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# 统计代写|多尺度模型代写Multilevel Models代考|INWS0016 Levels of aggregation and ecological fallacies

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## 统计代写|多尺度模型代写Multilevel Models代考|Levels of aggregation and ecological fallacies

When studying relationships among variables, there has often been controversy about the appropriate ‘unit of analysis’. We have alluded to this already in the context of ignoring hierarchical data clustering and, as we have seen, the issue is resolved by explicit hierarchical modelling.

One of the best known early illustrations of what is often known as the ecological or aggregation fallacy was the study by Robinson (1950) of the relationship between literacy and ethnic background in the United States. When the mean literacy rates and mean proportions of Black Americans for each of nine census divisions are correlated the resulting value is $0.95$, whereas the individual-level correlation ignoring the grouping is $0.20$. Robinson was concerned to point out that aggregate-level relationships could not be used as estimates for the corresponding individual-level relationships and this point is now well understood. In Chapter 3 , we discuss some of the statistical consequences of modelling only at the aggregate level.

Sometimes the aggregate level is the principal level of interest, but nevertheless a multilevel perspective is useful. Consider the example (Derbyshire, 1987) of predicting the proportion of children socially ‘at risk’ in each local administrative area for the purpose of allocating central government expenditure on social services. Survey data are available for individual children with information on risk status so that a prediction can be made using area based variables as well as child and household based variables. The probability $(\pi)$ of a child being ‘at risk’ was estimated by the following (single level) equation
$$\operatorname{logit}(\pi)=-6.3+5.9 x_1+2.2 x_2+1.5 x_3$$
where $x_1$ is the proportion of children in the area in households with a lone parent, $x_2$ is the proportion of households in each area which have a density of more than $1.5$ persons per room and $x_3$ is the proportion of households whose ‘head’ was born in the British ‘New Commonwealth’ or Pakistan. All these explanatory variables are measured at the aggregate area level and the response is the proportion of children at risk in each area. Although we can regard this analysis as taking place entirely at the area level (with suitable weighting for the number of children in each area), there are advantages in thinking of it as a 2-level model with each child being a level 1 unit and the response variable being the binary response of whether or not the child is at risk.

## 统计代写|多尺度模型代写Multilevel Models代考|Causality

In the natural sciences, experimentation has a dominant position when making causal inferences. This is both because the units of interest can be manipulated experimentally, typically using random allocation, and because there is a widespread acceptance that the results of experiments are generalisable over space and time. The models described in this book can be applied to experimental or non-experimental data, but the final causal inferences can differ. Nevertheless, most of the examples used are from non-experimental studies in the human sciences and a few words on causal inferences from such data may be useful.

If we wish to answer questions about a possible causal relationship between, say, class size and educational achievement, an experimental study would need to assign different numbers of level 1 units (students) randomly to level 2 units (classes or teachers) and study the results over a time period of several years. This would be time consuming and could create ethical problems. In addition to such practical problems, any single study would be limited in time and place, and require extensive replication before results could be generalised confidently. The specific context of any study is important; for example, the state of the educational system and the resources available at the time of the study. The difficulty from an experimental viewpoint is that it is practically impossible to allocate randomly with respect to all such possible confounding factors.

A further limitation of randomised controlled trails (RCTs) is that they cannot necessarily deal with situations where the composition of a higher level unit interacts with the treatment of interest, to affect the responses of lower level units. Thus, in schooling studies the size of class may affect the progress of students only when the proportion of ‘low achieving’ students is above a certain threshold. Randomisation will tend to eliminate classes with extreme proportions so that such effects may not be discovered. Goldstein (1998) looks at this case in more detail.

## 统计代写|多尺度模型代写Multilevel Models代考|Levels of aggregation and ecological fallacies

$$\operatorname{logit}(\pi)=-6.3+5.9 x_1+2.2 x_2+1.5 x_3$$

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