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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.


Minimum sample size required for correct identification of scaling category (in 95% of replicates) as a function of slope.Each point represents one of the 22 variables found to be allometric (with the required sample size plotted against the slope) for OLS (black) and RMA (grey). Minimum required sample sizes are small away from 1.00 (i.e., strongly allometric) and increase exponentially to vertical asymptotes as the slope approaches 1.00 (i.e., isometric). The relationship is best described by a hyperbolic function (solid line), with 95% confidence intervals indicated in grey. The fitted model includes both the RMA and OLS data. Simpler model (Eq. (1.1)) with two parameters (A). More complex model (Eq. (1.2)) with third term, allowing for error in y-axis (B). The vertical dashed line indicates a slope of 1.00.
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fig-6: Minimum sample size required for correct identification of scaling category (in 95% of replicates) as a function of slope.Each point represents one of the 22 variables found to be allometric (with the required sample size plotted against the slope) for OLS (black) and RMA (grey). Minimum required sample sizes are small away from 1.00 (i.e., strongly allometric) and increase exponentially to vertical asymptotes as the slope approaches 1.00 (i.e., isometric). The relationship is best described by a hyperbolic function (solid line), with 95% confidence intervals indicated in grey. The fitted model includes both the RMA and OLS data. Simpler model (Eq. (1.1)) with two parameters (A). More complex model (Eq. (1.2)) with third term, allowing for error in y-axis (B). The vertical dashed line indicates a slope of 1.00.

Mentions: There is a strong correlation between the slope of the relationship between two variables and the minimum number of specimens needed to determine a scaling category with 95% confidence. The further the slope deviates from 1.00 (either positively or negatively) the fewer specimens are needed to conclude allometry (Table 4 and Fig. 6). Conversely, as the slope becomes closer to 1.00, the number of specimens increases dramatically.


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

Brown CM, Vavrek MJ - PeerJ (2015)

Minimum sample size required for correct identification of scaling category (in 95% of replicates) as a function of slope.Each point represents one of the 22 variables found to be allometric (with the required sample size plotted against the slope) for OLS (black) and RMA (grey). Minimum required sample sizes are small away from 1.00 (i.e., strongly allometric) and increase exponentially to vertical asymptotes as the slope approaches 1.00 (i.e., isometric). The relationship is best described by a hyperbolic function (solid line), with 95% confidence intervals indicated in grey. The fitted model includes both the RMA and OLS data. Simpler model (Eq. (1.1)) with two parameters (A). More complex model (Eq. (1.2)) with third term, allowing for error in y-axis (B). The vertical dashed line indicates a slope of 1.00.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig-6: Minimum sample size required for correct identification of scaling category (in 95% of replicates) as a function of slope.Each point represents one of the 22 variables found to be allometric (with the required sample size plotted against the slope) for OLS (black) and RMA (grey). Minimum required sample sizes are small away from 1.00 (i.e., strongly allometric) and increase exponentially to vertical asymptotes as the slope approaches 1.00 (i.e., isometric). The relationship is best described by a hyperbolic function (solid line), with 95% confidence intervals indicated in grey. The fitted model includes both the RMA and OLS data. Simpler model (Eq. (1.1)) with two parameters (A). More complex model (Eq. (1.2)) with third term, allowing for error in y-axis (B). The vertical dashed line indicates a slope of 1.00.
Mentions: There is a strong correlation between the slope of the relationship between two variables and the minimum number of specimens needed to determine a scaling category with 95% confidence. The further the slope deviates from 1.00 (either positively or negatively) the fewer specimens are needed to conclude allometry (Table 4 and Fig. 6). Conversely, as the slope becomes closer to 1.00, the number of specimens increases dramatically.

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.