Advanced Topics

Ricardo G. Tomáz

Introduction

Most EpiExposure workflows can be completed using the default model specification introduced in previous sections

However, advanced applications often require additional flexibility regarding:

This section discusses these advanced topics and provides recommendations for robust DLNM analyses.

Alternative model engines

EpiExposure separates exposure-lag specification from model estimation.

Consequently, the same DLNM structure can be fitted using different regression engines.

Supported families by modeling engine

The modeling engines available in fit_epidlnm() and find_bestfit() differ in the response families exposed through the EpiExposure interface. The table below summarizes the current EpiExposure support. It describes the package interface rather than every family accepted natively by the underlying modeling packages.

Engine Beta Binomial Poisson Gamma Gaussian Negative binomial
glm No Yes Yes Yes Yes No
glmmTMB Yes Yes Yes Yes Yes Yes, NB2
gam Yes Yes Yes Yes Yes Yes, NB2
gamm No Yes Yes Yes Yes No
gls No No No No Yes No
spaMM Yes Yes Yes Yes Yes Yes, NB2
brms Yes Yes, Bernoulli Yes Yes Yes Yes, NB2
INLA Yes Yes Yes Yes Yes Yes, NB2
bdlnm Yes Yes Yes Yes Yes Yes, NB2

Note: Negative-binomial support refers to the quadratic-variance NB2 parameterization. The table describes the families currently exposed through the EpiExposure interface, not every family that may be available natively in the underlying modeling packages. For INLA-backed engines, availability also depends on the likelihoods provided by the installed INLA version.

Because family specification follows the same fit_epidlnm() interface across engines, the sections below do not repeat every supported engine-family combination. Instead, each example highlights a capability that differs meaningfully across engines, such as conventional random effects, spatial autocorrelation, or Bayesian inference.

Reference DLNM design

The examples below use the same cross-basis specification whenever possible. This allows the modeling engines to be compared without repeatedly rebuilding the exposure-lag design.

library(EpiExposure)
data("poisson_data")

head(poisson_data)
#>   epi_id block time    tmean  wetness      rain y
#> 1      1   B01    0 25.32685 14.55902 11.629805 5
#> 2      1   B01    1 25.72697 13.35332  0.000000 5
#> 3      1   B01    2 24.88694 13.00895  8.036259 5
#> 4      1   B01    3 26.04031 13.22694  0.000000 5
#> 5      1   B01    4 25.94690 15.30767  6.709351 5
#> 6      1   B01    5 24.85839 17.41708  8.826920 5
cb <- define_exposures(
  data = poisson_data,
  vars = c(
    "tmean",
    "rain",
    "wetness"
  ),
  max_lag = 85,
  df_var = 3,
  df_lag = 2
)

dat <- build_design(
  data = poisson_data,
  cb_templates = cb,
  groups = c(
    "epi_id",
    "block"
  )
)
dat_poisson <- prepare_response(
  data = dat,
  response = "y",
  family = "poisson"
)

Fixed-effect reference model

A generalized linear model provides the simplest fixed-effect specification. It is useful as a reference when random or spatial dependence is not required.

fit_glm <- fit_epidlnm(
  data = dat_poisson,
  model_engine = "glm",
  family = "poisson",
  epiexposure_spec = attr(
    cb,
    "spec"
  )
)
summary(fit_glm)
#> 
#> Call:
#> stats::glm(formula = y_model ~ 1 + cb_tmean_1 + cb_tmean_2 + 
#>     cb_tmean_3 + cb_tmean_4 + cb_tmean_5 + cb_tmean_6 + cb_rain_1 + 
#>     cb_rain_2 + cb_rain_3 + cb_rain_4 + cb_rain_5 + cb_rain_6 + 
#>     cb_wetness_1 + cb_wetness_2 + cb_wetness_3 + cb_wetness_4 + 
#>     cb_wetness_5 + cb_wetness_6, family = "poisson", data = dat_poisson)
#> 
#> Coefficients:
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  -3.303652   1.132719  -2.917 0.003539 ** 
#> cb_tmean_1    0.090028   0.013837   6.506 7.71e-11 ***
#> cb_tmean_2    0.012584   0.025493   0.494 0.621554    
#> cb_tmean_3    0.140732   0.043795   3.213 0.001312 ** 
#> cb_tmean_4    0.270869   0.097282   2.784 0.005363 ** 
#> cb_tmean_5   -0.027014   0.016290  -1.658 0.097252 .  
#> cb_tmean_6    0.162029   0.039149   4.139 3.49e-05 ***
#> cb_rain_1     0.091313   0.025531   3.577 0.000348 ***
#> cb_rain_2    -0.259121   0.033323  -7.776 7.48e-15 ***
#> cb_rain_3     0.078788   0.015872   4.964 6.91e-07 ***
#> cb_rain_4    -0.064189   0.022854  -2.809 0.004975 ** 
#> cb_rain_5     0.001674   0.036894   0.045 0.963806    
#> cb_rain_6     0.035621   0.050073   0.711 0.476846    
#> cb_wetness_1  0.051027   0.003629  14.062  < 2e-16 ***
#> cb_wetness_2 -0.086751   0.010892  -7.965 1.65e-15 ***
#> cb_wetness_3  0.049067   0.014053   3.492 0.000480 ***
#> cb_wetness_4 -0.098276   0.036715  -2.677 0.007434 ** 
#> cb_wetness_5  0.059851   0.009419   6.355 2.09e-10 ***
#> cb_wetness_6 -0.084802   0.018470  -4.591 4.40e-06 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for poisson family taken to be 1)
#> 
#>     Null deviance: 2866.42  on 519  degrees of freedom
#> Residual deviance:  955.26  on 501  degrees of freedom
#> AIC: 3133.2
#> 
#> Number of Fisher Scoring iterations: 4

