Hierarchical Bayesian models

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(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: 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 6.9e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.69 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1: 
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#> Chain 1: 
#> Chain 1:  Elapsed Time: 1.602 seconds (Warm-up)
#> Chain 1:                0.839 seconds (Sampling)
#> Chain 1:                2.441 seconds (Total)
#> Chain 1:
#> Warning: There were 819 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: 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        1.388515e-01 3.143483e-04 5.686656e-03  1.283237e-01
#> alpha2        1.971229e-01 4.240475e-04 7.614690e-03  1.841599e-01
#> alpha3        7.639700e-03 4.500291e-04 6.817731e-03  3.534974e-04
#> tau_alpha1    1.663986e-01 2.691150e-02 3.429053e-01  1.791474e-06
#> tau_alpha2    7.574213e-02 2.141102e-02 1.909780e-01  3.895651e-06
#> tau_alpha3    2.082875e-01 7.744633e-02 6.028339e-01  2.965544e-06
#> mu_alpha1    -1.855848e-01 3.037000e+00 4.764718e+01 -1.097078e+02
#> mu_alpha2     2.225335e-01 2.954215e+00 4.037329e+01 -9.986680e+01
#> mu_alpha3    -4.196577e+00 2.551721e+00 4.143201e+01 -1.027110e+02
#> sigma_alpha1  1.434457e+02 6.087243e+01 1.397043e+03  8.863586e-01
#> sigma_alpha2  9.138533e+01 1.633538e+01 5.341490e+02  1.235928e+00
#> sigma_alpha3  9.076527e+01 1.643673e+01 4.931546e+02  6.803698e-01
#> lp__         -2.536086e+03 3.070347e-01 3.741862e+00 -2.544322e+03
#>                        25%           50%           75%         97.5%      n_eff
#> alpha1        1.351990e-01  1.384015e-01  1.424784e-01  1.506125e-01  327.25899
#> alpha2        1.921341e-01  1.963159e-01  2.016397e-01  2.133971e-01  322.45975
#> alpha3        2.424724e-03  5.710295e-03  1.043535e-02  2.551799e-02  229.50835
#> tau_alpha1    3.664167e-04  7.883421e-03  1.445835e-01  1.272861e+00  162.35759
#> tau_alpha2    3.012210e-04  3.573543e-03  4.029680e-02  6.546563e-01   79.55953
#> tau_alpha3    4.317701e-04  4.822130e-03  8.921679e-02  2.160280e+00   60.58899
#> mu_alpha1    -6.208494e+00  1.048231e-01  5.859708e+00  1.104209e+02  246.14159
#> mu_alpha2    -7.215519e+00  3.118702e-01  9.501639e+00  8.932361e+01  186.76865
#> mu_alpha3    -9.924458e+00 -1.992227e-01  5.134974e+00  8.396866e+01  263.63647
#> sigma_alpha1  2.629909e+00  1.126270e+01  5.224112e+01  7.474100e+02  526.71797
#> sigma_alpha2  4.981555e+00  1.672828e+01  5.761791e+01  5.068755e+02 1069.21787
#> sigma_alpha3  3.347934e+00  1.440060e+01  4.812534e+01  5.808039e+02  900.19228
#> lp__         -2.538429e+03 -2.535551e+03 -2.533332e+03 -2.529930e+03  148.52535
#>                   Rhat
#> alpha1       1.0103855
#> alpha2       1.0026672
#> alpha3       0.9998278
#> tau_alpha1   1.0071998
#> tau_alpha2   1.0304386
#> tau_alpha3   1.0395851
#> mu_alpha1    1.0195686
#> mu_alpha2    0.9999242
#> mu_alpha3    0.9997145
#> sigma_alpha1 1.0030298
#> sigma_alpha2 0.9998938
#> sigma_alpha3 0.9997435
#> lp__         1.0038998
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(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 2.6e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.26 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: 5000 / 5000 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 0.364 seconds (Warm-up)
#> Chain 1:                0.161 seconds (Sampling)
#> Chain 1:                0.525 seconds (Total)
#> Chain 1:
#> Warning: There were 843 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.