Poverty in Perspective: a typology of poverty in Scotland

Annex B - Latent Class Analysis: Detailed methodology

Latent class models (Lazarsfeld and Henry, 1968) are frequently used to explore possible typologies and patterns of clustering in data sets. We theorise that the associations between the poverty indicators are a function of some underlying general variable that characterises experiences of poverty. This variable is a categorical, nominal variable that classifies members of the population in terms of those different experiences. It is not tangible and cannot be observed directly, hence it is termed a latent variable. By contrast, the survey items are termed observed or manifest variables.

The basic latent class model can be specified most simply as follows:

x is a categorical latent variable, with q unordered categories j = 1, … , q; and

y _i ( i = 1, ... , p) are p observed or manifest variables, where y _i has c _i categories s = 1, ... , c _i.

We model the probabilities of belonging to class j:

η _j = P( x = j), j = 1, … , q

and the conditional response probabilities:

π _is ( j) = P( y _i = s│ x = j),

that is, the probability of responding in category s to item i, given membership of latent class j.

As explained in the main report, we study the estimated conditional probabilities π _is ( j) to reach an interpretation of a latent class model. Having reached an interpretation for the latent classes, we might then be interested in the proportions of people expected to belong to each of them; this is given in the class probabilities, η _j.

The model can equivalently be presented as a regression model, where we regress multiple observed survey items on a single nominal latent variable. Since all of the survey items are categorical, this is a logistic regression model for s - 1 categories for each item:

where x _l ( l = 2, … , q) are dummy variables representing the latent classes. It is straightforward to extend this formulation to include covariates on the right hand side of the equation. We include covariates in the models in order to provide socio-demographic profiles for the classes, essentially using covariates as explanatory variables or predictors of class membership, as illustrated figuratively in the diagram below:

In latent variable modelling terminology the part on the left hand side in the diagram is broadly termed the 'structural' part of the model, in contrast to the item response probabilities (on the right hand side) which comprise the 'measurement' part of the model. We chose to model the covariates by treating them as 'active' (in the terminology of LatentGold software used to estimate the models), meaning that we estimate item response probabilities conditional on class, conditional on covariates. This procedure means that including covariates can affect the class compositions and distributions, since it assumes conditional independence between poverty indicators and covariates, given classes.

Overall we considered that it was an appropriate theoretical choice to include variables that are often spoken of as 'drivers' of poverty as active predictors of class membership, but we note that future research would be valuable to investigate whether the conditional independence assumption is reasonable, and if not, for which combinations of variables it might be violated ^[6] . We carefully compared models that treated the covariates as 'active' and as 'inactive' (where the structural part of the model is estimated separately from the measurement part of the model). Entropy and classification error statistics indicated that the models with active covariates are more successful than their equivalents with inactive covariates. We found that the class solutions were a little different in each approach, and more so for the younger life-stages. Some further scrutiny of modelling choices might therefore be useful in future research.

Our practical modelling strategy

When the researcher has no fixed ideas about the nature of clustering in the data, part of the challenge of latent class modelling is to decide how many classes should be used to best represent a set of data. For each life-stage we ran a series of models, beginning with a two-class solution, up to a seven-class solution, and compared those solutions to each other both in terms of overall fit statistics and in terms of interpretability of the classes. For overall fit statistics we used the standard AIC and BIC measures (where for a given data set, smaller numbers indicate better models). Figure B1 summarises AIC and BIC for these investigations. AIC decreased steadily with number of classes (as we would expect); BIC suggested between 2- and 4-class solutions for families with children, 3- and 5- class solutions for working age adults without children, and 4- and 5-class solutions for older people. Together these suggested to us that we should at least consider 5-, 6- and 7-class models in case they yielded additional insights in terms of substantive interpretation. As we increased the number of classes, the models tended to return solutions with some very small class sizes, for example with classes that were estimated to contain less than 10% the population in that life-stage. We scrutinised these smaller classes very carefully for their utility, paying particular attention to the uncertainty around their estimated parameters. Ultimately we have retained only one model (for the older age life-stage) that involves a notably small class; we chose this solution partly on the grounds of fit, and partly on the grounds of the added insights that the extra class brings to the analysis. So for the older-age group we present a five-class model, while we present four-class models for the other two life-stages. Tables B1- B3 present the logit estimates for these models. Tables B4- B6 present the estimated class and item response probabilities for the models ( ηˆ _j and πˆ _is ( j) respectively) with estimated standard errors.

Having settled on these models, we re-ran each one on a random sub-sample (of half of the respondents) of its respective life-stage in order to obtain an indicative evaluation of its robustness to sampling variation. For all life stages we found the models fitted on the random sub-sample to be close to their full-sample versions. We would note that the estimate of the class sizes are not very stable across models, so care should be taken not to read too much into those values: we would suggest instead that the utility of our analyses lies in uncovering patterns of clustering in the poverty indicators and naming those common profiles of experiences.

For each model fitted, we assessed the substantive interpretability of the classes it contained by inspecting conditional item response probabilities. Alongside this we scrutinised the utility of each item. We dropped a minimal number of variables from the models after assessing how well they enabled us to discriminate between (i.e. produce clear differentiable definitions of) latent classes. Discrimination power was assessed in terms of inferential statistics for the items' 'slope' coefficients (using the overall Wald statistic for each item, and the individual z-statistic for each estimated parameter, adopting p<0.05 and z>|1.96|as a rough guide). Using these, we decided to remove the items 'live more than 30 mins from a green space' and 'home has condensation problems' from all three life-stage analyses. In addition we dropped from the older age life-stage analysis the items 'overcrowded accommodation' and 'have had difficulties paying rent/mortgage in the last year'; respondents in this life-stage were extremely unlikely to report experiencing either of these. Aside from these differences, we used the same sets of variables for all three life stages - except in the case where individual items were obviously not appropriate for some groups (e.g. 'number of children' would not be a relevant variable to include for the 'working age without children' group). This was in order to be able to draw comparisons about relative risks of different poverty indicators across life stages.

The analyses were carried out in LatentGold (Vermunt and Magidson, 2016) software using Full Information Maximum Likelihood (FIML), allowing us to include a number of items that were not asked of the full sample. FIML handles missing values directly in the likelihood function - analogously estimating the model 'around the gaps' rather than either deleting cases listwise from the analysis, or imputing missing values. For the 22 items analysed, no case in the data set returned a complete set of answers; most contained between two and five missing values, but a few cases contained as many as 17 missing values. Most of the missingness in the data comes from the survey design: the data analysed are pooled from two waves, and include a few questions that were only asked in sub-samples in one of the waves. The analysis notably contains four items ^[7] that were only asked to a small section of respondents. Given that this missingness is due to survey design based on random allocation of respondents to different versions of the questionnaire, it seems not unreasonable to treat it as random and use FIML to deal with it. The items in question are not pivotal for the interpretation of the classes, but their inclusion adds depth to our substantive understanding of them. LatentGold deals with missingness on covariates in a different way, either deleting cases listwise or using a fairly simple imputation. Since missingness is very limited on covariates, we opted to use listwise deletion for these.

Figure B.1 AIC and BIC statistics for series of models within each life stage

Table B.1 Parameter estimates for 4-class latent class model, families with children