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Small sample sizes in the study of ontogenetic allometry; implications for palaeobiology.

Brown CM, Vavrek MJ - PeerJ (2015)

Bottom Line: Across a variety of subsampling techniques, used to simulate different taphonomic and/or sampling effects, smaller sample sizes gave less reliable and more variable results, often with the result that allometric relationships will go undetected due to Type II error (failure to reject the hypothesis).This may result in a false impression of fewer instances of positive/negative allometric growth in fossils compared to living organisms.No mathematically derived minimum sample size for ontogenetic allometric studies is found; rather results of isometry (but not necessarily allometry) should not be viewed with confidence at small sample sizes.

View Article: PubMed Central - HTML - PubMed

Affiliation: Royal Tyrrell Museum of Palaeontology , Drumheller, Alberta , Canada.

ABSTRACT
Quantitative morphometric analyses, particularly ontogenetic allometry, are common methods used in quantifying shape, and changes therein, in both extinct and extant organisms. Due to incompleteness and the potential for restricted sample sizes in the fossil record, palaeobiological analyses of allometry may encounter higher rates of error. Differences in sample size between fossil and extant studies and any resulting effects on allometric analyses have not been thoroughly investigated, and a logical lower threshold to sample size is not clear. Here we show that studies based on fossil datasets have smaller sample sizes than those based on extant taxa. A similar pattern between vertebrates and invertebrates indicates this is not a problem unique to either group, but common to both. We investigate the relationship between sample size, ontogenetic allometric relationship and statistical power using an empirical dataset of skull measurements of modern Alligator mississippiensis. Across a variety of subsampling techniques, used to simulate different taphonomic and/or sampling effects, smaller sample sizes gave less reliable and more variable results, often with the result that allometric relationships will go undetected due to Type II error (failure to reject the hypothesis). This may result in a false impression of fewer instances of positive/negative allometric growth in fossils compared to living organisms. These limitations are not restricted to fossil data and are equally applicable to allometric analyses of rare extant taxa. No mathematically derived minimum sample size for ontogenetic allometric studies is found; rather results of isometry (but not necessarily allometry) should not be viewed with confidence at small sample sizes.

No MeSH data available.


Allometric power plots illustrating the effect of sample size (random subsample) on the scaling trend of three representative variables.Variable 1, positively allometric (A); Variable 3, isometric (B); Variable 10, strongly negatively allometric (C). In all cases, white indicates isometry, green indicates positive allometry, red represents negative allometry, and grey indicated disagreement between OLS and RMA. The bars at the top represent the minimum sample size needed to achieve the same scaling trend as the entire dataset. For all variables see Fig. S2.
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fig-5: Allometric power plots illustrating the effect of sample size (random subsample) on the scaling trend of three representative variables.Variable 1, positively allometric (A); Variable 3, isometric (B); Variable 10, strongly negatively allometric (C). In all cases, white indicates isometry, green indicates positive allometry, red represents negative allometry, and grey indicated disagreement between OLS and RMA. The bars at the top represent the minimum sample size needed to achieve the same scaling trend as the entire dataset. For all variables see Fig. S2.

Mentions: The effect of sample size on the conclusions of scaling trends can be visualized through the use of allometric power plots (Fig. 5 and Fig. S2), which plot the proportion of subsample replicates resulting in the categorical results (i.e., positive allometry, isometry, and negative allometry) against sample size. For all variables, subsampling methods, and regression types, the trends at low sample sizes were dominantly isometry, and as the sample size increased the percentage of replicates that were isometric either decreased (for ‘true’ allometry) or increased (for ‘true’ isometry). The sample size at which 95% of the replicates had the same result as the ‘true’ trend (i.e., that of the entire sample), for which the true trend was allometric was determined and recorded. This represents the sample size required for 95% confidence in a conclusion of allometry for that particular variable. When the ‘true’ trend was isometric, however, the level at which 95% of the replicates resulted in the correct trend was more difficult to determine, as the smallest samples usually resulted in the correct conclusion.


