Sequence Analysis

Construction of the Muyil ceramic sequence

            Appendix 3 contains the supporting statistical analyses used to begin the process of constructing ceramic complexes for Muyil. At the start, all sherds were first tabulated by type and variety to obtain total counts and weights. These data are shown in a table at the beginning of Appendix 3. The sheer quantity of data appeared overwhelming, and so from this tabulation, several numerically well-represented types were chosen to analyze further. Since my intent was to perform an initial factor analysis to look for associations in the data, I limited the input data to types with counts of 10 sherds or more or weights of 100 g or greater. Where possible the greatest level of detail, the ceramic variety, was used. Data was also grouped to two higher levels, however, the ceramic type and the ceramic group, and, as necessary, varieties and types were combined to obtain a large enough sherd count to use the variable. Factor analysis was used to search for strong associations between sets of cases (pit levels) and variables (sherd types or ceramic groups.) I was particularly interested to find temporal factors, among others, and the factor analysis did succeed at this task. Raw sherd counts and raw sherd weights were separately analyzed several times by factor analysis. Use of sherd weights as input produced results similar to those of using sherd counts as input, but without comparability to the seriation charts of Appendix 2 which show proportions by count. Weights were, therefore, not used for further analyses. In all, 432 cases, each representing the data from one excavation level of one test pit and 329 categories of sherds and other material were used in the early analyses to search for patterning in the data.

            Over the course of repeatedly performing a factor analysis, several variables (sherd types and other categories) were deliberately removed from the analysis. Specifically, faunal material, lithics, and other non-ceramic artifacts were removed from consideration. Also, as mentioned above, sherd variables with fewer than 10 sherds were dropped from the analysis. Lastly, many variables which I came to believe were ill-suited to the analysis were removed from the input data. For example, many sherds had been classified in the laboratory as "red monochrome." Since this category includes red-slipped sherds which could not be identified to a particular ceramic group or a particular type due to their poor condition, I found that the factor which loaded strongly on this variable correlated strongly not only with cases containing Sierra Red from the Formative, but also with cases containing Mama Red from the Late Postclassic. This association, while perhaps useful for other purposes, was frustrating our search for temporal associations in the data, and so the category "red monochrome" was dropped from further analysis. Other similar categories (black monochromes, browns, whites) were also removed from the analysis for the same reason. Likewise, some categories of sherds were grouped, to raise the counts for the grouped variable, while eliminating variables with a very low counts. This process resulted in reducing the number of variables used in the later factor analyses down to 33 (from the original 329).

            A similar pruning process was applied to the cases also. (A case consists of the sherds from one level of one test pit.) The raw data was reviewed to remove cases with fewer than 10 sherds. Also, several of the original cases were mixed lots containing sherds from more than one level, such as a profile-cleaning operation, and we removed these. Finally, lots that had obviously mixed material from several eras, such as often occurs in Maya platform fill, were identified by a review of the individual seriation charts for the test pits and the test pit profiles. This resulted in paring the number of cases for the later analyses to 273 of the original 419.

            Data was analyzed several times with a factor analysis computer program (4M) from the BMDPÔ Statistical Software System (1985) using the Tulane Computing Services IBM 3081. This program was used to extract principal components and to produce a factor loadings table using varimax orthogonal rotation. The number of components extracted was based upon the number of factors whose eigenvalues (variance explained) was >1.0, i.e., where the variance explained by the factor was greater that the variance explained by a single variable. In the later factor analysis runs, after the number of cases and variables had been reduced, the number of factors settled to nine or ten, with the last two or three of these factors having variances close to 1.0 and loading heavily on a single variable.

            The last analysis (included in Appendix 3) produced nine factors. Seven of these had a loading of >0.70 on two or more variables, which indicates that more than one of the ceramic variables was responsible for the factor's ability to explain variance. The variance explained by the individual factors ranged from 6.4 down to 1.68. These seven factors together accounted for 75% of the variance in the data set used. The last two factors, with a variance explained of 1.18 and 1.11 respectively, accounted for an additional 7% of the variance. Excerpts from the analysis are shown in Appendix 3. On page 18 of the output from BMDP, I have blanked factor loadings whose value is <0.25 in order to highlight those variables that have heavy factor loadings. Each variable name (ceramic variable) begins with a two-letter code which identifies its known date (such as "EC" for Early Classic) at other sites, and therefore its anticipated date at Muyil. The codes, which were used for identification, not to assert the temporal sequence, are explained on page 18 of the computer output in Appendix 3. The individual factors clearly show a clustering of those ceramics at Muyil that have a similar age at other sites. For example, Factor 1 contains heavy factor loadings on eight variables. All eight of these variables had been previously coded with the prefix "PR" to indicate that they dated to the Formative or Protoclassic elsewhere in the Maya area (see Appendix 3, BMDP output page 18). Finally, the correlation matrix of the variables (sherd groupings) in the factor analysis was tested using Bartlett's Chi-square and found to be significant with p<0.001.

            A factor analysis with oblique rotation was performed to check the extent to which some factors might be correlated. There already appeared to be an overlap of ceramic variables in factors three and four in the orthogonal rotation analysis. The output from the oblique rotation factor analysis showed that: (a) all the same factors were produced as had been produced with orthogonal rotation, and in the same order (decreasing amount of variance explained), but with slightly different factor loadings; (b) factor four was not loaded above 0.25 on either Vista Alegre Striated or Encanto Striated. The result is to make this factor seem slightly later temporally than factor four of the orthogonal analysis; and (c) the only factor covariances greater than 0.10 were between factors three and four (0.30) and factors five and six (0.22) and these levels of covariance are not remarkably high. On the basis of this rather low correlation between factors, we returned to and continued to rely on the orthogonal rotation factor analysis. No output from the oblique rotation analysis has been included here or in Appendix 3.

