# Use Case 02: Simulation of trials with geographical spillover

Effects of settlement patterns, choices of cluster size and buffer widths, and the extent of spillover between arms on the outcomes of CRTs do not lend themselves to mathematical analysis. Simulations of trials are used to explore the effects of these variables on trial power and on the robustness of statistical methodologies.

Trials can be simulated using the `simulateCRT` function, which augments a `trial` data frame (created externally) or object of class `CRTsp` (created by package functions) with simulated outcome data. The input object must be given location information and both cluster and arm assignments (see Use Case 1) (or the package can generate these if the objective is purely simulation.

Information about the underlying spatial pattern of disease is used in the form of the intra-cluster correlation of the outcome, which is input to the simulation as variable `ICC_inp`, and of the `propensity`. The former takes a single value for the chosen design. The latter takes a positive real value for each location. In the case of malaria, `propensity` can be thought of as measuring exposure to infectious mosquitoes.

`ICC_inp` and `propensity` may either be estimated from other datasets or supplied by the user. The behaviour of the function depends on which variables are supplied, and the value of `generateBaseline`, as follows:

Data supplied by the user | Function behaviour
--- | ---
`propensity` supplied by user | Baseline data are created by sampling around `propensity`
Baseline data are supplied by user and `propensity` is not supplied | `propensity` is created from the baseline data
Neither baseline data nor `propensity` are supplied | `propensity` is generated using normal kernels, with the bandwidth adjusted to achieve the input value of the `ICC_inp` (after the further smoothing stage to simulate spillover (see below))

The effect of intervention is simulated as a fixed percentage reduction in the `propensity`. Contamination or spillover between trial arms is then modelled as a additional smoothing process applied to the intervention-adjusted `propensity` via a further bivariate normal kernel. In the case of mosquito borne disease this is proposed as an approximation to the effect of mosquito movement. The degree of spillover is specified either as a spillover interval with the `spillover_interval` parameter, or as `sd`, the bandwidth of the corresponding normal kernel. If both are provided then it is the value of `spillover_interval` that is used.

#### Example with baseline data provided as proportions

    library(CRTspat)
    set.seed(1234)
    example_locations <- readdata('example_site.csv')
    example_locations$base_denom <- 1
    library(dplyr)
    example_randomized <- CRTsp(example_locations) %>%
      aggregateCRT(auxiliaries = c("RDT_test_result", "base_denom")) %>%
      specify_clusters(h = 50, algorithm = 'NN') %>%
      randomizeCRT(matchedPair = FALSE)
    summary(example_randomized)

    ## ===============================CLUSTER RANDOMISED TRIAL ===========================
    ##
    ## Summary of coordinates
    ## ----------------------
    ##                Min.   : 1st Qu.: Median : Mean   : 3rd Qu.: Max.   :
    ##       x        -3.20    -1.39    -0.30    -0.07     1.25     5.16
    ##       y        -5.07    -2.84     0.19     0.05     2.49     6.16
    ## nearestDiscord -3.47    -0.49     0.03     0.14     0.73     3.30
    ##
    ## Total area (within  0.2 km of a location) :  27.6 sq.km
    ## Total area (convex hull) :  48.2 sq.km
    ##
    ## Locations and Clusters
    ## ----------------------                                          -
    ## Coordinate system                      (x, y)
    ## Locations:                                                      1181
    ## Available clusters (across both arms)                           24
    ##   Per cluster mean number of points                             49.2
    ##   Per cluster s.d. number of points                             3.9
    ## Cluster randomization:                      Independently randomized
    ## No power calculations to report          -
    ##
    ## Other variables in dataset
    ## --------------------------          RDT_test_result  base_denom

    plotCRT(example_randomized, map = TRUE, legend.position = c(0.8, 0.8))
      example2a <- simulateCRT(example_randomized,
               effect = 0.8,
               outcome0 = 0.5,
               generateBaseline = FALSE,
               baselineNumerator = "RDT_test_result",
               baselineDenominator = "base_denom",
               ICC_inp = 0.05, spillover_interval = 0.8)
    summary(example2a)

    ## ===============================CLUSTER RANDOMISED TRIAL ===========================
    ##
    ## Summary of coordinates
    ## ----------------------
    ##                Min.   : 1st Qu.: Median : Mean   : 3rd Qu.: Max.   :
    ##       x        -3.20    -1.39    -0.30    -0.07     1.25     5.16
    ##       y        -5.07    -2.84     0.19     0.05     2.49     6.16
    ## nearestDiscord -3.47    -0.49     0.03     0.14     0.73     3.30
    ##
    ## Total area (within  0.2 km of a location) :  27.6 sq.km
    ## Total area (convex hull) :  48.2 sq.km
    ##
    ## Locations and Clusters
    ## ----------------------                                          -
    ## Coordinate system                      (x, y)
    ## Locations:                                                      1181
    ## Available clusters (across both arms)                           24
    ##   Per cluster mean number of points                             49.2
    ##   Per cluster s.d. number of points                             3.9
    ## Cluster randomization:                      Independently randomized
    ## No power calculations to report          -
    ##
    ## Other variables in dataset
    ## --------------------------          RDT_test_result  base_denom  denom  propensity  num

    library(Matrix)
    examplemesh100 <- readdata("examplemesh100.rds")
    example2aanalysis <- CRTanalysis(trial=example2a, method = 'T')
    summary(example2aanalysis)

