Author: Aurora Hofman
Date: 2020-07-21

# Introduction to attrib

#### Aurora Hofman

#### 2020-07-21

    library(attrib)
    #> attrib 2021.1.2 https://folkehelseinstituttet.github.io/attrib
    library(data.table)

# Introduction

`attrib` provides a way of estimating what the mortality would have been if some given exposures are set to a reference value. By using simulations from the posterior distribution of all coefficients we can easily aggregate over time and locations while still estimating valid credible intervals.

This vignette will go through:

- how to use `fit_attrib` to fit the model to the data
- how to use `est_attrib` to estimate the mortality under different scenarios (i.e. when the exposures are at reference values and at observed values)
- some examples of usages of the resulting dataset

## Data example

We will use the datasets `fake_data_county` and `fake_data_nation`.

`fake_data_county` consists of fake mortality data for all counties of Norway on a weekly basis from 2010 until 2020. The dataset consists of the following features:

- location_code: Location code of the different counties
- week: Week number
- season: Years of the season
- year: Year
- yrwk: Year and week
- x: Number of weeks from the start of the season
- pop: Population size
- pr100_ili: Percentage of doctors consultations diagnosed with influenza like illnesses
- pr100_ili_lag_1: pr_100_ili lagged with one week
- temperature: Temperature
- temperature_high: number of heatwaves
- deaths: number of deaths

`fake_data_nation` is a similar dataset at the national level.

    data_fake_county <- attrib::data_fake_county
    data_fake_nation <- attrib::data_fake_nation
    head(data_fake_county, 5)
    #>    location_code week    season year    yrwk  x    pop pr100_ili
    #> 1:      county03    1 2009/2010 2010 2010-01 24 693494 0.5302766
    #> 2:      county03    1 2010/2011 2011 2011-01 24 693494 0.7234059
    #> 3:      county03    1 2011/2012 2012 2012-01 24 693494 1.7817489
    #> 4:      county03    1 2012/2013 2013 2013-01 24 693494 1.6000083
    #> 5:      county03    1 2013/2014 2014 2014-01 24 693494 1.6037883
    #>    pr100_ili_lag_1 temperature temperature_high deaths
    #> 1:       0.4496182 -0.94325234                0    103
    #> 2:       0.5359123  0.01462311                0     96
    #> 3:       1.8919261 -2.03893188                0    105
    #> 4:       1.3909804 -6.02035548                0     97
    #> 5:       1.3942666 -1.72146124                0    114

In this example we will look at the exposures `pr100_ili_lag_1` and `temperature_high` and calculate the attributable mortality due to these exposures.

# Fitting using fit_attrib

## County level

We want to estimate the attributable mortality due to ILI and heatwaves. `attrib` lets us fit models with both fixed and random effect and offsets using linear mixed models (LMM).

We use the `glmer` function from the `lme4` package. In practice, this means we must specify the response, offsets, the fixed effects, and the random effects. In our case we will model the response deaths as a function of:

- the fixed effects:
  - temperature_high
  - pr100_ili_lag_1
  - sin(2 * pi * (week - 1) / 52)
  - cos(2 * pi * (week - 1) / 52)
- the random effects:
  - (1|location_code)
  - (pr100_ili_lag_1|season)
- the offset:
  - log(pop)

    #response
    response <- "deaths"

    # fixed effects
    fixef_county <- " temperature_high +
      pr100_ili_lag_1 +
      sin(2 * pi * (week - 1) / 52) +
      cos(2 * pi * (week - 1) / 52)"

    #random effects
    ranef_county <- "(1|location_code) +
      (pr100_ili_lag_1|season)"

    #offset
    offset_county <- "log(pop)"

Now we fit the model using `fit_attrib`.

    fit_county <- fit_attrib(data_fake_county,
                              response = response,
                              fixef = fixef_county,
                              ranef = ranef_county,
                              offset = offset_county)

This results in the following fit:

    fit_county
    #> Generalized linear mixed model fit by maximum likelihood (Laplace
    #>   Approximation) [glmerMod]
    #>  Family: poisson  ( log )
    #> Formula: deaths ~ temperature_high + pr100_ili_lag_1 + sin(2 * pi * (week -
    #>     1)/52) + cos(2 * pi * (week - 1)/52) + offset(log(pop)) +
    #>     (1 | location_code) + (pr100_ili_lag_1 | season)
    #>    Data: data
    #>       AIC       BIC    logLik  deviance  df.resid
    #>  44645.63  44706.39 -22313.82  44627.63      6305
    #> Random effects:
    #>  Groups        Name            Std.Dev. Corr
    #>  location_code (Intercept)     0.000000
    #>  season        (Intercept)     0.000000
    #>                pr100_ili_lag_1 0.004719  NaN
    #> Number of obs: 6314, groups:  location_code, 11; season, 11
    #> Fixed Effects:
    #>                 (Intercept)             temperature_high
    #>                    -8.79616                      0.08188
    #>             pr100_ili_lag_1  sin(2 * pi * (week - 1)/52)
    #>                     0.02796                      0.01878
    #> cos(2 * pi * (week - 1)/52)
    #>                     0.07498
    #> convergence code 0; 0 optimizer warnings; 1 lme4 warnings

Note that fit has the added attributes `offset` (saving the offset name) and `fit_fix` (the coefficients of the linear model fitted on only the fixed effects). These are needed by `est_attrib` to create the dataset containing only the fixed effects.

