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A Tale of Two Margins: Exploring the Probabilistic Processes that Generate Prison Visits in the First Two Years of Incarceration



The study extends previous literature (Cochran in J Crim Justice 40:433–440, 2012, J Res Crime Delinq 51(2):200–229, 2014) by simultaneously examining two margins: the probability of receiving a visit and the number of visits a prisoner receives conditional on receiving any visits; adding a level of nuance to the exploration of prison visits.


A random sample of New York State prisoners admitted between 2000 and 2013 who served at least 24 months and had basic admission, release, and transfer data (N = 22,975) were selected. Visit patterns were derived using group-based trajectory models with a zero-inflated Poisson specification and up to a cubic polynomial on probability and count parameters.


The best fitting model had seven groups that displayed wide variation in the probability of a visit (in both directions). By contrast, the number of visits, conditional on receiving a visit, is relatively constant over time. Subsequent dual trajectory modeling of prison visits with distance from home county demonstrates that the dynamic patterns of probability of visit correspond with dynamic patterns of distance from home county.


We demonstrate that time variation in visitation occurs along the prevalence margin. Researchers interested in studying the longitudinal relationship between visits and outcomes should be attentive to this result. Additionally, characteristics of prisoners assigned to the trajectory groups using Posterior Probabilities of Assignment suggest that pre-prison factors (i.e. criminal record) and in-prison policy decisions (i.e. custody level) are associated with particular patterns of visits over time; highlighting the challenge to understanding the effect of visitation in studies without explicit causal identification strategies.

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  1. It is important to note here that this analysis does not indicate whether it would be beneficial to incentivize visits. However, the point about policy is relevant whether visits should be encouraged or discouraged.

  2. Several studies have established a link between distance from home and receipt of visits (Acevedo and Bakken 2001; Cochran et al. 2015; Jackson et al. 1997; Tahamont 2013).

  3. Visitation rates range from 17 to 25% in some studies of Florida (FL) samples (Cochran 2012 (25%); Cochran 2014 (17–25%); Mears et al. 2012 (24%)), to 40% in a nationally representative sample (Tahamont 2013) and 42% in a different FL sample (Bales and Mears 2008), to the high end of 61% in Minnesota (Duwe and Clark 2013), 76% self-reported visits in a survey of FL prisoners (Liu et al. 2016); and 79% of female prisoners in a maximum security prison in the northeastern U.S. (Acevedo and Bakken 2001).

  4. Family reunion visits, sometimes called “conjugal visits”, are the one exception to the general political palatability of visiting programs. These visits, during which inmates stay overnight in a semi-private setting with family members, were allowed in eight states in 2011 (Boudin et al. 2014), but by 2015 they were only allowed in four states.

  5. Visitation guidelines for New York State DOCCS are available at:

  6. Siennick et al. (2013) examined the temporal relationship between visitation and misconduct week by week in a sample of inmates who received one or more visits. This study provides some indirect information on patterns of prison visitation.

  7. This was the total sample across 10 cohorts modeled. For any single cohort modeled the sample ranged from N = 1485 for 8-month trajectory model down to N = 639 for the 17-month trajectory model.

  8. Similar to the Cochran (2012) paper, the second most common group following “no visitation” was the “near entry” with 4–15% of the sample experiencing this pattern (depending upon cohort length) of peaking probability of visitation at around 0.5 during the first half of the cohort (Cochran 2014). The “near release” group (4–9%) had their highest probability of receiving a visit during the second half of their stay, with a peak probability of around 0.4–0.5. Lastly, a “sustained visitation” group (3–7%) was also present across all cohorts, with a 0.9 probability of receiving a visit across several of the months of observation. Additional information on model fit was provided in the Cochran (2014) paper: average posterior probabilities of group membership (AvePP) and odds of correct classification (OCC) statistics all fell within the recommended guidelines (Nagin 2005).

  9. See Piquero (2008) for a summary of the literature on trajectories of criminal behavior and criminal justice contact as people age.

  10. Depending on the date of admission, a 28-day month might include visits during two different calendar months.

  11. We count visits by unique combination of visitor and visit date. For example, 3 unique visitors on 1 calendar date equals 3 visits, just as 1 unique visitor on 3 calendar dates equals 3 visits.

  12. Poisson models have difficulty fitting when the sample contains small numbers of people with wide variation in counts in a given period. Trajectory modelers have commonly dealt with this problem by top coding the data (Nagin 2005).

  13. Although these estimates can be used to then identify trajectories for individuals, which can then be explained in subsequent analyses, these individual trajectories are driven by levels of the outcome and the overall sample-based estimates of change as opposed to individual patterns of change over time (Bushway et al. 2009).

  14. We will deal with the problem of skewness by top coding the counts of visits at the 95th percentile, which turns out to be a maximum of five visits in a month. Standard Poisson models can easily handle counts within this range. However, we should keep in mind the cost when examining the following trajectories, which is that the trajectory for the group with the highest number of visits may be artificially lower, and more stable than it would otherwise be in the absence of the top coding.

  15. To be clear, individuals who have a non-zero probability of receiving a visit (not a true-zero) can still receive zero visits, according to the Poisson likelihood function, conditional on the rate of visits. As such, the logit captures the extra-Poisson probability of not receiving a visit.

  16. It is possible to choose more parsimonious forms of any given model, testing for lower orders of the polynomial for given trajectories. In this paper, we have chosen to stay with cubic form for all trajectories.

