Quantifying Patterns in Mortuary Practices: An Application of Factor Analysis and Cluster Analysis to Data From the Taosi Site, China

: In Chinese mortuary research, too much reliance on traditional qualitative typological analysis renders quantitative attributes of mortuary practice data ignored. Examining the Taosi cemetery one of the famous cemeteries of Neolithic China ( 2300 – 1900 BC ) , this study discusses the advantages and disadvan - tages of typology and digital methods. Extant qualitative research has classi ﬁ ed the Taosi burials into six vertical categories, representing a pyramidal social hierarchy. However, this approach solely relied on the labor expenditure principle, whose outcome was highly subjective and di ﬃ cult to verify. This study applies a multivariant analysis. Factor analysis is used to investigate the correlations within the mortuary data. The statistical factor scores quantify the di ﬀ erences between the combination of burial objects in di ﬀ erent tombs and allow clari ﬁ cation by cluster analysis to investigate their di ﬀ erent social meanings. The results reveal two - axial divisions in the Taosi cemetery: vertical strati ﬁ cation based on hierarchy and horizontal di ﬀ er - entiation based on social categories/identities. Compared with the simplistic typological description, such a quantitative method reveals the characteristics of each category more clearly, clari ﬁ es the classi ﬁ cation criteria, and extracts more detailed information about the society and its mortuary practices.


Introduction
The Late Neolithic period in China witnessed a high degree of social complexity (Liu, 2005;Liu & Chen, 2012).An obvious vertical social stratification and horizontal division of social identity emerged.This social complexity was reflected in the complexity and differentiation of the patterns of mortuary practices.Burials at various Late Neolithic sites in China have shown a tendency for diversification and stratification, as evidenced by differences in burial size, diversification of burial objects, and differentiation of mortuary rituals (Dong, Lin, Zhu, Luan, & Underhill 2019;Liu, 2003;Underhill, 2000Underhill, , 2002)).Such trends have led archaeologists to make certain classifications (Dai, 2019;Pearson, 1981;Yan, 2013;Yu, 2000).
In traditional archaeological research, the typology of different burial types basically applies the labor expenditure principle: the labor invested in a mortuary practice should positively relate to the social status of the deceased (Binford, 1971).In such a classification, which is influenced by the Binford (1962Binford ( , 1965Binford ( , 1971) ) and Saxe (1971) burial theories, the size of the burial or the number of grave goods becomes the main reference standard to classify and rank the different types of burials.Such an analytical approach has classified different burial classes to demonstrate vertical social stratification (O'Shea, 1984;Tainter, 1978).However, this kind of classification and the corresponding theory are increasingly being questioned because of the ignorance of various funerary practices of different societies and the symbolic meaning of certain burial goods (Hodder, 1980(Hodder, , 1982;;Pearson, 1982;Shanks & Tilley, 1982).The symbolic meaning and ideology of burials are important dimensions that indicate social will and different social groups, which are not discernible simply by the amount of labor invested (Hirth, 2003).Therefore, archaeologists are gradually paying attention to the symbolic meaning of special objects and culturally contextualizing certain mortuary patterns.
While researchers have begun combining the two theories to integrate mortuary analysis (Liu, 1999;Morris, 1991), most burial classifications in Chinese archaeology still apply the traditional typology based on the labor expenditure principle to determine political differentiation from top to bottom.Its main drawback is that it sets an empirical presupposition to seek vertical stratification of society (i.e., differences in labor input are a presupposed premise for political rank).Such a presupposition may be multiculturally maladaptive and detrimental for the discussion on cultural specificity and human agency mentioned above.
In order not to presuppose classification criteria and to get rid of the top-down approach, factor analysis within multivariate analysis can be helpful (McHugh, 1999;Vierra & Carlson, 1981).Factor analysis can assess all burial attributes as a whole and examine the potential relationships that exist among all variables in an integrated mortuary manner.Furthermore, combining this with cluster analysis can effectively classify the burials according to different potential relationships of variables.To examine the effectiveness of such a multivariate analysis method, this research examined the Taosi cemetery, a representative of the highly complex cemetery of the Neolithic period in China, to classify the different types of burials.Then, the social attributes were combined with the classification results to understand the social dimension of mortuary behavior.Finally, the results were compared with extant studies to discuss the advantages and disadvantages of multivariate analysis in the application of burial classification.
2 Multivariant Analysis: Factor Analysis and Cluster Analysis