## National level

We estimate the same as before But on a national level, meaning we remove the random effect (1|location_code) since we only have one location code. This gives the following features:

- the fixed effects:
  - temperature_high
  - pr100_ili_lag_1
  - sin(2 * pi * (week - 1) / 52)
  - cos(2 * pi * (week - 1) / 52)
- the random effects:
  - (pr100_ili_lag_1|season)
- the offset:
  - log(pop)

    # take in the fixed effects
    response = "deaths"
    fixef_nation <- "temperature_high +
      pr100_ili_lag_1 +
      sin(2 * pi * (week - 1) / 52) +
      cos(2 * pi * (week - 1) / 52)"

    #take in the random effects
    ranef_nation <- "(pr100_ili_lag_1|season)"

    # take in the offset
    offset_nation <- "log(pop)"

      fit_nation <- fit_attrib(data_fake_nation,
                               response = response,
                               fixef = fixef_nation,
                               ranef = ranef_nation,
                               offset = offset_nation)

# Using the sim function

The `sim` function can be used to generate simulations for all the rows in our data.

It first generates `n_sim` simulations from the posterior distribution of the coefficients from out fit before applying these coefficients on our dataset generating `n_sim` simulations and expected mortality for each line. This is quite generic. Hence if the goal is to compute attributable mortality or incident risk ratios we use `est_attrib` as shown in a later part of the vignette.

    n_sim <- 20
    sim_data <- sim(fit_nation, data_fake_nation, n_sim)
    head(sim_data[id_row == 1], 5)
    #>    week    season year    yrwk  x location_code     pop pr100_ili
    #> 1:    1 2009/2010 2010 2010-01 24         norge 5367580  1.160403
    #> 2:    1 2009/2010 2010 2010-01 24         norge 5367580  1.160403
    #> 3:    1 2009/2010 2010 2010-01 24         norge 5367580  1.160403
    #> 4:    1 2009/2010 2010 2010-01 24         norge 5367580  1.160403
    #> 5:    1 2009/2010 2010 2010-01 24         norge 5367580  1.160403
    #>    pr100_ili_lag_1 temperature_high deaths id_row sim_id sim_value
    #> 1:        1.100072                0    891      1      1  896.6544
    #> 2:        1.100072                0    891      1      2  902.2916
    #> 3:        1.100072                0    891      1      3  896.5612
    #> 4:        1.100072                0    891      1      4  896.2302
    #> 5:        1.100072                0    891      1      5  903.5675

We can see that we now have multiple expected mortalities for the same dataline. This is due to the coefficient simulations.

# Estimating attributable mortality using est_attrib

To estimate attributable mortality we simulate:

- the estimated mortality for observed exposures
- the estimated mortality for the exposures set to reference values

This is easily done using `est_attrib`.

We need to give the fit, the dataset, the exposures with reference values, and the number of simulations. `est_attrib` will then using the `arm::sim` function to generate simulations of the underlying posterior distribution. `attrib::sim` will then combine the simulated coefficients to estimate the modeled outcome (i.e. number of deaths) for each simulation.

    exposures <- list( "temperature_high" = 0, "pr100_ili_lag_1" = 0)
    n_sim <- 20
    est_attrib_sim_county <- attrib::est_attrib(fit_county,
                                            data_fake_county,
                                            exposures = exposures,
                                            n_sim = n_sim)

    est_attrib_sim_nation <- attrib::est_attrib(fit_nation,
                                            data_fake_nation,
                                            exposures = exposures,
                                            n_sim = n_sim)

