---
title: "APC Analysis with APCtools"
author: "Alexander Bauer, Maximilian Weigert"
date: "`r format(Sys.time(), '%d.%B %Y')`"
output:
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    toc: yes
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    %\VignetteIndexEntry{APC Analysis with APCtools}
    %\usepackage[utf8]{inputenc}
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---

```{r, echo=FALSE}
# global settings
knitr::opts_chunk$set(fig.width = 10)
```

This document gives an overview of the functionality provided by the R package
`APCtools`.

Age-Period-Cohort (APC) analysis is used to disentangle observed trends (e.g. of social, economic, medical or epidemiological data) to enable conclusions about the developments over three temporal dimensions:

* Age, representing the developments associated with chronological age over someones life cycle.
* Period, representing the developments over calendar time which affect all age groups simultaneously.
* Cohort, representing the developments observed over different birth cohorts and generations.

The critical challenge in APC analysis is that these main components are linearly dependent:
$$
cohort = period - age
$$

Accordingly, flexible methods and visualization techniques are needed to properly
disentagle observed temporal association structures.
The `APCtools` package comprises different methods that tackle this problem and
aims to cover all steps of an APC analysis.
This includes state-of-the-art descriptive visualizations as well as visualization
and summary functions based on the estimation of a generalized additive regression model (GAM).
The main functionalities of the package are highlighted in the following.

For details on the statistical methodology see
[Weigert et al. (2021)](https://doi.org/10.1177/1354816620987198) or
our corresponding [research poster](https://www.researchgate.net/publication/353852226_Visualization_techniques_for_semiparametric_APC_analysis_Using_Generalized_Additive_Models_to_examine_touristic_travel_distances).
The *hexamaps* (hexagonally binned heatmaps) are outlined in
[Jalal & Burke (2020)](https://doi.org/10.1097/EDE.0000000000001236).


## Load relevant packages

Before we start, let's load the relevant packages for the following analyses.

```{r packages, message = FALSE}
library(APCtools)
library(dplyr)    # general data handling
library(mgcv)     # estimation of generalized additive regression models (GAMs)
library(ggplot2)  # data visualization
library(ggpubr)   # arranging multiple ggplots in a grid with ggarrange()

# set the global theme of all plots
theme_set(theme_minimal())
```

## Example data

APC analyses require long-term panel or repeated cross-sectional data.
The package includes two exemplary datasets on the travel behavior of German
tourists (dataset `travel`) and the number of unintentional drug overdose deaths
in the United States (`drug_deaths`). See the respective help pages `?travel`
and `?drug_deaths` for details.

In the following, we will use the `travel` dataset to investigate if travel
distances of the main trip of German travelers mainly
change over the life cycle of a person (age effect), macro-level developments
like decreasing air travel prices in the last decades (period effect) or the
generational membership
of a person, which is shaped by similar socialization and historical experiences
(cohort effect).

```{r data preparation}
data(travel)
```

## Descriptive visualizations

Different functions are available for descriptively visualizing observed structures.
This includes plots for the marginal distribution of some variable of interest,
1D plots for the development of some variable over age, period or cohort,
as well as density matrices that visualize the development over all temporal
dimensions.


### Marginal distribution of one variable

The marginal distribution of a variable can be visualized using `plot_density`.
Metric variables can be plotted using a density plot or a boxplot, while
categorical variables can be plotted using a bar chart.

```{r, message=FALSE, warning=FALSE, fig.height=2.2}
gg1 <- plot_density(dat = travel, y_var = "mainTrip_distance", log_scale = TRUE)
gg2 <- plot_density(dat = travel, y_var = "mainTrip_distance", log_scale = TRUE,
                    plot_type = "boxplot")
gg3 <- plot_density(dat = travel, y_var = "household_size")

ggpubr::ggarrange(gg1, gg2, gg3, nrow = 1)
```

### 1D: One variable against age, period or cohort

Plotting the distribution of a variable against age, period or cohort
is possible with function `plot_variable`.
The distribution of metric and categorical variables is visualized using
boxplots or line charts (see argument `plot_type`)
and bar charts, respectively. The latter by default show relative frequencies,
but can be changed to show absolute numbers by specifying argument
`geomBar_position = "stack"`.

```{r, message = FALSE, fig.height=4}
plot_variable(dat = travel, y_var = "mainTrip_distance",
              apc_dimension = "period", plot_type = "line", ylim = c(0,1000))

plot_variable(dat = travel, y_var = "household_size", apc_dimension = "period")
```

### 2D: Density matrices

To include all temporal dimensions in one plot, `APCtools` contains function
`plot_densityMatrix`. In Weigert et al. (2021), this plot type was referred to
as *ridgeline matrix* when plotting multiple density plots for a metric variable.
The basic principle of a density matrix is to (i) visualize two of the temporal
dimensions on the x- and y-axis (specified using the argument `dimensions`), s.t.
the third temporal dimension is represented on the diagonals of the matrix, and
(ii) to categorize the respective variables on the x- and y-axis in meaningful
groups. The function then creates a grid, where each cell contains the distribution
of the selected `y_var` variable in the respective category.

