difficult-to-detect species require greater effort.

Observed richness is therefore influenced by several factors:

• The number of individuals recorded
• The number and size of sampling units
• The area covered by the survey
• The number of sampling dates or seasons
• The spatial arrangement of samples
• The detectability of different organisms
• The sampling equipment and field method
• The skill and taxonomic resolution of the observers

If two datasets differ greatly in these respects, comparing their raw species totals can produce the wrong ecological conclusion.

This is also why species richness and species diversity should not be treated as identical measurements. Richness counts species, whereas broader diversity measures also account for their relative abundances.

What Is a Species Accumulation Curve?

A species accumulation curve plots the cumulative number of species recorded against increasing sampling effort.

The horizontal axis represents effort, such as:

• Number of individuals
• Number of quadrats
• Number of transects
• Number of traps
• Number of survey visits
• Total area sampled

The vertical axis represents the cumulative number of species detected.

How the curve develops

At the beginning of a survey, the curve often rises steeply because many previously unrecorded species are encountered. As sampling continues, most common species have already been found, and each additional unit contributes fewer new species. The curve then begins to flatten.

A typical accumulation pattern has three broad stages:

1. Rapid initial increase: Most samples contain several species not recorded previously.
2. Slower accumulation: Common species have been detected, and new records are increasingly uncommon.
3. Possible plateau: Additional effort produces few new species.

A curve approaching a plateau suggests diminishing returns from additional sampling. It does not prove that every species in the community has been detected. Rare, seasonal or poorly detected species may remain unobserved.

Observed-order and randomized curves

A simple collector’s curve adds samples in the order in which they were collected. Its shape may depend heavily on that order. For example, beginning in a species-rich habitat patch can make the curve rise more rapidly than beginning in a relatively uniform patch.

For general comparisons, researchers commonly randomize the order of sampling units many times. The mean richness at each level of effort is then plotted, often with a confidence interval. Randomization reduces the influence of the original sample order.

Accumulation curve, rarefaction curve and richness estimator

What Is Rarefaction?

Rarefaction is a statistical interpolation method used to estimate the expected diversity in a smaller sample drawn from the observed data.

Suppose Site A contains 500 recorded individuals and Site B contains 1,500. Comparing their raw species totals gives Site B more opportunities to include uncommon species. Rarefaction can estimate the expected richness of Site B at 500 individuals, allowing both sites to be examined at the same number of observations.

Rarefaction does not simply remove the last 1,000 records or take one arbitrary subsample. It considers the possible ways a smaller sample could be drawn and calculates the expected number of species.

Individual-based rarefaction formula

For abundance data, the expected number of species in a rarefied sample of m individuals can be written as:

E(Sm) = Σ [1 − C(N − ni, m) / C(N, m)]

where:

• E(S_m) is the expected number of species at sample size m
• N is the total number of individuals in the original sample
• n_i is the abundance of species i
• C(a,b) represents the number of combinations of a objects taken b at a time

The term inside the brackets is the probability that species i appears at least once in a subsample of size m. Adding this probability across all species gives expected richness.

Rarefaction is interpolation

Traditional rarefaction estimates diversity at a sample size smaller than or equal to the observed sample. It is therefore an interpolation procedure: it works within the information already contained in the dataset.

Predicting what might be found after increasing the sample beyond its observed size is called extrapolation. Extrapolation depends more strongly on model assumptions, and its uncertainty usually increases as the prediction moves farther beyond the collected data.

Individual-Based and Sample-Based Rarefaction

The appropriate method depends on how the biodiversity data were collected.

Individual-based rarefaction

Individual-based rarefaction is used when the dataset contains the abundance of each species. The sampling effort is expressed as the number of individual organisms recorded.

Example abundance table:

This approach is useful for inventories in which individuals can be counted meaningfully, such as trees, insects, birds or benthic organisms.

Sample-based rarefaction

Sample-based rarefaction—also called incidence-based rarefaction—is used when the data record whether each species was present or absent in multiple sampling units.

A sampling unit could be a quadrat, transect, trap, pond, camera station or survey date.