Conventional random effects

When epidemics are grouped within experimental blocks, observations from the same block may share unmeasured conditions. A random intercept can account for this hierarchical dependence.

fit_glmm <- fit_epidlnm(
  data = dat_poisson,
  model_engine = "glmmTMB",
  family = "poisson",
  random_effect = "block",
  epiexposure_spec = attr(
    cb,
    "spec"
  )
)
summary(fit_glmm)
#>  Family: poisson  ( log )
#> Formula:          
#> y_model ~ 1 + cb_tmean_1 + cb_tmean_2 + cb_tmean_3 + cb_tmean_4 +  
#>     cb_tmean_5 + cb_tmean_6 + cb_rain_1 + cb_rain_2 + cb_rain_3 +  
#>     cb_rain_4 + cb_rain_5 + cb_rain_6 + cb_wetness_1 + cb_wetness_2 +  
#>     cb_wetness_3 + cb_wetness_4 + cb_wetness_5 + cb_wetness_6 +  
#>     (1 | block)
#> Data: dat_poisson
#> 
#>       AIC       BIC    logLik -2*log(L)  df.resid 
#>    2920.3    3005.4   -1440.2    2880.3       500 
#> 
#> Random effects:
#> 
#> Conditional model:
#>  Groups Name        Variance Std.Dev.
#>  block  (Intercept) 0.04283  0.207   
#> Number of obs: 520, groups:  block, 20
#> 
#> Conditional model:
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  -2.574612   1.158675  -2.222 0.026281 *  
#> cb_tmean_1    0.081602   0.014101   5.787 7.17e-09 ***
#> cb_tmean_2   -0.008901   0.025844  -0.344 0.730544    
#> cb_tmean_3    0.115294   0.044915   2.567 0.010260 *  
#> cb_tmean_4    0.170722   0.098491   1.733 0.083031 .  
#> cb_tmean_5   -0.023288   0.017047  -1.366 0.171890    
#> cb_tmean_6    0.119456   0.039905   2.994 0.002758 ** 
#> cb_rain_1     0.129022   0.026462   4.876 1.08e-06 ***
#> cb_rain_2    -0.239983   0.033977  -7.063 1.63e-12 ***
#> cb_rain_3     0.056071   0.016278   3.445 0.000572 ***
#> cb_rain_4    -0.057666   0.023722  -2.431 0.015063 *  
#> cb_rain_5    -0.038092   0.037558  -1.014 0.310473    
#> cb_rain_6     0.060427   0.052017   1.162 0.245368    
#> cb_wetness_1  0.052540   0.003729  14.091  < 2e-16 ***
#> cb_wetness_2 -0.093637   0.011114  -8.425  < 2e-16 ***
#> cb_wetness_3  0.053720   0.014580   3.685 0.000229 ***
#> cb_wetness_4 -0.093537   0.037422  -2.499 0.012437 *  
#> cb_wetness_5  0.055911   0.009722   5.751 8.87e-09 ***
#> cb_wetness_6 -0.090618   0.018671  -4.853 1.21e-06 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The term generated internally is equivalent to:

(1 | block)

This effect represents discrete group-level heterogeneity. It does not model spatial correlation as a function of distance.

