Fitting the logistic regression model

Reconviction in this study is treated as a "success", and model the probability of reconviction with a single explanatory factor – the predicted modal group

membership from the trajectory analysis.

Recall from the previous chapter that after fitting the GBTM models for both datasets in the previous chapter, the quartic polynomial negative binomial regression model (quartic4) and the B-spline negative binomial model with 4 degrees of freedom (B- Spline4) with 4 classes were chosen as the best fitting models. The posterior probabilities of the 4 class solution models were extracted and used in finding the most likely trajectory group for each offender. The justification for this is that the trajectories contain a summary of the criminal history variables, and therefore may produce simpler models. The assigned trajectory group was then used as a factor in the predictor variable in a logistic regression model, with a “reconviction within 2 years” indicator variable as the dependent variable. Logistic regression models were then repeated for “reconviction within 10 years” as the dependent variable. The predicted probabilities for each of the 4 offending trajectory groups were calculated using Equation ( 6.2 ). The results of the logistic regression models can be seen below in Table 6.1 and Table 6.2. In all of these tables, as common with the rest of this thesis, the first category is treated as the reference. In this case, class 1 is the largest category.

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Table 6.1 Logistic Regression estimates for OI datasetfor various time horizons and trajectory models

OI N = 4420 2 year Quartic4

Estimate Std. Error z value P-value

Intercept -2.77 0.12 -23.91 <0.0001

Class 1 – Low Rate Persistent 0.00

Class 2 – Low Rate Adolescent Limited -0.22 0.15 -1.48 0.14

Class 3 – High Rate Adolescent Peaked 1.03 0.18 5.75 <0.0001

Class 4 – High Rate Persistent 2.63 0.19 13.54 <0.0001

2 year B-Spline4

Estimate Std. Error z value P-value

Intercept -3.04 0.09 -33.0. <0.0001

Class 1 – Low Rate Adolescent Limited 0.00

Class 2 – Low Rate Persistent 0.38 0.16 2.4 0.0165*

Class 3 – High Rate Adolescent Peaked 1.45 0.16 9.12 <0.0001

Class 4 – High Rate Chronic 3.18 0.2 15.9 <0.0001

10 year Quartic4

Estimate Std. Error z value P-value

Intercept -1.66 0.07 -22.26 <0.0001

Class 1 – Low Rate Persistent 0.00

Class 2 – Low Rate Adolescent Limited 0.47 0.09 5.30 <0.0001

Class 3 – High Rate Adolescent Peaked 0.92 0.13 7.18 <0.0001

Class 4 – High Rate Persistent 2.44 0.18 13.32 <0.0001

10 year B-Spline4

Estimate Std. Error z value P-value

Intercept -1.26 0.05 -27.19 <0.0001

Class 1 – Low Rate Adolescent Limited 0.00

Class 2 – Low Rate Persistent -0.34 0.1 -3.57 <0.0001

Class 3 – High Rate Adolescent Peaked 0.70 0.11 6.36 <0.0001

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Table 6.2 Logistic regression estimates for the CCLS dataset for various time horizons and trajectory models

CCLS N = 4420 2 year Quartic4

Estimate Std. Error z value P-value

Intercept -1.56 0.05 -30.27 <0.0001

Class 1 – Low Rate Persistent 0.00

Class 2 – Adolescent limited 0.77 0.1 7.89 <0.0001

Class 3 – Late Starters 1.66 0.1 17.44 <0.0001

Class 4 – High Rate Persistent 2.66 0.13 19.91 <0.0001

2 year B-Spline4

Estimate Std. Error z value P-value

Intercept -1.56 0.05 -30.30 0.0001

Class 1 – Low Rate Persistent 0.00

Class 2 – Adolescent limited 0.78 0.01 7.94 <0.0001

Class 3 – Late Starters 1.68 0.1 17.65 <0.0001

Class 4 – High Rate Persistent 2.67 0.13 19.91 <0.0001

10 year Quartic4

Estimate Std. Error z value P-value

Intercept -0.04 0.04 -1.07 0.28

Class 1 – Low Rate Persistent 0.00

Class 2 – Adolescent limited 0.63 0.09 7.03 <0.0001

Class 3 – Late Starters 1.80 0.12 15.03 <0.0001

Class 4 – High Rate Persistent 2.34 0.19 12.38 <0.0001

10 year B-Spline4

Estimate Std. Error z value P-value

Intercept -0.05 0.04 -1.15 0.25

Class 1 – Low Rate Persistent 0.00

Class 2 – Adolescent limited 0.63 0.09 7.08 <0.0001

Class 3 – Late Starters 1.83 0.12 15.17 <0.0001

Class 4 – High Rate Persistent 2.34 0.19 12.36 <0.0001

From the above estimates in Table 6.1 and Table 6.2, an offender belonging to trajectory group 𝑘 has the estimated probability of reconviction equal to:

𝑃̂(𝑦 = 1) = 𝑒(𝛼+𝛽𝑘) 1 + 𝑒(𝛼+𝛽𝑘)

So for example, an OI offender who belongs to trajectory 𝑘 = 2, the estimated probability of ten-year reconviction from the b-spline model is equal to:

𝑃̂(𝑦 = 1) =_1+𝑒𝑒(−1.26−0.34)_{(−1.26−0.34)} = 0.167

Therefore the estimated probabilities of reconviction can be calculated for each trajectory class and each follow-up time.