By default, age and period are depicted on the x- and y-axis, respectively,
and cohort on the diagonals. The categorization is defined by specifying two of
the arguments `age_groups`, `period_groups` and `cohort_groups`.

```{r, fig.width=8, message=FALSE}
age_groups    <- list(c(80,89),c(70,79),c(60,69),c(50,59),
                      c(40,49),c(30,39),c(20,29))
period_groups <- list(c(1970,1979),c(1980,1989),c(1990,1999),
                      c(2000,2009),c(2010,2019))

plot_densityMatrix(dat              = travel,
                   y_var            = "mainTrip_distance",
                   age_groups       = age_groups,
                   period_groups    = period_groups,
                   log_scale        = TRUE)
```

To highlight the effect of the variable depicted on the diagonal (here: cohort),
different diagonals can be highlighted using argument `highlight_diagonals`.

```{r, fig.height=6.5, message=FALSE}
plot_densityMatrix(dat                 = travel,
                   y_var               = "mainTrip_distance",
                   age_groups          = age_groups,
                   period_groups       = period_groups,
                   highlight_diagonals = list("born 1950 - 1959" = 8,
                                              "born 1970 - 1979" = 10),
                   log_scale           = TRUE)
```

For metric variables it is further possible to use a color scale for
the density function (arguments `y_var_cat_breaks` and `y_var_cat_labels`)
to highlight local developments over the APC dimensions.

```{r, fig.height=6.5, message=FALSE}
dist_cat_breaks <- c(1,500,1000,2000,6000,100000)
dist_cat_labels <- c("< 500 km","500 - 1,000 km", "1,000 - 2,000 km", "2,000 - 6,000 km", "> 6,000 km")

plot_densityMatrix(dat              = travel,
                   y_var            = "mainTrip_distance",
                   age_groups       = age_groups,
                   period_groups    = period_groups,
                   log_scale        = TRUE,
                   y_var_cat_breaks = dist_cat_breaks,
                   y_var_cat_labels = dist_cat_labels,
                   highlight_diagonals = list("born 1950 - 1959" = 8,
                                              "born 1970 - 1979" = 10),
                   legend_title     = "Distance category")
```

Alternatively to plotting a conditional density in each matrix cell, metric
variables can also be visualized using boxplots by specifying `plot_type = "boxplot"`.
Categorical variables are automatically plotted using conditional bar plots.

```{r, fig.height=6.5, message=FALSE}
plot_densityMatrix(dat                 = travel,
                   y_var               = "household_size",
                   age_groups          = age_groups,
                   period_groups       = period_groups,
                   highlight_diagonals = list("born 1950 - 1959" = 8,
                                              "born 1970 - 1979" = 10))
```


### 2D: Heatmaps

Similar to density matrices, heatmaps simultaneously visualize all three APC
dimensions. Instead of individual distributions in separated categories, however,
a color scale is used to depict the mean value in a specific region.
When setting `bin_heatmap = FALSE`, the function `plot_APCheatmap` shows
the average observed values of a metric variable for each observed combination
of age and period values (if `dimensions = c("period","age")`).
The borders of groups along the diagonal can be marked using argument `markLines_list`.

```{r, fig.height=6.5, fig.width=8}
plot_APCheatmap(dat            = travel,
                y_var          = "mainTrip_distance",
                y_var_logScale = TRUE,
                bin_heatmap    = FALSE,
                markLines_list = list(cohort = c(1900,1920,1939,1946,
                                                 1966,1982,1994)))
```

By default, the heatmap is binned in five year blocks along the
x- and y-axis to focus on larger-scale developments. Note that the grid used
for binning can flexibly be defined with the argument `bin_heatmapGrid_list`.

```{r, fig.height=6.5, fig.width=8}
plot_APCheatmap(dat            = travel,
                y_var          = "mainTrip_distance",
                y_var_logScale = TRUE,
                markLines_list = list(cohort = c(1900,1920,1939,1946,
                                                 1966,1982,1994)))
```


### 2D: Hexamaps

As an alternative to classical heatmaps observed developments can also be
visualized using *hexamaps*, i.e. hexagonally binned heatmaps with rotated axes
to give all three APC dimensions similar visual weight. This resolves the
central problem of classical heatmaps that developments along the diagonal
dimension are visually much harder to grasp than developments along the x- or
y-axis. Function `plot_APChexamap` can be called similarly to `plot_APCheatmap`.
Note, however, that hexamaps are currently implemented using the base R plot
functions and not based on `ggplot2`.

```{r, fig.height=6.5}
plot_APChexamap(dat            = travel,
                y_var          = "mainTrip_distance",
                y_var_logScale = TRUE)
```


