Abstract
Freezing temperatures in fall, winter, and spring can cause damage to multiple perennial fruit crops including northern highbush blueberry (Vaccinium corymbosum). Predictive modeling for lethal temperatures allows producers to make informed decisions about freeze mitigation practices but is lacking for northern highbush blueberry grown in the Pacific Northwest. If buds are hardier than air temperatures, unnecessary use of propane heaters and/or wind machines is costly. In contrast, use of heaters and/or wind machines during freezing, damaging temperatures can minimize crop damage and potential yield loss. The objective of this study was to model cold hardiness across multiple cultivars of northern highbush blueberry grown in various regions in Washington, USA, and to generate predictive cold hardiness models that producers in the Pacific Northwest could use to inform freeze mitigation. Multiple years of experimental cold hardiness data were collected on four cultivars of northern highbush blueberry grown in western and eastern Washington, USA. Freeze chambers were used to reduce bud temperatures systematically, after which buds were dissected and bud survival was assessed. A generalized linear mixed model with a binomial response and logit link was fit to each cultivar to characterize the relationship between bud survival, freezer temperature, recent air temperatures, and growing degree days from fall acclimation to late winter/spring deacclimation. Model simulation was performed to obtain marginal-scale lethal temperature estimates. Model error estimation was performed using cross validation. Results show cultivar-specific cold hardiness models can be generated, and model development and use can help growers make more informed decisions regarding freeze protection that also minimizes costly applications of freeze protection when unnecessary. Furthermore, such models can be adapted to other blueberry growing regions and cultivars experiencing similar climactic conditions.
The United States is a historically important and significant global producer of blueberries (Vaccinium sp.) (International Blueberry Organization 2022). Washington, USA, and the broader Pacific Northwest region lead the United States in highbush blueberry production, with Washington alone producing 81,647 t of fruit in 2021 valued at $228 million and representing 27% of national production (US Department of Agriculture, National Agricultural Statistics Service 2022). Most of the Pacific Northwest industry cultivates northern highbush blueberry (Vaccinium corymbosum; hereafter, “blueberries”) for fresh and processed markets. However, despite the scale and significance of the industry, floral buds are susceptible to freeze injury, and this injury can reduce yields.
Freeze tolerance, more generally referred to as “cold hardiness,” is measured by lethal temperature on specific plant tissues such as floral buds, xylem, or phloem. In fruit crops, freezing temperatures (<0 °C) can affect negatively the floral buds and cellular tissues of which they are composed. These floral buds are important because they are the source of the fruit crop. Freezing temperatures can result in extracellular freeze damage whereby water freezes outside of plant cells, and can induce cellular damage through dehydration and osmotic stress, which can result in cell death [reviewed by Pearce (2001)]. Freezing temperatures can also result in intracellular damage to protoplasmic structures and cell death when water crystallizes into ice inside the cell (Levitt 1980). Depending on the cultivar and prior years’ crop, there may be 7 to 10 developing flower primordia within a single floral bud (Strik et al. 2017). Each flower primordia has the potential to develop into a blueberry fruit, rendering survival of those developing flower primordia important for maximizing yield.
Freeze events may be advective or radiative in nature, where advective freeze events typically coincide with windy conditions and radiative freeze events typically occur under calm and clear conditions (Smith 2019). Radiative freezes often occur with an inversion, when the air is colder on the ground and warmer higher in the atmosphere. Freeze events are commonly radiative in perennial crop systems such as orchards and vineyards (Evans and Alshami 2009). To mitigate damage during inversions in the Pacific Northwest, the operation of propane heaters produces both convective and radiant heat to raise field air and plant surface temperatures. In addition, wind machines mix warmer inversion air with cooler air on the ground layer, effectively elevating the average air temperature in a field. Freeze mitigation using propane wind machines and heaters can be costly. It is common practice for blueberry growers in parts of the Pacific Northwest to have 50 to 60 heaters/ha, each using 3.8 L/h of propane, and one wind machine per 4 ha using 3.1 L/h of propane (AgHeat 2023a, 2023b; Orchard-Rite 2023). This results in a minimum of 191 L/h/ha, or $76 ha/h, assuming a cost of $0.40/L for propane. With spring nightly freezes sometimes requiring cold mitigation for several hours two to four times a week, as well as fluctuations in the cost of propane, the overall expense for growers can be substantial.
