Comparing Various Approaches to Create a Wealth Index Using Mexico 2010 Census Microdata

Comparing Various Approaches to Create a Wealth Index Using Mexico 2010 Census Microdata Angira Mondal1, Lara L. Cleveland2 1University of Pennsylvania 2University of Minnesota – Twin Cities Abstract A wealth index is a measure to determine a households' aggregate living standard and socioeconomic position. In the absence of true information about household wealth, social sciences often construct proxy measures using information about household assets collected in population surveys. Such indices are used to position households in relative terms for studying differences by socioeconomic status. In literature, classical principal component analysis (PCA) has been extensively used to create different wealth indices. The aim of this study is to explore various modifications of the classical PCA approach such as PCA with tetrachoric correlation matrix, and Sparse PCA with Pearson and tetrachoric correlation matrix to create household wealth index based on household asset data from IPUMS International 2010 census microdata of Mexico. Spearman's rank correlation along with quintile based ranking of the households is used to compare the consistency of the indices derived using the various methods. Key Words: Census microdata, PCA, SPCA, Person correlation, tetrachoric correlation 1. Introduction Measurement of socioeconomic position (SEP) is crucial to estimate poverty, inequality in the society, social determinants of health, and basically most demographic analyses. Conventionally, household income or expenditure is used to measure the socioeconomic position. However, collecting income or expenditure data is cost intensive and challenging, especially, in low-income and middle-income countries. To address this issue, household assets have been widely used as proxy measure of wealth and to determine the socioeconomic position of a household. Census data is collected across the world providing us information regarding the population and their housing characteristics. An important feature of the census data is the large-scale representative data for all population groups of a country. To build the wealth index using household assets, we use household level census microdata from the IPUMS International database. IPUMS International has one of the largest archives of household level and person level census data from around the world and across time. Traditionally, classical PCA which is PCA with Pearson correlation has been used to build wealth index, such as Demographic and Health Surveys (DHS) wealth index, using household assets. In general, the assets considered are discrete variables. Hence, using Pearson correlation in this scenario violates a key assumption of the requirement of continuous data to build the Pearson correlation matrix. To address this issue of binary data and the Pearson correlation matrix, we investigate the use of PCA with tetrachoric correlation as compared to classical PCA. Additionally, we also explore another modification of the classical PCA which is sparse PCA (SPCA) with both Pearson and tetrachoric correlation. 2. Data The 2010 household census mircrodata for Mexico, available on the IPUMS International database has been used to build the household wealth index. The inclusion criteria consist of households that are in the universe and have known household assets where universe is defined as the households or dwellings that were asked the census questions. The data structure is such that each row represents a household and binary asset variables. There are a total of 2,778,337 households in the sample and we consider 21 assets. All the household asset variables considered are harmonized variables, i.e., these variables are comparable across time and different country censuses. A binary classification of the harmonized asset variables is created to classify the assets as improved vs unimproved asset based on the Joint Monitoring Programme (JMP) by WHO/UNICEF and the Multidimensional Poverty Index. Table 1 presents the binary variable categories. Table 1: Binary classification of assets into improved and unimproved asset Asset Improved asset Unimproved asset Automobile >=1 automobile No automobile Bathing Facility Has bathing facility No bathing facility Cell phone ≥1 cellular phone No cellular phone Computer ≥1 personal computer No personal computer Cooking fuel Piped, tanked or bottled petroleum gas Wood and other plant fuel, charcoal and other fuel such as alcohol, biogas, discarded water material, dung or manure, etc. Crowding ≤2 persons per room > 2 persons per room Electricity Has electricity No electricity Floor material Cement Unfinished (earth) material Internet Has internet connection No internet connection Kitchen Has a kitchen to prepare food No kitchen Phone ≥1 phone No phone Radio Has radio No radio Refrigerator Has refrigerator No refrigerator Roof material Masonry, concrete, clay tile, other tile Wood and other plant materials, cardboard, discarded or scrap material, asbestos, sheet metal Sewage Sewage system for public sewage disposal or septic tank for private sewage disposal Not connected to sewage disposal system Toilet Flush toilet No toilet or non-flush toilet or other unspecified type of toilet Trash Trash collected directly from the household or indirectly from garbage container or deposit Trash burned, buried, thrown into street, vacant land, common area, water bodies, canyon or gulley or communal refuse dump TV >= 1 television No television Wall material Brick, block, stone, cement, wood Waste, scrap, discarded material, cardboard sheet, reed, bamboo, palm, adobe, mud, metal or asbestos sheet Water supply Has clothes washing machine No washing machine 3. Methods To compare the wealth index derived using various methods, classical principal component analysis (PCA), PCA with tetrachoric correlation matrix, and Sparse PCA (SPCA) using Pearson and tetrachoric correlation matrix, were implemented. The index for each household is created using a linear combination of all the household assets and the weights assigned, where the weights correspond to the PCA or SPCA loadings derived from the first principal component which we know captures the maximum variability in the data, assuming that the wealth of a household accounts for the highest variability in the access of assets for the household. 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊ℎ 𝐼𝐼𝐼𝐼𝐼𝐼𝑊𝑊𝑥𝑥𝑖𝑖=𝑊𝑊1𝑥𝑥𝑖𝑖1+𝑊𝑊2𝑥𝑥𝑖𝑖2+⋯+𝑊𝑊𝑝𝑝𝑥𝑥𝑖𝑖𝑝𝑝 where 𝑊𝑊1, … , 𝑊𝑊𝑝𝑝 represents the PCA or SPCA weights for the 𝑝𝑝 assets, i.e., the loadings derived from the first principal component and 𝑥𝑥𝑖𝑖1, … , 𝑥𝑥𝑖𝑖𝑝𝑝 represents the value of assets 1, … , 𝑝𝑝 for household 𝑖𝑖 such that 𝑥𝑥𝑖𝑖𝑖𝑖 = 0 𝑜𝑜𝑜𝑜 1, 𝑗𝑗= 1, … , 𝑝𝑝 as the asset variables are binary. Classical PCA is conducted on the Pearson correlation matrix of the data where the data consists of binary asset variables and is standardized, i.e., 𝑥𝑥𝑖𝑖𝑖𝑖 =⎩ ⎪ ⎨ ⎪ ⎧ 1−𝑥𝑥𝚥𝚥 � 𝑠𝑠𝑖𝑖,𝑖𝑖𝑖𝑖 ℎ𝑜𝑜𝑜𝑜𝑠𝑠𝑊𝑊ℎ𝑜𝑜𝑊𝑊𝐼𝐼 𝑖𝑖 ℎ𝑊𝑊𝑠𝑠 𝑖𝑖𝑖𝑖𝑝𝑝𝑜𝑜𝑜𝑜𝑖𝑖𝑊𝑊𝐼𝐼 𝑊𝑊𝑠𝑠𝑠𝑠𝑊𝑊𝑊𝑊 𝑗𝑗 0−𝑥𝑥𝚥𝚥 � 𝑠𝑠𝑖𝑖,𝑖𝑖𝑖𝑖 ℎ𝑜𝑜𝑜𝑜𝑠𝑠𝑊𝑊ℎ𝑜𝑜𝑊𝑊𝐼𝐼 𝑖𝑖 ℎ𝑊𝑊𝑠𝑠 𝑜𝑜𝐼𝐼𝑖𝑖𝑖𝑖𝑝𝑝𝑜𝑜𝑜𝑜𝑖𝑖𝑊𝑊𝐼𝐼 𝑊𝑊𝑠𝑠𝑠𝑠𝑊𝑊𝑊𝑊 𝑗𝑗 where ,𝑗𝑗= 1, … , 𝑝𝑝 represents the binary assets, 𝑥𝑥𝚥𝚥 � and 𝑠𝑠𝑖𝑖 are the mean and standard deviation of asset 𝑗𝑗 for all the households respectively. Since, the data consists of binary asset variables, an important assumption of the data being continuous for Pearson correlation matrix, is violated. To resolve this issue, tetrachoric correlation is used to build the index and the results are compared to that of Pearson correlation. The original concept of tetrachoric correlation was proposed by Pearson (1900) and later it came to be known as tetrachoric correlation. Tetrachoric correlation estimates the correlation between two underlying latent continuous variables that are only observed dichotomously. In the case of the household assets, we assume the latent continuous variables to be the propensity to have the asset and the overall underlying continuous latent variable to be the wealth and socioeconomic position of the household. Another method that was used to create the index is SPCA. Zou, Hastie, and Tibshirani (2006, as cited in Xie et al., 2023) proposed this method that constructs principal components with sparse loadings by introducing a quadratic penalty and a lasso penalty (1-norm penalty) in the PCA objective function. For any 𝜆𝜆> 0 and 𝜆𝜆1,𝑖𝑖≥0, �𝛼𝛼,�𝛽𝛽󰆹�= �|𝑋𝑋𝑖𝑖−𝛼𝛼𝛽𝛽𝑇𝑇𝑋𝑋𝑖𝑖|2 𝑛𝑛 𝑖𝑖=1 +𝜆𝜆�|𝛽𝛽𝑖𝑖|2 𝑘𝑘 𝑖𝑖=1 +�𝜆𝜆1,𝑖𝑖|𝛽𝛽𝑖𝑖| 𝑘𝑘 𝑖𝑖=1 subject to 𝛼𝛼𝑇𝑇𝛼𝛼=𝐼𝐼𝑘𝑘, where 𝛼𝛼𝑝𝑝 × 𝑘𝑘, 𝛽𝛽𝑝𝑝 × 𝑘𝑘 and 𝑋𝑋𝑖𝑖 denotes the 𝑖𝑖𝑡𝑡ℎ row vector of 𝑋𝑋𝑛𝑛 × 𝑝𝑝. The SPCA loadings of each principal component is provided by each column of 𝛽𝛽󰆹. The asset weights is provided by the loadings of the first principal component by normalization of 𝛽𝛽󰆹1, i.e., 𝛽𝛽 �1 � 𝛽𝛽 �1�. 