    head(est_attrib_sim_county, 5)
    #>    location_code week    season year    yrwk  x    pop pr100_ili
    #> 1:      county03    1 2009/2010 2010 2010-01 24 693494 0.5302766
    #> 2:      county03    1 2010/2011 2011 2011-01 24 693494 0.7234059
    #> 3:      county03    1 2011/2012 2012 2012-01 24 693494 1.7817489
    #> 4:      county03    1 2012/2013 2013 2013-01 24 693494 1.6000083
    #> 5:      county03    1 2013/2014 2014 2014-01 24 693494 1.6037883
    #>    pr100_ili_lag_1 temperature temperature_high deaths id sim_id
    #> 1:       0.4496182 -0.94325234                0    103  1      1
    #> 2:       0.5359123  0.01462311                0     96  2      1
    #> 3:       1.8919261 -2.03893188                0    105  3      1
    #> 4:       1.3909804 -6.02035548                0     97  4      1
    #> 5:       1.3942666 -1.72146124                0    114  5      1
    #>    sim_value_exposures=observed sim_value_temperature_high=0
    #> 1:                     115.2805                     115.2805
    #> 2:                     115.0171                     115.0171
    #> 3:                     119.8199                     119.8199
    #> 4:                     118.3090                     118.3090
    #> 5:                     116.6153                     116.6153
    #>    sim_value_pr100_ili_lag_1=0
    #> 1:                    113.6265
    #> 2:                    113.3176
    #> 3:                    113.9784
    #> 4:                    113.6816
    #> 5:                    112.4611

We can see in the above dataset that the columns id, sim_id, sim_value_exposures=observed, sim_value_temperature_high=0, sim_value_pr100_ili_lag_1=0 are added to the previous set of columns. For each row in the original dataset we now have 20 rows, one for each of the simulations done by est_attrib. In each row we see the estimate of the number of deaths given a reference value for sim_value_temperature_high and sim_value_pr100_ili_lag_1.

To make the data processing easier later we convert the dataset from wide to long form and collapse the estimated mortality

    est_attrib_county_long<-data.table::melt.data.table(est_attrib_sim_county,
                                                      id.vars = c("location_code",
                                                                  "season",
                                                                  "x",
                                                                  "week",
                                                                  "id",
                                                                  "sim_id",
                                                                  "deaths",
                                                                  "sim_value_exposures=observed"),
                                               measure.vars = c("sim_value_temperature_high=0",
                                                                "sim_value_pr100_ili_lag_1=0"))
    data.table::setnames(est_attrib_county_long, "variable", "attr")

    head(est_attrib_county_long, 5)
    #>    location_code    season  x week id sim_id deaths
    #> 1:      county03 2009/2010 24    1  1      1    103
    #> 2:      county03 2010/2011 24    1  2      1     96
    #> 3:      county03 2011/2012 24    1  3      1    105
    #> 4:      county03 2012/2013 24    1  4      1     97
    #> 5:      county03 2013/2014 24    1  5      1    114
    #>    sim_value_exposures=observed                         attr    value
    #> 1:                     115.2805 sim_value_temperature_high=0 115.2805
    #> 2:                     115.0171 sim_value_temperature_high=0 115.0171
    #> 3:                     119.8199 sim_value_temperature_high=0 119.8199
    #> 4:                     118.3090 sim_value_temperature_high=0 118.3090
    #> 5:                     116.6153 sim_value_temperature_high=0 116.6153

We can see that the columns sim_value_temperature_high=0, sim_value_pr100_ili_lag_1=0 are now collapsed into the new column attr and value with attr describing which exposure we have and value giving the corresponding reference value.

    est_attrib_nation_long<-data.table::melt.data.table(est_attrib_sim_nation,
                                                      id.vars = c("location_code",
                                                                  "season",
                                                                  "x",
                                                                  "week",
                                                                  "id",
                                                                  "sim_id",
                                                                  "deaths",
                                                                  "sim_value_exposures=observed"),
                                               measure.vars = c("sim_value_temperature_high=0",
                                                                "sim_value_pr100_ili_lag_1=0"))
    data.table::setnames(est_attrib_nation_long, "variable", "attr")

# Compare the national data to data aggregated from county to national level.

We will now aggregate our two simulated datasets (one on a county level and one on a national level) to aid in comparison.

## Aggregate from county/weekly to national/seasonal

We proceed by aggregating the county dataset to the national/seasonal level. Afterwards we calculate the expected attributable mortality, `exp_attr`, by subtracting `value` (the simulated expected number of deaths given the reference value of the exposure) from the sim_value_exposures=observed.

To be able to separate this dataset from the other we add a tag.

    aggregated_county_to_nation <-  est_attrib_county_long[,.(
      "sim_value_exposures=observed" = sum(`sim_value_exposures=observed`),
      value = sum(value),
      deaths = sum(deaths)
    ), keyby = .(season, attr, sim_id)]

    # Add exp_attr, exp_irr and a tag.
    aggregated_county_to_nation[, exp_attr:= (`sim_value_exposures=observed` - value)]
    aggregated_county_to_nation[, tag := "aggregated_from_county"]

    head(aggregated_county_to_nation, 5)
    #>       season                         attr sim_id sim_value_exposures=observed
    #> 1: 2009/2010 sim_value_temperature_high=0      1                     44404.45
    #> 2: 2009/2010 sim_value_temperature_high=0      2                     44054.80
    #> 3: 2009/2010 sim_value_temperature_high=0      3                     44194.47
    #> 4: 2009/2010 sim_value_temperature_high=0      4                     44573.15
    #> 5: 2009/2010 sim_value_temperature_high=0      5                     44510.29
    #>       value deaths exp_attr                    tag
    #> 1: 43994.19  44421 410.2645 aggregated_from_county
    #> 2: 43656.55  44421 398.2478 aggregated_from_county
    #> 3: 43803.38  44421 391.0954 aggregated_from_county
    #> 4: 44144.29  44421 428.8643 aggregated_from_county
    #> 5: 44092.58  44421 417.7154 aggregated_from_county

