Parametrs characteristic of Atrypa Histrix, collected from various localities of the Cerro Gordo Member of the Upper Devonian Lime Creek Formation, were measured as: 1) Length, 2) Width, 3) Thickness, 4) Number of growth lines, and 5) Number of plications. The consistency of data obtained within a sample location was tested against similar data representing the total population by means of F-tests. Results suggested that the three sample locations represented at least two statistically random sample populations. These statistical differences in the single biologic population may be alternately interpreted as: 1) the population differed prior to preservation, or 2) is attributed to post depositional changes at the sample locations. F-tests of the parameter ratios increased resolution sufficiently to suggest a single sample population statistically characteristic of the one biologic population

Elwell, J. H.

Simon, D. E.

Name not available

Statistical Methods for Integrating Populations of a Fauna Collected at Various Localities

University of Northern Iowa

Proceedings of the Iowa Academy of Science Volume 76 Annual Issue Article 38 1969 Statistical Methods for Integrating Populations of a Fauna Collected at Various Localities D. E. Simon Iowa State University J. H. Elwell Iowa State University Let us know how access to this document benefits you Copyright ©1969 Iowa Academy of Science, Inc. Follow this and additional works at: https://scholarworks.uni.edu/pias Recommended Citation Simon, D. E. and Elwell, J. H. (1969) "Statistical Methods for Integrating Populations of a Fauna Collected at Various Localities," Proceedings of the Iowa Academy of Science, 76(1), 267-274. Available at: https://scholarworks.uni.edu/pias/vol76/iss1/38 This Research is brought to you for free and open access by the Iowa Academy of Science at UNI ScholarWorks. It has been accepted for inclusion in Proceedings of the Iowa Academy of Science by an authorized editor of UNI ScholarWorks. For more information, please contact scholarworks@uni.edu. Statistical Methods for Integrating Populations of a Fauna Collected at Various Localities D. E. S1MON1 AND J. H . ELWELL2 Abstract. Pararneten characteristic of Atrypa Histrix, collected from various localities of the Cerro Gordo Member of the Upper Devonian Lime Creek Formation, were measured as: 1) Length, 2) Width, 3) Thickness, 4) Number of growth lines, and 5) Number of plications. The consistency of data obtained within a sample location was tested against similar data representing the total population by means of F-tests. Results suggested that the three sample locations rep-resented at least two statistically random sample populations. These statis• tical differenc.es in the single biologic population may be alternately inter-preted as: 1 ) the population differed prior to preservaton, or 2) is at-tributed to post depositional changes at the sample locations. F-tests of the parameter ratios increased resolution sufficiently to suggest a single sample population statistically characteristic of the one biologic population. Brachiopods identified by Vondra ( 1) as the species Atrypa Histrix were collected from three exposures of the Upper Devonian Lime Creek Formation. The sample locations are identified on the Index Map, Figure 1, as Rockford Brick and Tile Quarry, Bird JOWA Fig. 1. Index may showing location of sampling sites. 1 Graduate teaching Assistant, Geology, Iowa State University, Ames, Iowa. 1 Instructor, Geology, Iowa State University, Ames, Iowa. 267 1Simon and Elwell: Statistical Methods for Integrating Populations of a Fauna CollecPublished by UNI ScholarWorks, 1969268 IOWA ACADEMY OF SCIENCE [Vol. 76 H ill and Roadcut. These locations are well known and it is gen-erally agreed that these brachiopod samples form one biologic pop-ulation. Several a ttributes of the brachiopod samples may be measured and recorded . The set of such attributes used in this study is shown in Figure 2. These data were compiled for 22 LENGTH THICKNESS GROWTH LINE~! -<-WIDTH ~ -~~ J ~ """f PLICATION TOP VIEW SIDE VIE\'.{ A TRYPA HISTRIX Fig. 2. Schema tic line drawing of Atrypa Histrix showing pa rameters measured . brachiopods from Rockford Brick and Tile Quarry, 36 from Bird Hill and 27 from Roadcut. Having generated these data we are concerned with what they represent. There are three possibil-ities: 1) they could represent only the specimens collected ; 2) they could represent different ecological or preservation conditions characteristic of each sampl·e location; or 3) they could be char-acteristic of the single biologic popula tion . The purpose of this study is to illustrate how the self-consistency of the data may be used to de termine a set of attributes that are characteristic of one statistical popula tion . Such a set would be characteristic of the single biologic popula tion . F-TESTS Let us consider how to test for the self-consistency in data be-tween sample locations. The criteria for acceptance as one useful statistical population are 1) samples may be considered as one random population ; 2 ) characteristic relationships be linear or con- 2Proceedings of the Iowa Academy of Science, Vol. 76 [1969], No. 1, Art. 38https://scholarworks.uni.edu/pias/vol76/iss1/381969] ST