Sample

Participants were part of a data-set of 2394 UK adult males convicted of sexual offences. All participants were assessed between 2003 and 2014 (with 65% of assessments occurring between 2007 and 2011, inclusive) for eligibility for the prison-based version of HMPPS Core Sex Offender Treatment Programme (SOTP). Approximately 97% of the sample was serving a sentence for a conviction between 1996 and 2012, with 77% convicted between 2003 and 2009. Ages ranged between 21 and 84 years (mean 41.8, s.d. = 12.7, median 42 years). Most of the sample was recorded as being in aggregate White ethnic categories (89.8%), with 5.9% classified in aggregate Black ethnic categories, 2.7% in aggregate Asian ethnic categories, 1.3% in aggregate ‘mixed-race’ ethnic categories, 0.2% in aggregate ‘other’ ethnic categories, with a further 0.1% not specified and 4.4% of the data missing. Most of the full sample (n = 1526: 68.0%) attended the Core SOTP (rolling or fixed formats), 362 (16.1%) attended a combination of the Core and Extended SOTP, 315 (14.0%) attended the Better Lives Booster (for individuals with intellectual difficulties) and 42 (1.9%) attended the Healthy Sexual Functioning programme (information about Core and Extended SOTP is given in Supplementary Material A, available at https://doi.org/10.1192/bjp.2023.126). Consequently, our sample was heterogeneous regarding executive functioning, sexual interests and victim types.

After removing duplicates and individuals for whom more than 10% of data were missing, a final data-set of 2245 participants remained for analysis. Participants were randomly allocated to a test (n = 1577: 70.2%) or validation (n = 668: 29.8%) data-set. Results from a simulation study have suggested that a minimum sample size of 500 should be sufficient to accurately identify a correct number of latent profiles.^{Reference Tein, Coxe and Cham14}

Ethics approval was not received for this human study because we used existing data collected by HMPPS for the purposes of service evaluation. All adult participants provided written informed consent for their data to be used in research. The project was approved by the Ministry of Justice National Research Committee (granted 5 May 2021).

Measure selection

The clinical data-set contained pre- and post-programme test scores on 92 scales from 17 psychological measures (see Supplementary Material A for descriptions), along with demographic, intervention and offence characteristics. Only pre-programme test scores were considered for inclusion. One scale was removed because of missing data (30% missing) and two further scales were removed because of perfect or near-perfect positive correlations with other study measures (Supplementary Material A). Any scales that were positively oriented were reverse scored so that higher scores were indicative of greater risk/impairment. Forty scales were removed because of a lack of theoretical support as psychologically meaningful risk factors.^{Reference Mann, Hanson and Thornton3}

Exploratory factor analysis (EFA) was employed to reduce the number of variables, with the aim of selecting one variable per theoretically plausible risk domain for use in the LPA. Three metrics were used to judge how many factors could plausibly be extracted from the data: (a) parallel analysis; (b) Velicer's minimal average partial (MAP) criterion; and (c) the very simple structure (VSS) criterion.^{Reference Revelle and Rocklin15} Parallel analysis revealed that the eigenvalues of seven components exceeded the associated simulated eigenvalue generated from random data. Both the MAP (smallest average squared partial correlation of 0.0129) and the VSS criterions (maximum correlation of 0.79) suggested four components. Since all three tests broadly indicated that additional value is limited beyond four factors (Supplementary Fig. S1, Supplementary Material A), EFA using a varimax rotation and a maximum likelihood solution was used to establish the fit of a four-factor model. Supplementary Table S1 (Supplementary Material A) provides the factor loadings per scale cluster. The root mean square of the residuals (RMSR) was 0.05 (d.f. corrected RMSR = 0.06), χ²(524) = 4.2, P < 0.001.

The EFA resulted in four classes of scales, approximating three of the four SARN domains of risk:^{Reference Thornton4} factor 1, socio-affective and emotional management; factor 2, sexual preoccupation/interests, child specific; factor 3, sexual preoccupation/interests, non-child specific; and factor 4, pro-offending thinking (adult and child). Finally, we constructed correlation matrices for each of the four factors, to ensure that we selected scales for inclusion in the LPA that (a) had a sufficiently high factor loading and (b) had a high average correlation with other scales and thus were a good exemplar of the underlying general construct.