Perennial plants in temperate climates with freezing winters go through a process of acclimation in preparation for cold winter months, and deacclimation in preparation for growth and flowering in the late winter and spring (Ehlenfeldt et al. 2012; Ferguson et al. 2011). Shortening daylengths and progressively cooler temperatures in the fall cause plants to enter an ecodormant state in which they acclimate to freezing temperatures. Endodormancy (or “deep dormancy”) occurs after acclimation is complete in midwinter and is the period when plant cold hardiness is at a maximum but can fluctuate based on cultivar, fall temperatures, and precipitation/irrigation (Bittenbender and Howell 1975). During this period of endodormancy, plants accumulate chilling hours or “chill units.” A certain number of these chill units need to accumulate for a given species or cultivar before the plant can break dormancy successfully and regrow when temperature conditions are suitable. Exact chill units or hours per cultivar of blueberry are unknown, but cultivars with more southern highbush (complex hybrids of V. corymbosum and other Vaccinium spp.) genetics can require as little as 200 chill units whereas northern highbush cultivars could require as high as 1000+ chill hours (Ehlenfeldt et al. 1995, 2012).
After chill units have been acquired, the plant reenters a state of ecodormancy. While in this ecodormant stage, floral buds deacclimate to freezing temperatures as daylength and temperatures increase from late winter to spring. In general, blueberries are cold hardy in the middle of winter to about –26 to –29 °C (Longstroth 2012). However, annual blueberry winterhardiness can fluctuate by as much as 6 to 8 °C, depending on preceding weather conditions (Hanson et al. 2007; Rowland et al. 2005). Knowing how floral bud hardiness fluctuates based on preceding weather conditions and how that influences overall floral bud hardiness is critical for defining threshold temperatures to implement freeze mitigation practices. However, the most critical time for freeze injury is late winter and early spring, when buds are deacclimating and air temperatures can fluctuate. Floral bud phenology also changes during this period, as well as associated freeze hardiness. Floral bud phenology is known to be better predicted through measuring heat accumulation compared with using calendar days (Logan et al. 1990). Heat accumulation is generally measured through growing degree days (GDDs) with crop-specific upper and lower thresholds for development.
Currently, blueberry growers in Washington determine relative cold hardiness by examining the external phenological stage of floral buds and compare it to predicted hardiness temperatures published from Michigan State University (Longstroth 2012). Forecasted weather is compared against estimates of cold hardiness. If temperatures are expected to fall below the expected hardiness for a given phenological stage, then growers will use wind machines and/or propane heaters for freeze mitigation. However, there is uncertainty in whether this method is accurate and applies to conditions in other blueberry-growing regions, and whether there is variation by cultivar.
To estimate hardiness more accurately, previous studies have developed models that consider cumulative temperature history. For instance, Fowler et al. (1999) created a cold hardiness model for cereal crops that bases the lethal temperature at which 50% of flower primordia within a bud will die (LT50) on the difference between the previous day’s LT50 and daily temperature; Ebel et al. (2005) developed a freeze risk model for Mandarin oranges (Citrus reticulata) using the number of hours in the past 500 h during which the temperature was <10 °C. Ferguson et al. (2011) estimated grapevine (Vitis vinifera) cold hardiness in Washington, USA, by combining the prior 1-d average temperature with the cumulative hardiness value for the season.
To predict fluctuating cold hardiness more accurately through environmental parameters, a multiyear experiment was conducted with the objective to create a cold hardiness model for multiple cultivars of blueberry grown in various regions in Washington that producers in the Pacific Northwest could use to inform freeze mitigation. Successful development and deployment of cold hardiness models can, in turn, allow for more data-driven decision making at the farm level to avoid the costs of under- or overprotecting the crop. Furthermore, such a model could be applied to other crops and regions that are threatened by freeze damage.
Materials and Methods
Study site and experimental design.
One-year-old blueberry lateral shoots were collected randomly from different rows in multiple fields in eastern and western Washington from 2015 to 2021 (Table 1). In total, there are 12, 15, 11, and 13 datasets, or unique combinations of field and year, for ‘Aurora’, ‘Draper’, ‘Duke’, and ‘Liberty’, respectively, from multiple years and different locations. These specific cultivars were selected with grower input because they are widely planted across the focal region and are important economically. Tissues subjected to the freeze assay were collected across individual bushes (i.e., the top and bottom) in an alternating manner in the early morning. Shoots were removed using pruning shears and were placed in sealable, plastic bags with a moistened paper towel around the stem end, and then in coolers with ice for transport to the laboratory. Weekly collections occurred in fields in eastern Washington and alternate weeks from fields in western Washington.