A TISTICAL METHODS 269 stant; and 3) characteristic relationships have a unique set of values for the biologic speci>es. The first criterion may be evaluated by means of the variance ratio, F. Consider repeated measure-ments of a single specimen length. The sample standard deviation per degree of freedom or mean square ( 2) would be a measure of the uncertainty in length expected for a single observation. F-tests or ratios of mean squares calculated for different groupings esti-mate differences in the uncertainty of observations between groups of samples. Identical groups would hav·e a ratio of one and the calculated value of F would be influenced by the number of ob-servations and the sample uncertainty. Values of F have been tab-ulated for samples of difforent sizes drawn from a single random population. Critical tabulated values, Snedecor and Cochran (2) are used to test the self-consistency of these data. Calculated values that exceed the tabulated value for samples drawn from a single random population only 5% of the time are identified by*. Highly significant values which exceed the tabulated 1 % value are iden-tified by**. In our testing of self-consistency we will assume that more than one statistical population is present if either tabulated value is exceeded by the calculated F. Table 1. Variance ratios of length, width, thickness, growth lines and plications between different groups. Total" Total• To talc Rockford Roo.dcut Bird Hill Length ·----------------- 1.07 4.64** 2.03** Width ------------------ 1.10 3.75** 1.86* Thickness ·------------· 1.14 3.20** l.881* Growth Lines Plications ------ 1.11 2.90** 2.61 ** 1.35 5.61** 2.13** a *5% significance level(21,84 df) = 1.68 **1 % significance level(21,84 df) = 2.09 h *5% significance level(84,27 df) = 1.75 **1 % significance level(84,27 df) = 2.23 ' *5% significance level(34,84 df) = 1.58 **1% significance. Ieve1(34·,84 df) = 1.90 d *5% significance level(21,62 df) = 1.73 * * 1 % significance level ( 21,62 elf) = 2 .1 7 l\1EASURED-Por1;LATIONS Rockford" Roadcut & Bird Hill 2.42** 2.11 * 1.85* 2.50** 2.03* Calculated F values for the attributes measured are shown in Table 1. From the * symbols shown we conclude that no measured parameter is characteristic of one statistical population. Since they represent more than one population, none of these measurements 3Simon and Elwell: Statistical Methods for Integrating Populations of a Fauna CollecPublished by UNI ScholarWorks, 1969270 IOWA ACADEMY OF SCIENCE [Vol. 76 may represent the single biologic population statistically. Because they lack self-consistency we reject these <lata as parameters char-acteristic of the biologic population. Since the acoeptance of multiple populations conflicts with the Symbol LIW LIT WIT GIP GIL GIW GIT PIL PIW PIT Table 2 Key to coding of comparisons Test Comparisons Length to Width Length to Thickness Width to Thickness Growth Lines to Plications Growth Lines to Length Growth Lines to Width Growth Lines to Thickness Plications to Length Plications to Width Plications. to Thickness Table 3. Variance ratios for the parameter ratios compared between LIW LIT TIW GIP GIL GIW GIT PIL P!W PIT groups. Tota!• Totalb Total 0 Rockford Roadcut Bird Hill .......................................... 1.02 1.24 1.10 ___ ,, ____________________________ 1.06 1.45 1.07 ------··------------·-----·---1.05 1.17 1.02 --------------·-··--------·----- 1.22 1.18 1.23 --------········-------------· 1.04 1.59 1.29 ·········--·---·------·-·----- 1.00 1.29 1.41 ·-----------------------·----- 1.03 1.11 1.38 -----------·------------·---·--· 1.26 1.19 1.05 -·--------------------·------- 1.22 1.48 1.14 """"""mom•••••••ooommm•••H••• 1.13 1.74 1.13 a * 5% significance level(21,84 df) = 1.68 ** 1 % significance level(21,84 df) = 2.09 b * 5% significance ]evel(84,27 df) = 1.75 ** 1 % significance level(84-27 df) = 2.23 0 * 5% significance level(34,84 df) = 1.58 ** 1 % significance level(34,84 df) = 1.90 d * 5% significance leve1(21,62 df) = 1.73 ** 1 % significance level ( 21,62 df) = 2.1 7 Rockfordd Roadcut & Bird 1-Iill 1.14 1.30 1.17 1.24 1.03 1.19 1.35 1.19 1.05 1.09 4Proceedings of the Iowa Academy of Science, Vol. 76 [1969], No. 1, Art. 38https://scholarworks.uni.edu/pias/vol76/iss1/381969] STATISTICAL METHODS 271 original assumption of a single biologic population, grouping of the measurements was tried in an attempt to find parameters char-acteristic of the single biologic population. The new ratio para-meters are shown in Table 2, which also seives as a key to the coding of parameters. Calculated F values for the ratio parameters are shown in Table 3. It should be noted that the same sample groupings that were significantly different in Table 1 as indicated by the * are now consistent with the assumption of a single statist-ical population. The absence of * symbols indicates that any of the ratios will satisfy the single statistical population criterion re-quired of the useful statistical parameters sought to describe the single biologic population. By changing the form of the parameter, and testing the self-consistency of the sample data, we have found candidates for useful parameter description. !61 :.a te • • 2-4 . . 12 . ... :t. I& .. . . ~ . . . • It a .. o· *. 