---
title: "Adopting Functional Perspectives"
author: "Hansjörg Neth, SPDS, uni.kn"
date: "2021 03 20"
output: 
  rmarkdown::html_vignette: 
    fig_caption: yes
vignette: > 
  %\VignetteIndexEntry{Functional Perspectives}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

# URLs:
url_riskyr_org <- "http://134.34.54.61:3838/spds/riskyr/" # Note: "http://riskyr.org/" redirects there.

# Load pkg:
library("riskyr")
```

<!-- Motto: -->

> The greatest value of a picture is when it forces us to notice what we never expected to see.  
> (John W. Tukey)^[Tukey, J.W. (1977), _Exploratory data analysis_. Reading, MA: Addison-Wesley. (p. vi).]


<!-- riskyr logo: -->

<a href="https://github.com/hneth/riskyr/">
<img src = "../inst/pix/riskyr_cube.png" alt = "riskyr" style = "width: 125px; float: right; border:20;"/>
</a>


<!-- Introduction: --> 

The key **riskyr** data structure essentially describes a network of dependencies. 
This is best illustrated by the network diagram (see examples of `plot_fnet()` in the [user guide](A_user_guide.html) and [data formats](B_data_formats.html)). 
However, sometimes it is instructive to view all possible values of a parameter as a function of some other variable. 
A functional perspective illustrates how the value of some variable (or its values) changes as a function of another (and their values). 


## Functions

The basic format of a function is $y = f(x)$, which illustrates how values of\ $y$ 
depend on values of\ $x$ given some function\ $f$. **riskyr** provides two functions for 
viewing parameters as a function of other parameters (and their values). 

<!-- 
$$
\begin{aligned}
y \ &= \ f(x)  &  \\
y \ &= \ f(\texttt{prev} \textrm{, from 0 to 1})  \textrm{ with } y \in \{\texttt{PPV}, \texttt{NPV}, \texttt{ppod}, \texttt{acc}\} & (1)  \\
y \ &= \ f(\texttt{sens} \times\ \texttt{spec} \textrm{, both from 0 to 1, for given value of } \texttt{prev}) \textrm{ with } y \in \{\texttt{PPV}, \texttt{NPV}, \texttt{ppod}, \texttt{acc}\} & (2) \\ 
\end{aligned}
$$
-->

## Curves as a function of prevalence

The `plot_curve()` function draws the curves (or lines) of selected parameters as a function of the prevalence (with `prev` ranging from\ 0 to\ 1) 
for a given decision process or diagnostic test (i.e., given values of\ `sens` and\ `spec`): 

$$y \ = \ f(\texttt{prev} \textrm{, from 0 to 1})  \textrm{ with } y \in \{\texttt{PPV}, \texttt{NPV}, \texttt{ppod}, \texttt{acc}\}  \ \ \ \ \ \   (1)$$

As an example, reconsider our original scenario (on mammography screening, see [user guide](A_user_guide.html)). Earlier, we computed a positive predictive value (PPV) of\ 7.8%. 
But rather than just computing a single value, we could ask: How do values of PPV develop as a function of prevalence? The `plot_curve()` function illustrates this relationship: 

```{r plot-curve-1, fig.align = "center", fig.width = 6, fig.height = 4.5, fig.show = 'hold', fig.cap = "Showing PPV and NPV as a function of prevalence (for a prevalance of 1% and given values of sensitivity and specificity) in the original mammography screening scenario."}
plot_curve(prev = .01, sens = .80, spec = (1 - .096), 
           what = c("prev", "PPV", "NPV"), 
           title_lbl = "Mammography screening", cex.lbl = .8)
```

The curves illustrate that values of `PPV` and `NPV` crucially depend on the prevalence value\ `prev` in the current population. In fact, they actually vary across their entire range (i.e., from\ 0 to\ 1), rendering any communication of their value utterly meaningless without specifying the current population's prevalence value. 

The dependency of\ `PPV` and\ `NPV` on\ `prev` can be illustrated by assuming a higher prevalence rate. For instance, if we knew that some woman was genetically tested and known to exhibit the notorious BRCA1 mutation, the prevalence value of her corresponding population (given a positive mammography result in a routine screening) is increased to about 60% (graph not shown here to save space, but try running the following code for yourself): 

```{r plot-curve-2, eval = FALSE, fig.align = "center", fig.width = 6, fig.height = 4.5, fig.show = 'hold', fig.cap = "Showing PPV and NPV as a function of prevalence (for an increased prevalence of 60% and given values of sensitivity and specificity)."}
high.prev <- .60   # assume increased prevalence due to BRCA1 mutation

plot_curve(prev = high.prev, sens = .80, spec = (1 - .096), 
           what = c("prev", "PPV", "NPV"), 
           title_lbl = "Mammography screening (BRCA1 mutation)", cex.lbl = .80)
```

This shows that ---\ given an increased prevalence value `prev` of 60%\ --- the positive predictive value\ `PPV` of a positive test result increases from\ 7.8% (in the standard population) to around\ 93% (given the BRCA1 mutation). 

In addition, the actual values of population and test parameters are often unclear. The `plot_curve()` function reflects this by providing an uncertainty parameter\ `uc` that is expressed as a percentage of the specified value. For instance, the following assumes that our parameter values may deviate up to\ 5% from the specified values and marks the corresponding ranges of uncertainty as shaded areas around the curves that assume exact parameter values. 

Both the notions of expressing probabilities as a function of prevalence and of uncertainty ranges for imprecise parameter estimates can be extended to other probabilities. The following curves show the full set of curves currently drawn by\ `plot_curve()`. In addition to the predictive values\ `PPV` and\ `NPV`, we see that the bias or proportion of positive decisions\ `ppod` and the overall accuracy\ `acc` also vary as a function of the prevalence\ `prev`:

