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Parametric Bayesian framework

Currently, serosv only has models under parametric Bayesian framework

Proposed approach

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

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

Where:

  • \(n = (n_1, n_2, ..., n_m)\) and \(n_i\) is the sample size at age \(a_i\)
  • \(y = (y_1, y_2, ..., y_m)\) and \(y_i\) is the number of infected individual from the \(n_i\) sampled subjects

Farrington

Refer to Chapter 10.3.1

Proposed model

The model for prevalence is as followed

\[ \pi (a) = 1 - exp\{ \frac{\alpha_1}{\alpha_2}ae^{-\alpha_2 a} + \frac{1}{\alpha_2}(\frac{\alpha_1}{\alpha_2} - \alpha_3)(e^{-\alpha_2 a} - 1) -\alpha_3 a \} \]

For likelihood model, independent binomial distribution are assumed for the number of infected individuals at age \(a_i\)

\[ y_i \sim Bin(n_i, \pi_i), \text{ for } i = 1,2,3,...m \]

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

\[ \alpha_j \sim \text{truncated } \mathcal{N}(\mu_j, \tau_j), \text{ } j = 1,2,3 \]

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

\[ P(\alpha|y) \propto \prod^m_{i=1} \text{Bin}(y_i|n_i, \pi(a_i, \alpha)) \prod^3_{i=1}-\frac{1}{\tau_j}\text{exp}(\frac{1}{2\tau^2_j} (\alpha_j - \mu_j)^2) \]

  • Where the flat hyperprior distribution is defined as followed:

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

    • \(\tau^{-2}_j \sim \Gamma(100,100)\)

The full conditional distribution of \(\alpha_i\) is thus \[ P(\alpha_i|\alpha_j,\alpha_k, k, j \neq i) \propto -\frac{1}{\tau_i}\text{exp}(\frac{1}{2\tau^2_i} (\alpha_i - \mu_i)^2) \prod^m_{i=1} \text{Bin}(y_i|n_i, \pi(a_i, \alpha)) \]

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 (\(\alpha_3 = 0\))

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

df <- mumps_uk_1986_1987
model <- hierarchical_bayesian_model(df, type="far3")
#> 
#> SAMPLING FOR MODEL 'fra_3' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 0.000129 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 1.29 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1: 
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#> Chain 1: Iteration: 5000 / 5000 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 15.482 seconds (Warm-up)
#> Chain 1:                21.15 seconds (Sampling)
#> Chain 1:                36.632 seconds (Total)
#> Chain 1:
#> Warning: There were 838 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: The largest R-hat is 1.14, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> 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        1.408781e-01 1.096470e-03 6.583110e-03  1.285038e-01
#> alpha2        1.995220e-01 9.879167e-04 8.615884e-03  1.836674e-01
#> alpha3        7.466994e-03 3.924748e-04 6.897456e-03  4.472780e-04
#> tau_alpha1    3.118564e-01 9.967545e-02 7.708859e-01  3.921300e-06
#> tau_alpha2    7.333867e-01 1.487363e-01 1.792088e+00  5.903764e-06
#> tau_alpha3    2.633947e+00 1.643591e+00 5.143803e+00  2.944859e-06
#> mu_alpha1     1.554005e+00 1.935997e+00 3.849420e+01 -9.265843e+01
#> mu_alpha2     6.597082e+00 2.973981e+00 3.641011e+01 -6.704919e+01
#> mu_alpha3     2.852824e+00 4.456617e+00 4.146922e+01 -9.728614e+01
#> sigma_alpha1  9.035379e+01 2.183061e+01 7.215499e+02  5.752691e-01
#> sigma_alpha2  6.576310e+01 1.425332e+01 3.635106e+02  3.671380e-01
#> sigma_alpha3  9.407694e+01 2.504552e+01 8.046322e+02  2.224100e-01
#> lp__         -2.534384e+03 6.344957e-01 4.222197e+00 -2.542987e+03
#>                        25%           50%           75%         97.5%
#> alpha1        1.357578e-01  1.412377e-01  1.463428e-01     0.1512404
#> alpha2        1.934689e-01  1.996487e-01  2.063040e-01     0.2170242
#> alpha3        3.152994e-03  4.954288e-03  9.662025e-03     0.0267386
#> tau_alpha1    9.859920e-04  7.117460e-03  1.271444e-01     3.0217457
#> tau_alpha2    4.173289e-04  3.002324e-02  3.368339e-01     7.4189809
#> tau_alpha3    8.832888e-04  4.686090e-02  2.341148e+00    20.2172677
#> mu_alpha1    -4.010137e+00  3.519832e-01  8.508863e+00    95.6694049
#> mu_alpha2    -2.239597e+00  3.401933e-01  8.064976e+00    95.5871103
#> mu_alpha3    -1.991155e+00  1.183101e-02  3.071295e+00   105.8255113
#> sigma_alpha1  2.804474e+00  1.185325e+01  3.184731e+01   505.0026217
#> sigma_alpha2  1.723029e+00  5.771268e+00  4.895091e+01   411.5637679
#> sigma_alpha3  6.535602e-01  4.619497e+00  3.364724e+01   582.7640733
#> lp__         -2.537377e+03 -2.534227e+03 -2.531187e+03 -2526.3895654
#>                    n_eff     Rhat
#> alpha1         36.046964 1.076246
#> alpha2         76.060465 1.050501
#> alpha3        308.854674 1.000146
#> tau_alpha1     59.814121 1.001727
#> tau_alpha2    145.172619 1.002282
#> tau_alpha3      9.794474 1.129788
#> mu_alpha1     395.349388 1.005220
#> mu_alpha2     149.888196 1.001482
#> mu_alpha3      86.584668 1.000960
#> sigma_alpha1 1092.448366 1.001546
#> sigma_alpha2  650.431856 1.000082
#> sigma_alpha3 1032.130451 1.005289
#> lp__           44.281239 1.034539
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

\[ \pi(a) = \frac{\beta a^\alpha}{1 + \beta a^\alpha}, \text{ } \alpha, \beta > 0 \]

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

\[ \text{logit}(\pi(a)) = \alpha_2 + \alpha_1\log(a) \]

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

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

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

The full conditional distribution of \(\alpha_1\) is thus

\[ P(\alpha_1|\alpha_2) \propto -\frac{1}{\tau_1} \text{exp} (\frac{1}{2 \tau_1^2} (\alpha_1 - \mu_1)^2) \prod_{i=1}^m \text{Bin}(y_i|n_i,\pi(a_i, \alpha_1, \alpha_2) ) \]

And \(\alpha_2\) 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(df, type="log_logistic")
#> 
#> SAMPLING FOR MODEL 'log_logistic' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5.8e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.58 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)
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#> Chain 1: Iteration: 4500 / 5000 [ 90%]  (Sampling)
#> Chain 1: Iteration: 5000 / 5000 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 4.359 seconds (Warm-up)
#> Chain 1:                4.504 seconds (Sampling)
#> Chain 1:                8.863 seconds (Total)
#> Chain 1:
#> Warning: There were 481 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.