GAM estimation

The gam engine can fit the DLNM design with mgcv. Beta and negative-binomial models use mgcv extended families and are fitted through the supported REML route by default.

data("nb2_data")

head(nb2_data)
#>   epi_id block time    tmean  wetness      rain y
#> 1      1   B01    0 28.51652 3.304398  9.234377 7
#> 2      1   B01    1 25.78990 5.600390 19.225955 7
#> 3      1   B01    2 26.86089 7.060567  0.000000 7
#> 4      1   B01    3 25.19746 5.064939  8.700396 7
#> 5      1   B01    4 25.80954 5.634410  0.000000 7
#> 6      1   B01    5 25.28982 6.813331  5.096005 7
dat_nb2 <- build_design(
  data = nb2_data,
  cb_templates = cb,
  groups = c(
    "epi_id",
    "block"
  )
)

dat_nb2 <- prepare_response(
  data = dat_nb2,
  response = "y",
  family = "negative_binomial"
)
fit_gam <- fit_epidlnm(
  data = dat_nb2,
  model_engine = "gam",
  family = "negative_binomial",
  epiexposure_spec = attr(
    cb,
    "spec"
  )
)
summary(fit_gam)
#> 
#> Family: Negative Binomial(3.312) 
#> Link function: log 
#> 
#> Formula:
#> y_model ~ 1 + cb_tmean_1 + cb_tmean_2 + cb_tmean_3 + cb_tmean_4 + 
#>     cb_tmean_5 + cb_tmean_6 + cb_rain_1 + cb_rain_2 + cb_rain_3 + 
#>     cb_rain_4 + cb_rain_5 + cb_rain_6 + cb_wetness_1 + cb_wetness_2 + 
#>     cb_wetness_3 + cb_wetness_4 + cb_wetness_5 + cb_wetness_6
#> 
#> Parametric coefficients:
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  -0.184097   2.303002  -0.080 0.936287    
#> cb_tmean_1    0.050136   0.028851   1.738 0.082253 .  
#> cb_tmean_2   -0.129826   0.052604  -2.468 0.013588 *  
#> cb_tmean_3    0.009794   0.089261   0.110 0.912633    
#> cb_tmean_4   -0.166707   0.200448  -0.832 0.405595    
#> cb_tmean_5   -0.052436   0.036715  -1.428 0.153242    
#> cb_tmean_6    0.012849   0.082609   0.156 0.876399    
#> cb_rain_1     0.198301   0.057344   3.458 0.000544 ***
#> cb_rain_2    -0.328951   0.071447  -4.604 4.14e-06 ***
#> cb_rain_3     0.027017   0.036851   0.733 0.463466    
#> cb_rain_4    -0.094688   0.050348  -1.881 0.060015 .  
#> cb_rain_5    -0.085088   0.085783  -0.992 0.321251    
#> cb_rain_6    -0.085474   0.106752  -0.801 0.423318    
#> cb_wetness_1  0.066901   0.006780   9.868  < 2e-16 ***
#> cb_wetness_2 -0.082650   0.021324  -3.876 0.000106 ***
#> cb_wetness_3  0.094385   0.025050   3.768 0.000165 ***
#> cb_wetness_4 -0.012930   0.072911  -0.177 0.859238    
#> cb_wetness_5  0.058337   0.016182   3.605 0.000312 ***
#> cb_wetness_6 -0.036181   0.044278  -0.817 0.413850    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> 
#> R-sq.(adj) =  0.467   Deviance explained =   55%
#> -REML = 1848.3  Scale est. = 1         n = 520

GAMM estimation

The gamm engine combines the fixed DLNM design with a conventional random intercept. In the current EpiExposure interface, random_effect is required.

fit_gamm <- fit_epidlnm(
  data = dat_poisson,
  model_engine = "gamm",
  family = "poisson",
  random_effect = "block",
  epiexposure_spec = attr(
    cb,
    "spec"
  )
)
#> 
#>  Maximum number of PQL iterations:  20

The fitted object contains two native components:

