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Hierarchical Bayesian models

Source: vignettes/hierarchical_model.Rmd
hierarchical_model.Rmd

Parametric Bayesian framework

Currently, serosv only has models under parametric Bayesian framework

Proposed approach

Prevalence has a parametric form π(ai,α)\pi(a_i, \alpha) where α\alpha is a parameter vector

One can constraint the parameter space of the prior distribution P(α)P(\alpha) in order to achieve the desired monotonicity of the posterior distribution P(π1,π2,...,πm|y,n)P(\pi_1, \pi_2, ..., \pi_m|y,n)

Where:

  • n=(n1,n2,...,nm)n = (n_1, n_2, ..., n_m) and nin_i is the sample size at age aia_i
  • y=(y1,y2,...,ym)y = (y_1, y_2, ..., y_m) and yiy_i is the number of infected individual from the nin_i sampled subjects

Farrington

Refer to Chapter 10.3.1

Proposed model

The model for prevalence is as followed

π(a)=1−exp{α1α2ae−α2a+1α2(α1α2−α3)(e−α2a−1)−α3a} \pi (a) = 1 - exp\{ \frac{\alpha_1}{\alpha_2}ae^{-\alpha_2 a} + \frac{1}{\alpha_2}(\frac{\alpha_1}{\alpha_2} - \alpha_3)(e^{-\alpha_2 a} - 1) -\alpha_3 a \}

For likelihood model, independent binomial distribution are assumed for the number of infected individuals at age aia_i

yi∼Bin(ni,πi), for i=1,2,3,...m y_i \sim Bin(n_i, \pi_i), \text{ for } i = 1,2,3,...m

The constraint on the parameter space can be incorporated by assuming truncated normal distribution for the components of α\alpha, α=(α1,α2,α3)\alpha = (\alpha_1, \alpha_2, \alpha_3) in πi=π(ai,α)\pi_i = \pi(a_i,\alpha)

αj∼truncated 𝒩(μj,τj),j=1,2,3 \alpha_j \sim \text{truncated } \mathcal{N}(\mu_j, \tau_j), \text{ } j = 1,2,3

The joint posterior distribution for α\alpha can be derived by combining the likelihood and prior as followed

P(α|y)∝∏i=1mBin(yi|ni,π(ai,α))∏i=13−1τjexp(12τj2(αj−μj)2) P(\alpha|y) \propto \prod^m_{i=1} \text{Bin}(y_i|n_i, \pi(a_i, \alpha)) \prod^3_{i=1}-\frac{1}{\tau_j}\text{exp}(\frac{1}{2\tau^2_j} (\alpha_j - \mu_j)^2)

  • Where the flat hyperprior distribution is defined as followed:

    • μj∼𝒩(0,10000)\mu_j \sim \mathcal{N}(0, 10000)

    • τj−2∼Γ(100,100)\tau^{-2}_j \sim \Gamma(100,100)

The full conditional distribution of αi\alpha_i is thus P(αi|αj,αk,k,j≠i)∝−1τiexp(12τi2(αi−μi)2)∏i=1mBin(yi|ni,π(ai,α)) P(\alpha_i|\alpha_j,\alpha_k, k, j \neq i) \propto -\frac{1}{\tau_i}\text{exp}(\frac{1}{2\tau^2_i} (\alpha_i - \mu_i)^2) \prod^m_{i=1} \text{Bin}(y_i|n_i, \pi(a_i, \alpha))

Fitting data

To fit Farrington model, use hierarchical_bayesian_model() and define type = "far2" or type = "far3" where

  • type = "far2" refers to Farrington model with 2 parameters (α3=0\alpha_3 = 0)