## Model-based analyses

After getting a descriptive overview of the data with the functions outlined
above, an established approach to analyze APC structures in more detail is to
use the semiparametric approach offered by generalized additive regression
models (GAMs). This regression-based approach offers several benefits:

* Compared to alternative regression-based APC approaches it circumvents the
issue of linear dependency of the APC dimensions by estimating a flexible
two-dimensional tensor product surface.
* The model separates the underlying smooth effects of age, period and cohort
from random variation in the data and allows for the subsequent visualization of
marginal age, period and cohort effects.
* While the above heatmaps and hexamaps can only be used to visualize the observed
distribution of *metric* variables, the regression-based approach also allows for estimating
and plotting the mean structure in settings with a categorical response variable,
for example by estimating a logistic regression model.
* The regression-based approach allows for accounting for further control
variables in the model. E.g., when analyzing how travel distances developed over
the last decades, each persons income can be included as a covariate in the model
estimation to account for income differences between travelers in different decades.

For estimating a regression model we utilize the function `gam` implemented in
package `mgcv`. For further methodological details see Weigert et al. (2021).

Let's fit two GAM models, one without and one with further control variables.
See `?travel` for an explanation of the used variables.
The tensor product surface is estimated with two marginal P-spline bases with
eight basis functions each.

```{r}
# GAM without covariates
model_pure <- gam(mainTrip_distance ~ te(age, period, bs = "ps", k = c(8,8)),
                  data = travel)

# GAM including covariates
model_cov  <- gam(mainTrip_distance ~ te(age, period, bs = "ps", k = c(8,8)) +
                    residence_region + household_size + s(household_income),
                  data = travel)

# create a named list of the two models, useful for some functions
model_list <- list("pure model"      = model_pure,
                   "covariate model" = model_cov)
```


### Heatmaps

Similary to the descriptive visualization outlined above, the tensor product
surface estimated with the regression model can be created with
`plot_APCheatmap`, by passing the model object as argument `model` to the
function.

```{r, fig.height=3, fig.width=8}
plot_APCheatmap(dat   = travel,
                model = model_pure)
```

### Hexamaps

The same applies to the hexamaps. By calling `plot_APChexamap` and specifying
the `model` argument, a hexamap of the estimated mean structure is created.

```{r, fig.height=6.5}
plot_APChexamap(dat   = travel,
                model = model_pure)
```

### Marginal APC effects

Based on the tensor product surface estimated through the regression model,
marginal effects can be extracted by taking the average of all values on the
surface along one dimension. Marginal effects can be visualized using
`plot_marginalAPCeffects`.

```{r}
plot_marginalAPCeffects(model    = model_pure,
                        dat      = travel,
                        variable = "age")
```

The marginal effects of multiple models can also be visualized in one plot
using `plot_jointMarginalAPCeffects`.

```{r}
plot_jointMarginalAPCeffects(model_list  = model_list,
                             dat         = travel,
                             vlines_list = list("cohort" = c(1900,1920,1939,
                                                             1946,1966,1982,
                                                             1994)))
```

### Partial APC plots

partial APC plots can be used to get deeper insights into a specific
temporal effect and the interrelations between the temporal dimensions.
These plots show the estimated effect for one temporal dimension dependent on the remaining two
dimensions. The mean marginal effect is marked as a bold blue line,
and one grayscale line is added for each partial effect,
that is, for the estimated differences when just focusing on a specific age, period or cohort group.

In the following example, the period effect is visualized, dependent on different
age groups and cohorts.
See [Weigert et al. (2021)](https://doi.org/10.1177/1354816620987198) for details
on the interpretation of partial APC plots.

```{r}
plot_partialAPCeffects(model    = model_pure,
                       dat      = travel,
                       variable = "period")
```


### APC summary table

The function `create_APCsummary` can be used to get a compact overview of the
magnitude of the individual APC effects. It extracts information about the
minimum and maximum values of the marginal APC effects and computes the overall
size of each effect.

```{r}
create_APCsummary(model_list = model_list,
                  dat        = travel)
```

### Covariate effect plots

Control variables in the regression models can be visualized using the functions
`plot_linearEffects` and `plot_1Dsmooth` which create an effect plot of all
linear effects in a model and a line plot of a one-dimensional smooth effect,
respectively.

```{r}
APCtools::plot_linearEffects(model_cov)
APCtools::plot_1Dsmooth(model_cov, select = 2)
```

### Model summary tables

The function `create_modelSummary` can be used to create publication-ready
tables of the full model results. It can either be applied to a list of multiple
models or a list of a single model, and returns a list with both a table for
all linear coefficients and all nonlinear estimates.

```{r}
summary_list <- create_modelSummary(model_list)
summary_list[[1]]
summary_list[[2]]
```