The expected richness at t sampling units can be expressed as:

E(St) = Σ [1 − C(T − Qi, t) / C(T, t)]

where:

• T is the total number of sampling units
• Q_i is the number of units in which species i occurs
• t is the standardized number of units

Incidence-based analysis preserves information about how species are distributed among sampling units. It is often preferable when abundance counts are unreliable or when the survey design is explicitly based on replicated plots, traps or visits.

Which method should you choose?

Sample Coverage: Comparing Completeness Instead of Size

Standardizing by equal sample size is useful, but equal-sized samples are not always equally complete.

Imagine sampling two communities with 500 individuals each. One community has relatively few, common species. The other has many uncommon species. A sample of 500 individuals might capture nearly all the diversity of the first community but only a fraction of the second.

Sample coverage estimates the proportion of individuals in the community that belong to species already represented in the sample. In practical terms, it describes how completely the community has been sampled.

• High estimated coverage indicates that most individuals belong to species already detected.
• Low coverage indicates that a meaningful portion of the community probably belongs to undetected species.

Many species represented by only one individual—called singletons—or by only one sampling unit—called uniques—usually indicate incomplete sampling.

Coverage-based rarefaction compares communities at the same estimated level of completeness rather than merely at the same number of individuals or sampling units. This can be more informative when communities differ greatly in species rarity and evenness.

Equal effort does not necessarily produce equal knowledge. Coverage-based comparison asks whether the samples represent their communities to a comparable degree.

Rarefaction with Hill Numbers

Rarefaction is often introduced as a method for comparing species richness, but the framework can also be applied to Hill numbers.

Hill numbers express diversity as an effective number of species:

• q = 0: Species richness. Every species receives equal weight, making the result sensitive to rare and undetected species.
• q = 1: Exponential Shannon diversity. Species are weighted according to abundance without disproportionately emphasizing dominant species.
• q = 2: Inverse Simpson concentration. Common and dominant species receive greater weight.

Examining all three orders gives a more complete picture of community structure.

For example, two sites could have similar rarefied richness but different q = 1 or q = 2 diversity. This would indicate that they contain similar numbers of species at the standardized level, but one site has a less even abundance distribution.

You can calculate Shannon diversity, Simpson diversity, species richness, evenness and related measures with the Decoding Biosphere Biodiversity Index Calculator. The calculator is useful for analysing observed alpha diversity; specialized software is still needed to produce formal rarefaction and extrapolation curves.

Worked Example: Comparing Two Forest Surveys

Consider two hypothetical forest surveys:

A direct comparison suggests that Forest B is richer because 31 species were recorded there, compared with 24 in Forest A. However, Forest B received twice as many quadrats.

Step 1: Preserve the incidence data

For every species, record the quadrats in which it occurred. A total richness value alone is insufficient because rarefaction requires information about the distribution of species among sampling units.

Step 2: Construct accumulation curves

Randomize the order of quadrats repeatedly and calculate the mean cumulative richness after 1, 2, 3 and subsequent quadrats. Add uncertainty intervals around the mean curves.

Step 3: Standardize the comparison

Use sample-based rarefaction to estimate the expected richness of Forest B at 10 quadrats. Compare this estimate with the richness of Forest A at the same effort.

Three outcomes are possible:

• If Forest B still has clearly greater expected richness at 10 quadrats, the difference is not explained solely by its larger sample.
• If the standardized values are similar, much of the raw difference may result from unequal sampling effort.
• If Forest A has greater expected richness at 10 quadrats, it accumulated species faster even though its final observed total was lower.

Step 4: Examine uncertainty and completeness

Compare confidence intervals and estimated coverage. Two samples standardized to 10 quadrats may still differ in completeness if species occur more sparsely or unevenly in one forest.

Overlapping confidence intervals are a useful visual warning that the evidence for a difference may be weak, but overlap alone is not a complete statistical test. Use an analysis appropriate to the survey design when a formal hypothesis test is required.

This example is illustrative. A real rarefaction result must be calculated from the underlying abundance or incidence matrix.

How to Interpret Species Accumulation and Rarefaction Curves

1. Compare curves at a common reference point

Do not compare only the final endpoints when the samples differ in size. Compare the curves at the same number of individuals, sampling units or estimated coverage.