## Aggregating the national model per season

For the national model we aggregate over seasons and create exp_attr in the same way as above.

    aggregated_nation <-  est_attrib_nation_long[, .(
      "sim_value_exposures=observed" = sum(`sim_value_exposures=observed`),
      value = sum(value),
      deaths = sum(deaths)
    ), keyby = .(season, attr, sim_id)]

    aggregated_nation[, exp_attr:= (`sim_value_exposures=observed` - value)]
    aggregated_nation[, tag:= "nation"]
    head(aggregated_nation, 5)
    #>       season                         attr sim_id sim_value_exposures=observed
    #> 1: 2009/2010 sim_value_temperature_high=0      1                     44322.82
    #> 2: 2009/2010 sim_value_temperature_high=0      2                     44394.29
    #> 3: 2009/2010 sim_value_temperature_high=0      3                     44758.00
    #> 4: 2009/2010 sim_value_temperature_high=0      4                     44500.63
    #> 5: 2009/2010 sim_value_temperature_high=0      5                     44318.66
    #>       value deaths exp_attr    tag
    #> 1: 44322.82  44421        0 nation
    #> 2: 44394.29  44421        0 nation
    #> 3: 44758.00  44421        0 nation
    #> 4: 44500.63  44421        0 nation
    #> 5: 44318.66  44421        0 nation

For simplicity we `data.table::rbindlist` the two datasets together.

    library(ggplot2)
    data_national<- data.table::rbindlist(list(aggregated_county_to_nation, aggregated_nation))

## Calculate simulation quantiles.

The next thing to do is to aggregate away the simulations. The benefits of having the simulations is the possibility it gives to efficiently compute all desired quantiles. For this example we will use the .05, .5 and .95 quantiles.

    # Quantile functins
    q05 <- function(x){
      return(quantile(x, 0.05))
    }
    q95 <- function(x){
      return(quantile(x, 0.95))
    }

We compute the quantiles for exp_attr in the following way.

    col_names <- colnames(data_national)
    data.table::setkeyv(data_national,
                        col_names[!col_names %in% c("exp_attr",
                                                    "sim_id",
                                                    "sim_value_exposures=observed",
                                                    "value",
                                                    "deaths")])

    aggregated_sim_seasonal_data_national<- data_national[,
                                       unlist(recursive = FALSE,
                                              lapply(.(median = median, q05 = q05, q95 = q95),
                                                                        function(f) lapply(.SD, f)
                                       )),
                                       by = eval(data.table::key(data_national)),
                                       .SDcols = c("exp_attr")]

    head(aggregated_sim_seasonal_data_national,5)
    #>       season                         attr                    tag
    #> 1: 2009/2010 sim_value_temperature_high=0 aggregated_from_county
    #> 2: 2009/2010 sim_value_temperature_high=0                 nation
    #> 3: 2009/2010  sim_value_pr100_ili_lag_1=0 aggregated_from_county
    #> 4: 2009/2010  sim_value_pr100_ili_lag_1=0                 nation
    #> 5: 2010/2011 sim_value_temperature_high=0 aggregated_from_county
    #>    median.exp_attr q05.exp_attr q95.exp_attr
    #> 1:        411.4074     389.1726     428.9511
    #> 2:          0.0000       0.0000       0.0000
    #> 3:        697.3777     500.6152     848.0875
    #> 4:       1195.2093    1083.6132    1368.1267
    #> 5:        433.6910     413.0803     452.7565

We can now see that we have credible intervals and estimates for attributable deaths for all exposures.

## Plot to compare the national with the aggregated county to national model

To be able to compare the two models we make a point range plot using ggplot2.

    q <- ggplot(aggregated_sim_seasonal_data_national[attr == "sim_value_pr100_ili_lag_1=0"],
                           aes(x = season, y = median.exp_attr, group = tag, color = tag))
    q <- q + geom_pointrange(aes(x = season, y = median.exp_attr, ymin = q05.exp_attr, ymax = q95.exp_attr), position = position_dodge(width = 0.3))
    q <- q + ggtitle("Attributable mortality due to ILI in Norway according to 2 models")
    q <- q +  scale_y_continuous("Estimated attributable mortality")
    q <- q +  theme(axis.text.x = element_text(angle = 90),axis.title.x=element_blank())
    q <- q +  labs(caption = glue::glue("Aggregated county model: Attributable mortality modeled on a county level before beeing aggregated up to a national level.\n National model: Attributable mortality modeled on a national level."))
    q