```{r plot-curve-3, fig.align = "center", fig.width = 6, fig.height = 4.5, fig.show = 'hold', fig.cap = "Curves that show PPV/NPV, ppod, and acc as a function of an prevalence (for given values of sensitivity and specificity) when assuming an increased prevalence of 60% and an uncertainty range of 5%."}
high.prev <- .60   # assume increased prevalence due to BRCA1 mutation

plot_curve(prev = high.prev, sens = .80, spec = (1 - .096), 
           what = c("prev", "PPV", "NPV", "ppod", "acc"), 
           title_lbl = "Mammography screening (BRCA1 mutation)", uc = .05, cex.lbl = .80)
```


## Planes as a function of sensitivity and specificity (given a prevalence)

The `plot_plane()` function draws a plane for a selected parameter as a function of sensitivity and specificity values (with\ `sens` and\ `spec` both ranging from\ 0 to\ 1) for a given prevalence\ `prev`: 

$$y \ = \ f(\texttt{sens} \times\ \texttt{spec} \textrm{, both from 0 to 1, for given value of } \texttt{prev}) \textrm{ with } y \in \{\texttt{PPV}, \texttt{NPV}, \texttt{ppod}, \texttt{acc}\}  \ \ \ \ \ \ \ (2)$$

Some examples (not shown here, but please try evaluating the following function calls):

```{r plot-plane-PPV, eval = TRUE, fig.align = "center", fig.width = 5.0, fig.height = 3.5, fig.show = 'hold', fig.cap = "Plane showing the positive predictive value (PPV) as a function of sensitivity and specificity for a given prevalence."}
plot_plane(prev = high.prev, sens = .80, spec = (1 - .096), what = "PPV",  
           title_lbl = "A. Mammography (BRCA1)", cex.lbl = .8)
```

Related plots (showing different probabilities) include:

```{r plot-planes-else, eval = FALSE, fig.width = 5, fig.height = 4, fig.show = 'asis', fig.cap = "Planes showing NPV, the proportion of positive predictions (ppod), and overall accuracy (acc), as a function of sensitivity and specificity for a given prevalence."}
plot_plane(prev = high.prev, sens = .80, spec = (1 - .096), what = "NPV",  
           title_lbl = "B. Mammography (BRCA1)", cex.lbl = .8)
plot_plane(prev = high.prev, sens = .80, spec = (1 - .096), what = "ppod", what_col = "firebrick", 
           title_lbl = "C. Mammography (BRCA1)", phi = 45, cex.lbl = .8)
plot_plane(prev = high.prev, sens = .80, spec = (1 - .096), what = "acc",  what_col = "forestgreen", 
           title_lbl = "D. Mammography (BRCA1)", cex.lbl = .8)
```

Overall, viewing conditional probabilities (like\ `PPV` or `NPV`, but also\ `ppod` or\ `acc`) as a function of other probabilities (e.g., `prev`, `sens`, `spec` or `fart`) often reveals unexpected relationships and can enable new insights. 


### Resources

The following resources and versions are currently available:

Type:                    | Version:           | URL:                           |        
:------------------------|:-------------------|:-------------------------------|
A. **riskyr** (R package): | [Release version](https://CRAN.R-project.org/package=riskyr)  | <https://CRAN.R-project.org/package=riskyr> |
    &nbsp;               | [Development version](https://github.com/hneth/riskyr/)         | <https://github.com/hneth/riskyr/> | 
B. **riskyrApp** (R Shiny code): | [Online version](`r url_riskyr_org`)                    | [http://riskyr.org/](`r url_riskyr_org`) | 
    &nbsp;               | [Development version](https://github.com/hneth/riskyrApp/)      | <https://github.com/hneth/riskyrApp/> | 
C. Online documentation: | [Release version](https://hneth.github.io/riskyr/)              | <https://hneth.github.io/riskyr/> | 
    &nbsp;               | [Development version](https://hneth.github.io/riskyr/dev/)      | <https://hneth.github.io/riskyr/dev/> | 


## Contact

<!-- uni.kn logo and link to SPDS: -->  
<a href="https://www.spds.uni-konstanz.de/">
<img src = "../inst/pix/uniKn_logo.png" alt = "spds.uni.kn" style = "width: 280px; float: right; border:15;"/> 
</a>

We appreciate your feedback, comments, or questions. 

- Please report any **riskyr**-related issues at <https://github.com/hneth/riskyr/issues/>.  

- Contact us at <contact.riskyr@gmail.com> with any comments, questions, or suggestions.  


## All riskyr vignettes

<!-- riskyr logo: -->
<a href="https://github.com/hneth/riskyr/">
<img src = "../inst/pix/riskyr_cube.png" alt = "riskyr" style = "width: 125px; float: right; border:20;"/>
</a>

<!-- Index of vignettes: -->

| Nr.  | Vignette | Content    |        
| ---: |:---------|:-----------|
| A. | [User guide](A_user_guide.html) | Motivation and general instructions | 
| B. | [Data formats](B_data_formats.html) | Data formats: Frequencies and probabilities | 
| C. | [Confusion matrix](C_confusion_matrix.html) | Confusion matrix and accuracy metrics |
| D. | [Functional perspectives](D_functional_perspectives.html) | Adopting functional perspectives |
| E. | [Quick start primer](E_riskyr_primer.html) | Quick start primer |

<!-- eof. -->
