knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
The quest for certainty is the biggest obstacle to becoming risk savvy. (Gerd Gigerenzer)^[Gigerenzer, G. (2014). Risk savvy: How to make good decisions. New York, NY: Penguin. (p. 21).]
A major challenge in mastering risk literacy is coping with inevitable uncertainty. Fortunately, uncertainty in the form of risk can be expressed in terms of probabilities and thus be measured and calculated  or "reckoned"  with (Gigerenzer, 2002). Nevertheless, probabilities are pretty big obstacles for everyone not trained as a statistician. As probabilities are notoriously difficult to communicate and understand, this vignette explains how probabilities can be defined and computed in terms of frequencies.
The problems addressed by riskyr
and the scientific discussion surrounding them is framed in terms of two representational formats: frequencies are distinguished from probabilities. (See the user guide for background and references.)
riskyr
reflects this division by distinguishing between the same two data types and hence provides objects that contain frequencies (specifically, a list called freq
) and objects that contain probabilities (a list called prob
). But before we explain their contents, it is important to realize that any such separation is an abstract and artificial one. It may make sense to distinguish frequencies from probabilities for conceptual and educational reasons, but both in theory and in reality both representations are intimately intertwined.
In the following, we will first consider frequencies and probabilities by themselves, but then show how both are related. As a sneak preview, the following network plot shows frequencies as its nodes and probabilities as the edges that link the nodes:
library("riskyr") # load the "riskyr" package plot_fnet(prev = .01, sens = .80, spec = NA, fart = .096, # 3 essential probabilities N = 1000, # 1 frequency area = "no", # all boxes have the same size p.lbl = "nam", # show probability names on edges title.lbl = "Mammography screening")
For our purposes, frequencies simply are numbers that can be counted  either 0 or positive integers.^[It seems plausible that the notion of a frequency is simpler than the notion of probability. Nevertheless, confusion is possible and typically causes serious scientific disputes. See Gigerenzer & Hoffrage, 1999, and Hoffrage et al., 2002, for different types of frequencies and the concept of "natural frequencies".]
The following 11 frequencies are distinguished by riskyr
and contained in freq
:
Nr. Variable  Definition 
:: :
 1.  N
 The number of cases (or individuals) in the population. 
 2.  cond.true
 The number of cases for which the condition is present (TRUE
). 
 3.  cond.false
 The number of cases for which the condition is absent (FALSE
). 
 4.  dec.pos
 The number of cases for which the decision is positive (TRUE
). 
 5.  dec.neg
 The number of cases for which the decision is negative (FALSE
). 
 6.  dec.cor
 The number of cases for which the decision is correct (correspondence of decision to condition). 
 7.  dec.err
 The number of cases for which the decision is erroneous (difference between decision and condition). 
 8.  hi
 The number of hits or true positives: condition present (TRUE
) & decision positive (TRUE
). 
 9.  mi
 The number of misses or false negatives: condition present (TRUE
) & decision negative (FALSE
). 
 10.  fa
 The number of false alarms or false positives: condition absent (FALSE
) & decision positive (TRUE
). 
 11.  cr
 The number of correct rejections or true negatives: condition absent (FALSE
) & decision negative (FALSE
). 
The frequencies contained in freq
can be viewed from two perspectives:
From the entire population to different parts or subsets (topdown):
Whereas N
specifies the size of the entire population, the other 10 frequencies denote the number of individuals or cases in some subset. For instance, the frequency dec.pos
denotes individuals for which the decision or diagnosis is positive. As this frequency is contained within the population, its numeric value must range from 0 to N
.
From four essential subgroups to various combinations of them (bottomup):
As the four frequencies hi
, mi
, fa
, and cr
are not further split into subgroups, we can think of them as atomic elements or four essential frequencies. All other frequencies in freq
are sums of various combinations of these four essential frequencies. This implies that the entire network of frequencies and probabilities (shown in the network diagram above) can be reconstructed from these four essential frequencies.
