FLAMES: Flexible Link function with AsyMptotES

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<div>
 <h3 style="text-align: justify; padding-top:50px">FLAMES: Flexible Link function with AsyMptotES</h3>
 <h4 style="text-align: justify; padding-top:0px">Estimating the SUS population in Brazil</h4>
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 <h5>
 Douglas R. Mesquita Azevedo 
 Marcos Oliveira Prates 
 Renato Martins Assunção
 
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 <img src="data:image/png;base64,#img/logo_ufmg.png" alt="ufmg" height="50"/>
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## Agenda

+ Motivation
+ Anomaly detection in SUS production
+ SUS population
+ Link functions in GLM models
+ Introducing the link functions with asymptotes
+ Simulation study
+ Estimating the SUS population in Brazil
+ FLAMES
+ Conclusion

---

## Motivation

Brazil manages the largest public health system in the world:
- more than .enfase[3.5 billion outpatient procedures] per year
- more than .enfase[12 million hospitalizations] per year
- nearly .enfase[5.18 billion dollars] per year on these services
- more than .enfase[220,000 establishments] provide services to the SUS, including hospitals, clinics, and laboratories

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## Motivation

The private health insurance market in Brazil is also very large due to the inefficiencies of the public health sector
- over .enfase[50 million people] use these services
- it is the .enfase[second largest] in the world (United States is the first ranked)

The .enfase[ANS] (Agência Nacional de Saúde) regulates health insurance in Brazil and also provides general statistics (by municipalities)

---
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## Motivation

- Automatic detection of anomalies in the payment system for service providers
- It is very .enfase[difficult to audit] the SUS system due to its proportions
- More than .enfase[30 million time series] to be analyzed
- We need the number of people using the service in each neighborhood (.enfase[SUS population]) which is not provided by ANS

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## Anomaly detection

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## Anomaly detection

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## Anomaly detection

Compare the number of procedures for a specific \{hospital, city, period\} with the expected number of procedures (based on country or state).

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## Anomaly detection

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## Anomaly detection

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## Anomaly detection

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## Anomaly detection

+ For each city, find the .enfase[SUS population] using data from ANS
+ For a given city, procedure and, period, find the .enfase[total number of procedures performed]
+ Find .enfase[Brazilian expected rate] as well as rates for each state
+ Compare standard rates with the rate of a specific city (conservative approach)
+ Ring a bell if we observe higher rates for a specific city
+ Last but not least: .enfase[find out which hospitals/labs are serving these cities]

---

## SUS population

+ ANS provides, for each city, the .enfase[number of private health insurance plans]
+ By difference, we can estimate the .enfase[number of people who depends on SUS]
+ It is .enfase[difficult to find anomalies in big populations] (São Paulo, Rio de Janeiro, Belo Horizonte, ...)
+ What is needed to have SUS populations in a more detailed areal space (neighborhood)?

.mathbox[
  We need to .enfase[learn] which .enfase[characteristics] are .enfase[associated] with our outcome: .enfase[the person is covered by a private health insurance]!
]

.mathbox[
  The .enfase[outcome is available] for at least .enfase[one level] for performance check!
]

---
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## SUS population - PNAD 2008

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## SUS population - PNAD 2008

+ Around 40,000 Brazilian households
+ Covariates:
  + Gender
  + Income
+ `$Y$`:
  + 0: The person is covered by a private health insurance
  + 1: The person is not covered by a private health insurance (he/she depends on SUS)
  
---
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## SUS population

+ .enfase[Individual level]: Learn the association between covariates and outcome (.enfase[PNAD])
  + We would like a very accurate model at this level
  + Our main interest is prediction
  + Models are stratified by state
+ .enfase[City level]: Compare predictions: proposed model and observed populations (.enfase[ANS])
  + Gender: proportion of males in each city (.enfase[IBGE])
  + Income: average income in each city (.enfase[IBGE])
+ .enfase[Neighborhood level]: Predict SUS population
  + Gender: proportion of males in each neighborhood (.enfase[IBGE])
  + Income: average income in each neighborhood (.enfase[IBGE])

---
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## SUS population - Individual level

---

## GLM

The Generalized Linear Model (GLM) is given by:

`$g(\mu) = X\beta$`
]

where:
- `$D$` is a probability distribution such as: Normal, .enfase[Bernoulli], Poisson, ...
- `$\mu \in (a, b)$` is a parameter (mean)
- `$\theta$` represents any other parameters
- `$g(.)$` is the .enfase[link function] that maps `$g(\mu):(a, b) \rightarrow \mathbb{R}$`

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## Binary regression

As `$\mu \in (0, 1)$`, any `$g(\mu): (0, 1) \rightarrow \mathbb{R}$` is a valid link function.