Four cultivars of northern highbush blueberries were collected over various years to assess cold hardiness in eastern [eastern Washington (E WA): West Prosser, East Prosser, Patterson, and Benton City] and western [western Washington (W WA): Mount Vernon 1, Mount Vernon 2, and Lynden] Washington, USA.
In eastern Washington, a programmable freezer was used to freeze buds, whereas in western Washington, a glycol bath [50% propylene glycol and 50% water (v/v)] was used (Polyscience Circulating Bath Model PP28R-30; Polyscience, Niles, IL, USA). The same freezing rates occurred in each laboratory. From each sample, three randomly selected shoots with a minimum of three floral buds per shoot were exposed to progressively colder temperatures starting at −1 °C. Samples were frozen slowly at the following rates to simulate a natural freeze event: –1 to –4 °C drop in temperature by lowering 1/2 °C every 30 min, –4 to –8 °C drop in temperature by lowering 1 °C every 30 min, and –8 to –28 °C drop in temperature by lowering 2 °C every 30 min (Arora et al. 2000). Samples were also thawed slowly to 1 °C, then 4 °C for 1 h, and then to 20 °C for 24 h to allow for oxidation of any freeze-damaged tissue. The top three apical floral buds from each of the three shoots at each temperature (n = 9/temperature) were dissected to determine individual flower mortality. The remaining 20 shoots were considered controls and were stored for 12 to 24 h with stems in water and in a refrigerator (4 °C) to determine any prior field damage. For each bud, the total number of live flowers, dead flowers, and phenological floral bud stage were recorded. Ovary tissue that was dark brown or black was scored as dead, whereas green ovary tissue was scored as living.
Statistical analysis.
Four cultivar-specific Bayesian generalized linear mixed models of bud cold hardiness were developed for ‘Aurora’, ‘Draper’, ‘Duke’, and ‘Liberty’ that use historical weather information to predict future bud cold hardiness. The model uses a latent state to quantify and adjust for the degree of bud death that already exists in the field. Bayesian model posterior estimation is accomplished with Hamiltonian Monte Carlo (HMC). Simulation-based inference is performed to quantify the lethal temperature (LT) representative of the population at large. Specifically, in contrast to estimating LT for the average field, a broader scope LT was obtained that represents average floral death across years, fields, and sampling dates for a given set of known weather history. This type of inference is more commonly termed “marginal inference” (Stroup 2013). Marginal LT was computed as the temperature at which the marginal survival probability is 1 – α (measured as a percentage). See Supplemental Material for a more detailed description of the statistical methodology.
An internal validation was conducted in line with recommendations from Harrell and Slaughter (2022) to use all available data at time of model publication. This was done to reduce uncertainty in internal validation assessment and to minimize error in predictions. Validation was performed using cross validation with folds defined by the unique datasets. For each holdout sample, the HMC algorithm was run for a total of 200 iterations, including 100 warmup iterations. A mean absolute probability error (MAPE) was computed as the average difference in probability between model predictions that exclude the holdout data and model estimates that include the holdout data. In this application, the MAPE estimates how different LT predictions will be from actual loss—in this case, averaging across the study design matrix X (e.g., an absolute error of 10% would occur if 90% survival was predicted, but 80% survival was observed in the holdout sample). A MAPE was used because of its close relationship with grower financial loss resulting either from freeze damage or excessive mitigation, compared with error in LT, which is related less directly. A MAPE of 8% indicates that, on average, one would observe 8% more survival or 8% more lethality than predicted [e.g., an LT of 10% kill (LT10) will have and 8% variation around that LT10, with some fields experiencing less or more damage].
All analyses were performed using R Statistical Software (R Foundation for Statistical Computing, Vienna, Austria). Chill portions were computed with the chillR R Package (ver, 0.72.5, R Foundation for Statistical Computing) (Luedeling 2022). HMC was performed using rstan (R package version 2, R Foundation for Statistical Computing).