# Comparing cumulative sums over seasons

When operating on the national level, we prefer to aggregate the county model to national level (instead of using the national model). This ensures consistent results at all geographical levels.

    aggregated_county_to_nation <-  est_attrib_county_long[, .(
      "sim_value_exposures=observed" = sum(`sim_value_exposures=observed`),
      value = sum(value),
      deaths = sum(deaths)
    ), keyby = .(season, x, week, attr, sim_id)]

    aggregated_county_to_nation[, exp_attr:= (`sim_value_exposures=observed` - value)]
    aggregated_county_to_nation[, exp_irr:= (`sim_value_exposures=observed` /value)]
    head(aggregated_county_to_nation,5)
    #>       season x week                         attr sim_id
    #> 1: 2009/2010 1   30 sim_value_temperature_high=0      1
    #> 2: 2009/2010 1   30 sim_value_temperature_high=0      2
    #> 3: 2009/2010 1   30 sim_value_temperature_high=0      3
    #> 4: 2009/2010 1   30 sim_value_temperature_high=0      4
    #> 5: 2009/2010 1   30 sim_value_temperature_high=0      5
    #>    sim_value_exposures=observed    value deaths exp_attr  exp_irr
    #> 1:                     789.3466 752.2463    816 37.10021 1.049319
    #> 2:                     785.3745 749.3336    816 36.04085 1.048097
    #> 3:                     786.4563 751.0825    816 35.37382 1.047097
    #> 4:                     792.8259 754.0530    816 38.77294 1.051419
    #> 5:                     793.8784 756.1073    816 37.77111 1.049955

Again we compute the quantiles.

    col_names <- colnames(aggregated_county_to_nation)
    data.table::setkeyv(aggregated_county_to_nation, col_names[!col_names %in% c("exp_attr", "exp_irr","sim_id", "exposures", "sim_value_exposures=observed", "value")])

    aggregated_county_to_nation_weekly <- aggregated_county_to_nation[,
                  unlist(recursive = FALSE, lapply(.(median = median, q05 = q05, q95 = q95),
                                                   function(f) lapply(.SD, f)
                  )),
                  by=eval(data.table::key(aggregated_county_to_nation)),
                  .SDcols = c("exp_attr", "exp_irr")]

We then estimate the cumulative sums of attributable mortality and corresponding credible intervals.

    aggregated_county_to_nation_weekly[, cumsum := cumsum(median.exp_attr), by = .( attr, season)]
    aggregated_county_to_nation_weekly[, cumsum_q05 := cumsum(q05.exp_attr), by = .( attr, season)]
    aggregated_county_to_nation_weekly[, cumsum_q95 := cumsum(q95.exp_attr), by = .( attr, season)]

    head(aggregated_county_to_nation_weekly, 5)
    #>       season x week                         attr deaths median.exp_attr
    #> 1: 2009/2010 1   30 sim_value_temperature_high=0    816    37.201790055
    #> 2: 2009/2010 1   30  sim_value_pr100_ili_lag_1=0    816     0.000000000
    #> 3: 2009/2010 2   31 sim_value_temperature_high=0    854    79.588212191
    #> 4: 2009/2010 2   31  sim_value_pr100_ili_lag_1=0    854     0.002235554
    #> 5: 2009/2010 3   32 sim_value_temperature_high=0    777    44.697266901
    #>    median.exp_irr q05.exp_attr q05.exp_irr q95.exp_attr q95.exp_irr
    #> 1:       1.049530 35.200399792    1.046738 38.782413467    1.051816
    #> 2:       1.000000  0.000000000    1.000000  0.000000000    1.000000
    #> 3:       1.105772 75.252401160    1.099759 83.005850286    1.110698
    #> 4:       1.000003  0.001599943    1.000002  0.002726044    1.000003
    #> 5:       1.059231 42.274743419    1.055891 46.607747380    1.061966
    #>          cumsum   cumsum_q05   cumsum_q95
    #> 1: 3.720179e+01 3.520040e+01 3.878241e+01
    #> 2: 0.000000e+00 0.000000e+00 0.000000e+00
    #> 3: 1.167900e+02 1.104528e+02 1.217883e+02
    #> 4: 2.235554e-03 1.599943e-03 2.726044e-03
    #> 5: 1.614873e+02 1.527275e+02 1.683960e+02