Factor Analysis
Factor analysis examines the dependencies within a correlation matrix of original variables and attributes a number of variables with intricate relationships to new dimensions called factors.These factors elucidated and explained in detail with regards to their potential relationships between variables (Baxter, 2015;Jolliffe, 2002;Shennan, 1988;VanPool & Leonard, 2011).In archeological research, factor analysis is widely used to investigate common factors representing underlying dimensions in archaeological attributes and to elucidate socio-cultural mechanisms behind such dimensions (McHugh, 1999).Factor analysis was applied to archaeological research in the 1960s and became particularly popular in the 1970s (Doran & Hodson, 1975;Vierra & Carlson, 1981).Especially in burial studies, factor analysis has been considered an important method.Factor analysis can summarize the common meanings of attributes in a burial through the structure of their interrelationships.These common meanings represented by factors may reflect certain social relationships or social meanings, such as gender and age divisions, evidence of chronological change, or evidence of vertical status differences (Baxter, 1994;Parry & McArdle, 1991;Tainter, 1975Tainter, , 1978)).For example, Vehik (1977) applied factor analysis to study the 24 burial mounds and cairns in southwest Missouri (AD 500-100).He observed that factor analysis allowed him to classify three temporal groups within the burial complex based on different factors, indicating different trade relationships.Furthermore, factor analysis of grave goods groups can also detect a fraction of the potential markers of different groups in delineating ethnic and social structures (Korokhina & Grechko, 2021).
There are also many researchers who have adopted similar research methods for use in mortuary analysis, such as principal component analysis (PCA) and correspondence analysis; both of which employ a similar dimensionality reduction thinking to represent complex relationships of variables in a manageable way (Greenacre, 1984;Baxter, 1994;He, 2019).PCA can condense the original complexities in data into several principal components through matrix transformations and investigate the linear combination of each principal component with the original variables (Shennan, 1988).However, each original variable has a certain loading in the principal component, and the distribution of the magnitude of these loadings is not clearly demarcated, making it difficult to clearly express which original variables are represented by each principal component.Therefore, the extracted principal components cannot clearly explain what they represent.Most results of PCA are used as mediators for other analyses (McHugh, 1999).CA is a method of studying categories data and can be seen as "an attempt to define new variables that explain as much as possible of the departure of a table from the form it would have If there were no association between rows and columns" (Baxter, 1994, p. 114).It can suggest the relationship between variables graphically but do not give a specific statistic to measure this correlation.
With such consideration, although the above two analysis methods can analyze the relationship between variables, they are inferior to factor analysis in terms of interpreting the meaning of the results.In particular, factor analysis can more clearly explain the social meaning of the common factors through the factor rotation (Jolliffe, 2002).Therefore, factor analysis is a more suitable method for burial research and utilized in this study.
In recent years, factor analysis has been used in studies on Chinese burials.Shelach (2001Shelach ( , 2016) ) used factor analysis in his study on the Chinese Bronze Age burials in Inner Mongolia.He concluded that status and gender identity were the most obvious differences among the burials and that differences in social status also shaped the different artifact combinations.In addition, Jaffe (2012) used factor analysis to examine Chinese Western Zhou tombs, and based on the variation and combination of factor scores, he found that there were burial groups with different orientations in the same cemetery.These groups represented the identities of local burial systems that were different from the central dynastic system of the Western Zhou.Therefore, factor analysis can provide a bottom-up approach to burial research, circumventing the bias associated with the top-down approach inherent in previous studies (Jaffe, 2012;Jaffe & Cao, 2018).Thus, factor analysis is appropriate to examine a complex set of burial data to determine their potential relationships and possible patterns.

Cluster Analysis
Cluster analysis is a common quantitative classification method based on internal similarity and partitioning between groups to categorize data into different groups (Rencher & Schimek, 1997).Archaeologists usually use cluster analysis to divide various archaeological data into groups of certain significance; it has been widely used in studies on burial data (Clark, 1969;Tainter, 1975;VanPool & Leonard, 2011).
However, the interpretation of the cluster results is surrounded by some controversy.Many archaeologists believe that cluster analysis may create communities that do not exist in the real world (Baxter, 2015).While this could happen sometimes, archaeologists must make appropriate adjustments to the clusters based on the cultural context, such as adjusting the criteria for classification thresholds based on the context, when interpreting results (Tainter, 1975).Despite such concerns, cluster analysis provides archaeologists with an operational quantitative classification method with clearly defined boundaries.
Based on the different functions and characteristics of the abovementioned methods, this study uses a combination to classify burials.Such a combination assesses burial behavior as a whole, comprehensively examines the correlations between burial attributes and extracts interlinked factors for cluster analysis.Such a method focuses on the possible common orientations and patterns contained in the data set with minimal human intervention, summarizes them, and creates an effective and quantifiable classification.