Results
Cultivar-specific cold hardiness models were generated with maximum chill portions (CPmax) estimated as 55. Fixed-effects model parameters with standard errors (SEs) are shown in Table 2. The parameters can be used to construct model predictions, but they are not interpreted easily. Variance components with SEs are shown in Table 3. To illustrate the difference between conditional and marginal LT, estimated (posterior median) survival curves for data collected between 15 and 25 GDD is shown in Fig. 1. The conditional LT10 is –20.5 °C and the marginal LT10 is –18.1 °C. The difference in this case is 2.4 °C. Note that at the conditional LT10, one would obtain an expected 24% floral death across all sampling dates considered here, whereas at the marginal LT10, one would obtain an expected 10% floral death across all sampling dates considered. The conditional LT10 will show smaller variability with regard to LTs of the sampling dates around it, but the marginal LT is expected to show better correspondence with the actual overall damage.
Fixed-effects Bayesian generalized linear mixed model parameters and SEs for modeling cold hardiness among four cultivars of northern highbush blueberry.i
Variance components parameters of the Bayesian generalized linear mixed model and SEs of northern highbush blueberry cultivars used for modeling cold hardiness in Washington, USA.i
Estimated (posterior median) cold hardiness survival curves for northern highbush blueberry floral buds with data collected between 15 and 25 growing degree days (GDD). This includes Benton City (eastern Washington, USA) on 14 Mar 2017 and 27 Jan 2020; Mt Vernon 1 (western Washington, USA) on 7 Jan 2019, 6 Jan 2020, and 22 Jan 2020; Mt Vernon 2 (western Washington, USA) on 6 Jan 2020; and West Prosser (eastern Washington, USA) on 27 Jan 2020. Points indicate the position of the lethal temperature (on the x-axis) at which 10% of flower primordia within a bud will die (LT10) for each sampling date. The thick dashed line shows the survival curve conditional on the dataset (uy×) and sampling date (ud) random effects set to the average of this sample. The conditional LT10 is extracted as the temperature for which the thick dotted line hits 90% survival (0.9 on the y-axis). The solid black line shows the average probability across all sampling dates in this sample (averaging all the thin dashed curves). Note that the marginal survival curve is not a logistic curve. The marginal LT10 is extracted as the temperature for which the marginal survival curve crosses 90% (0.9 on the y-axis).
Citation: HortScience 58, 11; 10.21273/HORTSCI17320-23
The estimated probability of flower primordia survival as a function of freezer temperature for an example dataset from 23 sampling dates from 2019 to 2020 is shown in Fig. 2. In this case, the probability of survival follows an expected trajectory in which buds acclimate to progressively colder temperatures, and deacclimate in late winter/early spring. The estimated marginal LT10 across the season for a sample weather profile in eastern Washington is shown in Fig. 3.
Estimated cold hardiness date-level probability of northern highbush blueberry flower primordia survival (line) and proportion of developing flower primordia surviving (dots) vs. freezer temperature for 23 sampling dates from the 2019–20 season for the “Old Prosser” AgWeatherNet station outside Prosser, WA, USA. Date are provided for each figure as year-month-date. The Z matrix is constructed with the dataset and dates from the fitted data, but integrates across unobserved temperature errors, shoots, and cans. Ideally, when characterizing historical survival probability curves, the Z matrix would also condition on temperature errors, but simulations of these were not retained in the fitted model.
Citation: HortScience 58, 11; 10.21273/HORTSCI17320-23
Posterior mean population-level cold hardiness lethal temperature at which 10% of flower primordia within a bud will die (LT10) vs. date based on weather near Benton City in eastern Washington, USA, from 2019 to 2020 for northern highbush blueberry cultivars Aurora (A), Draper (B), Duke (C), and Liberty (D). The gray ribbon shows 95% credible intervals for LT10. The blue ribbon shows temperature minimum and maximum.
Citation: HortScience 58, 11; 10.21273/HORTSCI17320-23
All cultivars show an instantaneous temperature drop of relatively similar magnitude (Table 4). On average, it is estimated that a drop of 1 °C in the instantaneous temperature will result in a 25% reduction in predicted flower survival if occurring at LT50 (Table 4), and a 9% reduction in predicted flower survival probability if occurring at LT10. There was strong evidence of a decrease in survival probability for all cultivars when the prior 2-d average temperature was warmer. On average, a drop of 1 °C in the 2-d average temperature corresponds to a 4% increase in predicted flower survival probability when other conditions are held constant, and predictions are proximal to LT50. Note that the effects and associations described are indicative of flower-level survival; overall field-level survival will not be affected as highly because all developing flowers will not experience LT50 at the same temperature.