- .enfase[logit]: `$g(\mu) = \log(\frac{\mu}{1-\mu})$`
- .enfase[probit]: `$g(\mu) = \Phi^{-1}(\mu)$`
- .enfase[cloglog]: `$g(\mu) = \log(-\log(1-\mu))$`
- .enfase[loglog]: `$g(\mu) = \log(-\log(\mu))$`
- .enfase[cauchit]: `$g(\mu) = F_{t_1}^{-1}(\mu)$`
- .enfase[robit]: `$g(\mu) = F_{t_{\nu}}^{-1}(\mu)$`

---
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## Binary regression

---

## Link functions with asymptotes

`$Y$`: .enfase[death due to breast cancer]
- the individual can heal `$P(Y = 1|X) \leq d \leq 1$`
- even in less dangerous scenarios we can observe deaths `$P(Y = 1|X) \geq c \geq 0$`

`$Y$`: .enfase[conversion in an e-commerce]
- the majority will not buy anything `$P(Y = 1|X) \leq d \leq 1$`

`$Y$`: .enfase[churn in a streaming service]
- We can observe `$P(Y = 1|X) \geq c \geq 0$`

`$Y$`: .enfase[the person has a private insurance plan]

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## Link functions with asymptotes

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## Link functions with asymptotes

Conventional modelling for binary outcomes:

.mathbox[
  `$Y_i|\mu_i \sim \text{Bernoulli}(Y_i; \mu_i)$`
  
  `$g(\mu_i|\mathbf{\beta}) = \beta_0 + \beta_1\times X_{1i} + \ldots + \beta_k\times X_{ki} = X\beta$`
]

For the aforementioned `$g(.)$` functions we have that:

E.g.: logit

.mathbox[
  `$g^{-1}(x) = \frac{exp(x)}{1 + \exp(x)} = \frac{1}{\exp(-x)+1} \implies g^{-1}(\infty) = 1$` and `$g^{-1}(-\infty) = 0$`
]

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## Link functions with asymptotes

.mathbox[
  `$Y_i|\mu_i \sim \text{Bernoulli}(Y_i; \mu_i)$`
  
  `$g^{-1}_{cd}(X\beta|c, d) = c + (d - c)\times g^{-1}(X\beta)$`
]

For the aforementioned `$g(.)$` functions we have that:

E.g.: logit

.mathbox[
  `$g_{cd}^{-1}(x) = c + (d-c)\frac{1}{\exp(-x)+1} \implies g^{-1}(\infty) = d$` and `$g^{-1}(-\infty) = c$`
]

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## Link functions with asymptotes

E.g. .enfase[cauchit] and .enfase[logit]:

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## Inference

The model is given by:

`$\mu_i | \beta, c, d, \nu = g_{cd}^{-1}(X\beta) = c + (d-c)g^{-1}(X\beta)$`
]

Priors:

`$c \sim \text{Beta}(a_c, b_c)$`

`$d|c \sim \text{Beta}(a_d, b_d)\mathbf{I}(d > c)$`

`$\nu \sim \text{Exp}(\lambda)$`

`$\lambda \sim \text{Uniform}(a_{\lambda}, b_{\lambda})$`
]

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## Inference

List of available link functions in .enfase[https://github.com/DouglasMesquita/FLAMES]:

---

## Simulation

1) the conventional model, with .enfase[c = 0] and .enfase[d = 1]

2) a model with a fixed minimum proportion of cases, implying on .enfase[c = 0.20] and .enfase[d = 1]

3) a model with a maximum proportion of noncases, implying on .enfase[c = 0] and .enfase[d = 0.95]

4) a model with a minimum and maximum proportion of cases and non-cases, respectively, implying on .enfase[c = 0.20] and .enfase[d = 0.95]

In all scenarios, we are taking `$\beta_0 = 0; \beta_1 = -1; \beta_2 = 0.5$`

For each model, we generated .enfase[100 data sets]

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## Simulation

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## SUS population - Model fit

Because of the very large demographic, economic and cultural differences in Brazil, .enfase[a model was fitted for each one of the 27 states], indexed by `$j = 1; ... ; 27$`.

Using the 2008 PNAD survey, let `$Y_{ji} = 1$`, if the `$i$`th person living at state `$j$` .enfase[is not covered by a private health insurance], and 0, otherwise.

`$\mu_{ji} | \beta, c, d = c_j + (d_j - c_j)g^{-1}(\beta_{0j} + \beta_{1j}\times \text{Income}_{ji} + \beta_{2j}\times \text{Gender}_{ji})$`
]

where,

+ `$g(.)$` is a link function
+ `$\text{Income}_{ji}$` is the income of that household (scaled)
+ `$\text{Gender}_{ji}$` is the gender of the household respondent

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## SUS population - Model fit

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## SUS population - Model fit

Model parameters estimates for the 10 most populous states in Brazil

Fit measures for MG under several models

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## SUS population - Model fit

With the estimates of `$\hat{c}_j$`, `$\hat{d}_j$`, `$\hat{\beta}_{0j}$`, `$\hat{\beta}_{1j}$` and `$\hat{\beta}_{2j}$` at hand, one can proceed to estimate .enfase[SUS population] at any level `$k \in j$` as