summary(fit_gamm)
#> 
#> Family: poisson 
#> Link function: log 
#> 
#> Formula:
#> y_model ~ 1 + cb_tmean_1 + cb_tmean_2 + cb_tmean_3 + cb_tmean_4 + 
#>     cb_tmean_5 + cb_tmean_6 + cb_rain_1 + cb_rain_2 + cb_rain_3 + 
#>     cb_rain_4 + cb_rain_5 + cb_rain_6 + cb_wetness_1 + cb_wetness_2 + 
#>     cb_wetness_3 + cb_wetness_4 + cb_wetness_5 + cb_wetness_6
#> 
#> Parametric coefficients:
#>               Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  -2.573106   1.180352  -2.180 0.029726 *  
#> cb_tmean_1    0.081602   0.014365   5.681 2.28e-08 ***
#> cb_tmean_2   -0.008901   0.026325  -0.338 0.735418    
#> cb_tmean_3    0.115297   0.045756   2.520 0.012051 *  
#> cb_tmean_4    0.170724   0.100318   1.702 0.089407 .  
#> cb_tmean_5   -0.023289   0.017367  -1.341 0.180525    
#> cb_tmean_6    0.119458   0.040642   2.939 0.003442 ** 
#> cb_rain_1     0.129018   0.026943   4.789 2.21e-06 ***
#> cb_rain_2    -0.239982   0.034612  -6.934 1.27e-11 ***
#> cb_rain_3     0.056073   0.016573   3.383 0.000772 ***
#> cb_rain_4    -0.057667   0.024167  -2.386 0.017396 *  
#> cb_rain_5    -0.038088   0.038251  -0.996 0.319851    
#> cb_rain_6     0.060423   0.052990   1.140 0.254717    
#> cb_wetness_1  0.052540   0.003798  13.832  < 2e-16 ***
#> cb_wetness_2 -0.093637   0.011321  -8.271 1.20e-15 ***
#> cb_wetness_3  0.053718   0.014853   3.617 0.000329 ***
#> cb_wetness_4 -0.093536   0.038125  -2.453 0.014490 *  
#> cb_wetness_5  0.055910   0.009904   5.645 2.77e-08 ***
#> cb_wetness_6 -0.090616   0.019022  -4.764 2.49e-06 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> 
#> R-sq.(adj) =  0.631   
#>   Scale est. = 1         n = 520
summary(
  fit_gamm,
  component = "lme"
)
#> Linear mixed-effects model fit by maximum likelihood
#>   Data: data 
#>        AIC     BIC    logLik
#>   524.4704 609.547 -242.2352
#> 
#> Random effects:
#>  Formula: ~1 | block
#>         (Intercept) Residual
#> StdDev:   0.2066749        1
#> 
#> Variance function:
#>  Structure: fixed weights
#>  Formula: ~invwt 
#> Fixed effects:  list(fixed) 
#>                    Value Std.Error  DF   t-value p-value
#> X(Intercept)  -2.5731056 1.1803515 482 -2.179949  0.0297
#> Xcb_tmean_1    0.0816015 0.0143649 482  5.680623  0.0000
#> Xcb_tmean_2   -0.0089008 0.0263249 482 -0.338115  0.7354
#> Xcb_tmean_3    0.1152970 0.0457561 482  2.519817  0.0121
#> Xcb_tmean_4    0.1707239 0.1003175 482  1.701835  0.0894
#> Xcb_tmean_5   -0.0232888 0.0173666 482 -1.341008  0.1805
#> Xcb_tmean_6    0.1194585 0.0406424 482  2.939258  0.0034
#> Xcb_rain_1     0.1290176 0.0269427 482  4.788593  0.0000
#> Xcb_rain_2    -0.2399821 0.0346118 482 -6.933535  0.0000
#> Xcb_rain_3     0.0560726 0.0165732 482  3.383323  0.0008
#> Xcb_rain_4    -0.0576668 0.0241673 482 -2.386149  0.0174
#> Xcb_rain_5    -0.0380883 0.0382508 482 -0.995753  0.3199
#> Xcb_rain_6     0.0604231 0.0529900 482  1.140273  0.2547
#> Xcb_wetness_1  0.0525396 0.0037983 482 13.832359  0.0000
#> Xcb_wetness_2 -0.0936366 0.0113209 482 -8.271142  0.0000
#> Xcb_wetness_3  0.0537184 0.0148528 482  3.616711  0.0003
#> Xcb_wetness_4 -0.0935359 0.0381246 482 -2.453426  0.0145
#> Xcb_wetness_5  0.0559104 0.0099040 482  5.645222  0.0000
#> Xcb_wetness_6 -0.0906163 0.0190219 482 -4.763798  0.0000
#>  Correlation: 
#>               X(Int) Xcb_t_1 Xcb_t_2 Xcb_t_3 Xcb_t_4 Xcb_t_5 Xcb_t_6 Xcb_r_1
#> Xcb_tmean_1   -0.879                                                        
#> Xcb_tmean_2   -0.730  0.756                                                 
#> Xcb_tmean_3   -0.945  0.821   0.673                                         
#> Xcb_tmean_4   -0.866  0.910   0.863   0.773                                 
#> Xcb_tmean_5   -0.381  0.210   0.420   0.559   0.147                         
#> Xcb_tmean_6   -0.664  0.655   0.596   0.513   0.845  -0.279                 
#> Xcb_rain_1     0.012 -0.028  -0.016  -0.002  -0.021   0.022  -0.013         
#> Xcb_rain_2     0.032 -0.074  -0.071  -0.021  -0.077   0.021  -0.056  -0.076 
#> Xcb_rain_3    -0.063  0.022  -0.051   0.052  -0.001   0.036  -0.027  -0.662 
#> Xcb_rain_4    -0.004  0.061   0.057   0.001   0.057  -0.015   0.032   0.033 
#> Xcb_rain_5    -0.034  0.001  -0.048   0.035  -0.017   0.040  -0.041  -0.602 
#> Xcb_rain_6     0.005  0.052   0.031  -0.015   0.040  -0.033   0.028   0.050 
#> Xcb_wetness_1  0.002 -0.062  -0.036  -0.048  -0.047  -0.018  -0.016   0.054 
#> Xcb_wetness_2 -0.023 -0.011   0.002  -0.038  -0.014  -0.035  -0.023   0.002 
#> Xcb_wetness_3 -0.198 -0.043  -0.035   0.007  -0.027   0.023  -0.003  -0.045 
#> Xcb_wetness_4 -0.048 -0.047   0.000  -0.046  -0.052   0.019  -0.059   0.015 
#> Xcb_wetness_5 -0.122  0.006  -0.013   0.009   0.010  -0.008   0.026  -0.097 
#> Xcb_wetness_6 -0.033 -0.002  -0.014  -0.002  -0.043   0.016  -0.060   0.048 
#>               Xcb_r_2 Xcb_r_3 Xcb_r_4 Xcb_r_5 Xcb_r_6 Xcb_w_1 Xcb_w_2 Xcb_w_3
#> Xcb_tmean_1                                                                  
#> Xcb_tmean_2                                                                  
#> Xcb_tmean_3                                                                  
#> Xcb_tmean_4                                                                  
#> Xcb_tmean_5                                                                  
#> Xcb_tmean_6                                                                  
#> Xcb_rain_1                                                                   
#> Xcb_rain_2                                                                   
#> Xcb_rain_3     0.040                                                         
#> Xcb_rain_4    -0.605  -0.011                                                 
#> Xcb_rain_5     0.025   0.938   0.020                                         
#> Xcb_rain_6    -0.669  -0.010   0.866  -0.004                                 
#> Xcb_wetness_1  0.046  -0.102   0.058  -0.108   0.027                         
#> Xcb_wetness_2 -0.045   0.053  -0.011   0.072  -0.012  -0.109                 
#> Xcb_wetness_3  0.029   0.113  -0.097   0.108  -0.055  -0.149   0.122         
#> Xcb_wetness_4  0.048  -0.035  -0.045  -0.016  -0.036   0.381   0.470   0.106 
#> Xcb_wetness_5  0.017   0.153  -0.122   0.148  -0.073  -0.579   0.203   0.785 
#> Xcb_wetness_6  0.051  -0.076  -0.008  -0.061   0.007   0.358   0.128  -0.118 
#>               Xcb_w_4 Xcb_w_5
#> Xcb_tmean_1                  
#> Xcb_tmean_2                  
#> Xcb_tmean_3                  
#> Xcb_tmean_4                  
#> Xcb_tmean_5                  
#> Xcb_tmean_6                  
#> Xcb_rain_1                   
#> Xcb_rain_2                   
#> Xcb_rain_3                   
#> Xcb_rain_4                   
#> Xcb_rain_5                   
#> Xcb_rain_6                   
#> Xcb_wetness_1                
#> Xcb_wetness_2                
#> Xcb_wetness_3                
#> Xcb_wetness_4                
#> Xcb_wetness_5 -0.123         
#> Xcb_wetness_6  0.732  -0.433 
#> 
#> Standardized Within-Group Residuals:
#>         Min          Q1         Med          Q3         Max 
#> -2.81800137 -0.78615982 -0.07432645  0.70191859  5.10243548 
#> 
#> Number of Observations: 520
#> Number of Groups: 20