4. Results To conduct SPCA with Pearson and tetrachoric correlation, 5-fold cross-validation was used to select the quadratic penalty parameter from the range 10∧𝑠𝑠𝑊𝑊𝑠𝑠(−6, 0, 𝑏𝑏𝑏𝑏= 1) and lasso penalty (1-norm) parameter from the range 𝑠𝑠𝑊𝑊𝑠𝑠(0.001,1,𝑏𝑏𝑏𝑏= 0.2) . The optimal parameters chosen for these two methods such that highest PVE is achieved are 0.001 for 1-norm penalty and 1 for quadratic penalty. To perform all the analyses R 4.0.2 has been used. We observe that across the various methods implemented, the top three assets with the highest weight assigned remain consistent – bathing facility, access to toilet and cooking fuel. The proportion of households that have access to bathing facility, toilet and cooking fuel are 46%, 46% and 63% respectively. However, when the asset weights beyond the top three are considered, we observe that the order changes depending on the type of correlation matrix and method. For example, in the case of tetrachoric correlation, we observe that access to internet has the 4th highest weight, but it is 14th in the order of asset weights when Pearson correlation is used to form the index. Figure 1 illustrates the asset weights along with the proportion of households that have access to the assets. Figure 1: Comparison of asset weights using PCA and SPCA with Pearson and Tetrachoric correlation (with chosen parameters) and proportion of households than have access to an asset. 00.2 0.4 0.6 0.8 1 Bathing Facility Toilet Cooking Fuel Internet Computer Refrigerator TV Washer Electricity Trash disposal Sewage System Phone Cellular phone Floor Automobile Roof Water supply Crowding Radio Wall Kitchen Asset Weights 00.5 1 Proportion of households that have access to the asset Comparing PCA and SPCA, we observe that the asset weights for SPCA are lower in magnitude than those for PCA. However, for the chosen parameters, SPCA does not discount any of the assets considered, i.e., we do not get a sparse loading vector and even though the asset weight is quite small, i.e., none of the assets is assigned the weight of zero. The chosen parameters based on maximum PVE, applies a small penalty for creating sparsity and strong penalty that shrinks the parameters. We also considered the parameters which have the next highest PVE, i.e., 1-norm parameter of 0.401 with quadratic penalty parameter of 1 and 1-norm parameter of 0.401 with quadratic penalty parameter of 0.1, but we do not get sparse loadings for both these cases. When we look through the ranges considered, we conclude that sparsity is introduced with a cost to the PVE. Additionally, the PVE for PCA and SPCA with tetrachoric correlation is 57%, compared to 33% for Pearson correlation. We also note that if we choose parameters for SPCA with the intension to prioritize sparsity over PVE, the PVE decreases from 57% and hence PVE for SPCA with tetrachoric correlation becomes less than PCA with tetrachoric correlation. We observe that tetrachoric correlation is more computationally intensive as compared to Pearson correlation. Also, SPCA is more computationally intensive as compared to PCA. Additionally, performing the 5-fold cross validation to select the optimal parameters adds to the computational load. Thus, based on the PVE, lack of sparse loadings for SPCA and the trade-off with PVE and computational time and intensity, we conclude that PCA with tetrachoric correlation performs the best out of the four methods considered. Figure 2: Heat map with number and percentage of households representing the rank assigned by PCA with tetrachoric correlation, SPCA with Pearson correlation and SPCA with tetrachoric correlation compared to the PCA with Pearson correlation. Based on the asset weights computed, for each of the four methods, wealth index is computed for each household and a quintile ranking of the index is created such that rank 1 represents the poorest household and rank 5 