We can then plot the estimated cumulative attributable mortality over influenza seasons in Norway

    library(ggplot2)
    q <- ggplot(
      data = aggregated_county_to_nation_weekly[
        season %in% c(
          "2015/2016",
          "2016/2017",
          "2017/2018",
          "2018/2019",
          "2019/2020"
        ) &
        attr == "sim_value_pr100_ili_lag_1=0"
      ],
      aes(
        x = x,
        y = cumsum,
        group = season,
        color = season,
        fill = season
      )
    )
    q <- q + geom_line()
    q <- q + geom_ribbon(
      data = aggregated_county_to_nation_weekly[
        season %in% c("2019/2020") &
        attr == "sim_value_pr100_ili_lag_1=0"
      ],
      aes(
        ymin = cumsum_q05,
        ymax = cumsum_q95
      ),
      alpha = 0.4,
      colour = NA
    )
    q <- q + scale_y_continuous("Estimated cumulative attributable mortality")
    q <- q + ggtitle("Estimated cumulative attributable mortality over influenza seasons in Norway")
    q

We can also plot the estimated weekly attributable mortality in Norway

    q <- ggplot(
      data = aggregated_county_to_nation_weekly[attr == "sim_value_pr100_ili_lag_1=0"],
      aes(x = x, y = cumsum, group = season)
      )
    q <- q + geom_line(
      data = aggregated_county_to_nation_weekly[
        season != "2019/2020" &
        attr == "sim_value_pr100_ili_lag_1=0"
      ],
      aes(
        x = x,
        y = median.exp_attr,
        group = season
      ),
      color = "grey"
    )
    q <- q + geom_line(
      data = aggregated_county_to_nation_weekly[
        season == "2019/2020" &
        attr == "sim_value_pr100_ili_lag_1=0"
      ],
      aes(
        x = x,
        y = median.exp_attr,
        group = season
      ),
      color = "blue"
    )
    q <- q + geom_ribbon(
      data = aggregated_county_to_nation_weekly[
        season == "2019/2020" &
        attr == "sim_value_pr100_ili_lag_1=0"
      ],
      aes(
        x = x,
        ymin = q05.exp_attr,
        ymax = q95.exp_attr
      ),
      fill = "blue",
      alpha=0.4
    )
    q <- q + scale_y_continuous("Estimated attributable mortality")
    q <- q + ggtitle("Estimated mortality due to ILI per week")
    q

# Incident rate ratio

Until now we have focused on estimating attributable mortality. Now we will investigate computing the incident rate ratio (IRR) for pr100_ili_lag_1. To do this we will use the fit made by `fit_attrib` on the county dataset but we will change the values for pr100_ili_lag_1 to 1 (IRRs are generally expressed as the effect of the exposure changing from 0 to 1).

    data_fake_county_irr <- data.table::copy(data_fake_county)
    data_fake_county_irr[, pr100_ili_lag_1 := 1]
    head(data_fake_county_irr, 5)
    #>    location_code week    season year    yrwk  x    pop pr100_ili
    #> 1:      county03    1 2009/2010 2010 2010-01 24 693494 0.5302766
    #> 2:      county03    1 2010/2011 2011 2011-01 24 693494 0.7234059
    #> 3:      county03    1 2011/2012 2012 2012-01 24 693494 1.7817489
    #> 4:      county03    1 2012/2013 2013 2013-01 24 693494 1.6000083
    #> 5:      county03    1 2013/2014 2014 2014-01 24 693494 1.6037883
    #>    pr100_ili_lag_1 temperature temperature_high deaths
    #> 1:               1 -0.94325234                0    103
    #> 2:               1  0.01462311                0     96
    #> 3:               1 -2.03893188                0    105
    #> 4:               1 -6.02035548                0     97
    #> 5:               1 -1.72146124                0    114

Then we can set the reference value to zero and hence obtain the IRR for the given exposure.

    exposures_irr = c(pr100_ili_lag_1 = 0)

Now we use `est_attrib` to create the simulations.

    est_attrib_sim_county_irr <- attrib::est_attrib(
      fit_county,
      data_fake_county_irr,
      exposures = exposures_irr,
      n_sim = 100
    )
    head(est_attrib_sim_county_irr, 5)
    #>    location_code week    season year    yrwk  x    pop pr100_ili
    #> 1:      county03    1 2009/2010 2010 2010-01 24 693494 0.5302766
    #> 2:      county03    1 2010/2011 2011 2011-01 24 693494 0.7234059
    #> 3:      county03    1 2011/2012 2012 2012-01 24 693494 1.7817489
    #> 4:      county03    1 2012/2013 2013 2013-01 24 693494 1.6000083
    #> 5:      county03    1 2013/2014 2014 2014-01 24 693494 1.6037883
    #>    pr100_ili_lag_1 temperature temperature_high deaths id sim_id
    #> 1:               1 -0.94325234                0    103  1      1
    #> 2:               1  0.01462311                0     96  2      1
    #> 3:               1 -2.03893188                0    105  3      1
    #> 4:               1 -6.02035548                0     97  4      1
    #> 5:               1 -1.72146124                0    114  5      1
    #>    sim_value_exposures=observed sim_value_pr100_ili_lag_1=0
    #> 1:                     117.1848                    114.1279
    #> 2:                     116.3623                    113.8233
    #> 3:                     115.4643                    112.9495
    #> 4:                     116.3412                    113.6468
    #> 5:                     116.4120                    114.1933

We see we have obtained values for the reference of the exposure in the same way as before. The difference is that we changed the dataset before running est_attrib. This means we will now be observing the difference between `pr100_ili_lag_1=0` and `pr100_ili_lag_1=1`.