The default summary describes the fixed/population GAM component. The component = "lme" option displays the mixed-model component.

For non-Gaussian responses, mgcv::gamm() uses a PQL-based fitting route. Binary responses should therefore be treated cautiously.

Spatially autocorrelated effects with spaMM

Consider a regional study of a foliar rice disease conducted at spatially distributed field locations. Daily mean temperature, leaf wetness duration, and rainfall are recorded during the 86 days preceding the final assessment. The outcome is the final number of lesions per plant.

The cross-basis terms estimate non-linear and delayed environmental effects. A Matérn random effect accounts for residual geographic similarity among nearby fields after those measured environmental effects have been considered.

Spatially dependent fields

data("st_poisson")

head(st_poisson)
#>   epi_id block time    x_coord    y_coord    tmean  wetness     rain y
#> 1      1   B01    0 -0.4210796 -0.1649767 24.06189 10.81782 0.000000 4
#> 2      1   B01    1 -0.4210796 -0.1649767 24.82615 11.15336 0.000000 4
#> 3      1   B01    2 -0.4210796 -0.1649767 26.11794 12.20862 0.000000 4
#> 4      1   B01    3 -0.4210796 -0.1649767 26.17299 11.70761 6.440981 4
#> 5      1   B01    4 -0.4210796 -0.1649767 26.39594 13.08992 0.000000 4
#> 6      1   B01    5 -0.4210796 -0.1649767 26.23450 14.78350 0.000000 4

cb_spatial <- define_exposures(
  data = st_poisson,
  vars = c(
    "tmean",
    "rain",
    "wetness"
  ),
  max_lag = 85,
  df_var = 3,
  df_lag = 2
)

dat_spatial <- build_design(
  data = st_poisson,
  cb_templates = cb_spatial,
  groups = c(
    "epi_id",
    "x_coord",
    "y_coord"
  )
)

dat_spatial <- prepare_response(
  data = dat_spatial,
  response = "y",
  family = "poisson"
)
fit_spatial <- fit_epidlnm(
  data = dat_spatial,
  model_engine = "spamm",
  family = "poisson",
  random_effect = NULL,
  spatial_effect = c(
    "x_coord",
    "y_coord"
  ),
  spatial_structure = "matern",
  spatial_group = NULL,
  epiexposure_spec = attr(
    cb_spatial,
    "spec"
  )
)
summary(fit_spatial)

