Assessing Risk-Based Policies for Pretrial Release and Split Sentencing in Los Angeles County Jails

Court-mandated downsizing of the CA prison system has led to a redistribution of detainees from prisons to CA county jails, and subsequent jail overcrowding. Using data that is representative of the LA County jail system, we build a mathematical model that tracks the flow of individuals during arraignment, pretrial release or detention, case disposition, jail sentence, and possible recidivism during pretrial release, after a failure to appear in court, during non-felony probation and during felony supervision. We assess 64 joint pretrial release and split-sentencing (where low-level felon sentences are split between jail time and mandatory supervision) policies that are based on the type of charge (felony or non-felony) and the risk category as determined by the CA Static Risk Assessment tool, and compare their performance to that of the policy LA County used in early 2014, before split sentencing was in use. In our model, policies that offer split sentences to all low-level felons optimize the key tradeoff between public safety and jail congestion by, e.g., simultaneously reducing the rearrest rate by 7% and the mean jail population by 20% relative to the policy LA County used in 2014. The effectiveness of split sentencing is due to two facts: (i) convicted felony offenders comprised ≈ 45% of LA County’s jail population in 2014, and (ii) compared to pretrial release, split sentencing exposes offenders to much less time under recidivism risk per saved jail day.

the 4986 defendants with pretrial release interviews, 390 were not released. Assuming that all 2981 defendants without pretrial release interviews were detained, we estimate that (1- an average of these conditional probabilities over the two studies [1,2], we assume that the probability of release for a defendant of low, medium or high risk is 0.667, 0.500 and 0.333, respectively. Therefore, the conditional probabilities of pretrial detention given risk category are P (D|L)=0.333, P (D|M)=0.500 and P (D|H)=0.667.
Having derived the pretrial detention probabilities conditioned on risk, we now compute the pretrial detention probabilities conditioned on charge. By the top right box on page 79 of [3], the probability that a felon in LA County was released is 0.23 and the probability that a defendant with a non-felony in LA County was released is 0.66. Letting N denote nonfelony and F denote felony, the conditional probabilities of pretrial detention given charge are P (D|N)=0.34 and P (D|F)=0.77. Using the risk proportions in Table 1 of the main text, we find that the probability of being detained is P (D)=0.549(0.667)+0.281(0.500)+0.170(0.333) =0.563.
Our goal now is to derive a mathematical expression for the conditional probability of pretrial detention given the charge type C (either N or F) and risk category R (either L, M or H), which we denote by P (D|C,R), in terms of the probabilities P (D|R), P (D|C) and P (D) calculated above. By our assumption that C and R are independent, it follows that P (C|R)=P (C) and P (C|R,D)=P (C|D). By the definition of conditional probability, we have P (C|D)=P (C,D)/P (D) and P (C|R,D)=P (C,D|R)/P (D|R). Substituting the right sides of these two equations into P (C|R,D)=P (C|D) yields P (C,D|R)/P (D|R)=P (C,D)/P (D).
Substituting P (D|C)P (C) for P (C,D) in this equation and rearranging gives Finally, we find that = P (D, C|R) P (C) because C and R are independent, (D) Substituting the numerical values above into the right side of (E) and then noting that the conditional probability of release given crime type and risk category is 1-P (D|C,R), we get the conditional release probabilities in Table 2 of the main text.

B Pretrial Release After Recidivism
According to Table 2 of [3], 20-21% of felons in LA County receive financial release. Table 19 in the Appendix of [4] states that 19% of felons receive financial release in LA County, confirming the numbers in [3]. When averaged over the 75 largest counties in the U.S., Table 6 in [4] states that 33% of felons receive financial release and Table 8

C Interarrival Times
The arraignments under custody from the LA County Sheriff's Department's Year in Review reports (e.g., [5]) for the years 2008-2012 give an average of 350.4/day with no obvious longterm trend (Table A). There is little variability over the time of year (page 36 of Appendix C in [3]), and we ignore the fact that fewer arraignments occur over the weekend (page 34 of Appendix C in [3]). We estimate the squared coefficient of variation (i.e., the variance divided by the square of the mean) of the interarrival times to be 0.465 from the data in Table A by assuming that the times between consecutive arrivals are independent and identically distributed. The Erlang distribution is characterized by its shape parameter and scale parameter. The Erlang shape parameter is the reciprocal of the squared coefficient of variation, and so we set the shape parameter equal to two. The mean of an Erlang distribution is the product of its shape and scale parameters. Because the arrival rate of 350/day includes recidivists, who account for slightly less than half of the total number of arrivals, we adjust the rate of new arrivals to be 193/day so that the total arrival rate is 350/day. This yields a shape parameter of 1/395.

