Regression Analysis

The next step requires fitting a linear regression model with the OA values of the amenities that showed relevant associations with IMD as independent variables and IMD as dependent one. This requires a four-step process: (i) normalising variables to meet the assumption of linear regression; (ii) scaling the normalised variables to obtain comparable results; (iii) addressing possible collinearity among variables;

(iv) performing stepwise linear regression to avoid model over fit (there might be hundreds of OA values that can potentially be regressed against IMD). I provide more details for each step next.

First, normalisation is required as it is an assumption of linear regression that candidate variables are normally distributed. In case these are not, various transfor-mations can be applied to normalise them (e.g., exponentiation, logarithmic

trans-66 Chapter 4. Amenities and Socio-economic Deprivation

formation). Second, scaling is necessary as the OA values may have different mag-nitudes and may thus be hard to compare and interpret, if regressed untransformed against IMD. Scaling consists in calculating the standard scores (i.e., z scores) of the normalised OA values. This can be done through the following formula:

z= X− µ σ

where X represents the raw value, µ the mean of the population, and σ its standard deviation.

The third step requires testing relevant OA values for collinearity. This phe-nomenon occurs when two or more candidate variables for a regression model are strongly correlated among each other. If strongly collinear variables are used in a regression, it is likely that the resulting fit and regression coefficients are inflated or show unexpected signs. To check for this issue, I compute the Variance Infla-tion Factors (VIFs) associated with each candidate variable. Let reg be a regression model with predictor variables v1, vi, ..., vn. The VIF of the variable vi is obtained by, first, performing linear regression with v_i as dependent variable and the other variables as independent ones v1, vi− 1, v_i+ 1, ..., v_n, and, second, by using the overall model fit (i.e., R² value) obtained at the previous step in the following for-mula:

V IF= 1 1 − R².

If a variable has a strong linear relation with at least another one, its correlation coefficient is likely to be close to 1 and the VIF related to that variable large. A VIF equal to or greater than 10 is a sign of a collinearity issue [88]. If the candidate variables do not show VIFs above 10, they can be used as independent variables in a linear regression model with IMD as dependent one. Conversely, if candidate variables have VIFs equal to or greater than 10, one can implement a stepwise procedure that, first, excludes the candidate variable with the highest VIF and then repeats the same process until none of the variable has a VIF equal to or greater than 10. At the end of this procedure, candidate variables should be devoid of collinearity

4.3. Approach 67 and can be regressed against IMD.

The final step requires performing stepwise linear regression between the rele-vant OA values and IMD. This technique is necessary as there might be hundreds of OA values that can potentially be regressed against IMD. A simple linear regression would output a model with too many coefficients that would be hard to interpret and plausibly run into the issue of over fitting. Conversely, a stepwise linear regression outputs a parsimonious model that relies on the smallest number possible of re-gression coefficients. Having obtained a model through this technique, a reliability check has to be performed on the outcomes of such model. In the previous sec-tion, I explained how spatial autocorrelation can bias the outcomes of a correlation analysis. The same phenomenon can also affect the outcome of a stepwise linear re-gression by causing over-inflated rere-gression coefficients or unexpected signs. This is caused by not considering the information derived from the spatial dependency of observations located close to each other in space. For this reason, it is necessary to check whether spatial autocorrelation is present in the outcome of a regression too.

To this end, I use a technique renowned in spatial studies, the Moran’s test [89], that checks whether residuals are spatially autocorrelated. The outputs of this test are two. An index (I) that varies between -1 and 1, which can be interpreted similarly to a Pearson’s correlation coefficient, and a p-value that measures the statistical signif-icance of the test. A negative Moran’s I generally occurs when the spatial tendency is such that dissimilar values cluster together (dispersed pattern). Conversely, it is positive when similar values are located near each other (clustered pattern). If the output of this test shows that there is no statistical evidence that spatial autocor-relation is present in the residuals, the outcome of the regression can be accepted and interpreted. Conversely, if this is not the case, one should consider regression techniques that incorporate the spatial information in their equations. One of these techniques – and the one adopted in this study – is the Spatial Autoregressive (SAR) model, a type of regression that accounts for the proximity of observations in space by including a spatial weighting matrix in the equation [90]. More specifically, the

68 Chapter 4. Amenities and Socio-economic Deprivation general formula for a SAR model is:

Y_n= λWnY_n+ X_nβ + ε_n,

where Y_nis the dependent variable; W_nis an n by n spatial weighting matrix⁹applied to the variable Y_n, with λ being a spatial autoregressive parameter estimated from the data. X_n is the independent variable, β is the regression coefficient associated with X_n, and ε_n is the error term. In practice, this model expresses the concept that the value of a variable at a given location is associated with the values of the same variable measured at nearby locations, reflecting a sort of interaction effect.

To ascertain that the SAR model accounts for all the spatial autocorrelation present in the data, I utilise again the Moran’s test on the residuals of the SAR model. If the Moran’s I is close to 0 or is not statistically significant (p-value >0.05), the SAR model can be accepted and interpreted. Otherwise, it should be rejected.

I present in the next section the results of the analysis performed on the three UK urban areas under study (i.e., Greater London, Greater Manchester, and West Midlands).