represents the richest household. We observe that when we compare PCA with tetrachoric correlation to PCA with Pearson correlation, SPCA with Pearson correlation and SPCA with tetrachoric correlation, we find that 10%, 10% and 21% of the households are assigned a consecutive rank respectively. Additionally, Figure 2 illustrates that in cases when a consecutive rank is assigned PCA with tetrachoric correlation assigns a lower rank as compared to PCA with Pearson correlation, i.e., a household assigned rank 2 by PCA with Pearson correlation is assigned rank 1 in this case. However, we observe that when PCA with tetrachoric correlation assigns households as rank 5, i.e., the richest households, about 1% of the households are assigned rank 4 by SPCA with tetrachoric correlation. Spearman’s rank correlation is used to compare the quintile rankings created by the four methods and we observe that 𝜌𝜌≥0.95 as shown in Table 2. The heatmaps along with the high correlation between the ranking of the households derived using the four methods show high consistency across the methods. Table 2: Spearman’s rank correlation (𝝆𝝆) for the wealth index quintile rankings PCA with tetrachoric correlation SPCA with tetrachoric correlation PCA with Pearson correlation SPCA with Pearson correlation PCA with tetrachoric correlation 1 SPCA with tetrachoric correlation 0.95 1 PCA with Pearson correlation 0.97 0.95 1 SPCA with Pearson correlation 0.97 0.95 1 1 5. Discussion 5.1 Conclusion Traditionally, PCA with Pearson correlation is used for building a wealth index. To address the violation of the assumption of using Pearson correlation for a dataset with binary variables, we compare the results of PCA and SPCA with Pearson correlation to that of PCA and SPCA with tetrachoric correlation. We observe that higher PVE is achieved by tetrachoric correlation (57%) as compared to Pearson correlation (33%) irrespective of whether PCA or SPCA is implemented. Additionally, SPCA with the chosen parameter based on cross-validation, does not provide us with sparse loadings. Comparing PCA with tetrachoric correlation to SPCA with tetrachoric correlation we find that even though the PVE is 57% for both, SPCA with tetrachoric correlation is more computationally intensive. We conclude based on the lack of sparse loadings by using SPCA, higher PVE and the computational intensity, PCA with tetrachoric correlation performs the best and is chosen as the final model. The heatmaps where even if rank assigned by two methods is different, it is always the consecutive rank, and high Spearman’s rank correlation (≥0.95) between the quintile ranking of the household wealth index, shows high consistency between all the methods. 5.2 Future work To substantiate that PCA with tetrachoric correlation performs the best amongst the methods considered, these methods will be implemented to census microdata from various developing countries. It would also be evaluated if there is a consistency in the household assets that are assigned the maximum weight across countries and develop a global optimal set of asset variables for developing wealth index. 6. Acknowledgment The authors wish to acknowledge the statistical office that provided the underlying data making this research possible: National Institute of Statistics, Geography, and Informatics, Mexico. The research has been supported by National Science Foundation (SES 1852842, 2345439; 09/1408/29), National Institute of Child Health and Human Development (R01HD047283; 07/21-06/26) and National Institute on Aging (R01AG062601; 08/19-07/29). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Science Foundation or National Institutes of Health. 7. References Alkire, S. and Jahan, S. (2018). ‘The New Global MPI 2018: aligning with the Sustainable Development Goals’, HDRO Occasional Paper, UNDP. Chasekwa, B., Maluccio, J. A., Ntozini, R., Moulton, L. H., Wu, F., Smith, L. E., Matare, C. R., Stoltzfus, R. J., Mbuya, M. N. N., Tielsch, J. M., Martin, S. L., Jones, A. D., Humphrey, J. H., Fielding, K., & SHINE Trial Team (2018). Measuring wealth in rural communities: Lessons from the Sanitation, Hygiene, Infant Nutrition Efficacy (SHINE) trial. 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