Small sample sizes in the study of ontogenetic allometry; implications for palaeobiology.

Brown CM, Vavrek MJ - PeerJ (2015)

Allometric power plots illustrating the effect of sample size (random subsample) on the scaling trend of three representative variables.Variable 1, positively allometric (A); Variable 3, isometric (B); Variable 10, strongly negatively allometric (C). In all cases, white indicates isometry, green indicates positive allometry, red represents negative allometry, and grey indicated disagreement between OLS and RMA. The bars at the top represent the minimum sample size needed to achieve the same scaling trend as the entire dataset. For all variables see Fig. S2.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4358694&req=5

fig-5: Allometric power plots illustrating the effect of sample size (random subsample) on the scaling trend of three representative variables.Variable 1, positively allometric (A); Variable 3, isometric (B); Variable 10, strongly negatively allometric (C). In all cases, white indicates isometry, green indicates positive allometry, red represents negative allometry, and grey indicated disagreement between OLS and RMA. The bars at the top represent the minimum sample size needed to achieve the same scaling trend as the entire dataset. For all variables see Fig. S2.
Mentions: The effect of sample size on the conclusions of scaling trends can be visualized through the use of allometric power plots (Fig. 5 and Fig. S2), which plot the proportion of subsample replicates resulting in the categorical results (i.e., positive allometry, isometry, and negative allometry) against sample size. For all variables, subsampling methods, and regression types, the trends at low sample sizes were dominantly isometry, and as the sample size increased the percentage of replicates that were isometric either decreased (for ‘true’ allometry) or increased (for ‘true’ isometry). The sample size at which 95% of the replicates had the same result as the ‘true’ trend (i.e., that of the entire sample), for which the true trend was allometric was determined and recorded. This represents the sample size required for 95% confidence in a conclusion of allometry for that particular variable. When the ‘true’ trend was isometric, however, the level at which 95% of the replicates resulted in the correct trend was more difficult to determine, as the smallest samples usually resulted in the correct conclusion.

Bottom Line: Across a variety of subsampling techniques, used to simulate different taphonomic and/or sampling effects, smaller sample sizes gave less reliable and more variable results, often with the result that allometric relationships will go undetected due to Type II error (failure to reject the hypothesis).This may result in a false impression of fewer instances of positive/negative allometric growth in fossils compared to living organisms.No mathematically derived minimum sample size for ontogenetic allometric studies is found; rather results of isometry (but not necessarily allometry) should not be viewed with confidence at small sample sizes.

View Article: PubMed Central - HTML - PubMed

Affiliation: Royal Tyrrell Museum of Palaeontology , Drumheller, Alberta , Canada.

ABSTRACT
Quantitative morphometric analyses, particularly ontogenetic allometry, are common methods used in quantifying shape, and changes therein, in both extinct and extant organisms. Due to incompleteness and the potential for restricted sample sizes in the fossil record, palaeobiological analyses of allometry may encounter higher rates of error. Differences in sample size between fossil and extant studies and any resulting effects on allometric analyses have not been thoroughly investigated, and a logical lower threshold to sample size is not clear. Here we show that studies based on fossil datasets have smaller sample sizes than those based on extant taxa. A similar pattern between vertebrates and invertebrates indicates this is not a problem unique to either group, but common to both. We investigate the relationship between sample size, ontogenetic allometric relationship and statistical power using an empirical dataset of skull measurements of modern Alligator mississippiensis. Across a variety of subsampling techniques, used to simulate different taphonomic and/or sampling effects, smaller sample sizes gave less reliable and more variable results, often with the result that allometric relationships will go undetected due to Type II error (failure to reject the hypothesis). This may result in a false impression of fewer instances of positive/negative allometric growth in fossils compared to living organisms. These limitations are not restricted to fossil data and are equally applicable to allometric analyses of rare extant taxa. No mathematically derived minimum sample size for ontogenetic allometric studies is found; rather results of isometry (but not necessarily allometry) should not be viewed with confidence at small sample sizes.

No MeSH data available.