A high-resolution analysis of process improvement: use of quantile regression for wait time.

Organizational process improvement often emphasizes reductions in variation; mean performance may remain unchanged if the effect is primarily a reduction in extreme values (high quantiles). Such data may be best analyzed by modeling the temporal regression relationship at the 0.95th quantile or 95th percentile (cases that exceed 95 percent of the distribution) controlling for the independent variables. A conventional linear regression analysis assumes the relationship observed for the mean holds at the different quantiles of the response variable. The estimated temporal mean trend basically determines the shape of temporal trends at individual quantiles. The resulting trend estimates cannot be necessarily true or close to the true temporal trend at a particular quantile of interest. Least-squares analysis is not valid when addressing a performance improvement goal stated as “95 percent of patients have less than 10 days between first contact and first treatment.” Quantile regression, on the other hand, estimates the 95th percentile directly and allows us to investigate the question without constraining the trend at 95 percent to be of a certain shape. System-level studies of the performance measure “wait time to treatment” are rare in the U.S. health care system despite the importance of this measure. The need for systematically monitored and analyzed process measures is particularly pressing within the substance abuse treatment field (Garnick et al. 2002). Patients seeking treatment for alcohol and drug use disorders often encounter waiting lists and experience delays between the day they request care and the day they enter care. The delays reflect a fixed capacity for new patients and limited financing for expanding access to care (Hoffman and McCarty 2012). The organization and delivery of care, however, can also inhibit access to care and discourage patients from attending addiction treatment appointments (Stark, Campbell, and Brinkerhoff 1990; Hser et al. 1998; Ford et al. 2007). NIATx was the first widespread application of process improvement techniques to substance abuse treatment (http://www.niatx.net) (Capoccia et al. 2007). Participating agencies were trained to use a simplified version of the Institute for Healthcare Improvement's hospital improvement support system (http://www.ihi.org). A cross-site evaluation of NIATx participants examined: (1) change in days between first contact and first treatment and (2) the percentage of clients that began treatment and completed the first four units of care. Thirteen agencies that began participation in NIATx in August 2003 reduced days to treatment 37 percent—from 19.6 to 12.4 days across all levels of care. Retention in care also improved; the proportion of clients who completed a first session of care and returned for a second and third session of care increased 18 percent between the first and second session (72–85 percent) and 17 percent between the first and third session of care (62–73 percent) (McCarty et al. 2007). A subsequent analysis of 14 outpatient and intensive outpatient treatment programs within the second NIATx cohort replicated the reduction in wait time and the improvement in retention and noted that the first cohort sustained the gains during a 20-month follow-up (Hoffman et al. 2008). Previous analyses of wait time to treatment within the Network for the Improvement of Addiction Treatment (NIATx) employed a conventional least-squares regression analysis and estimated rates of change in the mean of the monthly averaged outcome variables (McCarty et al. 2007; Hoffman et al. 2008). Analysis of mean function is useful, but other aspects of the response distribution can also be of interest. Patient wait times may have unequal variation due to complex interactions between variables or unobserved exogenous noises that are not accounted for in the regression model. Figure 1 shows distributions of patient wait time in a NIATx program. The distribution of all wait times across the 15-month study period appears in Figure 1a. Figure 1b presents wait time distributions for month 1 (first), month 8 (mid), and month 15 (last) and illustrates change in wait time distributions over time. This variation in the distributions implies that more than one single slope describes the temporal trends in wait time from the first contact to the first treatment. In the presence of such heterogeneity, conventional least-squares regression models may underestimate, overestimate, or fail to detect important changes occurring locally at a certain quantile of data, because it focuses on changes in the means (Terrell et al. 1996; Cade, Terrell, and Schroeder 1999). Figure 2 provides expository guidelines where quantile regression is desirable over the conventional least-squares regression analysis. Figure 1 The Distributions of Wait Time to Treatment in Program 8. (A) Density Plot of All Wait Time. (B) Density Plot of Wait Time in the First (n = 39), Mid (n = 49) and Last Months (n = 33) Figure 2 Guidelines How to Choose between Least-Squares Regression and Quantile Regression Quantile regression (Koenker and Bassett 1978) extends the concept of percentile or quantile in the univariate analysis to regression and estimates regression relationships specific to a certain percentile of the response variable controlling for the independent variables. In its simplest application, quantile regression examines median regression and describes changes in the center of the distribution. As the median is robust to outliers in the univariate analysis, median regression is also robust to outliers. Median, however, is just one of the many quantiles that can be examined. A series of quantile regression for different percentiles estimates multiple rates of change, providing a more robust and detailed description of the regression relationships that can be overlooked by other regression methods. Quantile regression is well established and available in common statistical software (e.g., proc quantreg procedure in SAS; quantreg package in R). Applications include economics (Hendricks and Koenker 1992; Buchinsky 1994; Manning, Blunmberg, and Moutlton 1995; Poterba and Rueben 1995); biology, especially for growth curve and duration models (Cade, Terrell, and Schroeder 1999; Koenker and Geling 2001); medicine, especially for reference charts (Cole 1988; Cole and Green 1992; Wei et al. 2006); environmental modeling (Pandey and Nguyen 1999); and infrastructure studies (He, Simpson, and Wang 2000). More examples are available in Koenker (2005).