## Abstract

There is increasing awareness in the literature that the more commonly utilised approaches to measure wealth from assets are problematic. The theoretical foundations of sum-scores are weak and principal component analysis (PCA) is an inappropriate statistical method to apply to categorical data, akin to using a linear OLS regression with categorical data. Latent trait modelling (LTM) offers a statistically superior approach to measure household wealth. There are powerful arguments for using LTM: it takes into account the categorical nature of asset data; it is explicit about the assumptions underpinning the model, and it allows for inferences to be made about the broader population. This article applies LTM to three Malawian Demographic and Health Surveys, and compares results to those of a PCA approach. While the correlation is moderately high, indicating that a similar concept is being measured, results from LTM reflect the characteristics of the asset data and therefore represents a statistically superior measure of household wealth. Further research that draws on LTM methods to calculate wealth indices is to be encouraged.

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## Notes

- 1.
The terms ‘wealth index’ and ‘asset index’ are used interchangeably in the literature. Here I use the former, for consistency.

- 2.
Specifically, correlations are high where transitory shocks to expenditure are small and where random measurement error in expenditure is small, and in settings where individually consumed goods are the main component of expenditures (Filmer and Scott 2012). In settings where the opposite is true, results are quite different.

- 3.
PCA weights for household assets are derived and then used to construct a weighted aggregate index.

- 4.
LTM is also referred to as item response theory (IRT).

- 5.
I have not come across a normative approach to calculating wealth.

- 6.
See Szeles and Fusco (2013) for a discussion on item selection, as an important step in LTM.

- 7.
In the applied example here, I do not rank the categories, rather I treat variables with multiple categories as nominal, rather than ordinal.

- 8.
Such tests examine the extent to which results from the observed data could occur by random chance.

- 9.
For example, making inferences to the broader population, model diagnostics, including goodness-of-fit tests and statistical significance.

- 10.
Additionally, compared with PCA and MCA, another advantage of LTM is that the model can be used to specify a single, or several latent variables, depending on the underlying theory. In PCA and MCA the number of components (or dimensions) derived depends on the number of observed variables. Thus a PCA using four items will have three principal components, the first of which is often interpreted to be wealth, despite the second component perhaps explaining substantial variation in the items.

- 11.
The authors refer to it as item response theory (IRT).

- 12.
Documentation claims LTM is possible in Stata13: http://www.stata.com/stata13/generalized-sem/ (although not personally verified).

- 13.
Differences in terminology reflect disciplinary differences. The 'latent trait' terminology is used mostly in social psychology, and was coined by Lazarsfeld (1954), Skrondal and Rabe-Hesketh (2007). IRT is used mostly in educational research, first used by Lord (1980) for ability testing (Skrondal et al. 2007).

- 14.
A model with too many latent traits, given a number of items, is unidentified.

- 15.
Fixing the latent trait is necessary to identify the scale of the latent variable. Here the variance of the trait is fixed at 1. This could alternatively be done by fixing a parameter in the measurement model (e.g. setting the loading for variable ‘owning a flush toilet that is not shared' equal to 1). This would then scale the latent variable according to this variable's scale.

- 16.
For a binary item, this multinomial model is: \(\pi_{j1} (\eta ) = P(y_{j} = 1|\eta ) = \frac{{\exp (\tau_{j1} + \lambda_{j1} \eta )}}{{1 + \exp (\tau_{j1} + \lambda_{j1} \eta )}}\). The ordinal relationship would be modelled if the item were ordinal.

- 17.
If the question is coded one when answered correctly, and zero if answered incorrectly.

- 18.
When items are coded as one when the household is deprived of them, and zero when the household owns them.

- 19.
These are calculated using the default approach in Mplus. See Brown and Maydeu-Olivares (2012) for a detailed technical discussion of how latent trait scores are calculated.

- 20.
There are two types of uncertainty that could emerge from the estimation of scores. The first are predictive standard errors which do not take into account any uncertainty in estimates of the model parameters (e.g. loadings) but only the variation in responses (even for the same individual) over hypothetical repeated responses. These are produced by MPlus. The second takes into account the uncertainty in the parameters (as a result of sampling). A reasonable way of determining this type of uncertainty is with non-parametric boot-strapping.

- 21.
((

*O*−*E*)^{2}/*E*) where*O*is the observed frequency, and*E*the expected frequency. - 22.
While MPlus does not produce these fit statistics, marginal residuals can be calculated with R (see Kuha et al. 2012).

- 23.
A

*p*-value below 0.05 means we can reject the null hypothesis of no association at the five per cent significance level. - 24.
They continue stating that “[a]s land quality is highly heterogeneous, it is not clear that these few purchases or sales can be used to value land owned by all households. It should also be noted that the quantity of land owned may not always be a good measure of wealth. The amount of income that a rural household can generate will depend not only on the quantity of land it owns, but also whether the household can rent in additional land, whether the land is irrigated, whether it is flat, slightly or steeply sloped, and the type of soil” (383).

- 25.
MEASURE DHS stands for Monitoring and Evaluation to Assess and Use Results Demographic and Health Surveys. This project is implemented by the consultancy firm IFC International.

- 26.
Imputation was not necessary, as the number of missing values was minor.

- 27.
Comparing individual surveys where the survey design is taken into account separately (in each survey) is problematic (see discussion in Sect. 3 on iner-temporal comparisons), hence the choice of using a pooled sample.

- 28.
However, when analysing wealth a single DHS survey, it is possible to take weights and survey design into account with MPlus.

- 29.
Clustering tends to increases standard errors, whereas stratification has the opposite effect (Groves et al. 2009).

- 30.
AIC: With land—578,871.427; Without land—543,870.672

BIC: With land—579,297.302; Without land—544,278.803.

- 31.
Had it been possible to account for survey design, these results would allow for inference to the broader population, and

*p*-values could be interpreted as follows: there is evidence that in the population, the latent variable, wealth, is associated with all the items (*p*< 0.01). Such conclusions are possible to draw when analysing wealth in a single DHS survey, rather than making comparisons over time. - 32.
Unit difference between categories starting with zero (e.g. 0, 1, 2, 3).

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## Acknowledgments

I gratefully acknowledge the invaluable support from Dr. Sally Stares and Dr. Jouni Kuha, as well as Dr. Enrique Delamonica and an anonymous reviewer for their valuable suggestions. I take full responsibility for any remaining errors. I am grateful to MEASURE DHS for providing me with the DHS data and promptly addressing my queries. This work was supported by the Economic and Social Research Council (ESRC) through a studentship grant (1 + 3).

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Vandemoortele, M. Measuring Household Wealth with Latent Trait Modelling: An Application to Malawian DHS Data.
*Soc Indic Res* **118, **877–891 (2014). https://doi.org/10.1007/s11205-013-0447-z

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### Keywords

- Asset index
- Wealth index
- Latent trait modelling
- Principal component analysis
- Malawi
- DHS