    ##
    ## =====================CLUSTER RANDOMISED TRIAL ANALYSIS =================
    ## Analysis method:  T
    ## Link function:  logit
    ## Model formula:  arm + (1 | cluster)
    ## No modelling of spillover
    ## Estimates:       Control: 0.386 (95% CL: 0.301 0.478)
    ##             Intervention: 0.191 (95% CL: 0.131 0.269)
    ##                 Efficacy: 0.506 (95% CL: 0.351 0.784)
    ## Coefficient of variation: 49.27 (95% CL: 36.54 77.525)
    ##
    ## P-value (2-sided):  0.001224732

    plotCRT(example2aanalysis)
    example2aINLA <- CRTanalysis(trial=example2a,
                     method = 'INLA', link='logit', cfunc = 'Z',
                     clusterEffects = FALSE, spatialEffects = TRUE,
                     requireMesh = TRUE, inla_mesh = examplemesh100)
    plotCRT(example2aINLA, map = TRUE, fill = 'prediction',
      showClusterBoundaries = TRUE, legend.position = c(0.8, 0.8))

Fig 2.1 Map of allocations of clusters to arms

Fig 2.2 Plot of data by distance to other arm

Fig 2.3 Smoothed outcome from geostatistical model

#### Example with infectiousness proxy surface generated externally

    set.seed(1234)
    # Simulate a site with 2000 locations
    new_site <- CRTsp(geoscale = 2, locations=2000, kappa=3, mu=40)
    # propensity surface generated as an arbitrary linear function of x the co-ordinate
    new_site$trial$propensity <- 0.5*new_site$trial$x - min(new_site$trial$x)+1
    library(dplyr)
    example2b<- CRTsp(new_site) %>%
       specify_clusters(h = 40, algorithm = 'NN') %>%
       randomizeCRT(matchedPair = FALSE) %>%
       simulateCRT(effect = 0.8,
                   outcome0 = 0.5,
                   generateBaseline = TRUE,
                   ICC_inp = 0.05,
                   spillover_interval = 0.5)

    ##
    ## =====================    SIMULATION OF CLUSTER RANDOMISED TRIAL    =================

    ## Estimating the smoothing required to achieve the target ICC of 0.05

    ##
    tol: 0.005 loss = 0.0034482803603185

    ##
    bandwidth: 0.0117436284570214  ICC = 0.108722060252673 loss = 0.0034482803603185

    summary(example2b)

    ## ===============================CLUSTER RANDOMISED TRIAL ===========================
    ##
    ## Summary of coordinates
    ## ----------------------
    ##                Min.   : 1st Qu.: Median : Mean   : 3rd Qu.: Max.   :
    ##       x        -8.73    -5.16    -1.03     0.00     5.17    11.26
    ##       y        -9.54    -4.42    -0.57     0.00     4.56    10.44
    ## nearestDiscord -4.19    -0.93     0.00     0.02     0.96     3.94
    ##
    ## Total area (within  0.2 km of a location) :  181 sq.km
    ## Total area (convex hull) :  396 sq.km
    ##
    ## Locations and Clusters
    ## ----------------------                                          -
    ## Coordinate system                      (x, y)
    ## Locations:                                                      2000
    ## Available clusters (across both arms)                           50
    ##   Per cluster mean number of points                             40
    ##   Per cluster s.d. number of points                             0
    ## Cluster randomization:                      Independently randomized
    ## No power calculations to report          -
    ##
    ## Other variables in dataset
    ## --------------------------          denom  propensity  num  base_denom  base_num

    results2b <- CRTanalysis(example2b, method = 'GEE')

    ##     (Intercept) armintervention
    ##      -0.6322523      -0.4504233

    summary(results2b)

    ##
    ## =====================CLUSTER RANDOMISED TRIAL ANALYSIS =================
    ## Analysis method:  GEE
    ## Link function:  logit
    ## Model formula:  arm
    ## No modelling of spillover
    ## Estimates:       Control: 0.347 (95% CL: 0.292 0.406)
    ##             Intervention: 0.253 (95% CL: 0.216 0.294)
    ##                 Efficacy: 0.271 (95% CL: 0.061 0.433)
    ## Coefficient of variation: 50.367 (95% CL: 40.51 67.273)
    ## Intracluster correlation: 0.109
    ##

    plotCRT(example2b, map = TRUE, fill = 'clusters', showClusterLabels = TRUE, maskbuffer = 0.5)

Fig 2.4 Map of clusters in simulated trial

#### Example with baseline generated from user-provided values of the overall initial prevalence and ICC