            The second step in the analysis was to examine the individual cases (one level of one pit) that showed high case scores (typically >1.0) on one or more of the factors. These cases with high scores were examined by test pit by test pit to see whether, as we suspected, the factors themselves would form a seriation. A seriation of these scores would indicate that the factors had a temporal meaning. Such a seriation was indeed apparent in the data.

            In the discussion that follows, we have supplied labels for the factors, such as "Late Postclassic" for factor two. These names evolved out of the analysis, not in advance of it, by observing not only the seriation of the factors but also the ceramic types within the factors, which, in the main, were known to occur within a particular time period at other sites. We supply them at this point in the discussion because they aid in understanding the argument: the factors from the factor analysis seriate within the test pits. They represent sets of ceramic types or ceramic groups which themselves seriate individually and which are co-occuring within one level.

Figure 3 Seriation of factors for test pit 11

            Test pit 11 (Figure 3) provides one of the clearest examples of the seriation of factors. For example, by examining the cases with high factor scores, one may observe high scores for factor seven (Early Classic) in the lowest levels (10 - 12), high factor scores for factor five (Late Classic) in levels 7 - 9, followed by high factor scores for factors four (Terminal Classic/Early Postclassic) and eight (Ticul Thin-slate: Xelha variety) in levels 1 - 4.

            As is generally true for the factor scores for other test pits, a few cases have high scores for one or two factors, and the scores for the other factors are small negative numbers — indicating a slight bias against that factor within that level of the pit. Of the nine factors, the graphic examples given here include only the three to five most important factors, i.e., those with large positive or negative factor scores. (but see Appendix 3, BMDP pages 21-26, for the complete set of nine factor scores for each case.)  

Figure 4 Seriation of factors for test pit 5  

           Several additional examples will clarify the point being made here about the seriation of factors. In test pit 5 (Figure 4), the factor analysis highlights levels 13-15 by the high factor scores for factor one (Formative) for these levels. No other levels (cases) and no other factors participate to a significant degree. Factor three, the factor with the next highest factor scores, is shown in the illustration for comparison.  

Figure 5 Seriation of factors for test pit 1  

            Test pit 1 (Figure 5) gives another factor seriation example which shows the Early Classic factor seven appearing lowest (level 10), Late Classic factor three appearing in the middle of the pit (level 7), and two other factors (four and eight, from the Terminal Classic and Early Postclassic) appearing uppermost, in levels 1 and 2.  

Figure 6 Seriation of factors for test pit 12  

            Test pit 12 (6) provides a clear example of levels that, based upon their factor scores, contain Late Classic material at the lowest level (level 8, factor five) and Terminal Classic/Early Postclassic material at upper levels (levels 0, 1, 2, factor four). Other factors and other levels have negative factor scores.

Figure 7 Seriation of factors for test pit 24  

            Test pit 24 (Figure 7) is an example of a Late Postclassic ceremonial location at which the most numerous sherds recovered are of Chen Mul Modeled censers. This type predominates in factor two, and levels one and two of the test pit indeed have high factor scores for factor two in levels 1 and 2.  

Figure 8 Seriation of factors for test pit 15  

           Test pit 15 (Figure 8) shows the Early Classic factor seven in level 6 occurring lower in the stratigraphy than the Late Classic factor three in levels 2 and 3. In both cases the factor scores are rather low, and this has the effect of magnifying the also-low negative factor scores for Terminal Classic/Early Postclassic factors four and six in the figure.

Figure 9 Seriation of factors for test pit 10  

           In test pit 10 (Figure 9), there are high factor scores for factor seven in levels 9 and 11, accompanied by much lower scores for factor one for the Protoclassic. Above these, in levels 5 and 6, factor six for the Terminal Classic makes its first appearance with positive factor scores.  

Figure 10 Seriation of factors for test pit 6  

            In test pit 6 (Figure 10), factor one (Formative) is most strongly represented in level 9. Much higher in the test pit at level 3, factor two (Late Postclassic) and factor four (Terminal Classic/Early Postclassic) have positive factor scores (accompanied by negative scores for factors three, seven, and eight.)

Figure 11 Seriation of factors for test pit 13 

            Test pit 13 (Figure 11) gives an illustration of a high factor score for factor three (Late Classic) in level six, with much lower factor scores for this same factor in all higher levels. It also shows higher factor scores for factor four (Terminal Classic/Early Postclassic) in levels 1-5 (above factor three.)

            Based on this seriation, which, as has been shown,  occurs in many of the test pits, and the known dates of the ceramics at other sites, we preliminarily identified the factors in this way for further study:

                 Our judgment of how these factors relate to one another in time is shown in the accompanying diagram (Figure 12).  

Figure 12 Chronological chart of the ceramic factors  

             The next step in the establishment of the ceramic complexes was the seriation of test pit excavation levels across the site. We began with the cases used in the factor analysis, which had provided the hoped-for insights into patterning in the data, and continued to prune out some cases (with mixed material) and some variables which contained a very low percentage of the total sherds within a level (<10%). Using this process and arranging both variables and test pit levels to show the oldest material in the lower left and the youngest material in the upper right, we prepared a seriation chart (Figure 12). Factor 2 (Chen Mul Modeled censers) was removed from near the top of the table to condense it, and because this ceremonial variety is so readily recognized when it occurs. This chart was used to suggest which test pits to use for the final seriation.

              This seriation (Table IV), in which the vertical black bars show the presumed temporal extent of one variable, divides into the following eras:  

© Copyright 2000-2008 Walter R. T. Witschey   Page last updated Wednesday, April 02, 2008