D Time to Arraignment
Arraignment data from LA County in 2008 states that 4% of defendants were arraigned within 24 hours of arrest, 56.4% were arraigned within 48 hours, 70.4% were arraigned within 72 hours, and 95% were arraigned within 96 hours (page 63 of [3]). The maximum likelihood estimate for the lognormal parameters from these data yield the values in Table 3 of the main text.

E Time to Recidivism
Let the cumulative distribution function (CDF) F (t; Θ, j) be the probability that an offender recidivates within t time units of being released, given that he is of risk category j (where j = 1, 2, 3 correspond to low, medium and high risk) under parameter set Θ. These CDFs are specified for our five models in Table B. Let N j be the number of offenders in the CSRA cohort [6] with risk category j, where N = 3 j=1 N j . Let N jk be the number of offenders in the CSRA cohort with risk category j that did not recidivate within k − 1 years but did recidivate within k years. The data from [6] are N = 110, 313, N 1 = 18, 768, N 2 = 31, 024, The estimation results (Table C) suggest that the lognormal models outperform the proportional hazards models (as measured by the negative log-likelihood values), and the split is statistically significant in the lognormal model only if we incorporate risk-associated heteroskedasticity. We use the split lognormal model with heteroskedasticity, and the three probability density functions (one for each risk category) are pictured in Fig

F Failure To Appear
We use a validation study of COMPAS for predicting failure to appear in court in Broward County, FL [7]. COMPAS's 10 risk scores were aggregated into CSRA's three risk categories, where scores 1-4, 5-7 and 8-10 correspond to low, medium and high risk (page 20 of [7]). Table 13 in [7] gives failure-to-appear probabilities for the three risk groups at six different follow-up periods, but does not provide information on the time from arraignment to disposition. Hence, we cannot attempt to estimate (e.g., via logistic regression) whether the failure-to-appear probability depends on the arraignment-to-disposition delay. To minimize right-censoring (i.e., due to a case disposition date that is far in the future), we use their largest follow-up period of 12 months; these data are reproduced in Table D. Because the failure-to-appear probability is slightly higher for medium risk than high risk (the same is true for follow-up periods of three months and six months in Table 13 in [7]), we combine medium risk and high risk in our analysis. Hence, we assume that the failure-to-appear probability is 0.117 for low risk and is [456(0.180)+163(0.172)]/(456+163)=0.178 for both medium risk and high risk.

G Time from Arraignment to Case Disposition
Because the time from arrest to arraignment is typically much smaller than the time from arraignment to disposition, we use arrest-to-disposition time data from pages 55-56 in [3], which appear in Table E, to estimate the time from arraignment to disposition, using both gamma and lognormal distributions. Lognormal provides a better fit (using the Kolmogorov distances in Table E) for felonies in pretrial release, and gamma (with an increasing failure rate for those released and a decreasing failure rate for those in custody) provides a better fit for felonies under pretrial custody and for all non-felonies.

H Case Disposition Probabilities
The 14 case disposition probabilities in Table 4 in the main text are derived in three groups.
Ignoring the ongoing cases in Table 4 of [3] and combining acquitted and dismissed cases, we estimate the four proportion dismissed values that appear in Table 4 in the main text (e.g., for felons on pretrial release, (1635+46)/(12,154-1168)=0.153).
We denote the proportion of non-felony charges that are put on probation after pretrial release, put on probation after pretrial custody, receive a jail sentence after pretrial release and receive a jail sentence after pretrial custody by p 1 , p 2 , p 3 and p 4 , respectively. We jointly solve for these four unknowns using four equations (K)-(P). The first two equations use the proportion dismissed values, 0.207 and 0.052, in Table 4 in the main text: The  Table 4 in the main text.
The remaining six case disposition probabilities are derived jointly. We denote the proportion of felons who are put on probation after pretrial release, put on probation after pretrial custody, receive a jail sentence after pretrial release, receive a jail sentence after pretrial custody, are sent to prison after pretrial release, and are sent to prison after pretrial custody by p 5 , p 6 , p 7 , p 8 , p 9 and p 10 , respectively. We jointly solve for these six unknowns using six equations (K)-(P). The first two equations use the proportion dismissed values, 0.153 and 0.069: The third equation states that the proportion of felons that get probation is, using the last four columns in Table 1  Simultaneously solving (K)-(P) gives the proportions in Table 4 in the main text.

