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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.


The effect of sample size on the frequency of false allometry (‘Type I error’) (green), false isometry (‘Type II error’) (red), and wrong sign error (blue) in the random subsample replicates of A. mississippiensis for OLS (A) and RMA (B).Solid lines represent the mean of all 22 variable replicates and dotted lines represent one standard deviation of all 22 variable replicates (derived from the Alligator subsampling). For relative comparison, the mean sample sizes for the literature review of allometric studies of fossil and extant, (and vertebrae and invertebrate) allometric studies are indicated with the vertical bars.
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fig-7: The effect of sample size on the frequency of false allometry (‘Type I error’) (green), false isometry (‘Type II error’) (red), and wrong sign error (blue) in the random subsample replicates of A. mississippiensis for OLS (A) and RMA (B).Solid lines represent the mean of all 22 variable replicates and dotted lines represent one standard deviation of all 22 variable replicates (derived from the Alligator subsampling). For relative comparison, the mean sample sizes for the literature review of allometric studies of fossil and extant, (and vertebrae and invertebrate) allometric studies are indicated with the vertical bars.

Mentions: The relative rates of false allometry, false isometry, and sign error change drastically as a function of the sample size. Figure 7 illustrates the relative dominance of these errors as the sample size increases, for both OLS (A) and RMA (B) in the random subsample. In both cases, the false isometry (‘Type II error’) rate is consistently very high (mean >50% when n < 12) for small samples, and decreases as the sample size increases. In contrast, false allometry (‘Type I error’) and sign error rates are low, and very low respectively, (mean = ∼10% when n < 12, and mean <3% when n > 12), with false allometry rate changing little in response to increased sample size, and sign error not being a factor at sample sizes greater than twenty.


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

Brown CM, Vavrek MJ - PeerJ (2015)

The effect of sample size on the frequency of false allometry (‘Type I error’) (green), false isometry (‘Type II error’) (red), and wrong sign error (blue) in the random subsample replicates of A. mississippiensis for OLS (A) and RMA (B).Solid lines represent the mean of all 22 variable replicates and dotted lines represent one standard deviation of all 22 variable replicates (derived from the Alligator subsampling). For relative comparison, the mean sample sizes for the literature review of allometric studies of fossil and extant, (and vertebrae and invertebrate) allometric studies are indicated with the vertical bars.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig-7: The effect of sample size on the frequency of false allometry (‘Type I error’) (green), false isometry (‘Type II error’) (red), and wrong sign error (blue) in the random subsample replicates of A. mississippiensis for OLS (A) and RMA (B).Solid lines represent the mean of all 22 variable replicates and dotted lines represent one standard deviation of all 22 variable replicates (derived from the Alligator subsampling). For relative comparison, the mean sample sizes for the literature review of allometric studies of fossil and extant, (and vertebrae and invertebrate) allometric studies are indicated with the vertical bars.
Mentions: The relative rates of false allometry, false isometry, and sign error change drastically as a function of the sample size. Figure 7 illustrates the relative dominance of these errors as the sample size increases, for both OLS (A) and RMA (B) in the random subsample. In both cases, the false isometry (‘Type II error’) rate is consistently very high (mean >50% when n < 12) for small samples, and decreases as the sample size increases. In contrast, false allometry (‘Type I error’) and sign error rates are low, and very low respectively, (mean = ∼10% when n < 12, and mean <3% when n > 12), with false allometry rate changing little in response to increased sample size, and sign error not being a factor at sample sizes greater than twenty.

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.