The Uneasy Relationship Between Statistical Methods and Anthropological Research. Dr. Dwight Read Dept. of Anthropology and Dept. of Statistics UCLA

Transcription

1 The Uneasy Relationship Between Statistical Methods and Anthropological Research Dr. Dwight Read Dept. of Anthropology and Dept. of Statistics UCLA

2 Introduction I willl begin by characterizing statistics as finding patterning in aggregated data The probability basis for inferential statistics that relates sample statistics to population parameters is introduced next, using William Feller s notion of a conceptual experiment Although descriptive adequacy is obtained by defining an experiment defined by random sampling of a population, the latter does not characterize the processes of interest that structured the data brought forward for analysis Instead, an experiment needs to be defined by referring to a theory for the structuring processes; I exemplify this approach with archaeological data The example makes evident the problem that arises with data sets that are heterogeneous due to being the consequence of more than one data structuring process This leads to the double-bind problem that removing heterogeneity in data cases requires already having variables that measure the same process and removing heterogeneity in variables requires already having data cases that are the consequence of the same process A resolution of the double-bind problem is illustrated using a iterative procedure and leads to statistical analysis that gives rise to novel insights into the properties of the archaeological data set being analyzed

3 Qualitative Patterning Observed on Individual Cases A type refers to a category of ceramics that shares a consistent, specific and unique combination of physical attributes (such as paste type, color of decoration, kind of glazing, etc.). (Florida Museum of Natural History) Stamnos is a lidded storage jar for liquids that was standardized during the red-figure period. It is glazed inside. It has a short, stout neck, a wide, flat rim, and a straight body that tapers to a base. Horizontal handles are attached to the widest part of the jar.

4 Quantitative Patterning Observed on Individual Cases During experiments I ran every test many times and noticed just a little deviation between results. I considered those deviation to be negligibly small (statistically insignificant) so final comparison made from just one execution of test case for every language without gathering results of multiple tests and comparing their average. (

5 Pattern in the Aggregate a correlation [r = 0.547] is not predictive for individual cases. It is strictly a statistical statement about how two variables are related in aggregate. (emphasis in original;

6 What is Statistics? we can say that Statistics is the science concerned with the summary of data, trying to find regularities in these data ( emphasis added) Statistics has to do with phenomena where patterning is found in the aggregate, as opposed to phenomena where patterning is found on individual cases.

7 Implications of Patterning in the Aggregate Patterning in the aggregate specific implies that change in the aggregate may change the patterning. Therefore, statistical analysis begins by specifying the aggregate over which patterning is to be found. The aggregate over which patterning is to be found is typically called a population. A population must be well-defined; that is for any entity or observation, we must be able to determine if it is a member of the population or not.

8 Example of Defining a Population Archaeological Topic of Interest: Patterning in stone projectile points (arrowheads) Data set of interest: Projectile points made by the inhabitants of a Paleo-Indian site, 4Ven39, in Ventura County (occupied around 1400 AD). Method for Data Recovery: Excavate a sample of 2m x 2m grid squares placed randomly over the occupation area Data Set Brought Forward for Analysis: 64 projectile points recovered from the excavation Population Definition 1: Define the population to be the 64 projectile points recovered from the excavation

9 Projectile Points from 4Ven39

10 Measurements

11 Determine Patterning: Tip Angle (Sharpness) Projectile Points from 4Ven39 µ = 40.3 σ = 12.15

12 Population Definition 2 Though the population of recovered projectile points is welldefined, the archaeologist wants to know about all points, not just the ones that were recovered. Population 2: All projectile points in the site 4Ven39 Population 2 cannot be recovered in its entirety, so we use inferential statistics to infer values for population parameters from sample statistics. The sample consists of the recovered projectile points and for this sample the sample statistics are: x = 40.3,s =12.21

13 Statistical Inference From the sample statistics, the sample size n = 64, the assumption that the sample is a random sample from Population 2 due to the way the data were recovered, and the apparent normality of the distribution of data values, we conclude, with 95% confidence, that the true value for µ is in the interval [ x 1.53, x 1.53] = [37.3, 43.3]

14 Statistical Inference and Probability Inferential statistics is given a probability foundation (from Feller, W. An Introduction to Probability Theory and Its Applications) by defining a sample space Ω for a conceptual experiment, E, with outcomes when E is performed as follows: An experiment E determines a sample space Ω consisting of all possible, indecomposable outcomes for E. For a finite sample space, a value, p i, 0 p i 1, is assigned to each outcome, o i, in Ω, where i = 1. The value p i is interpreted as the probability of o i occurring when the experiment E is performed. A random variable is a mapping from Ω to the real numbers.

15 Connection Between Probability Theory and a Population Experiment E: Select an object from the (finite) population P randomly. Space Ω is determined by the objects in P Assign p = 1/N as the probability for each outcome in Ω, where N is the number of objects in P. A measurement X over the objects in P, such as the angle of a point, defines a mapping from Ω to R, hence is a random variable defined over the space Ω. Probabilities are assigned to the value x the measurement X may take on by letting o = {o i X(o i ) = x}. Then the probability that X takes on the value x is given by Prob(o) = Σ Prob(o i ) = Σp i.

