ANÁLISIS DESCRIPTIVOS

Having decided that multilevel models would provide the greatest insight, due to the hierarchical nature of the data under analysis, models were developed using MLwiN 2.1 software (Rasbash et al., 2009). The models were initially based on the dataset developed in Chapter 5, Cluster Analysis, of this thesis.

This dataset provided a hierarchical structure consisting of forty-one Police Force Areas (PFA‟s) in six derived clusters. Further models were developed also consisting of forty-one PFA‟s but grouped within nine distinct regional clusters. Both models analysed the effect of all Fixed Penalty Notices (FPN_1000‟s) and only speeding related Fixed Penalty Notices

(FPN_G16_1000‟s) on the Killed and Seriously Injured (KSI) accident rate, the dependent variable in all models.

In developing the models one must be aware of distributional concerns. As the data under analysis are aggregated counts the preferred method of analysis is Poisson regression or Negative Binomial regression, as detailed in Chapter 4, Section 4.1. In this case there is significant overdispersion present in the Poisson models, therefore Negative Binomial models are used in order to account for the overdispersion.

6.2.1 Multilevel Models using FPN’s

Two sets of clusters were used to develop two-level multilevel models, with PFA‟s as the level one variable and the derived clusters as the level two variable. In this section multilevel models based on derived clusters are presented. Two variables are used to construct multilevel models,

ZFPN_1000‟s and ZFPN_G16_1000s, which represent standardised values of FPN_1000‟s and FPN_G16_1000‟s. In Table 6.2.1.1 the results of model development are shown – a null model, a variance components model and a third model, giving the effects of ZFPN_1000‟s on the KSI rate.

In the variance components model, Table 6.2.1.1, there is a statistically significant random variation, at the 5% level, between the derived clusters as

well as a statistically significant fixed effect. The significance, for fixed effects, is derived by dividing the parameter estimate by its standard error. If the ratio is greater than 1.96 then the result is statistically significant. Significance tests for random effects, variances, follow the same calculation but the resulting p-value should be divided by two (see Snijders and Bosker, 1999). This does not apply to covariance which is simply the ratio of covariance estimate divided by the standard error estimate.

Table 6.2.1.1: Multilevel Negative Binomial Models of Effect of ZFPN_1000’s on Derived Clusters

The third and final model, in Table 6.2.1.1, details the results when the effects of FPN‟s are added. From previous analysis it was expected that ZFPN_1000‟s would be associated with a decrease in the KSI rate and here it can be seen that this is indeed the case with a significant fixed effect associated with ZFPN_1000‟s. Here, as with the variance components model, there is

significant random variation between clusters. The marked variation between clusters is expected as the clusters were developed in order to produce groups of Police Force Areas (PFA‟s) that have maximum variation between clusters and minimum variation within clusters. There is, however, no significant random variation, at the 5% level, between clusters associated with the effect of ZFPN_1000s.

The variation in the fixed effect of enforcement on different clusters is detailed in Figure 6.2.1.1. In general terms Figure 6.2.1.1 maps the variation between and within clusters. Each line represents the fixed effect of the enforcement variable, ZFPN, on the log of KSI rates for each individual cluster of PFA‟s and the position of each line, compared to all others, is a measure of the variation between clusters. The slope and gradient of each line is a measure of the variation, between PFA‟s, within each cluster. With the exception of Cluster 4 none of the other clusters has any significant effects at the 5% level, related to enforcement – see Table 6.2.1.2.

Figure 6.2.1.1: Effect of Enforcement – ZFPN_1000’s – on Derived Clusters

This is not surprising as the clusters were developed using KSI rates and ZFPN_1000s and the lack of a statistically significant variation between

clusters, in relation to ZFPN_1000s, indicates that the clusters are well defined and following the general trend identified in Chapter 4 – where increasing levels of police enforcement are linked to decreasing KSI rates. In Cluster 4 there is a statistically significant effect in relation to the effect of enforcement – ZFPN_1000‟s. This effect goes against the general trend of increased

enforcement leading to decreasing KSI rates and is most probably an artefact of the clustering algorithm, see Figure 5.2.1 in Chapter 5, where a group of six PFA‟s has been clustered together. If Cluster 4 is grouped with Cluster 5 then this effect, which is counter-intuitive in light of all other evidence, disappears.

Alternatively, as ZFPN‟s are at low levels, in Cluster 4, it may be that

increasing ZFPN may be in response to increasing KSI accidents. This is an area requiring further investigation, which is beyond the scope of this thesis.

Table 6.2.1.2: Parameter Estimates and p-values for Fixed Effects of ZFPN_1000’s on Derived Clusters

Models based on Derived Clusters

Fixed Effect Parameter Estimate

Standar d Error

Parameter Estimate /

Standard Error p-value Cluster 4 with ZFPN_1000‟s Effect 1.528 0.576 2.653 0.004 Cluster 6 with ZFPN_1000‟s Effect -0.519 0.366 -1.418 0.080 Cluster 3 with ZFPN_1000‟s Effect -0.297 0.281 -1.057 0.150 Cluster 2 with ZFPN_1000‟s Effect 0.078 0.181 0.431 0.334 Cluster 5 with ZFPN_1000‟s Effect -0.094 0.404 -0.233 0.408 Cluster 1 with ZFPN_1000‟s Effect 0.034 0.362 0.094 0.462

6.2.2 Multilevel Models using ZFPN_G16_1000’s

The methodology used, to develop the multilevel models in this section, is identical to that used in Section 6.2.1.1. Here the enforcement variable is ZFPN_G16_1000‟s, speeding related fixed penalty notices.

Table 6.2.2.1: Multilevel Negative Binomial Models of Effect of ZFPN_G16_1000’s on Derived Clusters

Once more three models are developed, see Table 6.2.2.1; a null model, a variance components model and a third model, looking at the effects of ZFPN_G16_1000‟s on the KSI rate. From the results of the variance

components model one can see a statistically significant random variation, at the 5% level, between clusters in relation to KSI rates. When the effect of enforcement is added, ZFPN_G16_1000‟s, a significant fixed effect is found indicating an increase in the number of ZFPN_G16_1000‟s leads to a

decrease in the KSI rates. No significant random variation, at the 5% level, between clusters is found relating to the effect of ZFPN_1000s.

Two clusters have a statistically significant fixed effect related to

ZFPN_G16_1000‟s; Clusters 4 and 5 – see Table 6.2.2.2. There is significant variation between clusters and this is shown in Figure 6.2.2.1.

Table 6.2.2.2: Parameter Estimates and p-values for Fixed Effects of ZFPN_G16_1000’s on Derived Clusters

Models based on Derived Clusters

Fixed Effect Parameter

Estimate S.E.

Parameter Estimate / Standard

Error p-value Cluster 4 with ZFPN_G16_1000‟sEffect 0.952 0.421 2.261 0.012 Cluster 5 with ZFPN_G16_1000‟sEffect 0.379 0.214 1.771 0.038 Cluster 6 with ZFPN_G16_1000‟sEffect -0.189 0.151 -1.252 0.106 Cluster 3 with ZFPN_G16_1000‟sEffect -0.119 0.179 -0.665 0.253 Cluster 2 with ZFPN_G16_1000‟sEffect 0.083 0.133 0.624 0.266 Cluster 1 with ZFPN_G16_1000‟sEffect 0.018 0.234 0.077 0.469

Figure 6.2.2.1: Effect of Enforcement – ZFPN_G16_1000’s – on Derived Clusters