Archaeological Background
The Taosi cemetery is located in Shanxi Province in the Central Plains of China (Figure 1a), dating back to 2300-1900 BC (Zhongguo, Shanxi, & Linfen, 2015).It is considered one of the most complex societies of the Late Neolithic period in China (Liu & Chen, 2012).The Taosi society has been divided into three periods according to the pottery typology (Table 1).The settlement at Taosi has witnessed dynamic socio-political changes within the three periods mentioned in Table 1.
Between the early and middle periods, there was a time characterized by the concentration and development of political power.It reached its peak in the middle period.In the late period, due to possible natural disasters or wars, the regime was destroyed, and the entire Taosi society dissolved (He, 2018).The Taosi site depicts a complex social structure and a regular settlement layout corresponding to different social functions, including a possible palace area, craft area, ritual area, and a dedicated burial area for the community.
The Taosi cemetery is located in the southeast of Taosi, far from the main population activity area (Figure 1b).So far, 1,426 graves have been discovered (Zhongguo et al., 2015).All are earthen pit vertical burials, highly uniform in shape, with more than 80% of them oriented between 110 and 160°, indicating shared burial customs and same clan identification (He, 2018).The Taosi cemetery has a well-planned layout (Figure 1c) and is considered to be a communal cemetery of Taosi society (Gao, 2007).However, there is considerable inner variation within these tombs.Most of the burials are small (under 1 m 2 ) and do not have any grave goods.Only 265 burials possessed funerary objects and these varied greatly in type and number (Dai, 2019).According to stratigraphy and pottery typology, these burials belong to the early and late periods of the Taosi site, and no Taosi inhabitants were buried in this cemetery in the middle period.
The original report classified the burials in the Taosi cemetery into five types, wherein the first three types were further divided into different subclasses (Appendix 1).Based on such classification, Miyamoto (2005) concluded that the Taosi cemetery was broadly divided into three classes: large, small, and medium, defining different social classes, from the elite to general commoners.Dai (2019) delineated the possible differences in burial categories based on gender.Previous studies have recognized the Taosi cemetery as a large communal cemetery.The obvious differences in grave scale and the number of grave goods may represent the different hierarchical social units.They also argued that there is a strict hierarchical order between different ranks of graves and that the complete ritual system is reflected in the burial system.
The traditional classification of Taosi burials relied on the principles of labor input to attain vertical social stratification (Miyamoto, 2005); however, an excessive focus on social stratification ignores the possible horizontal social identity differentiation reflected in the burial pattern.While some burials included only jade and stone tools, others included more weapons.The original classification only defined these as differences in social hierarchy; however, such an explanation is not comprehensive or convincing.Such a classification does not provide a comprehensive picture of possible burial behavior and horizontal social differentiation.Different social groups may have different attitudes toward death and burials, and people may use different identity markers in mortuary practice based on their preference (Pearson, 2003).Therefore, the presence or absence of specific burial objects and the different tendencies observed in relation to the combination of burial objects can express not just social rank but also different social identities.The classification requires an approach capable of considering more horizontal differentiation than the traditional top-down approach.
In the face of large internal variation of Taosi burials, a multivariate analysis could contribute to the investigation of the potential markers of different social groups and provide a more comprehensive classification (VanPool & Leonard, 2011).