2. Look at the slope

A steep slope means additional sampling is still producing many new species. A shallow slope means new species are being added slowly.

If the curve is still rising sharply at the final observed point, the inventory is probably incomplete.

3. Examine confidence intervals

Confidence intervals show uncertainty around expected diversity. A small visual difference between curves may not represent a reliable ecological difference when uncertainty is large.

4. Check whether the comparison uses interpolation or extrapolation

The observed portion of a curve is usually shown separately from its extrapolated portion. Extrapolated estimates should be interpreted more cautiously because they depend on estimates of undetected diversity.

5. Inspect more than species richness

Species richness is highly sensitive to rare species. Compare q = 0, q = 1 and q = 2 Hill-number curves when abundance or incidence data permit.

6. Consider the ecological scale

A curve produced from individuals pooled across an entire landscape answers a different question from one produced from standardized quadrats within a habitat. Always interpret the curve in relation to the spatial and temporal scale of sampling.

Recommended Workflow for a Fair Biodiversity Comparison

1. Define the biological question. Decide whether the goal is to compare habitats, seasons, treatments, restoration stages or geographic regions.

2. Use comparable field methods. Keep sampling equipment, quadrat size, survey duration, season and observer procedures as consistent as possible.

3. Retain the original data structure. Store species abundances or species-by-sampling-unit incidence data, not only total richness.

4. Plot accumulation curves. Use randomized sample order and include uncertainty intervals.

5. Assess sample completeness. Examine rare species, curve slopes and estimated sample coverage.

6. Choose a fair standardization. Compare samples at equal size, equal coverage or both.

7. Report Hill numbers where appropriate. Present richness alongside Shannon and Simpson effective diversity.

8. Separate interpolation from extrapolation. Clearly identify observed, rarefied and predicted sections of each curve.

9. Report uncertainty. Include confidence intervals and document the number of randomizations or bootstrap replications.

10. Explain the sampling design. State what the x-axis represents and whether the analysis is abundance-based or incidence-based.

Software commonly used

The R package iNEXT produces sample-size-based and coverage-based rarefaction and extrapolation curves for Hill numbers. It accepts both abundance and incidence data.

The R package vegan includes functions for species accumulation curves and ecological community analysis.

Researchers should document the software, package version, diversity order, endpoint, confidence level and data type used in the analysis.

Common Mistakes to Avoid

Comparing raw richness from unequal samples

A site with more individuals, quadrats or survey visits normally has more opportunities to accumulate species. Standardize before drawing conclusions.

Using only the final points of accumulation curves

Endpoints are not directly comparable when effort differs. Compare curves at a shared reference size or coverage.

Calling extrapolation rarefaction

Rarefaction interpolates to a smaller sample. Extrapolation predicts beyond the observed sample.

Ignoring sample coverage

Equal sample sizes may represent different proportions of their respective communities. Coverage provides important information about inventory completeness.

Mixing incompatible sampling methods

Rarefaction cannot correct fundamental differences such as pitfall traps at one site and visual surveys at another. Differences in detectability may remain even after numerical standardization.

Ignoring spatial and temporal heterogeneity

Pooling samples from different seasons or habitat types may produce a curve that reflects habitat turnover rather than local richness alone.

Assuming a flat curve proves complete sampling

A temporarily shallow slope may result from clustered sampling, seasonal absence or low detection probability. Biological knowledge and survey design remain essential.

Extrapolating too far

Long-range predictions can be unstable, especially for species richness. Keep extrapolation limited, show uncertainty and avoid presenting model estimates as observed records.

Discarding rare species as errors

Singletons and uniques strongly influence richness estimation and coverage. Verify possible identification or data-entry errors, but do not remove valid rare observations simply because they complicate the analysis.

Using rarefaction to repair biased sampling

Rarefaction standardizes sample size or completeness. It cannot make an unrepresentative sample representative, correct inconsistent taxonomy or remove detection bias automatically.

When Rarefaction May Not Be Enough

Rarefaction is powerful, but some research questions require additional methods.

