SMALL AREA ESTIMATES OF INCOME: MEANS, MEDIANS

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1 SMALL AREA ESTIMATES OF INCOME: MEANS, MEDIANS AND PERCENTILES Alison Whitworth (1), Kieran Martin (2), Cruddas, Christine Sexton, Alan Taylor Nikos Tzavidis (3), Marie Keywords: Model based estimates of income, Average household income, Median household income, Quantiles for income, Empirical best predictor. 1. INTRODUCTION The Office for National Statistics (ONS) currently produces model based estimates of mean household income and the proportion of households below the nationally defined poverty line for Middle Layer Super Output Areas (MSOAs) in England and Wales, (ONS, 2010 [1] ). The estimates have limitations in meeting user needs though because the mean income provides little information about the distribution across households and can be inflated by the relatively small number of households with very large incomes. Estimates of the median income (together with other quantiles) are considered to be more useful and would better inform user requirements. The current methods used by ONS for the published small area income estimates, model survey data in terms of area level aggregates and cannot be easily modified to estimate median income. Recent advances in small area estimation, for example the methods developed by Molina and Rao (2010) [2], provide a flexible approach by using simulation techniques based upon modelled parameters to obtain estimates for the whole population. The simulated estimates are then used to derive measures for the distribution. This paper explores an application of the method provided by Molina and Rao, applying it to 2001 Census data from the North West and South East regions of England to provide estimates of medians, means and quantiles of income at MSOA level. The method is assessed by measuring the precision of the estimates, and comparing them against the published estimates as well as independent proxy data for income such as the indices of deprivation. 2. METHODS 2.1. Current approach for estimates of household income The current approach described in ONS (2010), is to estimate the area (MSOA) level relationship between the survey variable and auxiliary variables by regressing individual responses from the Family Resource Survey (FRS) on area values of the covariates. The FRS is the survey with the largest sample that includes suitable questions on income and aims to interview all adults in a selected household. In 2001 a final sample size of 23,790 households was achieved. The auxiliary variables are generally average values of proportions relating to all individuals or households in the area and are based on administrative or census data with coverage in all the areas being modelled. 1 2 Office for National Statistics, Segensworth Road, Titchfield, Hants This paper reports on part of an ongoing project within the Small Area Estimation Branch, ONS, to improve the current published estimates of income and poverty. The initial development and programming was undertaken by Dr Kieran Martin who has since left the team. 3 Dept of Social Statistics & Demography, Southampton University, Hants

2 The model for income is: ln(y ir ) = α + βx k(ir) + u r + e ir ; y ir is weekly income for household i in postcode sector (PCS) r; X k(ir) is the population mean for the covariate in MSOA k that household i in PCS r falls within; α and β are the regression parameters for intercept and slope respectively; u r is the area level residual assumed to have expectation 0 and variance σ v 2 ; e ir is the individual within area residual, with expectation 0 and varianceσ e 2. Once the model has been fitted the model parameters are applied to the covariate values for each area to obtain the target estimates. While the model is constructed only on responses from sampled areas, the relationships identified are assumed to apply nationally Empirical best predictor (EBP) approach The EBP approach starts by fitting a standard mixed effects model which relates the observed survey household income to a set of covariates common to both the survey and census. The estimated parameters of the distribution of the out of sample income can then be obtained using the estimated parameters from the fitted model. The next step is to produce the EBP of the statistic we are interested in, for example MSOA household median income. For each out of sample household in the population a fixed number of estimates of household income are simulated through random sampling of the estimated conditional distribution, and the median household income is found for each MSOA for each of these simulations. The EBP of median household income for an MSOA is then obtained by averaging over all of the medians obtained from each of the simulations for that MSOA. The final stage is to obtain an estimate of the variation associated with the estimated median. A parametric bootstrap sampling technique is applied which samples the distribution of the fitted model to produce a large number of bootstrap census populations where every household has an estimated income. The true median household income is obtained for each MSOA in each bootstrap population. From each bootstrap population a new sample is drawn and the EBP estimation process applied to obtain a bootstrap estimate of the MSOA median household income. The MSE can then be calculated from the true median household incomes and the estimated median household incomes. Since estimates of income are obtained for individual households at each stage, it is relatively easy to obtain whatever small area statistic is required, for example mean household income, median household income, proportion of households in poverty, etc. For the application presented in this paper the method was applied to produce estimates of household income at MSOA level using the FRS individual household level covariates, 2001 Census household level covariates and MSOA level covariates sourced from the 2001 Census, the DWP administrative data and other sources of administrative data. (Details of the covariates considered are included in the paper.) Household income was defined as net household weekly income (adjusted for household size and composition after housing costs.