This specification generates:

Matern(1 | x_coord + y_coord)

The Matérn term is distinct from a conventional random intercept. It represents autocorrelation that changes continuously with the distance between locations.

Independent spatial fields by year

If the disease process is expected to have a different spatial realization in each growing season, year can define independent Matérn fields:

data("spatial_year")

head(spatial_year)
#>      epi_id location_id year block block_id time    x_coord  y_coord    tmean
#> 1 L001_2022        L001 2022   B01 2022_B01    0 -0.6698125 0.622697 24.38006
#> 2 L001_2022        L001 2022   B01 2022_B01    1 -0.6698125 0.622697 24.53832
#> 3 L001_2022        L001 2022   B01 2022_B01    2 -0.6698125 0.622697 24.90186
#> 4 L001_2022        L001 2022   B01 2022_B01    3 -0.6698125 0.622697 26.04159
#> 5 L001_2022        L001 2022   B01 2022_B01    4 -0.6698125 0.622697 23.60107
#> 6 L001_2022        L001 2022   B01 2022_B01    5 -0.6698125 0.622697 25.79864
#>     wetness      rain y
#> 1  4.903199 12.904436 3
#> 2  6.781023  7.651138 3
#> 3 11.195548  0.000000 3
#> 4 10.530333  0.000000 3
#> 5 10.322681  0.000000 3
#> 6 12.433350  0.000000 3
dat_spatial_year <- build_design(
  data = spatial_year,
  cb_templates = cb,
  groups = c(
    "epi_id",
    "x_coord",
    "y_coord",
    "year",
    "location_id",
    "block_id"
  )
)

dat_spatial_year <- prepare_response(
  data = dat_spatial_year,
  response = "y",
  family = "poisson"
)
fit_spatial_year <- fit_epidlnm(
  data = dat_spatial_year,
  model_engine = "spamm",
  family = "poisson",
  random_effect = "block_id",
  spatial_effect = c(
    "x_coord",
    "y_coord"
  ),
  spatial_structure = "matern",
  spatial_group = "year",
  epiexposure_spec = attr(
    cb_spatial,
    "spec"
  )
)
summary(fit_spatial_year)

The resulting spatial term is:

Matern(1 | x_coord + y_coord %in% year)

In this model, (1 | block_id) represents conventional experimental heterogeneity, whereas the Matérn term represents spatially autocorrelated residual variation within each year.

Independent spatial fields across epidemiological replicates

The grouping variable in a Matérn term can identify independent repetitions of an entire epidemiological experiment. Consider a field study conducted at 130 fixed locations. The same host, pathogen, experimental layout, and observation period are used in four independent repetitions.

Each repetition contains an 86-day exposure profile followed by one final disease assessment at each location. Temperature, rainfall, and leaf wetness can differ among repetitions, but the temporal positions remain comparable: the most recent observation corresponds to lag zero, and the oldest observation corresponds to lag 85.

Each combination of location and repetition represents a separate epidemic:

L001_Replicate_1

L001_Replicate_2

L001_Replicate_3

L001_Replicate_4

The cross-basis terms use the exposure profile from each epidemic to estimate the shared non-linear and delayed environmental associations. However, the residual spatial pattern of disease is not required to remain in the same locations across repetitions.

Independent residual spatial fields can be specified as:

Matern(1 | x_coord + y_coord %in% epidemic_replicate)

The model therefore allows each repetition to produce a different realized map of residual disease intensity. For example, areas with high residual disease in the first repetition do not need to remain high in the second repetition.

The Matérn parameters remain shared among repetitions. Consequently, all repetitions contribute jointly to estimation of the general magnitude, smoothness, and spatial scale of residual dependence, while retaining separate realized spatial fields.