0 ·o " 8 la • 20 ~ 2'8 LEl'UlTtt lll>lllt Fig. 3. Plot of length versus width. CORRELATION COEFFICIENT TESTS For the statistical population to satisfy the second criterion, a 5Simon and Elwell: Statistical Methods for Integrating Populations of a Fauna CollecPublished by UNI ScholarWorks, 1969272 IOWA ACADEMY OF SCIENCE [Vol. 76 linear relationship characteristic of the population is required. A plot of the ratio numerator vs the denominator should be a straight line if this requirement is to be met by the ratio. The correlation coefficient (r) is a numerical measurement of the abil-ity of these data to fit a straight line. The absolute value of (r) will lie between zero, a perfect circle, and 1, a straight line. The larger the absolute value of (r) the better the data fit a straight line. (2) For the purposes of indicating a linear relationship in the popula-tion sampled, an arbitrary limit of 0.8 was selected. Values of (r) greater than 0.8 will satisfy the linear relationship required of use-ful parameter ratios. THE LINEAR RELATIONSHIP A plot of Width vs Length is shown to illustrate a parameter ratio relationship in Figure 3. The (r) value is accepted as evidence that the ratio satisfies the linear relationship required of a useful statistical population. In passing, it should be noted that if the line passes through the origin the ratio may be characterized by a single constant, the slope of the line. However, if the line does not pass through the origin both a slope and an intercept are required to characterize the linear relationship. Such constants serve as ex-pected values of the parameters characteristic of the single biologic population. New sample locations which did not generate these expected constants would be suspected to contain a different biol-ogic population. Table 4 Correlation coefficients r as calculated for the various comparisons within different groups. Roadcut & Comparison Rockford Roadcut Bird Hill Bird Hill Total L/W 0.914 0.897 0.953 0.938 0.960 L/T 0.778 0.791 0.657 0.710 0.854 WIT 0.612 0.716 0.704 0.718 0.836 G/P 0.529 0.534 0.698 0.602 0.791 G/L 0.806 0.494 0.703 0.618 0.8r51 G/W 0.708 0.347 0.638 0.535 0.797 G/T 0.861 0.476 0.566 0.516 0.823 P/L 0.655 0.442 0.887 0.670 0.834 P/W 0.559 0.319 0.776 0.576 0.779 P/T 0.640 0.576 0.469 0.370 0.726 6Proceedings of the Iowa Academy of Science, Vol. 76 [1969], No. 1, Art. 38https://scholarworks.uni.edu/pias/vol76/iss1/381,969] STATISTICAL METHODS 273 A USEFUL p ARAMETER SET Correlation coefficients calculated for each parameter ratio at each sample location and for the total population are shown in Table 4. The individual sample locations are presented to show that the ( r) calculated for the total population is larger than the in-dividual (r's) for each parameter ratio. This qualitatively supports the concept of a single statistical population for ratio attribute in the biologic population sampled. To a first approximation several of the parameter ratios for the total population form a set that appear to satisfy the linear relationship required of a useful para-meter. The set characteristic of the single biologic population in-cludes ratios of L/W, G/L, W /T and L/T all which have a cor-relation coefficient of 0.8 or higher. The constants representing this set of parameters as slopes or intercepts are statistically char-acteristic of the biologic species. CONCLUSIONS 1. The selected set of slopes and intercepts may be used as ex-pected values for the biologic population. 2. Systematic deviation from the expected values may reveal anom-alies in certain sample location related to environment or pre-servation. 3. Whatever the paleontologist's interest, these methods will enable parameters to be tested and grouped so that the parameter re-ported will indeed represent the statistdal population he desires to study. 4. The major contribution of the paper is the compilation of com-mon statistical tests into a procedure for the statistical deter-mination of what population each parameter estimates. REFERENCES CITED VONDRA, CARL F. Associate Professor of Geology, Iowa State University Personal communication. SNEDECOR, GEORGE W. & WILLIAM G. CoCHRAN. Statistical Methods. Sixth Edition. The Iowa State University Press, Ames, Iowa, 1967. Appendix I Statistical formulas (Snedecor and Cochran, 1967). 1. X = observation 2. n = number of obse.rvations 3. X =mean 4. SSA= ~XA2 = ~(XA - X) 2 =Sum of Squares 5. d.f. = n-1 = Degrees of Freedom SS 6. MSA = A/d.f. =Mean Square 7Simon and Elwell: Statistical Methods for Integrating Populations of a Fauna CollecPublished by UNI ScholarWorks, 1969274 7. 8. 9. IOWA ACADEMY OF SCIENCE Fe AB = MS class A = F-test ' MS class B * F•>FT, exceeded by random population 5% ** Fc>FT, exceeded by random population 1 % SSAB = ~XAXB = { (:$XAXB) - (~XA) n (:$XB)} r = :$XAX:B/ ~XA2~XB2 = Correlation Coefficient "A measure of the degree of linearity: ± = perfect fit." [Vol. 76 8Proceedings of the Iowa Academy of Science, Vol. 76 [1969], No. 1, Art. 38https://scholarworks.uni.edu/pias/vol76/iss1/38

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Statistical Methods for Integrating Populations of a Fauna Collected at Various Localities

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