Do water savings persist? Using survival models to plan for long-term responses to extreme drought

The water consumption data used in this analysis are bi-monthly billing data for 19 324 single- and multi-family customers served by Mesa Water. Customers are this study's primary unit of analysis. We present results for customers in four residence types: single-family residences without outdoor space, single-family residences with indoor and outdoor space, multi-family residences without outdoor space, and multi-family residences with indoor and outdoor space. Billing data were cleaned and converted to pro-rated bi-monthly amounts prior to analysis (more details are provided in the SI).

Conservation and rebound were detected for each customer individually with the consumption change detection method presented in Bolorinos et al [24], which uses multiple, unsupervised change detection from Frick et al [42]. Figure 1 shows how we used change detection results to identify conservation and rebound. The change detection procedure fits a step function to consumption residuals. A reduction in the step function corresponds to a 'conservation' event. Follow-up time for a conserving residence begins with the first conservation detected on or after January–February 2014 and ends either with a rebound outcome, or on December 2018 (if no rebound is detected). Rebound outcomes are triggered by increases in the step function. We consider two types of rebound: 'any' rebound is triggered by any consumption increase; 'effective' rebound is triggered when cumulative consumption increases reverse at least 75% of a conservation's event's water savings. The time from a conservation to a rebound event defines its survival time (if no rebound event is detected, observations are censored at the end of the follow-up period). Note that, for any individual customer, the size and location of consumption changepoints—and thus conservation survival time—may be somewhat uncertain. Nonetheless, we assume this uncertainty is random, such that its effect on standard errors is reduced by large sample sizes (our final analysis includes 14 876 customers; the smallest customer category consists of 239 customers).

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Conservation survival times were analyzed with an accelerated failure time model, a common approach to analyzing censored time-to-event data [43]. Our model treats the logarithm of survival time for each customer $({t_i}$) as a linear function of baseline covariates and a parametric error ${\in _i}$ (1), which captures unexplained variation in conservation survival times. The model assumes that survival times are log-normally distributed (1). It is generally not possible to test this assumption since we do not observe survival times greater than 5 years; however, our large sample sizes mean that standard errors are still asymptotically valid.

The covariates in (1) are customer type (${\text{CustTyp}}{{\text{e}}_j}$, with fixed-effect ${\alpha _j}$), a set of customer-level variables ${X_{ik}}$—whose effects ${\beta _j}$ vary by customer type $j$, and a set of block-group and precinct level variables ${Z_{il}}$, assumed to have similar effect on all residence types. Summary statistics for all customer- and block-group-level covariates are given in the supplement

$$\begin{align}\log \left( {{t_i}} \right) & = \mathop \sum \limits_j {\alpha _j}{\text{CustType}}{_{ij}} +\! \mathop \sum \limits_j\! \mathop \sum \limits_k {\beta _{jk}}{\text{CustType}}{_{ij}}{X_{ik}} \nonumber\\ & \quad + \mathop \sum \limits_l {\gamma _l}{Z_{il}} + {\in _i}.\end{align} \tag{ 1 }$$

Customer-level covariates (${X_{ik}}$) are the relative size of a conservation event (as a fraction of 2002–2012 baseline water use), the logarithm of 2002–2012 baseline water use, an indicator flagging any occupancy change during follow-up, and an indicator flagging conservation events that coincided with a water use efficiency rebate given by Mesa Water (details on the two indicators are given in the supplement).

We also include five neighborhood-level covariates (${Z_{il}}$). Four are averages of U.S. American Community Survey's 2013–2019 annual estimates of median age, median household income, percentage of married families, and percentage of owner-occupied housing units. These are measured by U.S. census block groups, statistical sub-divisions of 600–3000 people the U.S. census bureau uses to tally census results [44]. The fifth ${Z_{il}}\,$ covariate is the share of votes received by Hillary Clinton in the 2016 presidential election, measured by U.S. electoral precincts (or voting districts) [45], independently-defined sub-divisions state and local governments use to administer federal elections [46].

Note that neighborhood variables were chosen among a larger subset of demographic characteristics. Many demographic characteristics are highly correlated (e.g. median home value, educational attainment, and median household income) and we simplified our analysis by selecting covariates that measure qualitatively different demographic features. We show in the SI that in-sample predictive performance of our subset of variables is similar to a model with all predictors, so they capture most meaningful cross-sectional variation in survival times.

