Estimating Mean Length of Stay in Prison: Methods and Applications

Abstract

This paper examines various approaches to studying the mean length of stay in prison. The literature contains a wide range of estimates of this quantity. The discrepancies that appear in these estimates and in the conclusions reached from them have been the subject of several reviews. We build upon that work, using the life table as the gold standard, to demonstrate the inaccuracy of common measures such as the ratio of the population size to the annual number of entrances or the mean length of time served by those exiting in a particular period. This demonstration is conducted in two parts. One part uses model populations with constant growth rates; the second part relies upon simulated prison populations with time-varying rates of entrance and exit. In addition, we introduce two new indirect measures that are more accurate than several existing indirect measures and that are relatively easy to use. The new measures are based on the entrance rate or the exit rate and adjust for the growth rate of the prison population.

Appendices

Appendix A: Sources and Processing of Data

The data used come from the Bureau of Justice Statistics. Using the 1995 Adult Correctional Census, in conjunction with the National Correctional Reporting Program’s admissions and exits, we projected the prison population of the 29 states—Alabama, California, Colorado, Hawaii, Illinois, Kentucky, Maryland, Michigan, Minnesota, Mississippi, Missouri, Nebraska, New Hampshire, New Jersey, New York, North Carolina, North Dakota, Ohio, Oklahoma, Oregon, Pennsylvania, South Carolina, Tennessee, Texas, Utah, Virginia, Washington, West Virginia, and Wisconsin—forward to mid-1997. Because no data exist concerning the duration-specific structure of the current population for each state, we adopted the duration-distribution of persons housed inside prison from the 1997 National Survey of Inmates in State and Federal Correctional Facilities. This distribution was national and not restricted to the 29 states. We excluded those housed in federal facilities and women.

After obtaining the duration-specific distribution of the population, we used the 1997 National Corrections Reporting Program Release to produce duration-specific exit rates for men. We exclude persons whose exit was a result of death or escape. Additionally, our estimates do not include time served prior to the current admission. That is, if the person spent time in jail for the current offense and it was applied to the sentence length, we do not consider that. We do this to avoid confusion in the denominator and numerator. Since the survey provides the admission date to prison of the sentenced person, the numerator should reflect the same.

We calculated exit rates for all offenses as well as one offense-specific category—murder. The calculation of the duration-specific rate, _n M _x , was the quotient of the number of exits in the duration group x to x + n and the mid-year population. The mid-year population was used to approximate the number of persons exposed to the risk of an exit. Thus the following equation represents the duration specific rate:

$$ {}_nM_x =\frac{{}_nD_x }{{}_nN_x }, $$

(A.1)

where _n D _x  = the number of exits occurring in the duration group of x to x + n and _n N _x  = the mid-year population for the duration category of x to x + n, our estimation of person-years lived in this portion of the analysis.

The same process was used for murder. The 1997 National Survey of Inmates in State and Federal Correctional Facilities was used to find the duration-specific structure of those in prison for murder, and then the release file was used to select those who were released for murder (includes murder and non-negligent manslaughter).

Appendix B: Derivations

In this section of the appendix we derive the new indirect estimators of the mean length of stay in prison based on stable population assumptions. Because it allows for population growth or decline, this class of populations is far more general than the class of stationary populations typically assumed in indirect indicators. Thus, the indirect estimators should be applicable to a much wider range of populations.

Because the entrance rate, exit rate, growth rate, and duration-composition are constant over time in a stable population, we can make use of several basic equations to express the relationships between various population components (Coale 1972; Preston et al. 2001). The expression for the entrance rate is

$$ b=\frac{1}{\int_0^\infty {\hbox{e}^{-ra}p(a)\hbox{d}a} } $$

(B.1)

where r = growth rate of the stable population and p(a) = probability of surviving from entry to duration a.

We expand e^−ra in a Taylor expansion through the first term and substitute into Eq. (B.1), giving

$$ b\approx \frac{1}{\int_0^\infty {[1-ra]p(a)\hbox{d}a} } $$

(B.2)

Recognizing that the integral of p(a) is mean length of stay upon entrance, e ⁰₀ , we rearrange this expression to give

$$ \hbox{e}_0^0 \approx \frac{1}{b[1-rA_P ]}, $$

(B.3)

where A _P is the mean duration of the stable population. Equation B.3 is the “corrected” entrance rate estimator of the mean length of incarceration. The correction involves the growth rate of the prison population and the mean length of imprisonment to date for those currently in prison.

The expression for the exit rate of a stable population is

$$ d=\frac{\int_0^\infty {\hbox{e}^{-ra}p(a)\mu (a)\hbox{d}a} }{\int_0^\infty {\hbox{e}^{-ra}p(a)\hbox{d}a} }, $$

(B.4)

where μ(a) = exit rate at duration a.

Differentiating this expression with respect to r and simplifying gives

$$ \frac{d}{\hbox{d}r}d\approx d[A_D -A_P ], $$

(B.5)

where A _D is the mean duration at exit for those exiting prison in a particular year or period. Alternatively,

$$ \frac{d}{\hbox{d}r}(\ln \;d)\approx A_D -A_P. $$

(B.6)

Thus, the proportionate error in d as an estimator of d ₀, the exit rate of the stationary population and the reciprocal of mean length of stay upon entrance, is r[A _D  − A _P ]. So the corrected estimator of mean length of stay upon entrance is

$$ \hbox{e}_0^0 \approx \frac{1}{d[\hbox{e}^{-r(A_D -A_P)}]}. $$

(B.7)

Equation B.7 is the corrected exit rate estimator of the mean length of incarceration. The correction involves the growth rate of the population as well as the difference between the mean duration at exit for those exiting in any particular period and the mean length of imprisonment for those currently in prison.