Health Systems Research and Dynamical Modeling Department of Epidemiology & Public Health

R: weapon of choice for simulation of a step wedge design

Step-wedge design – cluster-randomized design

Time1 2 3 4 5O X X X XO O X X XO O O X XO O O O X

• Time of crossover is randomized; crossover is unidirectional

• Outcome should be measured on each unit at each time step

• Multiple observations per unit

• Individuals at each time could be same (cohort) or different (cross-sectional)

Brown and Lilford BMC Medical Research Methodology 2006 6:54

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Reasons for choosing a step-wedge design

• Logistical or financial - cannot introduce the intervention in all units at once

• Systematically evaluate new program - Evaluate the community effectiveness of an intervention previously shown to be efficacious in an individually randomized trial or in a different setting

• To study the effect of time on intervention effectiveness (i.e., seasonality, time since introduction)

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Example: elimination of malaria from Rusinga Island, Kenya

Solarmal technology: odour-baited mosquito trap

Battery, charger + timer

Light switch

Mosquito Trap

Solar Panel

Connection for charging mobile phone

Solar panel system

Takken et al., University of Wageningen, the Netherlands13 September 2102

Possible comparisons in cluster-randomized trials of health interventions

Non-Inter-vention

Intervention Remote Baseline

DIRECT

OVERALL

INDIRECT

DIRECT AND INDIRECTIntervention phase

Baseline

Pre-intervention

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Non-interventionIntervention Remote

OVERALL WITH CONCURRENT COMPARATOR

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Step Wedge Design 8

Direct effect:

Incidence measures compared between intervened and non-intervened zones

Direct effect:

Incidence measures compared between intervened and non-intervened zones

Contemporaneous overall effect

Incidence measures between intervened and those close to intervention against those remote from intervention

Contemporaneous overall effect

Incidence measures between intervened and those close to intervention against those remote from intervention

Measures of efficacy

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Step Wedge Design 9

SolarMal design constraints

• 4000 traps, with 700 day rollout period. 24,000 individuals

• Cluster size ~ 60 traps – the number of traps that can be installed in one week

• Number of individuals receiving the intervention at each time step is the sum of the number of individuals per household per cluster

First look at the general problem of step wedge designs; then apply the theoretical results to SolarMal

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Three intervention rollout schema

Oil drop – starting in the lower left hand corner, to the upper right hand corner

Completely randomized design – randomly selecting clusters to receive the intervention

Hierarchical design – randomly select one of 9 sections, filling in those clusters first, then randomly selecting the next section

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The ‘Oil spill’ design?

t=0Oil drop

The ‘Oil spill’ design?

t=0Random

The ‘Oil spill’ design?

t=0Hierarchical

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Questions:

Which design has the most power – i.e., produces the narrowest confidence intervals? How does the potential bias rank across designs?

If clustering is helpful, to what degree?

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Each design evaluated for a suite of initial conditions and constraints

• Initial clinical case rate of 0.30, 0.50, 0.80

• Intervention efficacy of 0.1, 0.3, 0.5, and 0.8

• Community radius of 0.5, 1.0, 1.5, 2.0, 2.5

Power measure: the confidence interval width for the two efficacy measures for a given design and set of initial conditions

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Simulation

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Step Wedge Design 17

Simulation physical layout [ mimic Rusinga Island]

9 districts, each containing 9 clusters, for a total of 81 clusters

1000 houses are randomly located across the grid

3000 individuals are randomly assigned to the 1000 houses

Four stages to the simulation

Initialization t ≤10Burn-In 11< t ≤ 39Intervention 40 <t ≤ 121Post-Intervention t >121

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Simulation stage

Initialization (t=1-9) InfectionMatrix <- as.integer( runif(endInitial*NIndividuals) < initialPrevalence)

During this time we start to build the disease reservoir (t ≥10)reservoirx <- 0.1*InfectedNeighbors[t-6, ] + 0.2*InfectedNeighbors[t-7, ] + 0.4*InfectedNeighbors[t-8, ] + 0.2*InfectedNeighbors[t-9, ] + 0.1*InfectedNeighbors[t-10, ]

reservoir = rbind(reservoir, reservoirx , deparse.level=0)

Spacing of clinical episodes is two months +/- two weeks - We model this with a distributed lag – reservoir is a weighted average of previous clinical cases amongst neighbors