data("spatial_replicate")

head(spatial_replicate)
#>             epi_id location_id epidemic_replicate block        block_id time
#> 1 L001_Replicate_1        L001        Replicate_1   B01 Replicate_1_B01    0
#> 2 L001_Replicate_1        L001        Replicate_1   B01 Replicate_1_B01    1
#> 3 L001_Replicate_1        L001        Replicate_1   B01 Replicate_1_B01    2
#> 4 L001_Replicate_1        L001        Replicate_1   B01 Replicate_1_B01    3
#> 5 L001_Replicate_1        L001        Replicate_1   B01 Replicate_1_B01    4
#> 6 L001_Replicate_1        L001        Replicate_1   B01 Replicate_1_B01    5
#>     x_coord    y_coord    tmean   wetness rain y
#> 1 0.7586476 -0.4480224 23.89557  6.798144    0 5
#> 2 0.7586476 -0.4480224 24.45162  7.355172    0 5
#> 3 0.7586476 -0.4480224 24.49880  8.565241    0 5
#> 4 0.7586476 -0.4480224 24.68348 12.894026    0 5
#> 5 0.7586476 -0.4480224 25.65530 14.958210    0 5
#> 6 0.7586476 -0.4480224 23.98108 13.466749    0 5

cb_replicate <- define_exposures(
  data = spatial_replicate,
  vars = c(
    "tmean",
    "rain",
    "wetness"
  ),
  max_lag = 85,
  df_var = 3,
  df_lag = 2
)

dat_spatial_replicate <- build_design(
  data = spatial_replicate,
  cb_templates = cb_replicate,
  groups = c(
    "epi_id",
    "x_coord",
    "y_coord",
    "epidemic_replicate",
    "location_id",
    "block_id"
    
  )
)

dat_spatial_replicate <- prepare_response(
  data = dat_spatial_replicate,
  response = "y",
  family = "poisson"
)
fit_spatial_replicate <- fit_epidlnm(
  data = dat_spatial_replicate,
  model_engine = "spamm",
  family = "poisson",
  random_effect = "block_id",
  spatial_effect = c(
    "x_coord",
    "y_coord"
  ),
  spatial_structure = "matern",
  spatial_group = "epidemic_replicate",
  epiexposure_spec = attr(
    cb_replicate,
    "spec"
  )
)
summary(fit_spatial_replicate)

The conventional random intercept and the spatial term represent different sources of dependence. The block_id effect accounts for discrete experimental heterogeneity, whereas the Matérn term accounts for continuous residual similarity among nearby locations within each epidemiological repetition.

An epidemiological replicate must represent an independent repetition of the complete disease process. Repeated assessments made during the progression of a single epidemic should not be treated as independent replicates.

Gaussian generalized least squares

The gls engine is restricted to Gaussian outcomes with the identity link and does not support the EpiExposure random-intercept interface.

data("gaussian_data")

head(gaussian_data)
#>   epi_id block time    tmean  wetness     rain       y
#> 1      1   B01    0 29.16636 11.58879  0.00000 12.4501
#> 2      1   B01    1 27.30294 12.71218  0.00000 12.4501
#> 3      1   B01    2 26.86088 12.74422  0.00000 12.4501
#> 4      1   B01    3 26.29974 11.92657 10.12782 12.4501
#> 5      1   B01    4 26.67772 13.41525  0.00000 12.4501
#> 6      1   B01    5 26.63974 15.59525  0.00000 12.4501
cb_gaussian <- define_exposures(
  data = gaussian_data,
  vars = c(
    "tmean",
    "rain",
    "wetness"
  ),
  max_lag = 85,
  df_var = 3,
  df_lag = 2
)

dat_gaussian <- build_design(
  data = gaussian_data,
  cb_templates = cb_gaussian,
  groups = c(
    "epi_id"))

dat_gaussian <- prepare_response(
  data = dat_gaussian,
  response = "y",
  family = "gaussian"
)
fit_gls <- fit_epidlnm(
  data = dat_gaussian,
  model_engine = "gls",
  family = "gaussian",
  epiexposure_spec = attr(
    cb_gaussian,
    "spec"
  ),
  method = "REML"
)
summary(fit_gls)

Note: The INLA and B-DLNM commands in the following sections are shown but not evaluated during vignette construction because these models may require substantial computational time. Users can execute the code interactively after installing the optional packages INLA, bdlnm, and sn.

MCMC estimation with brms

The brms engine provides posterior sampling for the fixed DLNM coefficients and any requested hierarchical components.

fit_brms <- fit_epidlnm(
  data = dat_poisson,
  model_engine = "brms",
  family = "poisson",
  random_effect = "block",
  epiexposure_spec = attr(
    cb,
    "spec"
  ),
  chains = 4,
  iter = 2000,
  seed = 123
)
summary(fit_brms, eval=FALSE)

Bayesian DLNM with bdlnm

The bdlnm engine provides a Bayesian implementation specifically designed for distributed lag linear and non-linear models. Internally, the native bdlnm package fits the model using Integrated Nested Laplace Approximation through INLA::inla() and then draws samples from the approximate posterior distribution using INLA::inla.posterior.sample(). 【1-fe3db1】

Unlike the direct inla engine, which receives the epidemic-level cb_* columns as conventional fixed predictors, the bdlnm engine receives the original named crossbasis objects in the model formula. The fitted object stores the underlying INLA model, the basis objects, posterior coefficient draws, and summaries of those draws. 【1-fe3db1】