Materials and Methods
The sample in this research includes complete information on all burial attributes (N = 207).Each burial complex has been listed in the supplement material with detailed information.They date back to the early and late periods of Taosi society.There has been a controversy about the application of dummy data in factor analysis (VanPool & Leonard, 2011).To avoid the bias caused by this controversy, the study used only the burial attributes of quantitative variables.Furthermore, the author standardized the data, especially the number of jades.The jade decorations excavated from burials had inconsistent measurement standards in reports, and some samples that could be attributed to sets of decorations were described and counted separately but others were not (Zhongguo et al., 2015).Therefore, to avoid such counting confusion, in this article, jade decorations that could be grouped into sets, such as head ornaments (described in the original report as multiple head decorative jade pieces), were counted as a unified group and numbered 1 in the study.All the statistical analysis was done in SPSS (IBM SPSS Statistics for Mac, Version 26.0.0.0).
Before conducting factor analysis, it is essential to select the variables and test the model (Chen, 2005).First, not all variables in burials are correlated; some of them are occasionally caused by human will, which can cause interference in the statistical analysis.To examine whether these attributes are correlated with the overall pattern or not, this study used correlation analysis to filter them (Appendix 2).The resulting screened attributes were broadly classified into two categories.
(1) Variables related to labor investment including the area, the total number of grave goods, and the number of different types of burial objects.Bones, bronze items, and mandibula were excluded from the research data due to their weak relation with other variables.
(2) The so-called functional burial objects that may represent identification and burial rituals (Liu, 1999(Liu, , 2003)).This includes the number of weapons, tools, drinking vessels, cooking vessels, serving vessels, decoration and musical instruments, and so on.Variables in this category were strongly related and included in the factor analysis.Second, the research must ensure that the component extraction is scientific.The basic logic of factor analysis' is to examine the performance of each variable in a model that contains common factors.To determine whether the model building is scientific requires testing.In this study, the Kaiser-Meyer-Olkin (KMO) and Bartlett's test of sphericity were applied.In this study, the value of KMO is 0.8, which fulfills the general criteria for model construction (the general criterion is above 0.6 and higher values prove that the combination of variables chosen is more suitable for factor analysis) (Chen, 2005;VanPool & Leonard, 2011) and the Bartlett's test is also significant.The results revealed that the selection of variables and model was appropriate to conduct factor analysis (Table 2).

Factor Analysis Results
Factor analysis extracted three common factors (whose eigenvalue values are larger than 1) from the burial sample, explaining 60.38, 15.36, and 9.65% of the relationship among variance and more than 85% in total (Figure 2).Furthermore, this study utilizes rotation of factors to elucidate clearer social meaning of the factors extracted (Baxter, 1994;Jolliffe, 2002;VanPool & Leonard, 2011).VARIMAX is used to do rotation, which is effectively used in most cases of previous research (McHugh, 1999).
After rotating the factors and eliminating relatively uncorrelated attributes (Figure 2b and Table 3), it was observed that factor 1 embodied mainly indicators of labor inputs, such as area and wood vessels for eating and drinking functions.Therefore, factor 1 is likely to depict the tomb owner's financial power and their ability to feast.Factor 2 is mainly related to stone tools, especially stone weapons and the total number of grave goods.It is implied that the number of stone tools is closely related to the total number and factor 2 mainly represents a military force.Factor 3 contains variables with special functions, such as jade and musical instruments.Furthermore, the area factor has been included in factor 3, indicating a relationship with labor input.Jade and musical instruments from Neolithic China were considered a symbol of power in Neolithic China.Most noble burials were accompanied by musical instruments or large amounts of jade to represent their political nobility or religious status (He, 2018;Liu & Chen, 2012).Therefore, factor 3 could represent superiority in the realms of the political and religious.

Cluster Results
Factor scores that influenced each burial were the basis for conducting cluster analysis (Figure 3).Euclidean distance (the most common type of distance applied in clustering studies) and average linkage clustering method (which can avoid over-compression or expansion of the variable space to some degree) were utilized in this cluster analysis (He, 2019).When the threshold value was greater than 5, the clustering results could be divided into three large categories (Table 4), wherein cluster A consisted of four relatively independent burials.The discriminant analysis has proved the validation of this clustering result (Chen, 2005).The specific result is shown in Appendix 3.These burials displayed the largest labor input characteristics, that is, the largest burial area, a wide variety of burial objects, and numerous styles.In the previous classification of burials, cluster A burialswith the highest statuswere those of the rulers of Taosi society (Zhongguo et al., 2015).Cluster B contained few burials and displayed a large labor input, especially in terms of pottery and woodwares, which may be related to special status.Cluster C contained the largest number of burials and was significantly different from A and B. This group has a relatively small labor input, which implies that this category differs from A and B in terms of rank or social identity.The differentiation of the three clusters is consistent with previous research on the pyramidal structure of Taosi society (Miyamoto, 2005).This means that burials at the top represent the minority of the society and contain the most complex social information that allows them to exist independently from other burials.
To further examine the groups in the non-elite burial category, cluster C was examined separately.Cluster C was divided into seven groups when the threshold changed to around 1. A consistency test  measured whether the above grouping was scientific.Table 5 indicates that the mean and median values were significantly different between the groups, which proved that group C was statistically correct.

