Charles W. Harper

A FORTRAN IV program for comparing ranking algorithms in quantitative biostratigraphy

Abstract A program for evaluating the performance of competing ranking algorithms in stratigraphic paleontology is presented. The program (1) generates a hypothetical, and thus known, succession of taxa in time and (2) simulates their succession in strata at several local sample sites. If desired, (1) and (2) may be repeated for several (=50 or 100 for example) iterations and the local site data for each sent to two user routines for inferred rankings (inferred succession of events in time). First data for first and last occurrences (fads and lads) taken together, then data for for lads-only, then data for fads-only is sent. For each submission of data to a user routine, Kendall rank correlation coefficients and Spearman coefficients are computed comparing the inferred rankings generated by the user routine with the known succession of events in time. The performance of two competing ranking algorithms may be compared by (1) obtaining for each submitted dataset the differences between corresponding Kendall (and/or Spearman) coefficients computed for the two algorithms, and (2) testing the observed differences for statistical significance. A simple two-sided t -test may be used to test whether the observed mean difference between two corresponding coefficients differs significantly from zero; if c t -tests are performed, the level of significance of each should be set to alpha/ c to obtain a maximum experimentwise error rate of less than alpha. The program is used to compare three ranking algorithms provided by Agterberg and Nel (1982a, b) as well as to determine whether the algorithms work as well for datasets combining lads and fads vs datasets for lads-only or fads-only. Agterberg and Nels Presorting algorithm performed better than their Ranking or Scaling algorithm. All three performed slightly but significantly better on data for lads-only or fads-only as opposed to combined data.

A Fortran 90 program for statistical testing of alleged thickening andsor thinning upward patterns in sequences of strata

A Fortran 90 program, STRATA_PATTERN_TEST, is designed to test sedimentary strata for the significance of vertical patterns. The null hypothesis of no pattern is tested against: (1) The alternative of a single thickening (or thinning) upward sequence, the program tests the null hypothesis using a parametric and/or randomization test based on Kendall’s S, or equivalently Kendall’s Tau. (2) The alternative of trends in g subsequences recognized a-priori, the program carries out parametric tests (where available) and randomization tests using one of 10 test statistics, each a weighted sum of the g Tau coefficients calculated for the individual subsequences. (3) The alternative of trends in g subsequences recognized post-hoc, i.e. purely on the basis of observed thickness patterns, the program carries out randomization tests using a family of 13 test statistics, each equal to the maximum value of the appropriate test statistic (defined for subsequences recognized a-priori) that is attainable by partitioning the total sequence of beds into 1, 2, … up to g subsequences. (4) The hybrid alternative g subsequences recognized a-priori, one or more exhibiting a symmetric pattern recognized post-hoc, the program carries out randomization tests. Reasons for implementing the program in Fortran 90 rather than Fortran 77 are briefly discussed.

Population size and taxonomic diversity in geologic time: a brief look at two density-dependent models

Density-dependent models of changes in the population size of a fossil species in geologic time, or the taxonomic diversity of a fossil group in geologic time, can be formulated using (1) the logistic equation model or (2) a piecewise linear/stochastic model ofGrenfell et al. One may test (or at least get an indication) whether or not one of the two types of models holds for a particular estimate of population size (or taxonomic diversity) Xt by plotting the value of the estimate as a function of time plus a scattergram of Xt+1 versus Xt for all pairs of estimates of population size (or taxonomic diversity) {Xt, Xt+1} successive in time.KurzfassungDichte-abhängige Modelle von Wechseln in den Populationsgrößen einer fossilen Art im Laufe geologischer Zeit oder die taxonomische Diversität einer Fossilgruppe im Laufe geologischer Zeit kann man formulieren durch (1) logistische Gleichungen oder (2) durch ein stückweise lineares/stochastisches Modell vonGrenfell et al. Man mag ausprobieren, ob eines der beiden Modelle geeignet ist, die bestimmte Populationsgröße (oder taxonomischen Diversität) Xt abzuschätzen, indem der Schätzwert als Funktion der Zeit in Verbindung mit einem Streudiagramm von Xt+1 versus Xt für alle Paare von Schätzungen der Populationsgrößen (oder taxonomischen Diversität) {Xt, Xt+1} in einheitlicher Folge aufgetragen wird.