.mathbox[
`$\hat{\text{Pop}}_{jk} = [\hat{c}_j + (\hat{d}_j - \hat{c}_j)\times g^{-1}(\hat{\beta}_{0j} + \hat{\beta}_{1j}\times \text{Income}_{jk} + \hat{\beta}_{2j}\times \text{Gender}_{jk})]\times n_{jk}$`
]

where

+ `$\text{Income}_{jk}$` is the average income (scaled) of region `$jk$`
+ `$\text{Gender}_{jk}$` is the proportion of males of region `$jk$`
+ `$n_{jk}$` is the total population of region `$jk$`

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## SUS population - Model check

Estimated SUS population in comparison to the ANS data in the logarithm scale for MG and SP

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## SUS population - Model prediction

Neighborhoods in Belo Horizonte and its respective income average, estimated SUS population, and estimated proportion of SUS users

---

## FLAMES

Available at: https://github.com/DouglasMesquita/FLAMES

```r
devtools::install_github("DouglasMesquita/FLAMES")

out_arms <- mcmc_bin(
 data = df,
 formula = y ~ x1 + x2,
 nsim = 2000,
 burnin = 5000,
 lag = 10,
 type = "cloglog", # logit, probit, robit, cauchit
 sample_c = TRUE,
 sample_d = TRUE,
 method = "ARMS" # metropolis
)
```

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## FLAMES

```r
summary(out_arms)
```

```
## $Call
## mcmc_bin(data = bd, formula = f, nsim = nsim, burnin = burnin, 
##     lag = lag, type = type, sample_c = TRUE, sample_d = TRUE, 
##     method = "ARMS")
## 
## $Coeficients
##                   mean std_error   lower_95   upper_95
## (Intercept)  0.0239149 0.1957455 -0.3387205  0.4002764
## X           -1.6048333 0.5181484 -2.6397816 -0.7724575
## 
## $`Other parameters`
##                  mean  std_error  lower_95  upper_95
## c parameter 0.3174518 0.08798908 0.1299549 0.4642634
## d parameter 0.9320752 0.02038448 0.8968980 0.9750762
## 
## $`Fit measures`
##        DIC  -2*LPML     WAIC
## 1 1058.437 1058.637 1058.437
```

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## FLAMES

```r
summary(out_arms)
```

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## FLAMES

```r
par(mfrow = c(2, 2))
plot(out_arms, ask = F)
```

---

## Conclusions

- Demographic information is useful to .enfase[guide public health decision making] but is often .enfase[not available at all desired geographic levels].
- As a workaround, a .enfase[predictive model] can be built based on relevant .enfase[covariates available in the desired levels].
- We propose a .enfase[flexible link function framework] for GLM models, capable of accommodating .enfase[asymmetry, heavy tail, and asymptotes].
- The proposed methodology is used to determine the .enfase[Brazilian SUS population] for the neighborhoods of the main cities in Brazil.
- This local information is vital for the .enfase[InfoSAS project], and it is important for public agencies since it allows for local policies aimed to the population that actually takes advantage of the system.

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## Conclusions

- The PNAD survey was used to .enfase[link] the average .enfase[income] and .enfase[percentage of males] with the .enfase[health insurance status] by individuals. 
- The model was .enfase[fitted separately] for each Brazilian state given the cultural and social economic diversity of Brazil.
- It was possible to verify that the model is capable of estimating SUS population in an .enfase[unbiased and precise] way (based on ANS data).
- Finally, the .enfase[link function with asymptotes] was capable of .enfase[adapting] to the .enfase[asymmetry] and characteristics observed in the data.
- We were able to .enfase[transfer the acquired knowledge] from the individual and municipality levels to .enfase[estimate the SUS population] of each Brazilian .enfase[neighborhood].
- The .enfase[FLAMES R package] is available to fit the proposed class of .enfase[flexible link functions using Bayesian inference].

---

## References

.enfase[Carvalho O., Meira W. Jr., Prates M. (2016)]. **Infosas: um sistema de mineração de dados para controle da produção do sus**. _Revista do TCU, 137, 52-59_.

.enfase[Nagler J. (1994)]. **An alternative estimator to logit and probit**. _American Journal of Political Science, 38, 230–255_.

.enfase[Chen MH., Dey D. and Shao QM. (1999)]. **A new skewed link model for dichotomous quantal response data**. _Journal of the American Statistical Association 99, 1172–1186_.

.enfase[Bazán JL., Bolfarine H. and Branco MD. (2010)]. **A framework for skew-probit links in binary regression**. _Communications in Statistics 39, 678–697_.

.enfase[Jiang X., Dey DK., Prunier R., et al. (2013)]. **A new class of flexible link functions with application to species co-occurrence in cape floristic region**. _Annals of Applied Statistics, 7, 2180–2204_.

.enfase[Bazán JL., Romeo JS. and Rodrigues J. (2014)]. **Bayesian skew-probit regression for binary response data**. _Brazilian Journal of Probability and Statistics 28, 467–482_.

.enfase[Li D., Wang X., Lin L., et al. (2016)]. **Flexible link functions in nonparametric binary regression with Gaussian process priors**. _Biometrics 72, 707–719_.

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