Further Explanation and Possible Social Meaning of Classifications
The results presented above constitute a preliminary classification of burials in the Taosi cemetery, and the different factor composition in each group represents the different types of burials and their respective characteristics.
Cluster A: The highest class of burials.As mentioned earlier, this cluster is characterized as having the largest area and the richest variety of burial objects and styles (Table 4).Furthermore, the most obvious feature that distinguishes it from other types is the presence of musical instruments.These also included a large number of weapons and feasting-related vessels.As mentioned in previous studies, these burials belong to the rulers of Taosi society.The significant variance of the factor scores of cluster A implies people's different preferences, irrespective of their social class.
Cluster B: This group is characterized by extremely large factor 1 values, with relatively small scores for factors 2 and 3.This indicates that group B is characterized by a large number of pottery and woodwares, especially cuisine-related vessels, while other kinds of burial objects, especially jade and decorations, were rare.The members in this cluster belonged to the early period of Taosi society.These burials are also of a large size.This group indicates that there was a group of elites in the early period who were honored with only feasting-related vessels (Figure 4).
Cluster C: This group has the largest number of tombs, including different trends of grave assemblages, which may represent different vertical and horizontal divisions.This group was further divided into seven subgroups (Table 6).
C-1: Compared to the burials in group B, group C1 embodies the opposite characteristics.This group has a very small factor 1 score and large factor 3 score.This represents a high number of jade vessels, especially jade ornaments, and jade tools (Figure 4c).This group does not contain pottery or woodware vessels.Such preferences indicate that this group of tombs emphasize the significance of jade decoration in mortuary practice.
C-2 and C-3: These groups have similar factor score combinations, with relatively high factor 2 and low factor 3 scores (Figure 4b and c).This indicates that they favored more weapons and stone tools.They contain more weapons than other groups in cluster C and represent the same group of people who shared the same burial preferences.As for the differences between these groups, the scores of factor 1 are higher for C-2, which may indicate that the tomb owners of C-2 were somewhat richer than those of C-3.
C-4: Based on the comparison of grave goods, this group may represent a statistical error and have no practical meaning.Quantifying the Mortuary Practices  1239 C-5, C-6, and C-7: These groups have a small variation in factors and represent small-scale burials.Only factor 3 scores have depicted differences among these three groups (Figure 4c).C-5 included a certain number of jade decorations and no cuisine vessels.This combination of burial goods is similar to C-1, where the scale is smaller.C-6 and 7 contain only a few burial artifacts.
It can be observed that among the subgroups of cluster C, C-2 and 3 and C-1 and 5 have a consistent tendency of factor composition; therefore, they can be integrated into one group with minor internal  differentiation based on the number of particular objects.C-6 and 7 represent the poorest burials as evidenced by the simple composition of burial objects (Figure 5).In summary, the Taosi burials can be divided into three major types, that is, A, B, and C, which correspond to large differences in labor inputs that supports the social stratification represented in previous studies.However, type C was further divided into four major categories according to different combinations and quantities of burial objects.Each of them represents a different burial tendency and social group.