3 3. RESULTS The median weekly household income for MSOAs in the North West and South East of England was 299 in The median weekly income for the 25th percentile was 197; so on average the bottom 25% of households within MSOAs had a weekly household income of 197 or less. Conversely on average the top 25% of households within MSOAs had a weekly household income of 454 or more. The median weekly household income for MSOAs in the North West was lower than that for the South East (not shown). Table 1. Summary statistics for the MSOA percentiles of household income Percentile Minimum Median Maximum ,905 Table 2 gives summary statistics for the coefficients of variation (CVS). All CVs of variation remain small, indicating that a high degree of accuracy can be achieved. As might be anticipated, the percentiles with the most data, such as the median, have the lowest coefficient of variation, while for the 2.5th and 97.5th percentiles the coefficient of variation is larger. Table 2. Summary statistics for the coefficients of variation for the MSOA percentile income estimates Minimum Median Maximum 0.05 Quartile Quartile st 3 rd If more percentiles are calculated they can be used to produce visualisations of the income distribution in an MSOA. Figure 1 shows, for example, that for the MSOA in Reading there was a high peak in the proportion of households with income at approximately 250 per week and a steep decline in the distribution for higher incomes (with virtually no households above 1,500 per week), whereas for the MSOA in Chiltern the peak in the distribution was much less acute and at a higher income at around 350 to 400 per week. For Chiltern there was a more gradual decline in the proportion of households with higher income and some have an income greater than 1,500 per week.

4 Figure 1. Income distribution across four different MSOAs 4. CONCLUSIONS This application provides a practical demonstration of the method and provides proof of concept that estimates have been derived for the mean, median and quantiles of income as well as for poverty, all under one estimation model. Assessment of the estimates was promising; the coefficients of variation indicated that the estimates are precise and the low income and poverty estimates were significantly different from the higher ones. Although not directly comparable, assessment of the EBP estimates against proxy estimates gave assurance that model bias is relatively small. A limitation of the method however, is that individual level census data is required and so estimates can only be derived in census years. The usefulness of an individual set of estimates and how they can be used alongside the series of official estimates must be established. They will only provide a viable alternative to the current approach if the method can be extended to provide estimates between census years Future work The next step is to apply the method to all areas of England and Wales and use 2011/2012 data to provide updated estimates. Benchmarking the updated EBP estimates against direct estimates of income at region and country level for England and Wales (as undertaken for the official estimates) will ensure consistency in outputs across geographies and will facilitate a full assessment including more direct comparison between estimates produced using the EB method and the official estimates. The diagnostic plots for the model indicated that the underlying modelling assumptions may not be fully valid; residual analysis demonstrated a marked deviation away from the normal distribution. An alternative transformation of the response variable or an alternative model may prove more appropriate for the data. Finally a feasibility exercise should be undertaken to assess the potential to extend the approach to update the Census covariate data used in the model so that updated estimates could be derived between census years. This would provide a means for a continuous series of estimates with the flexibility to meet user requirements.

5 REFERENCES [1] ONS Validation report: Model-based estimates of households in poverty for Middle Layer Super Output Areas in England and Wales, ONS publication: /2008. [2] I. Molina, and J. N. K. Rao, Small area estimation of poverty indicators. The Canadian Journal of Statistics 38, (2010),