We now aggregate to the national seasonal level.

    aggregated_county_to_nation_sim_irr <-  est_attrib_sim_county_irr[, .(
      "sim_value_exposures=observed" = sum(`sim_value_exposures=observed`),
      "sim_value_pr100_ili_lag_1=0"= sum(`sim_value_pr100_ili_lag_1=0`),
      deaths = sum(deaths)
    ), keyby = .(season, sim_id)]

Here we generate the IRR:

    aggregated_county_to_nation_sim_irr[, exp_irr:= (`sim_value_exposures=observed`/`sim_value_pr100_ili_lag_1=0`
    )]
    head(aggregated_county_to_nation_sim_irr,5)
    #>       season sim_id sim_value_exposures=observed sim_value_pr100_ili_lag_1=0
    #> 1: 2009/2010      1                     45147.76                    43970.06
    #> 2: 2009/2010      2                     45155.71                    43429.33
    #> 3: 2009/2010      3                     45373.61                    43733.29
    #> 4: 2009/2010      4                     45022.49                    43300.55
    #> 5: 2009/2010      5                     44741.08                    43339.94
    #>    deaths  exp_irr
    #> 1:  44421 1.026784
    #> 2:  44421 1.039752
    #> 3:  44421 1.037507
    #> 4:  44421 1.039767
    #> 5:  44421 1.032329

Now we can compute the quantiles:

    col_names <- colnames(aggregated_county_to_nation_sim_irr)
    data.table::setkeyv(
      aggregated_county_to_nation_sim_irr,
      col_names[!col_names %in% c("exp_irr", "sim_id", "sim_value_exposures=observed", "sim_value_pr100_ili_lag_1=0")]
    )

    aggregated_county_to_nation_irr <- aggregated_county_to_nation_sim_irr[,
      unlist(recursive = FALSE, lapply(.(median = median, q05 = q05, q95 = q95), function(f) lapply(.SD, f))),
      by = eval(data.table::key(aggregated_county_to_nation_sim_irr)),
      .SDcols = c("exp_irr")
    ]
    aggregated_county_to_nation_irr[, tag := "aggregated"]

    aggregated_county_to_nation_irr
    #>        season deaths median.exp_irr q05.exp_irr q95.exp_irr        tag
    #>  1: 2009/2010  44421       1.032638    1.024027    1.040034 aggregated
    #>  2: 2010/2011  43062       1.027866    1.019445    1.035782 aggregated
    #>  3: 2011/2012  43431       1.026895    1.018240    1.034338 aggregated
    #>  4: 2012/2013  43228       1.028879    1.020966    1.037237 aggregated
    #>  5: 2013/2014  43328       1.025117    1.017164    1.033269 aggregated
    #>  6: 2014/2015  42951       1.027720    1.019838    1.035632 aggregated
    #>  7: 2015/2016  43736       1.022287    1.014470    1.029842 aggregated
    #>  8: 2016/2017  43821       1.033221    1.024841    1.041368 aggregated
    #>  9: 2017/2018  43716       1.031375    1.022904    1.038913 aggregated
    #> 10: 2018/2019  43326       1.026813    1.019242    1.033944 aggregated
    #> 11: 2019/2020  43572       1.030668    1.021900    1.037652 aggregated

Now we compare the resulting values for IRR with the ones obtained by `coef(fit_county)$season` and the 90 percent credible interval computed manually using the standard deviation given by summary(fit_county) for pr100_ili_lag_1.

    coef_fit_county <- data.table::as.data.table(coef(fit_county)$season)
    col_names_coef <- c("pr100_ili_lag_1")
    coef_irr_data <- coef_fit_county[, ..col_names_coef]
    coef_irr_data[, irr := exp(pr100_ili_lag_1)]
    coef_irr_data[, q05 := exp(pr100_ili_lag_1 - 1.645 *0.003761)]  # 0.003761 is the standard deviation from coef(fit_county)
    coef_irr_data[, q95 := exp(pr100_ili_lag_1 + 1.645 *0.003761)]
    coef_irr_data[, tag := "from_coef"]
    coef_irr_data
    #>     pr100_ili_lag_1      irr      q05      q95       tag
    #>  1:      0.03191717 1.032432 1.026064 1.038839 from_coef
    #>  2:      0.02728599 1.027662 1.021323 1.034039 from_coef
    #>  3:      0.02649399 1.026848 1.020515 1.033221 from_coef
    #>  4:      0.02850534 1.028916 1.022569 1.035301 from_coef
    #>  5:      0.02478480 1.025094 1.018772 1.031456 from_coef
    #>  6:      0.02723395 1.027608 1.021270 1.033986 from_coef
    #>  7:      0.02211952 1.022366 1.016060 1.028711 from_coef
    #>  8:      0.03244756 1.032980 1.026609 1.039390 from_coef
    #>  9:      0.03068968 1.031165 1.024805 1.037565 from_coef
    #> 10:      0.02624064 1.026588 1.020256 1.032959 from_coef
    #> 11:      0.02986668 1.030317 1.023962 1.036711 from_coef