Discussion
6.1 Social Differentiation in Taosi Society as Reflected Through the Burials: Differences in Social Hierarchy and Identity The pyramidal social structure has been confirmed by the burial classification structure observed in this study.In Taosi society, there was a clear vertical hierarchy of burials, evidenced by the vastly different labor inputs (Table 4).In type A, which represents the rulers of Taosi society, burials included the richest resources.The wealthy status is reflected by the total amount of burial objects and the large burial area.This mortuary tradition is similar to the other Neolithic chiefdoms in China such as the Liangzhu society (Qin, 2013;Renfrew & Liu, 2018) and Shandong Longshan societies (Underhill, 2000).Interestingly, unlike these chiefdoms, the Taosi rulers preferred to be buried with sets of cuisine vessels, including drinking and serving vessels, and kitchen-related items.During the Neolithic period, feasting was considered an effective way for the elite to demonstrate their political or economic power and negotiate social relations (Bray, 2003;Dietler & Hayden, 2010).Therefore, the presence of complete cuisine sets represents the superb feasting ability of the rulers and their high social status.Besides feasting goods, musical instruments became another significant ritualistic object in these burials, symbolizing the status of the chief in Taosi society.Burials in the next highest rank, that is, group B, have lesser total labor input than group A. They included a large number of wood and pottery vessels, which could be equal to group A burials.This proves the importance of feasting capacity in the society.Sets of pottery and wood cuisine goods became a common feature of elite burials.Unlike group A, group B did not have jade decorative objects and musical Quantifying the Mortuary Practices  1241 instruments.In traditional Chinese burial studies, jade and musical instruments are commonly regarded as prestigious goods.They are highly ritualistic and closely associated with religion such as the Liangzhu Culture.The difference between the two types of artifacts of groups A and B also highlights the rank differentiation among the elites.The ruler possessed the power of religious rituals in addition to enjoying great feasting abilities and secular dominance.The secondary elite possessed secular power and wealth but had no advantage in terms of religious rituals.Group C is clearly characterized by smaller and a lesser number of funerary objects.This represents a group of non-elite individuals and occupies 93% of all burials.This group represents the dominant group in Taosi society.However, in contrast to the previous approach, the existence of diverse identities of inhabitants beyond the elite group is also reflected through this classification.
Compared to the large size and various grave goods, burials categorized under group C with low labor expenditure display their own identity through different burial assemblages.Different burial assemblages were not due to the diverse ethnic groups of Taosi cemetery.The highly consistent burial orientation and shape of the Taosi cemetery, with a clear arrangement plan mentioned in previous studies, indicates that the Taosi cemetery was a communal cemetery.Furthermore, the human bone evidence proves that there was no obvious migration from other distant areas (Zhongguo et al., 2015).Therefore, social identity sufficiently explains the different preferences for burial assemblages.Groups C-2 and C-3 represent that section of the society which tended to be buried with weapons; they could have belonged to a militaryrelated group.Based on the area and different pottery, there was a division in power and status among this group, wherein burials of senior warriors included certain financial resources, as well as people who generally did not have the financial means to feast.Burials in groups C-1 and C-5 did not attach importance to the ability to feast and did not include serving or drinking vessels.Instead, they attached great importance to jade and decorations.Similarly, a hierarchical division is also observed based on the different amounts of jade and the burial size.As mentioned above, some jade is closely related to religion in Neolithic China, and jade decorations also represent a certain level of social power and special social status (Qin, 2013).So the owner of this group may be a group of people who had certain religious power in the Taosi society.
The classification of burials proves that there was a two-axial social differentiation in the Taosi society (Figure 5).The vertical differentiation of social status represents the power stratification, that is, from the rulers to the elites to the commoners.Horizontal differentiation represents the division of different social identities.The three categories of burials were weapons-orientated, jade-orientated, and small burials.Furthermore, there are internal divisions within these groups.Such a diverse composition of burial types and group differentiation represents a high degree of complexity in Taosi society.