Approximate differences in predicted primordia-level probability of death between two similar buds experiencing differing temperature conditions near the lethal temperature at which 50% of flowers within a bud are expected to die.i
All random effects variance components seemed to show substantial effects on survival probability (Table 5), which suggests clustering effects are non-negligible, and systematic correlation appears across buds, shoots, temperatures, sampling dates, and datasets. Higher levels of variation across field-specific effects are representative simply of the degree to which freeze damage appears to occur in the controls.
Standard deviations in the probability of flower primordia survival attributed to random-effect variance component sources of the design.i
The cross-validation average MAPE was 0.11 for ‘Duke’, 0.12 for ‘Liberty’, 0.08 for ‘Aurora’, and 0.10 for ‘Draper’. The MAPE for predicted survival categories is shown in Table 6, and the distribution of observed survival probability grouped by predicted probability is shown in Fig. 4.
Mean absolute difference in probability between the full data model and limited data models from the cross validation design.i
Extended boxplots showing the distribution of observed (e.g., measured) cold hardiness survival probability (on the x-axis) for different ranges of model-predicted survival probability (on the y-axis) for northern highbush blueberry cultivars Aurora (A), Draper (B), Duke (C), and Liberty (D) with the dataset for which predictions are made held out. The extended boxplots show intervals containing 50% (thickest), 80% (medium), and 90% (thinnest) of the survival probabilities.
Citation: HortScience 58, 11; 10.21273/HORTSCI17320-23
Discussion
A hierarchical Bayesian binomial logistic regression model was constructed that accounted for probability of survival in the field and in the freezer for four blueberry cultivars. Although previous research (Longstroth 2012) has shown a relationship between the phenological floral bud stage and floral bud hardiness, our prior (unpublished) work showed that including the bud stage in addition to temperature history did not improve the model significantly. Therefore, our model ultimately chose to use GDDs and not bud stage due to GDDs’ continuous nature and lack of need for data collection by growers. We are unaware of any other models of cold hardiness for blueberries that can predict cold hardiness based on weather forecasts, making this work foundational for others wishing to build predictive models to inform freeze mitigation in this or closely related species. This model can be used either to supplement freezer data or as a replacement in the absence of freezer data collection.
Our model makes use of data that include whether floral buds are frozen and cut, as compared with other work, such as Ferguson et al. (2011), for which differential thermal analysis (DTA) was used to assess LTs. Although DTA is less resource intensive and gives information more directly on the temperature at which an individual developing flower freezes, other studies (Flinn and Ashworth 1994) and our preliminary comparison of methods (unpublished) showed that DTA was not suitable for highbush blueberry. The unreliability in DTA for blueberry may be a result of the small tissue size of excised floral buds, cooling rate, inability of blueberry floral buds to supercool under natural conditions, and extraorgan freezing in bud scales or bracts (Flinn and Ashworth 1994).
Our model represents a shift in how prior flower death in the field is considered, compared with the work by Rowland et al. (2005), in which LTs were not estimated following substantial death in control shoots. We adjust for death in the controls by modeling directly the proportion of death in the field as a latent state. This methodology allows for the use of bud dissection data collected after freeze events. Because the proportion of death in the controls increases for a particular sampling date, the inclusion of this latent state tends to result in increased survival probabilities and greater uncertainty in the log odds.
Our methodology also differs from others in the intended scope of inference. In our modeling approach, we focus on predictions of survival probability and LT that reduce the error in predicted loss across the population of fields and years from the Pacific Northwest. As compared with Ferguson et al. (2011), our approach adjusts explicitly for the nonlinear relationship between LT and survival. We estimated variability from the dataset, date, temperature, shoot, and bud level, and integrated across these components to obtain mean survival across the population as well as associated LT. Alternatively, approaches that fail to consider the different variance components and the nonlinear link may fail to adjust for a heavily skewed survival distribution.
We conducted an internal cross validation to produce error in probability for different ranges of model-predicted probability. This error can be used to anticipate how much actual death on the field level will likely vary from the predicted proportion of death. Assuming a particular LT is hit precisely (e.g., the temperature goes down to LT10), this error estimates the variation we tend to observe from 10% death. We found that at LT10, the probability of 40% or more of the developing flowers in a field dying at that temperature is relatively low (<5% for all cultivars). However, there is substantial variability in what different fields will experience; for instance, it would not be surprising to see 25% death in the developing flowers in a field at LT10. Examining the error associated with both the mean LT and MAPE may better characterize grower financial loss or risk. The LT ± error helps growers decide on a temperature to initiate mitigation measures. However, if LT10 is predicting 10% loss, some fields may experience no loss or significantly more. The variability in predicted loss might be translated directly into revenue loss (e.g., losing 50% instead of 10% of the buds might be estimated as a $1 million revenue loss, depending on the size of the farm), and errors in the positive direction (meaning less damage from a less sensitive field) could correspond to economic loss resulting from unnecessary mitigation and fuel expenses. Averaging and minimizing lost revenue resulting from prediction error across growers is one way to try and minimize the overall loss experienced by growers.