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Step Wedge Design 19

Simulation stage

Burn in ((t ≥10 < 40) Pure ARMAfor (t in (endInitial+1):endOfBurnIn) { reservoirx = 0.1*nir[t-6, ] + 0.2*nir[t-7, ] + 0.4*nir[t-8, ] + 0.2*nir[t-9, ] + 0.1*nir[t-10, ]

infected[t,] = rbinom(reservoirx,1,reservoirx) # strange syntax of rbinom nir[t,] = sapply(neighborHs, function(X)mean(infected[t,X])) # tracking the mean case rate in each neighborhood

}

nir = non-intervened case rate – the intervention hasn’t yet been introduced

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Pure ARMA, Initial Prevalence = 0.4

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Simulation stage

Intervention (t=40-121) Maintenance (t>121)

infectedx =

(runif(1:NIndividuals) < ((1-efficacy*CoverageMatrix[t,])*(1-exp(-reservoirx*underlying))))

where underlying infection rate = -log(initialPrevalence/(1-initialPrevalence)) coverage = percent of the individual’s neighbors who have had the intervention

At time points greater than t > 121, the coverage is unit, and the simulation reverts to an Arma process at the new disease level.

Base case simulation

Eff

ica

cy

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Time point

40 60 80 100 120 140

efficacy=0.8 radius=1 initialPrevalence=0.5

PLOT Eff_I Eff_IIa Eff_IIb Eff_III eff_IV efficacy

Simulation Machinery

Simulation grid

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RandomizeDistribution sequence for cluster within district is assigned

RandomizeDistribution sequence for cluster within district is assigned

Houses X, Y coordinates 1000 houses across a 9 km2 grid. Corresponding district & cluster assignments are drawn from data in randomize

IndividualsEach house contains at least one individual

DistanceDistances between all individuals (not houses)

NeighborsStructured list of all neighbors within radius R of a given individual.

Houses X, Y coordinates 1000 houses across a 9 km2 grid. Corresponding district & cluster assignments are drawn from data in randomize

IndividualsEach house contains at least one individual

DistanceDistances between all individuals (not houses)

NeighborsStructured list of all neighbors within radius R of a given individual.

Intervention list The time-step in which each house (and the individuals within it) receives the intervention

Intervention list The time-step in which each house (and the individuals within it) receives the intervention

Simulation machinery: design-specific data containers

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Simulation machinery: data containers

Each row is a single time step, each column an individual

InfectionMatrix Values are 1 or 0 depending on the infection status of the individual

InfectedNeighbors Values are a percentage of infected neighbors for that individual.

ReservoirValues are a weighted mean of the number of infected neighbors at timesteps in the past, namely t-10 to t-6.

InterventionMatrix Values are 1 or 0 depending on the intervention status of the individual.From time-steps 0-39, Intervention matrix has value 0; for times after timestep 121,the end of rollout, all values are 1.

CoverageMatrix Values are a percentage of intervened neighbors for each individual.From time-steps 0-39, coverage matrix has value 0; for times after timestep 121,the end of rollout, all values are 1.

Each row is a single time step, each column an individual

InfectionMatrix Values are 1 or 0 depending on the infection status of the individual

InfectedNeighbors Values are a percentage of infected neighbors for that individual.

ReservoirValues are a weighted mean of the number of infected neighbors at timesteps in the past, namely t-10 to t-6.

InterventionMatrix Values are 1 or 0 depending on the intervention status of the individual.From time-steps 0-39, Intervention matrix has value 0; for times after timestep 121,the end of rollout, all values are 1.

CoverageMatrix Values are a percentage of intervened neighbors for each individual.From time-steps 0-39, coverage matrix has value 0; for times after timestep 121,the end of rollout, all values are 1.

Main simulation loop

To evaluate a range of community radii, efficacies, prevalences for given design

for i1 in (1 : 3)

radius = 0.5*i1

for i2 in (1 : 3)

efficacy=-0.1+0.3*i2

for i3 in (1 : 3)

initialPrevalence=-0.1+0.3*i3

}

}

}

Rinse, repeat, 2000 times.

Preliminary Results

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Summary

Problem lent itself to vectorized implementation

R scaled well to mainframe

Model can accommodate actual coordinates for Rusinga island

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Thank you !

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Simulation design for solar mal, possibly too specific

Community effects