  • type = "far3" refers to Farrington model with 3 parameters (α3>0\alpha_3 > 0)

df <- mumps_uk_1986_1987
model <- hierarchical_bayesian_model(age = df$age, pos = df$pos, tot = df$tot, type="far3")
#> 
#> SAMPLING FOR MODEL 'fra_3' NOW (CHAIN 1).
#> Chain 1: Rejecting initial value:
#> Chain 1:   Log probability evaluates to log(0), i.e. negative infinity.
#> Chain 1:   Stan can't start sampling from this initial value.
#> Chain 1: 
#> Chain 1: Gradient evaluation took 6.7e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.67 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1: 
#> Chain 1: 
#> Chain 1: Iteration:    1 / 5000 [  0%]  (Warmup)
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#> Chain 1: Iteration: 4500 / 5000 [ 90%]  (Sampling)
#> Chain 1: Iteration: 5000 / 5000 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 7.165 seconds (Warm-up)
#> Chain 1:                6.956 seconds (Sampling)
#> Chain 1:                14.121 seconds (Total)
#> Chain 1:
#> Warning: There were 568 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess

model$info
#>                       mean      se_mean           sd          2.5%
#> alpha1        1.401214e-01 3.547730e-04 6.258881e-03  1.285380e-01
#> alpha2        1.993669e-01 4.755861e-04 8.363286e-03  1.848245e-01
#> alpha3        8.856149e-03 3.568605e-04 7.071400e-03  4.349792e-04
#> tau_alpha1    1.655955e-01 3.058673e-02 3.947079e-01  1.762069e-06
#> tau_alpha2    1.324341e-01 2.370415e-02 2.930593e-01  2.488436e-06
#> tau_alpha3    7.120083e-01 1.464684e-01 1.637538e+00  3.348306e-06
#> mu_alpha1     5.309619e+00 2.830580e+00 5.072469e+01 -1.011749e+02
#> mu_alpha2     4.377174e+00 4.306267e+00 5.440375e+01 -1.063688e+02
#> mu_alpha3     9.083521e-01 2.602170e+00 3.312617e+01 -8.185717e+01
#> sigma_alpha1  1.438473e+02 3.692646e+01 1.158127e+03  8.186990e-01
#> sigma_alpha2  1.074171e+02 2.011728e+01 7.564462e+02  9.330434e-01
#> sigma_alpha3  1.890636e+02 9.517393e+01 5.132818e+03  3.995507e-01
#> lp__         -2.535748e+03 4.167295e-01 4.060532e+00 -2.544257e+03
#>                        25%           50%           75%         97.5%      n_eff
#> alpha1        1.356498e-01  1.399529e-01  1.441737e-01  1.532028e-01  311.23776
#> alpha2        1.933265e-01  1.992242e-01  2.047900e-01  2.172946e-01  309.23995
#> alpha3        3.306703e-03  7.023647e-03  1.243968e-02  2.624763e-02  392.65737
#> tau_alpha1    2.035679e-04  5.030110e-03  1.052841e-01  1.491946e+00  166.52741
#> tau_alpha2    2.822231e-04  4.927917e-03  8.502226e-02  1.148673e+00  152.84887
#> tau_alpha3    7.431981e-04  1.625552e-02  4.147886e-01  6.264065e+00  124.99567
#> mu_alpha1    -6.110784e+00  3.366331e-01  1.043346e+01  1.227509e+02  321.13506
#> mu_alpha2    -6.218693e+00  2.605688e-01  1.018331e+01  1.451005e+02  159.60841
#> mu_alpha3    -3.304551e+00  5.251649e-02  3.947081e+00  8.569737e+01  162.05831
#> sigma_alpha1  3.081901e+00  1.409975e+01  7.008830e+01  7.534708e+02  983.64134
#> sigma_alpha2  3.429523e+00  1.424519e+01  5.952645e+01  6.339306e+02 1413.89599
#> sigma_alpha3  1.552697e+00  7.843314e+00  3.668159e+01  5.465076e+02 2908.54511
#> lp__         -2.538286e+03 -2.535437e+03 -2.532915e+03 -2.528247e+03   94.94175
#>                   Rhat
#> alpha1       1.0003681
#> alpha2       1.0002450
#> alpha3       1.0064494
#> tau_alpha1   1.0035818
#> tau_alpha2   1.0137683
#> tau_alpha3   1.0023225
#> mu_alpha1    1.0024548
#> mu_alpha2    1.0050142
#> mu_alpha3    1.0061848
#> sigma_alpha1 0.9997586
#> sigma_alpha2 1.0009646
#> sigma_alpha3 0.9997631
#> lp__         1.0214395
plot(model)
#> Warning: No shared levels found between `names(values)` of the manual scale and the
#> data's fill values.