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Table 6.3 Predicted probabilities of recidivism for Offenders Index and CCLS for various time horizons and trajectory models

OI (group sizes are in brackets Quartic4/B-Spline4) 2 year 10 year

Quartic4 B-Spline4 Quartic4 B-Spline4

Class 1 – Low Rate Persistent (43%) (Quartic) / Low Rate Adolescent Limited (55%) (B-Spline)

0.059 0.046 0.161 0.221

Class 2 – Low Rate Adolescent Limited (42%) (Quartic) / Low Rate Persistent (31%) (B-Spline)

0.048 0.065 0.235 0.167

Class 3 – High Rate Adolescent Limited (11%/11%) 0.149 0.169 0.324 0.365

Class 4 – High Rate Persistent (4%/4%) 0.464 0.535 0.687 0.732

CCLS (group sizes are in brackets Quartic4/B-Spline4) 2 year 10 year

Quartic4 B-Spline4 Quartic4 B-Spline4

Class 1 – Low Rate Persistent (60%/60%) 0.174 0.173 0.490 0.489

Class 2 – Adolescent Limited (17%/17%) 0.313 0.313 0.643 0.643

Class 3 – Late Starters (15%/15%) 0.527 0.531 0.853 0.857

Class 4 – High Rate Persistent (8%/8%) 0.750 0.751 0.909 0.909

From Table 6.3 the OI offending group (England & Wales) most likely to be

reconvicted within two years in the quartic4 model is Class 4 (High Rate Persistent) which is nearly 8 times more likely than Class 1 (Low Rate Persistent). For the B- Spline4 model the probabilities are slightly higher than the Quartic4 model (for Class 4 only). The probability of reconviction for Class 1 and Class 2 are both low and similar percentages ranging between 5-7% probability of reconviction within two years. This would be expected for the ‘adolescent limited’ group as once offenders reach adulthood, it is anticipated that the majority of offenders will reduce and desist from offending (Agnew, 2006, Moffitt, 1993).

For the 10 year follow up, the overall probabilities of reconviction are much higher due to the lengthier time span increasing the chances of being arrested and

reconvicted. However, what is not known is whether the chances of reconviction are higher within the first five years or after so the probabilities should be interpreted with this in mind. The probabilities of reconviction for Class 4 are still the highest with

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(69/73%) chances of being reconvicted within 10 years. The other offending group probabilities have all increased substantially with all of them being more than double the probability of being reconvicted within 2 years. However, the lowest probabilities still belong to the Class 1 and Class 2. In general, all the predicted probabilities for each group differ between the two models for both the 2 year and 10 year follow up periods. The offending group sizes also differ between the two models which may explain some of the differences in the predicted probabilities.

The CCLS analysis show much higher probabilities of reconviction for both follow up periods in comparison to the OI. The offending group with the highest likelihood of reconviction in 2 years is Class 4 (the High Rate Persistent group, the same result as the OI) which is not surprising. The chance of reconviction for this group is extremely likely being 75% within 2 years and just over 90% in 10 years. The lowest likelihood for reconviction is Class 1 (the Low Rate Persistent group) this is still much higher when compared to the OI equivalent group. Unlike the OI, the Quartic4 and B- Spline4 model probabilities are virtually identical.

The big differences in reconviction probabilities between the two datasets could be due to a number of reasons. As already mentioned in Chapter 3, it is very difficult to directly compare conviction statistics from different countries and it should be done with caution. It is highly likely that there are some differences between the

reconviction probabilities between the two datasets, even though their four offending trajectory groups are of similar classifications, they are still slightly different and vary in size. The OI has more offenders belonging to its lower rate offending groups and the CCLS has more offenders belonging to its higher rate offending groups. As noted by Farrington and Zara (2015) “Recidivism in European countries is likely to reflect different legal systems and criminal laws adopted in each country, making

interpretation of reoffending and conviction data and comparison between countries complex”. It could be that the England and Wales have better deterrence strategies

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than the Netherlands therefore reducing the amount of offenders that are likely to reoffend. The Netherlands may have better strategies when it comes to targeting repeat offenders and therefore gain more reconvictions. In the Dutch criminal justice system, juvenile offences tend to be dealt with away from court and the police usually opt for diversionary measures when dealing with young offenders, leading to a big reduction in the number of court orders placing juveniles in institutions (Junger-Tas and Block, 1988). Therefore, offenders that are dealt with in court in the Netherlands are more likely to be for more serious offences. Another possible reason why the Netherlands reconviction probabilities are higher could be down to the way the dataset has been structured. As previously mentioned in chapter 3, each offences in the CCLS dataset, the charge for the highest pusnishment was coded. This could mean that there are more serious offenders in the CCLS dataset sample, compared to the OI sample, who are more likely to reoffend. This was previously observed from exploring the datasets in Chapter 3. It is already known that the CCLS dataset has higher mean number of offences than the OI dataset. Therefore it is expected that the recidivism possibilities will be higher to some groups in the CCLS.