In EpiExposure, this engine is integrated into the same prediction and uncertainty contract used by the other engines. Posterior fixed-effect draws are retained for draw-by-draw propagation through summarise_effects(), predict_outcomes(), and other downstream functions.

cb_bdlnm <- define_exposures(
  data = poisson_data,
  vars = c(
    "tmean",
    "rain",
    "wetness"
  ),
  max_lag = 85,
  df_var = 3,
  df_lag = 2
)


dat_bdlnm <- build_design(
  data = poisson_data,
  cb_templates = cb_bdlnm,
  groups = c(
    "epi_id",
    "block"
  )
)

dat_bdlnm <- prepare_response(
  data = dat_bdlnm,
  response = "y",
  family = "poisson"
)

bdlnm_bases <- attr(
  dat_bdlnm,
  "epiexposure_bdlnm_basis_objects",
  exact = TRUE
)

The model can then be fitted through the harmonized EpiExposure interface:

fit_bdlnm <- fit_epidlnm(
  data = dat_bdlnm,
  model_engine = "bdlnm",
  family = "poisson",
  random_effect = "block",
  random_effect_prior = list(
    prec = list(prior = "pc.prec",param = c(1,0.01))),
  epiexposure_spec = attr(dat_bdlnm, "epiexposure_spec"),
  sample.arg = list(n = 1000, seed = 123))
summary(fit_bdlnm)
fit_bdlnm$model
#fit_bdlnm$coefficients
fit_bdlnm$coefficients.summary


fit_bdlnm$model$summary.random
fit_bdlnm$model$summary.hyperpar

fit_bdlnm$model$summary.random$.epiexposure_re_index

head(fit_bdlnm$model$marginals.hyperpar)

The random_effect = "block" argument is translated internally to an independent latent effect of the form

f(block, model = "iid")

for INLA-backed engines. When supplied, random_effect_prior is passed directly to the hyper argument of the latent effect specification, allowing custom hyperprior definitions for the random-effect precision. For example,

f(block, model = "iid", hyper = random_effect_prior)

The sample.arg list controls posterior coefficient sampling performed by the native bdlnm engine. A nonzero seed provides reproducible posterior sampling. The native package uses 1,000 posterior samples by default, although the number of samples can be changed according to the required precision and available computational resources.

For deterministic EpiExposure predictions, posterior means of the fixed/population coefficients define the central expected response. When uncertainty is requested, the stored joint posterior coefficient draws are propagated through the inverse link one draw at a time.

The resulting intervals describe uncertainty in the population-level expected response. They do not include observation-level noise or group-specific random effects, even when random effects are included in the fitted model.

For both model_engine = "inla" and model_engine = "bdlnm", random- effect hyperparameters can be inspected through the underlying INLA fit (e.g., summary.hyperpar), allowing users to evaluate posterior distributions of random-effect precision parameters.

Direct INLA versus B-DLNM

Both engines rely on INLA for Bayesian inference and support the same likelihood families, random effects, and hyperprior specifications. The main difference lies in how the DLNM basis is represented and what is returned to the user.

fit_inla <- fit_epidlnm(
  data = dat_poisson,
  model_engine = "inla",
  family = "poisson",
  random_effect = "block",
  random_effect_prior = list(
      prec = list(
      prior = "pc.prec",
      param = c(1,0.01))),
  epiexposure_spec = attr(cb, "spec"))

Outputs shared by INLA and B-DLNM

Fixed-effect posterior summaries:

fit_inla$summary.fixed

fit_bdlnm$model$summary.fixed

Random-effect posterior summaries:

fit_inla$summary.random

fit_bdlnm$model$summary.random

Posterior distributions of hyperparameters:

fit_inla$marginals.hyperpar

fit_bdlnm$model$marginals.hyperpar

Marginal likelihood:

fit_inla$mlik

fit_bdlnm$model$mlik

For both engines, random-effect hyperparameters can be inspected through the INLA summaries. For example:

fit_inla$summary.hyperpar

fit_bdlnm$model$summary.hyperpar

Outputs specific to B-DLNM

names(fit_bdlnm)
names(fit_inla)

Original DLNM basis objects:

fit_bdlnm$basis

Posterior coefficient draws:


dim(fit_bdlnm$coefficients)

head(fit_bdlnm$coefficients)

Posterior coefficient summaries:


fit_bdlnm$coefficients.summary

Model objects

class(fit_inla)

class(fit_bdlnm)

class(fit_bdlnm$model)

The direct INLA engine is useful when users want a conventional INLA representation of the epidemic-level design matrix and direct access to standard INLA outputs. The B-DLNM engine is useful when users want to retain the native DLNM representation and work directly with posterior coefficient samples, fitted basis objects, and Bayesian DLNM workflows.