Log-logistic

Proposed approach

The model for seroprevalence is as followed

π(a)=βaα1+βaα,α,β>0 \pi(a) = \frac{\beta a^\alpha}{1 + \beta a^\alpha}, \text{ } \alpha, \beta > 0

The likelihood is specified to be the same as Farrington model (yi∼Bin(ni,πi)y_i \sim Bin(n_i, \pi_i)) with

logit(π(a))=α2+α1log(a) \text{logit}(\pi(a)) = \alpha_2 + \alpha_1\log(a)

  • Where α2=log(β)\alpha_2 = \text{log}(\beta)

The prior model of α1\alpha_1 is specified as α1∼truncated 𝒩(μ1,τ1)\alpha_1 \sim \text{truncated } \mathcal{N}(\mu_1, \tau_1) with flat hyperprior as in Farrington model

β\beta is constrained to be positive by specifying α2∼𝒩(μ2,τ2)\alpha_2 \sim \mathcal{N}(\mu_2, \tau_2)

The full conditional distribution of α1\alpha_1 is thus

P(α1|α2)∝−1τ1exp(12τ12(α1−μ1)2)∏i=1mBin(yi|ni,π(ai,α1,α2)) P(\alpha_1|\alpha_2) \propto -\frac{1}{\tau_1} \text{exp} (\frac{1}{2 \tau_1^2} (\alpha_1 - \mu_1)^2) \prod_{i=1}^m \text{Bin}(y_i|n_i,\pi(a_i, \alpha_1, \alpha_2) )

And α2\alpha_2 can be derived in the same way

Fitting data

To fit Log-logistic model, use hierarchical_bayesian_model() and define type = "log_logistic"

df <- rubella_uk_1986_1987
model <- hierarchical_bayesian_model(age = df$age, pos = df$pos, tot = df$tot, type="log_logistic")
#> 
#> SAMPLING FOR MODEL 'log_logistic' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 1.7e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.17 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1: 
#> Chain 1: 
#> Chain 1: Iteration:    1 / 5000 [  0%]  (Warmup)
#> Chain 1: Iteration:  500 / 5000 [ 10%]  (Warmup)
#> Chain 1: Iteration: 1000 / 5000 [ 20%]  (Warmup)
#> Chain 1: Iteration: 1500 / 5000 [ 30%]  (Warmup)
#> Chain 1: Iteration: 1501 / 5000 [ 30%]  (Sampling)
#> Chain 1: Iteration: 2000 / 5000 [ 40%]  (Sampling)
#> Chain 1: Iteration: 2500 / 5000 [ 50%]  (Sampling)
#> Chain 1: Iteration: 3000 / 5000 [ 60%]  (Sampling)
#> Chain 1: Iteration: 3500 / 5000 [ 70%]  (Sampling)
#> Chain 1: Iteration: 4000 / 5000 [ 80%]  (Sampling)
#> Chain 1: Iteration: 4500 / 5000 [ 90%]  (Sampling)
#> Chain 1: Iteration: 5000 / 5000 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 1.115 seconds (Warm-up)
#> Chain 1:                6.829 seconds (Sampling)
#> Chain 1:                7.944 seconds (Total)
#> Chain 1:
#> Warning: There were 174 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess

model$type
#> [1] "log_logistic"
plot(model)
#> Warning: No shared levels found between `names(values)` of the manual scale and the
#> data's fill values.