  17. The other three methods for assessing robustness to start values recommended by Sweeten and Hannula (2015) were either inappropriate for ZIP models (i.e. individual trajectory model start values) or too computationally intensive given our large sample size (i.e. bootstrapping start values, random effects start values).

  18. We top-coded the count of visits per month at the 95th percentile (5 visits) to correct for skewness in the visit counts and improve the fit of the Poisson models. As such, the trajectory of visit counts for Group 7 may be both lower and more stable than it would be without top coding.

  19. Monthly distance from home county was computed by taking a daily driving distance between the facility in which the inmate resided and county of commitment. Within each 28-day “month” these daily distances were combined into a weighted average based on the number of days in each facility. Google Maps API was used to capture the driving distance between the ZIP code centroid for each prison facility and the county centroid for home county.

  20. It is important to remember that the model groups and trajectories are not “fixed realities,” but rather a simplification of the population distribution that shows patterns over time (Nagin 2005, 173). This simplification can support further insight if we assign each individual in the sample to the group for which they have the highest posterior probability of group membership and then use those assignments to describe the characteristics of the individuals who are assigned to each group (this is sometimes referred to as the classify-analyze approach, see Nagin 2005).

  21. We have often heard the assertion that the inmates incarcerated closer to their homes will receive more visits from anti-social associates than those incarcerated farther away. While it is possible that anti-social visitors are also less motivated or have less resources (two things that are strongly related to the visitation rate) that is not necessarily the case. It is straightforward to envision a criminally motivated visitor who is highly motivated to visit and has resources to do so (i.e. someone who is trying to bring drugs into a prison for sale).

  22. Descriptive statistics on demographics, criminal record, and some prison experience variables for individuals assigned to the seven groups using PPA are available from the authors upon request.

  23. At the recommendation of a reviewer, we conducted a secondary analysis, splitting the sample into two groups: maximum security (which allows daily visits) and medium/minimum security (which have weekend/holiday visits). For the medium/minimum security sample, the 7-group trajectory model converged on patterns of probability, counts, and group sizes that were nearly identical to the main model (not surprisingly since these were the bulk of the cases on which the original model was estimated). For the maximum security sample, the picture was largely the same, with the exception of smaller estimated group size for the low/no visit group (the estimated group sizes across the remaining groups, including the highest visit groups, increased as a result). Results from these models are available from the authors upon request. We feel it is important to highlight that these “reverse” analyses still suffer from the same selection issues as the original descriptive analyses, in that individuals assigned to maximum security at admission are qualitatively different from those assigned to medium/minimum facilities.


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The authors would like to thank the staff at both DCJS and DOCCS for their assistance. All errors remain our own.


These data are provided by the New York State Division of Criminal Justice Services (DCJS) and New York State Department of Corrections and Community Supervision (DOCCS). The opinions, findings, and conclusions expressed in this publication are those of the authors and not those of DCJS or DOCCS. Neither New York State nor DCJS nor DOCCS assume liability for its contents or use thereof.

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Correspondence to Audrey Hickert.

Appendix: Robustness to Start Values

Appendix: Robustness to Start Values

In the text, we reference a working paper by Sweeten and Hannula (2015) that reviews the recent GTM literature with respect to probing the robustness of these models to the choice of start values. In this appendix, we provide a detailed description of our testing using the “stepping stones” method as well as a brief description of additional tests we conducted to probe the robustness of our seven-group model to the default start values selected by the traj plugin for Stata.

In the stepping down method one group is dropped at a time from the original eight-group model (“Original parameters” for the step down and step up methods were obtained using default start values for the Stata traj plugin). We use the remaining seven parameters as start values for a seven-group model. We ran two sets of stepping down models: one where all seven group sizes were set to be equal (14% each) and one where the size of the dropped group was automatically added to the first remaining group in the model (e.g., if an 8% size group was dropped, that amount was added to the first remaining category, such that a 7% group became a 15% group). In the equal group size stepping down models, two of the eight models resulted in the same BIC as the original seven-group, with coefficients similar to three decimals places for most parameters. The remaining six of eight models resulted in a worse fit (lower BIC values). In the unequal group size stepping down models, four of the eight models resulted in the same BIC as the original seven-group, with coefficients similar to three decimals places for most parameters (the remaining four resulted in a worse fit).

In the stepping up method, we add one group to the original six-group parameters (again obtained from default start values for the Stata traj plugin) and reduce the largest group percentage by 100/7 to carve out 14% for the added group. We used parameters from each of the eight groups in the eight-group model to add to the six group parameters for the stepping up method. All eight models from this stepping up method resulted in worse fit (lower BIC values) than the original seven-group model.

As an additional check, we ran two sets of 15 models with coefficients from a random matrix of start values [mean = 0, SD = 1]. We selected these values as most of the original parameters from the best fitting seven-group model were within this range. The first set of models ran with these random start values had equal group sizes (14% each), while the second had group sizes set near the original seven-group model (39, 26, 8, 7, 7, 7, 6%). All 15 of the equal group size models resulted in the same BIC as the original seven-group, with coefficients similar to three decimals places for most parameters. Twelve of the 15 unequal group size models resulted in the same BIC as the original seven-group, with coefficients similar to three decimals places for most parameters. One resulted in a worse fit (lower BIC values), while the remaining two failed to converge.

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Hickert, A., Tahamont, S. & Bushway, S. A Tale of Two Margins: Exploring the Probabilistic Processes that Generate Prison Visits in the First Two Years of Incarceration. J Quant Criminol 34, 691–716 (2018).

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  • Zero-inflated Poisson (ZIP)
  • Group-based trajectory model (GTM)
  • Prison visit