The accelerated failure time model in (1) was fit to each rebound outcome (any rebound and effective rebound) using the 'survreg' function in R's 'survival' package [47]. Since baseline rebound risk varies primarily with calendar time, the model was stratified by the time of each observation's conservation event (i.e. the distribution of ${\in _i}$ in (1) differs for observations with conservation events in each of the 12 bi-months from January–February 2014 to November–December 2015). Bootstrap resampling was used to compute standard errors of all coefficient and survival probability estimates.

We compare the 2012–2016 drought to the less-severe 2007–2009 drought by running a parallel consumption change detection procedure as used to detect conservation and rebound in 2014–2018. The training period was set to January 2002 to December 2006 and the analysis period was set to January 2007 to December 2013. As before, we ensured that the power of the change detection algorithm was the same for all time-periods by restricting our analysis to January 2008–December 2012.

We analyzed conservation survival in both droughts by comparing time to rebound of conservation detected in 2008–2009 and conservation detected in 2014–2015, following customers for 3–5 years (i.e. until December 2012 and December 2018, respectively). Note that the earlier drought's follow-up period is capped at 5 years. This is because less than 5 years separated it from the severe 2012–2016 drought, such that consumption rebound was less likely after year 5.

Each follow-up period begins in January to account for seasonality of responses to dry conditions; years were chosen to correspond to roughly similar hydrologic stages of the two droughts (details given in the SI). We note that it is virtually impossible to draw clean comparisons of two 5 year periods, either hydrologically—the two droughts were not of equal duration—or generally, since other external, time-specific consumption drivers also differentially affected water consumption (e.g. the 2007–2009 great recession, which reduced household incomes and thus could have dampened water demand). This analysis should thus be treated as a simple descriptive comparison of conservation and rebound during two intrinsically different droughts.

When comparing both droughts, we examine mean raw and structural consumption residuals for each customer type during each 5 year follow-up period (January 2008–December 2012, January 2014–December 2018). Raw consumption residuals are errors from the consumption change detection procedure's initial prediction step [24]. Structural consumption residuals are the mean step function fit to each customer's raw consumption residuals by change detection and capture consumption increases and decreases detected during follow-up [24].

Table 1 summarizes detected conservation and rebound by residence type. We detected conservation among 43%–57% of customers in 2014 or 2015. Median consumption reductions range from 22% of baseline (multi-unit residences without outdoor space), to 36% of baseline (single-unit residences). Of conserving customers, roughly 42% subsequently increased consumption before January 2019 (i.e. rebounded). Median rebound size is similar to conservation, ranging from 26% to 42% of baseline. On average, detected rebound reverses 84% of prior customer savings.

Figure 2 plots relative conservation and rebound size (relative to 2002–2012 baseline consumption) for rebounding customers. The locally estimated scatterplot smoothing (LOESS) smoother fit to the two variables suggests relative rebound size levels off with the size of prior savings, particularly for values larger than 50%.

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Figure 3(A) plots a Kaplan–Meier curve for survival against effective rebound, stratified by residence type. Kaplan–Meier curves show the fraction of a population with no outcome at each time-step in their follow-up. The figure shows that—5 years after first conserving—only 25% of customers had effectively rebounded (i.e. increased consumption to an extent that reversed 75% or more of prior savings). Survival probabilities were significantly higher among customers in single-unit residences with outdoor space than in multi-unit residences with outdoor space, with lowest survival in multi-unit residences without outdoor space.

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As figure 3(B) shows, the curves fit by accelerated failure times models imply long survival times: for the whole sample, time to any rebound averaged 4.6–4.8 years; time to effective rebound averaged 7.6–8.3 years. For the average customer, this survival implies it has taken 5 years for water use to increase detectably after 2014–2015—and will take roughly 8 years for water use to effectively rebound to prior levels.

Figure 4 displays coefficients and 95% confidence intervals for the effect of customer-level variables on 2014–2015 conservation survival time, as measured by any or effective rebound. Since the water use behaviors of the four residence types are inherently different, we model the effects of customer-level variables on them separately. For single- and multi-unit customers with outdoor space, the results show large conservation events (as a fraction of baseline water consumption) generally have a substantially longer time to effective rebound, reinforcing our previous finding that rebounding customers who conserved more were less likely to reverse prior savings. This effect is generally inverted, however, for any rebound: a 10% increase in relative conservation is associated with a 2%–3% decrease in time to any rebound—probably reflecting a greater likelihood that water consumption will adjust upwards after a drastic reductions.

