Methods for investigating zoning effects

Mark Tranmer

CCSR

Allowing for ‘area effects’

• Suppose we have some area level information

• Such as: aggregate information for a particular set of ‘areal units’ e.g. wards; EDs; Output Areas; Districts

• Or individual level data with area indicators.

Allowing for these effects in our analyses

• Then we might say ‘great! I’ll fit a multilevel model’ – especially if we have individual level data with area indicators.

• Or we might calculate correlations etc at the area level from aggregate area level data.

But …

• If we calculate area level correlations because we want to make inferences about individuals that live in those areas but we only have area level data… problem: ecological fallacy

• So let’s suppose we can actually do an analysis using individual level data with area indicators … e.g. a multilevel model. Hence simultaneously allowing for individual and area level effects.

• Does that solve the problem?

No, because …

• What do we mean by an ‘area’?

• Modifiable Areal Unit Problem (MAUP)

• Analyses that involve areas are affected by

The average population size of those areas: ‘scale effects’

No, because …

• Once we choose a particular scale, they are also affected by the way in which those areas are defined. I.e. the choice of boundaries: ‘Zoning effects’.

• Also: Scope effects? What is the overall region of study? This will have implications for the extent of variation.

Zoning effects example

• Suppose we have a region that contains a 9 areal units of equal population, and we want to make a ward from three of these contiguous units.

Zoning effects example

• Ward A1

Zoning effects example

• Ward B1

Zoning effects example

• Overlay wards A1 and B1

Zoning effects example

• We can also do the same thing for the other wards: e.g.

I’m interested in developing a statistical framework to investigate these effects

• I think a cross-classified multilevel model might be the way to tackle the problem

• What I hope to do is to find a way to assess the nature and extent of zoning effects at a particular scale.

Two level model(s)

ijAjAijA euy 2)( uAjAuVar

2)( eAijAeVar

ijBjBijB euy 2)( uBjBuVar

2)( eBijBeVar

Cross-classified model

),(),(),( jBjAijBjAjBjAjBjAi euuuy

2)( uAjAuVar 2)( uBjBuVar 2

),( )( uABjBjAuVar

2)( )( eABjAjBieVar

How to test this idea

• Simulated data: I set up a simulation study• I generated some simulated data for a normally

distributed variable. Each of the 9 cells in the grid has a different (but known) mean and within each of the 9 cells I set the variance to be equal (25).

• So I aimed to simulate complex between-cell variation (whilst knowing the procedure I had applied to induce that variation).

I assumed these zonings

ResultsTwo level models * Variance component estimates

      Ward Indiv

A, person     427 366

B, person     176 616

Cell,person     642 150

 

Cross-classified models

Estimated parameter:

Var(A) Var(B) Var(A*B) Var(Indiv)

A,B,cell,person 333 4 309 150

Conclusion

• I think we have a framework for investigating the causes of zoning effects

• It seems to work for simulated data, though I have yet to fully work out what these results mean

• Can anyone suggest to me some real data that investigate using this methodology.