The Advantages and Disadvantages of Multivariate Analysis
The multivariate method provides a clear classification logic and investigates the patterns underlying burial behavior.In the traditional method, the Taosi cemetery's size is used as the primary criterion to classify major categories.The burials and assemblages of different grave objects are added to the already classified framework.Due to this, when an assemblage did not completely correspond with a category, they were forcibly included to produce a specific type of result.Such bias explains why the classification boundary of the traditional method was blurred, and burials in the same category could not meet the classification interpretation.Factor and cluster analyses could detect mortuary patterns hidden in the variables without any presupposition and follow scientific, statistical logic.
Furthermore, compared to the traditional method, multivariate analysis can reveal more detailed information about mortuary practice and society.Although there is a certain connection between burial area and social rank, the size of the burial can only explore the division of social structure in a broad sense.Burials with obvious gaps can be divided by the attribute of area.Results of groups A and B are consistent with previous traditional results.However, when it comes to non-elite burials, the size attribute and the number of burial goods cannot fully explain the complexity of burials because burials have demonstrated a proclivity for different burial assemblages.Therefore, the traditional qualitative method is not suitable to classify small burials because much attention is paid to the area factor, which neglects burial goods and their combinations.This is where the results of the two methods differ.The multivariant method is better for exploring the different group compositions of non-elite burials.The groupings do not have any preconceptions or interventions that allow empirical judgments to misinterpret the data in certain societies.Furthermore, instead of the simple top-bottom grouping, the research also considered the principle of labor input along with the symbolic principle to determine the potential patterns of mortuary practice.
However, this method has a few disadvantages.Quantitative analysis is an examination of numerical values and is less sensitive to too small changes.Such drawbacks have also been pointed out by statisticians (Dash, Liu, Scheuermann, & Tan, 2003;Halkidi, Batistakis, & Vazirgiannis, 2001).The algorithm of hierarchical cluster analysis due to inefficient validation and lack of accuracy, mainly in the form of the lower clusters in small sized usually are too close to other clusters, which then increases the uncertainty of the clustering results.Often, complex burial information and burials with a larger number of burial objects can be clearly classified.Therefore, all the A-class burials were returned separately due to the unequal information of large and small burials.If the classification is conducted, there is further analysis of small data in conjunction with the archaeological context.This is why the threshold of clustering analysis becomes smaller due to small burials to examine more subtle changes.
Furthermore, the interpretation of clustering results cannot be separated from the traditional combination of qualitative analysis and archaeological contexts.In interpreting the results, the remains are archaeologically rich enough to examine the social information behind the groupings.Although quantitative classification can be informative, the purpose of the classification is not only to divide but also to explain the social information behind the burial behavior.For example, when sets of eating and drinking vessels are understood as embodying powerful feasting capacities associated with political regulation, jade burial groups with specific religious power and other information are inseparable from the special social meaning embedded in the previous qualitative examination.In quantitative methods, especially clustering processes, there are no predefined categories and no examples to show what kind of ideal relationships should exist between the data (Halkidi et al., 2001;Kaufman & Rousseeuw, 2009), so the validation of the statistical results in archaeology research must be done with specific social practices and cultural context as verification.Therefore, future research must remember that neither quantitative nor qualitative methods alone can explain complete burial behavior.Therefore, an effective combination of both is required.

Conclusion
This study applied multivariate analysis to classify the burials in Taosi cemetery, dating back to Neolithic China.The burials were divided into three large social groups based on different labor investments in mortuary practice.These groups represented the stratification of Taosi society, which was possibly related to wealth and political power.Moreover, the factor and cluster analysis observed the different preferences of burial assemblages within the non-elite group, indicating their possible different social identities.The result reveals two-axial divisions in the Taosi cemetery, which implies the complexity of Taosi society.This classification overcomes the drawbacks of traditional typology classification, clarifies the classification criteria, and extracts more detailed information about mortuary practice and society.
Of course, the multivariate statistical approach is not perfect.Some questions, such as how to standardize variables with different units before analysis, how to test whether the statistical results exist in the real world, and how to address the lack of sensitivity to small samples, need to be further explored and answered in future studies.
In conclusion, as discussed above, the qualitative and quantitative methods cannot be separated.Therefore, future studies must use a blend of these methods to conduct further research on burials.
Quantifying the Mortuary Practices  1243

Figure 1 :
Figure 1: The location of the Taosi site (a) and Taosi cemetery (b) and (c) the layout of the Taosi cemetery in the early period.

Figure 2 :
Figure 2: (a) The eigenvalues of factor analysis and (b) the component plot in rotated space.

Figure 3 :
Figure 3: The cluster results are based on factor scores for each burial (seven groups have too many members; to better display the results appropriately and save space, some members have been omitted from the figure; the completed results have been included in the supplementary materials).

Figure 4 :
Figure 4: Variation of factor scores in each burial group (a) factor 1; (b) factor 2; and (c) factor 3 (because of the large factor variation, they are not shown in the box plots to influence the comparison between the small values of other groups).

Figure 5 :
Figure 5: Final classification of burials in Taosi cemetery and possible implicated social structure.

Table 2 :
KMO and Bartlett'stest on factor analysis of data from Taosi

Table 3 :
Rotated significant component loadings of the variables Quantifying the Mortuary Practices  1237

Table 4 :
Average values of selected burial variables in different groups

Table 5 :
Homogeneity of variances among the seven sub-types of cluster C

Table 6 :
Averages of the selected burial variables in the subgroups of group C