A total of six subscales were selected for inclusion in the LPA: one from the socio-affective and emotional management factor (Impulsive carelessness:^{Reference D'Zurilla, Nezu and Maydeu-Olivares16} factor loading FL = 0.88, mean r (r ^m) = 0.61); one from the sexual preoccupation/interests, child-specific factor (Child molest: fantasy:^{Reference Nichols and Molinder17} FL = 0.81, r ^m = 0.53); two from the sexual preoccupation/interests, non-child-specific factor, representing interests (Rape: fantasy:^{Reference Nichols and Molinder17} FL = 0.73, r ^m = 0.43) and preoccupations (Sexual obsession:^{Reference Nichols and Molinder17} FL = 0.58, r ^m = 0.40); and two from the pro-offending thinking factor, one non-child specific (Rape myth acceptance:^{Reference Burt18} FL = 0.74, r ^m = 0.55) and one child specific (Sex with children:^{Reference Mann, Webster, Wakeling and Marshall19} FL = 0.78, r^m = 0.54).

Socially desirable responding

Given concerns about the tendency towards socially desirable responding, we used the Balanced Inventory of Desirable Responding (BIDR)^{Reference Paulhus20} to assess the extent to which response bias was problematic in the current sample compared with normative values. Reassuringly, earlier work with a subset of our sample showed that the extent of socially desirable responding was relatively small, and its impact on self-report measures was lower than expected.^{Reference Mathie and Wakeling21}

Latent profile analysis

Model-based clustering is based on the theory that data are derived from a mixture of underlying probability distributions.^{Reference Fraley and Raftery22} The most popular approach is the Gaussian mixture model, where each observation is assumed to be distributed as one of k multivariate-normal distributions, where k is the number of ‘mixture components’ or profiles.^{Reference Scrucca, Fop, Murphy and Raftery23} We estimated the optimum number of latent profiles using standardised raw scale scores and estimating profiles as finite mixture models. Data were analysed using R (version 3.5.1 for Windows) and primarily a combination of the ‘tidyLPA’ (version 1.0.8) and ‘mclust’ (version 5.4.6) packages. Gaussian finite mixture models were estimated using the expectation–maximisation (EM) algorithm (starting with the expectation step) for model-based hierarchical agglomerative clustering.^{Reference Scrucca, Fop, Murphy and Raftery23}

It is recommended that a range of criteria are used for identifying the correct number of profiles.^{Reference Tein, Coxe and Cham14} Although the Bayesian information criterion (BIC)^{Reference Schwarz24} is used as the default, we also used the Akaike information criterion (AIC),^{Reference Akaike25} consistent AIC (CAIC),^{Reference Bozdogan26} sample size adjusted BIC (SABIC)^{Reference Sclove27} and integrated complete-data likelihood criterion (ICL)^{Reference Biernacki, Celeux and Govaert28} to judge model estimation. Larger log-likelihood values (BIC, SABIC, AIC, CAIC or ICL) relative to the previous model indicate better fit. Entropy and minimum probability were also assessed,^{Reference Celeux and Soromenho29} with values closer to 1 indicative of better fit. Finally, we examined findings from the bootstrap likelihood ratio test (BLRT),^{Reference McLachlan30} with larger changes in log-likelihood relative to the previous model indicative of better fit. We calculated estimates for one to nine profiles, generated via two models that presumed either equal variances (as opposed to allowing variances to vary) and covariances fixed to zero (Model 1 in tidyLPA) or equal variances and equal covariances (Model 3 in tidyLPA).

Validity of latent profiles

The replicability of the final profile solution was validated using the validation data-set (30% of the total sample). To establish criterion-related evidence, a series of linear regression models and tests of association were used to examine whether profile allocation was associated with theoretically plausible criminological outcomes, including intelligence, assessed using various versions of the Wechsler Adult Intelligence Scale or Wechsler Abbreviated Scale of Intelligence,^{Reference Wechsler31–Reference Wechsler33} age, socially desirable responding, prior convictions (sexual, violent and non-sexual-non-violent), risk level (via the Risk Matrix 2000/S)^{Reference Thornton, Mann, Webster, Blud, Travers and Friendship34} and index offence codes that were child specific, related to indecent images of children (IIOC) or female specific. For each outcome, mean differences in scale score between profiles were calculated and plotted alongside Cohen's d effect sizes for each pairwise contrast.