I Post-sentence Jail Terms
We fit mixture (of pretrial release and pretrial custody) gamma and lognormal distributions for jail sentences for non-felonies using data on page 129 in [3] and Fig. 25 in Appendix C in [3], and for jail sentences for felonies using data in Chart 3 in [10]. The data (reproduced in where R and D denote the populations that are in pretrial release and pretrial detention, respectively, and w R and w D are weights proportional to the sizes of these two populations. Because 66% of non-felonies and 23% of felonies receive pretrial release ( §A), and 55.8% of defendants are charged with felonies (Table 3 of  where R and D are subscripts on the parameters in (Q) to denote the released or detained populations. We use a constrained maximum likelihood approach, where we require the mean jail sentence for non-felonies to be 2.78 times longer for those undergoing pretrial detention relative to those receiving pretrial release, and the mean jail sentence for felonies to be 2.36 times longer for those undergoing pretrial detention relative to those receiving pretrial release (page 10 of [8]). Hence, setting r = 2.78 for non-felonies and r = 2.36 for felonies, and setting F (0; Θ) = 0 and F (∞; Θ) = 1, we solve subject to e µ D +σ 2 The results appear in Table G and Fig. B, and we adopt the distribution with the lower negative log-likelihood: gamma for non-felonies and lognormal for felonies.

Model
Parameter Set CDF (Sign Constraint) lognormal β(−), µ, σ(+) split lognormal β(−), µ,σ(+), δ(+) split lognormal with heteroskedasticity β(−), µ,σ(+), δ(+), γ(−) δ proportional hazards β(+), λ 1 − e (−λt) e βx split proportional hazards β(+), λ, δ(+) δ 1 − e (−λt) e βx  Table C: Results from the maximum likelihood estimation in (F), with 95% confidence intervals in parentheses for the two models that provide the best fit. NLL is negative loglikelihood, and p-value is the likelihood ratio test of the enhanced model relative to the base (lognormal or proportional hazards) model.  Table D: Raw data for estimation of the failure-to-appear probabilities, reproduced from Table 13 in [7]. The sample size is the number of defendants in each risk category who did not recidivate during their first 12 months under pretrial release. The failure-to-appear probability is the proportion of these defendants who failed to appear for a court date during their first 12 months of pretrial release.   1  0  1827  1  0  4501  2  2  1829  2  8  2001  3  6  440  3  12  1167  4  8  1535  4  18  333  5  31  939  5  24  167  6  61  363  6  30  83  7  91  379  7  36  83   Table F: Raw data for estimation of the length of jail sentences. N i is the number of offenders whose jail sentence was in the interval [a i , a i+1 ). We have a 8 = ∞ for non-felonies (page 129 in [3]) and a 8 = 252 (Chart 3 in [10]). The population values for felonies were derived by multiplying the probabilities in [3] and [10] times the respective sample sizes of 7310 and 8336, and then rounding to the nearest integer.     Fig. B(b) in the main text, the optimal policy along the different points on the curve. Each policy is denoted by a pair of numbers, where the first number corresponds to the pretrial release for non-felonies (left column of Table A in the main text) and the second number corresponds to the pretrial release for felonies (middle column of Table A in the main text).  Table A, the optimal (i.e., optimizing over the remaining 16 options in Table A) tradeoff curves of the annual rearrest rate vs. (a) the mean jail population and (b) mean jail overcrowding, restricting to policies that treat felonies at least as strictly (with respect to pretrial release) as non-felonies of the same risk category. The circle denotes the status quo policy for LA County in early 2014.  Table A, the optimal (i.e., optimizing over the remaining 16 options in Table A) tradeoff curves of the annual rearrest rate vs. (a) the mean jail population and (b) mean jail overcrowding, restricting to policies that treat felonies at least as strictly (with respect to pretrial release) as non-felonies of the same risk category. The circle denotes the status quo policy for LA County in early 2014.