Add the correct seasons to the data.

    coef_irr_data <- cbind(season = aggregated_county_to_nation_irr$season, coef_irr_data)
    coef_irr_data
    #>        season pr100_ili_lag_1      irr      q05      q95       tag
    #>  1: 2009/2010      0.03191717 1.032432 1.026064 1.038839 from_coef
    #>  2: 2010/2011      0.02728599 1.027662 1.021323 1.034039 from_coef
    #>  3: 2011/2012      0.02649399 1.026848 1.020515 1.033221 from_coef
    #>  4: 2012/2013      0.02850534 1.028916 1.022569 1.035301 from_coef
    #>  5: 2013/2014      0.02478480 1.025094 1.018772 1.031456 from_coef
    #>  6: 2014/2015      0.02723395 1.027608 1.021270 1.033986 from_coef
    #>  7: 2015/2016      0.02211952 1.022366 1.016060 1.028711 from_coef
    #>  8: 2016/2017      0.03244756 1.032980 1.026609 1.039390 from_coef
    #>  9: 2017/2018      0.03068968 1.031165 1.024805 1.037565 from_coef
    #> 10: 2018/2019      0.02624064 1.026588 1.020256 1.032959 from_coef
    #> 11: 2019/2020      0.02986668 1.030317 1.023962 1.036711 from_coef

rbindlist the two datasets together.

    total_data_irr <- data.table::rbindlist(list(coef_irr_data, aggregated_county_to_nation_irr), use.names = FALSE)
    total_data_irr[, pr100_ili_lag_1 := NULL]
    total_data_irr
    #>        season      irr      q05      q95        tag
    #>  1: 2009/2010 1.032432 1.026064 1.038839  from_coef
    #>  2: 2010/2011 1.027662 1.021323 1.034039  from_coef
    #>  3: 2011/2012 1.026848 1.020515 1.033221  from_coef
    #>  4: 2012/2013 1.028916 1.022569 1.035301  from_coef
    #>  5: 2013/2014 1.025094 1.018772 1.031456  from_coef
    #>  6: 2014/2015 1.027608 1.021270 1.033986  from_coef
    #>  7: 2015/2016 1.022366 1.016060 1.028711  from_coef
    #>  8: 2016/2017 1.032980 1.026609 1.039390  from_coef
    #>  9: 2017/2018 1.031165 1.024805 1.037565  from_coef
    #> 10: 2018/2019 1.026588 1.020256 1.032959  from_coef
    #> 11: 2019/2020 1.030317 1.023962 1.036711  from_coef
    #> 12: 2009/2010 1.032638 1.024027 1.040034 aggregated
    #> 13: 2010/2011 1.027866 1.019445 1.035782 aggregated
    #> 14: 2011/2012 1.026895 1.018240 1.034338 aggregated
    #> 15: 2012/2013 1.028879 1.020966 1.037237 aggregated
    #> 16: 2013/2014 1.025117 1.017164 1.033269 aggregated
    #> 17: 2014/2015 1.027720 1.019838 1.035632 aggregated
    #> 18: 2015/2016 1.022287 1.014470 1.029842 aggregated
    #> 19: 2016/2017 1.033221 1.024841 1.041368 aggregated
    #> 20: 2017/2018 1.031375 1.022904 1.038913 aggregated
    #> 21: 2018/2019 1.026813 1.019242 1.033944 aggregated
    #> 22: 2019/2020 1.030668 1.021900 1.037652 aggregated
    #>        season      irr      q05      q95        tag

    q <- ggplot(
      data = total_data_irr,
      aes(
        x = season,
        group = tag,
        color = tag
      )
    )
    q <- q +  geom_pointrange(
      aes(
        y = irr,
        ymin = q05,
        ymax = q95
      ),
      position = position_dodge(width = 0.3)
    )
    q <- q + theme(axis.text.x = element_text(angle = 90),axis.title.x=element_blank())
    q <- q + labs(y = "Incident risk ratio")
    q <- q + ggtitle("Incident risk ratio for ILI per season")
    q

As we can see these intervals are very similar.

The benefit of the simulated approach is that this process will be equally easy no matter the complexity of what we want to compute the IRR for. We do not have to take into account the variance-covariance matrix at any stage.