# Adopting Functional Perspectives

#### Hansjörg Neth

#### 2018 02 10

> The greatest value of a picture is when it forces us to notice what we never expected to see.
> (John W. Tukey)1

The `riskyr` data structures essentially describe networks of dependencies. This is best illustrated by the network diagram (see examples of `plot_fnet` in the user guide and data formats). However, sometimes it is instructive to view all possible values of some parameter as a function of others. A functional perspective illustrates how the value of some parameter (or its values) changes as a function of other parameters (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 2 functions for viewing parameters as a function of other parameters (and their values).

## Curves as a function of prevalence

The function `plot_curve` 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). 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:

    plot_curve(prev = .01, sens = .80, spec = (1 - .096),
               what = c("prev", "PPV", "NPV"),
               title_lbl = "Mammography screening", cex.lbl = .8)

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.

The curves illustrate that values of `PPV` and `NPV` crucially depend on the value of prevalence `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 `prev`.

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):

    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 proportion of positive decisions `ppod` and the overall accuracy `acc` also vary as a function of prevalence `prev`:

    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)

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%.

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

The function `plot_plane` draws a plane for a selected parameter as a function of sensitivity and specificity (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):

    plot_plane(prev = high.prev, sens = .80, spec = (1 - .096), what = "PPV",
               title_lbl = "A. Mammography (BRCA1)", cex.lbl = .8)

Plane showing the positive predictive value (PPV) as a function of sensitivity and specificity for a given prevalence.

Related plots (showing different probabilities) include:

    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`/`NPV`, but also `ppod` and `acc`) as a function of other probabilities (e.g., `prev`, `sens`, `spec` or `fart`) often provides unexpected relationships and 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
| Development version | https://github.com/hneth/riskyr
B. `riskyrApp` (R Shiny code): | Online version | http://riskyr.org
| Development version | https://github.com/hneth/riskyrApp
C. Online documentation: | Release version | https://hneth.github.io/riskyr
| Development version | https://hneth.github.io/riskyr/dev

## Contact

We appreciate your feedback, comments, or questions.

- Please report any `riskyr`-related issues at https://github.com/hneth/riskyr/issues.

- Email us at contact.riskyr@gmail.com if you want to modify or share this software.

## All riskyr vignettes

Nr. | Vignette | Content
--- | --- | ---
A. | User guide | Motivation and general instructions
B. | Data formats | Data formats: Frequencies and probabilities
C. | Confusion matrix | Confusion matrix and accuracy metrics
D. | Functional perspectives | Adopting functional perspectives
E. | Quick start primer | Quick start primer

1. Tukey, J.W. (1977), Exploratory data analysis. Reading, MA: Addison-Wesley. (p. vi).↩