• Imperfect detection: Occupancy models or repeated-detection models may be needed when species are present but frequently missed.
• Strong spatial turnover: Alpha, beta and gamma diversity should be analysed separately.
• Environmental effects: Regression, ordination or multivariate community models may be needed to relate diversity to habitat variables.
• Taxonomic, functional and phylogenetic differences: Species counts alone may not capture ecological similarity among communities.
• Before–after or treatment studies: The analysis should respect replication, pairing and temporal dependence.

Rarefaction answers a specific question about diversity under standardized sampling. It should be part of the analysis—not a substitute for an appropriate study design.

Frequently Asked Questions

What does a species accumulation curve show?

It shows how the cumulative number of recorded species changes as sampling effort increases. It can help researchers evaluate inventory completeness and determine whether additional sampling is likely to detect many more species.

What is the main difference between accumulation and rarefaction curves?

An accumulation curve describes species discovery as effort increases. A rarefaction curve estimates expected diversity after standardizing to smaller sample sizes. The curves may look similar, but they are produced for different analytical purposes.

Does rarefaction estimate the total number of species?

Not by itself. Rarefaction estimates expected diversity within the range of the observed sample. Extrapolation and asymptotic estimators are used to estimate diversity beyond the collected sample.

Can rarefaction compare samples collected with different methods?

Only with great caution. It can standardize numerical effort, but it cannot eliminate biases caused by different equipment, seasons, observer skills, taxonomic resolution or detection probabilities.

Should biodiversity samples be compared at equal size or equal coverage?

Both views can be informative. Equal-size comparison controls the number of individuals or sampling units, while equal-coverage comparison controls estimated completeness. Coverage-based comparison is often preferable when communities differ substantially in rarity and evenness.

Can rarefaction be used with Shannon and Simpson diversity?

Yes. Modern frameworks use Hill numbers to rarefy and extrapolate species richness, exponential Shannon diversity and inverse Simpson concentration.

What data are needed for rarefaction?

You need species abundances or incidence data across multiple sampling units. A single total species count does not contain enough information for a reliable rarefaction analysis.

How many randomizations should be used?

There is no universal number for every study. Use enough permutations or bootstrap replications for the mean curve and uncertainty intervals to stabilize, and report the number used. Software defaults should be checked rather than accepted without documentation.

Conclusion

Biodiversity samples cannot be compared fairly using raw species totals alone when sampling effort or completeness differs.

Species accumulation curves show how quickly species are discovered. Rarefaction standardizes diversity to a common sample size, while coverage-based methods compare samples at a similar level of completeness. Extrapolation can indicate what additional sampling might reveal, but uncertainty increases beyond the observed data.

The strongest analysis combines standardized field methods, accumulation curves, sample coverage, uncertainty intervals and multiple Hill numbers. Together, these tools help distinguish genuine ecological differences from differences created by sampling.

Continue Learning

• Calculate Shannon, Simpson and species-richness measures
• Species Richness vs Species Diversity
• Biodiversity and Conservation Learning Hub
• Ecology and Environment Learning Hub
• Types of Ecosystems
• Environmental Science Study Hub
• Environmental Science Dictionary

References

1. Gotelli, N. J. and Colwell, R. K. (2001). Quantifying biodiversity: procedures and pitfalls in the measurement and comparison of species richness. Ecology Letters, 4, 379–391. https://doi.org/10.1046/j.1461-0248.2001.00230.x
2. Chao, A. and Jost, L. (2012). Coverage-based rarefaction and extrapolation: standardizing samples by completeness rather than size. Ecology, 93, 2533–2547. https://doi.org/10.1890/11-1952.1
3. Chao, A., Gotelli, N. J., Hsieh, T. C., Sander, E. L., Ma, K. H., Colwell, R. K. and Ellison, A. M. (2014). Rarefaction and extrapolation with Hill numbers: a framework for sampling and estimation in species diversity studies. Ecological Monographs, 84, 45–67. https://doi.org/10.1890/13-0133.1
4. Hsieh, T. C., Ma, K. H. and Chao, A. (2016). iNEXT: an R package for rarefaction and extrapolation of species diversity (Hill numbers). Methods in Ecology and Evolution, 7, 1451–1456. https://doi.org/10.1111/2041-210X.12613