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For single family customers with indoor and outdoor space, two other customer characteristics also significantly associate with conservation survival. First, higher baseline water consumption associates with greater survival against effective rebound, meaning conserving customers with higher baseline consumption were more likely to continue conserving water (which may reflect the impact of customer affluence, roughly proxied of average water use). Second, occupancy changes negatively associate with time to effective rebound, suggesting occupancy changes lead to other changes (e.g. landscaping, remodeling) that increase consumption.

Figure 4 shows water use efficiency rebates during follow-up are not significantly associated with survival against either type of rebound. This may be due to the small number of efficiency rebates in 2015–2019: only 30 among 6062 single indoor and outdoor customers with detected conservation in 2015–2016. Indeed, there were not enough observations to estimate the impact of efficiency rebates on indoor-only single- and multi-unit customers (hence their omission from figure 4).

Figure 5(A) shows how neighborhood-level demographic covariates correlate with conservation survival time—after accounting for the above customer-level effects. We included four demographic covariates measured at the level of U.S. census block-groups: median household income, median age, share of owner-occupied residences, and share of married households. In addition, we included the precinct-level share of votes received by Hillary Clinton, the 2016 U.S. Democratic presidential candidate.

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Results show median household income and the share of married households do not correlate significantly with time to either type of rebound. However, higher owner-occupied residence share and lower median age are both associated with greater survival against both types of rebound. As showcased in figure 5(B), the precinct-level share of Democratic votes associates positively with survival time against effective rebound, suggesting cultural and political factors were important in determining behaviors relevant to environmental actions and therefore conservation persistence.

We compare conservation survival in California's most recent (2012–2016) drought to the less severe 2007–2009 drought. Specifically, we compare conservation events detected in 2014 and 2015 to ones detected in 2008 or 2009, following conserving customers for 3–5 years.

Conservation was somewhat more persistent in 2008–2009 than it was in 2014–2015 (figure 5(A)). Average implied survival against effective rebound is 9.8–11.5 years for 2008–2009 conservation, 2–3.5 years longer than the survival time observed 2014–2015 conservation (panel (B)); survival against any rebound tells a similar story: survival times are 0.5–1 years longer in 2008–2009 than in 2014–2015. This suggests heightened public awareness and media coverage in 2014–2015 lead to increased but less persistent conservation than in the 2007–2009 drought. Note that this comparison does not isolate the effects of public awareness associated with severe droughts—the 2007–2009 economic recession may have limited water consumption growth (and rebound) after 2009, for example. At the very least, it provides no evidence that 2014–2015 conservation persisted longer than in 2008–2009.

Figure 5(C) explores one potential reason for the difference in conservation persistence. The figure shows changes in average raw and structural components of each customer type's water use residual. The raw component is the deviation of consumption from its predicted value obtained in the first step of consumption change detection. The structural component is the step function fit to the raw residual in the second step of consumption change detection, capturing all conservation and rebound events detected during the 5 year drought period.

Figure 5(C) shows that 2012–2016 drought's climax produced steep consumption reductions among customers with outdoor water use. Short-term consumption responses captured by raw residual changes (plotted as dotted, colored lines) shows precipitous declines into mid-2015, when California introduced the first statewide mandatory outdoor watering restrictions. A sizable share of these reductions disappears by year's end, however. More moderate declines in structural residuals persist for longer but are partially reversed over the next two years, such that structural savings 5 years after the drought's beginning are similar in 2012–2016 as in 2008–2012. Thus, although the 2012–2016 drought produced relatively deep consumption reductions among customers with outdoor watering, these were likely associated with more temporary behaviors (e.g. compliance with outdoor watering restrictions) that ended as dry conditions eased.

The evolution of consumption residuals is somewhat different among customers without outdoor watering. In single-unit residences, there is a less pronounced trough in raw and structural consumption residuals in mid-2015, after which residuals rise slightly before declining once more post-2017. The structural consumption residual change of customers in multi-unit residences reaches its minimum in mid-2016, with only a slight increase by the end of 2019. In both cases, water savings after 5 years are indistinguishable from those achieved after January 2008.