There are potential limitations of the study executed and used to generate our model. One limitation is that chill thresholds used in our study are not based on empirical data, but rather are estimated from prior research conducted in New Jersey and Maryland using diverse blueberry genotypes including northern highbush blueberry (Rowland et al. 2005). Although Rowland et al. (2005) used ‘Duke’, the other cultivars included in this study were not part of this original work, and this cultivar variation may affect chill thresholds. Diploid blueberry mapping populations have been shown to vary in chilling requirements (Rowland et al. 2014) and this variation could be cultivar specific and may affect chill thresholds. Unfortunately, cultivar-specific chill requirements have not been established for the cultivars evaluated in this study and are not provided by breeding programs upon cultivar release. Therefore, prior research was the best option to inform our model. The parameters of the chill model were meant to be tailored to specific plants in specific locations, and we use the default parameter values developed for warm winters in Israel (Luedeling and Brown 2011). However, determining cultivar-specific chilling requirements is recommended to fine-tune chill thresholds and subsequent cold hardiness models for key commercial cultivars.
Mitigating freeze or cold injury is a challenge that necessitates a balance between applying protective measures and economic losses from failing to do so. Our model presents an opportunity to provide growers with information needed to educate themselves regarding this balance, and can be adapted to other growing regions that cultivate blueberries. With the expansion of the global blueberry production area, there is an opportunity to fine-tune and adapt our model for other cultivars and regions of production.
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Supplemental Material
Description of statistical model. Similar to previous modeling efforts, we use temperature history to make predictions. Each model accounts for chill portions (CPs) in the fall, growing degree days (GDDs) in the spring, and recent 48-hour temperatures. The fall process of acclimation is characterized with CPs from the dynamic chill model (Luedeling and Brown 2011), whereas the late-winter to early-spring period of deacclimation is characterized with GDDs. Chill was computed with the dynamic model, with parameters left at their defaults as done by Luedeling and Brown (2011). GDDs associated with a given weather station were calculated on an hourly basis using the following equation:
CPs were computed using a start date of 1 Sep. CPmax was set at the average CPs accumulated on 1 Jan across years and sites from this experiment.
The freezer-specific effects, represented by
We impose the following weakly informative prior distributions: u ∼ normal(0, σ),
The HMC algorithm was run for a total of 1000 iterations, including 800 warmup iterations. Convergence was assessed with the potential scale reduction statistic
Model parameters for freezer temperature and past 2-d average temperature are interpreted using posterior medians to estimate the maximum difference in 2eloping flower survival probability that would correspond to a unit difference in the predictor, as suggested in Gelman et al. (2021). We also computed the standard deviation (SD) in survival probability attributable to each variance component, conditional on a median survival of 50%. We simulated normal random variates with the SD equal to the random effect SD, applied the logistic link, and computed the SD. Standard errors of the estimates are propagated similarly.
Except at the lethal temperature at which 50% of flower primordia within a bud will die (LT50), g(Xiβ) differs substantially from Eu[gi ] as a result of the nonlinear link function (see Fig. 1). Eu[gi] should be chosen if we are interested in providing predictions that reduce the error in the percentage crop loss when averaging across years, fields, and so on. We recommend the use of the summary Eu[gi] as a decision aid tool because it estimates a more applicable outcome compared to g(Xiβ), and further accounts for the effect of uncertainty in β on the survival probability.
Credible intervals can be extracted from the distribution of Eu[gi]. When estimating survival probabilities from the design, Xi and Zi may come fully or in part from the rows of X and Z. Alternatively, for making predictions for years, sites, and/or sampling dates not in the design, Xi and Zi come from the prediction design matrices X* and Z*. For instance, if the dataset, date, and temperature errors come from Z, whereas shoot and bud information come from Z*, we would be making predictions of overall loss for the entire field from which the buds were collected, on the day of collection, assuming that field were to experience the experimental temperature imposed on the collected buds.
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