16 Connection Between Probability Theory and a Population (example) Sample of excavated points Population, P, of projectile points made by inhabitants of 4Ven39 Experiment E: randomly select a point from P Space Ω of outcomes: projectile points made by inhabitants of 4Ven39 Probability of an outcome = 1/N, where N is the number of projectile points they made Random variable X: angle of the tip of a projectile point. Use inferential statistics to connect sample statistics to population parameters

17 Description versus Understanding The experiment E, of randomly selecting observations from population P, enables statistical description of any population with parameter values either computed directly when the entire population is accessible (e.g., P = {excavated projectile points}), or by inference from values obtained using a sample from the population (e.g., P = {all projectile points in the site 4Ven39}). Problem: E does not relate to the process(es) by which projectile points were made, hence parameter values do not characterize those processes.

18 Experiment Based on Process The experiment E of interest to the archaeologist is the process of an artisan at 4Ven39 making a projectile point. The outcomes of E are the projectile points that the artisans made. Archaeologists make the following assumptions: (1) the projectile points are part of a group s cultural repertoire, meaning that the artisans had shared normative values for some of the attributes of the projectile points and (2) different types of projectile points correspond to different normative values. Statistical analysis should lead to estimation of the normative values for those attributes that are part of the cultural repertoire of the artisans who manufactured the projectile points.

19 Examples of Normative Values (1) Qualitative: The points either have a leaf shape or a triangular shape. Other shapes are possible, but were not produced at 4Ven39. leaf shape triangular shape (2) Quantitative: The angle of the points is unimodal, approximately normal, suggesting that there was a single, normative value for the tip angle.

20 Statistical Model For Quantitative Normative Values population of possible manufacture errors sum of random sample of manufacture errors Quantitative Dimension Actual Value Normative Value Via the Central Limit Theorem, the frequency distribution for a random variable corresponding to a dimension under normative control will have a uni-modal, approximately normal distribution.

21 Not all Dimensions are Under Normative Control Distance from base of point to maximum width. Bar chart does not match the pattern for a dimension under normative control. Appears to be made up of two patterns: (1) 0 distance and (2) non-zero distances, with a unimodal distribution and normative value of 9 mm x = 4.2,s = 0.6

22 Distance to Maximum Width Leaf shape Triangular shape non-zero distance zero distance Dimension does not apply to triangular points

23 Heterogeneous Data Sets The normative values for the leaf shape points will differ, for some dimensions, from the normative values for the triangular shape points. This implies that we need to divide the original data set into subsets for which all members of a subset share the same dimensions and normative values.

24 Heterogeneous Dimensions The distance to maximum width is not under cultural constraint for the triangular points. This implies that we need to determine the dimensions for which there is normative control.

25 Heterogeneous Data Sets: Cluster Analysis? Include the length dimension in the cluster analysis

26 Cluster Analysis: Problem

27 Heterogeneous Dimensions: Principal Component Analysis?

28 Two Binds Bind 1: To subdivide the data set into homogeneous subsets, we need to know the dimensions for which there is normative control. Bind 2: To subdivide the dimensions into those for which there is normative control, we need to know the homogeneous subsets.

29 Double-Bind Problem Well-defined variables Not well-defined variables Well-defined data set Data set for statistical analysis Principal component analysis Not well defined data set Cluster analysis Data set brought forward for analysis

30 An Iterative Solution to Double-Bind Problem Examine each variable for a multi-modal distribution Subdivide data set by the modes of the multi-modal distribution Repeat on each subset until no further subdivisions are possible Examine pairs of variables for a multi-modal distribution Subdivide data set by the modes of the multimodal distribution Repeat on each subset until no further subdivisions are possible And so on.

31 Bimodal Distribution: Base Height Data Set S 1 (concave points) Data Set S (64 projectile points) Data Set S 2 (convex points)

32 Bimodal Distribution: Maximum Width Data Set S 1 Data Subset S 11 (narrow points) Data Subset S 12 (wide points)

33 Scattergram Plot (Length versus Tip Angle: S 11 and S 12 ) S 111 (broken, retouched) S 112 (intact) S 121 (broken, retouched) S 122 (intact)

34 Intact and Re-Sharpened Projective Points resharpened wide intact resharpened narrow

35 Characterization of Concave Projectile Points

36 Conclusion Insightful use of statistical analysis of data requires concordance between the conceptual basis of statistical methods and the structuring processes for the data brought forward for analysis. Heterogeneous data sets are typical since research goals are aimed at elucidating the processes structuring the phenomena of interest, hence welldefined data sets and well-defined sets of variables are not known in advance. Methods for doing this kind of pre-analysis of data are at a preliminary stage and the iterative method illustrated here is largely heuristic and does not yet have a well-developed formal foundation.