The following relationships hold among the 11 frequencies:
The population size N
can be split into several subgroups by classifying individuals by four different criteria:
Depending on the criterion used, the following relationships hold:
$$ \begin{aligned} \texttt{N} &= \texttt{cond.true} + \texttt{cond.false} & \textrm{(a)}\ &= \texttt{dec.pos} + \texttt{dec.neg} & \textrm{(b)}\ &= \texttt{dec.cor} + \texttt{dec.err} & \textrm{(c)}\ &= \texttt{hi} + \texttt{mi} + \texttt{fa} + \texttt{cr} & \textrm{(d)}\ \end{aligned} $$
Similarly, each of the subsets resulting from using the splits by condition, by decision, or by their correspondence, can also be expressed as a sum of two of the four essential frequencies. This results in three different ways of grouping the four essential frequencies:
(a) by condition (corresponding to the two columns of the confusion matrix):
$$ \begin{aligned} \texttt{N} \ &= \ \texttt{cond.true} & +\ \ \ \ \ &\texttt{cond.false} & \textrm{(a)} \ \ &= \ (\texttt{hi} + \texttt{mi}) & +\ \ \ \ \ &(\texttt{fa} + \texttt{cr}) \ \end{aligned} $$
(b) by decision (corresponding to the two rows of the confusion matrix):
$$ \begin{aligned} \texttt{N} \ &= \ \texttt{dec.pos} & +\ \ \ \ \ &\texttt{dec.neg} & \ \ \ \ \ \textrm{(b)} \ \ &= \ (\texttt{hi} + \texttt{fa}) & +\ \ \ \ \ &(\texttt{mi} + \texttt{cr}) \ \end{aligned} $$
(c) by the correspondence of decisions to conditions (corresponding to the two diagonals of the confusion matrix):
$$ \begin{aligned} \texttt{N} \ &= \ \texttt{dec.cor} & +\ \ \ \ \ &\texttt{dec.err} & \ \ \ \ \textrm{(c)} \ \ &= \ (\texttt{hi} + \texttt{cr}) & +\ \ \ \ \ &(\texttt{mi} + \texttt{fa}) \ \end{aligned} $$
It may be tempting to refer to instances of dec.cor
and dec.err
as "true decisions" and "false decisions".
However, this would invite conceptual confusion, as "true decisions" actually include cond.false
cases (cr
) and "false decisions" actually include cond.true
cases (mi
).
The notions of probability is as elusive as ubiquitous (see Hájek, 2012, for a solid exposition of its different concepts and interpretations). For our present purposes, probabilities are simply numbers between 0 and 1. These numbers are defined to reflect particular quantities and can be expressed as percentages or fractions of other numbers (frequencies or probabilities).
The following 10 probabilities are distinguished by riskyr
and contained in prob
:
Nr. Variable  Name  Definition 
: : : :
 1.  prev
 prevalence  The probability of the condition being TRUE
. 
 2.  sens
 sensitivity  The conditional probability of a positive decision provided that the condition is TRUE
. 
 3.  mirt
 miss rate  The conditional probability of a negative decision provided that the condition is TRUE
. 
 4.  spec
 specificity  The conditional probability of a negative decision provided that the condition is FALSE
. 
 5.  fart
 false alarm rate  The conditional probability of a positive decision provided that the condition is FALSE
. 
 6.  ppod
 proportion of positive decisions  The proportion (baseline probability or rate) of the decision being positive (but not necessarily TRUE
). 
 7.  PPV
 positive predictive value  The conditional probability of the condition being TRUE
provided that the decision is positive. 
 8.  FDR
 false detection rate  The conditional probability of the condition being FALSE
provided that the decision is positive. 
 9.  NPV
 negative predictive value  The conditional probability of the condition being FALSE
provided that the decision is negative. 
 10.  FOR
 false omission rate  The conditional probability of the condition being TRUE
provided that the decision is negative. 
Note that there are only two nonconditional probabilities:
prev
(1.) only depends on features of the condition.ppod
(6.) only depends on features of the decision.The other eight probabilities are conditional probabilities:
prev
and features of the decision.ppod
and features of the condition.The following relationships hold among the conditional probabilities:
sens
and miss rate mirt
are complements:$$
\texttt{sens} = 1  \texttt{mirt}
$$
 The specificity spec
and false alarm rate fart
are complements:
$$
\texttt{spec} = 1  \texttt{fart}
$$
 The positive predictive value PPV
and false detection rate FDR
are complements:
$$
\texttt{PPV} = 1  \texttt{FDR}
$$
 The negative predictive value NPV
and false omission rate FOR
are complements:
$$ \texttt{NPV} = 1  \texttt{FOR} $$
It is possible to adapt Bayes' formula to define PPV
and NPV
in terms of prev
, sens
, and spec
:
$$ \texttt{PPV} = \frac{\texttt{prev} \cdot \texttt{sens}}{\texttt{prev} \cdot \texttt{sens} + (1  \texttt{prev}) \cdot (1  \texttt{sens})}\ \ \ \texttt{NPV} = \frac{(1  \texttt{prev}) \cdot \texttt{spec}}{\texttt{prev} \cdot (1  \texttt{sens}) + (1  \texttt{prev}) \cdot \texttt{spec}} $$
Although this is how the functions comp_PPV
and comp_NPV
compute the desired conditional probability, it is difficult to remember and think in these terms. Instead, we recommend thinking about and defining all conditional probabilities in terms of frequencies (see below).
The easiest way to think about, define, and compute the probabilities (contained in prob
) are in terms of frequencies (contained in freq
):
 Nr. Variable  Name  Definition  as Frequencies 
: : : : : 
 1.  prev
 prevalence  The probability of the condition being TRUE
.  prev
= cond.true
/N

 2.  sens
 sensitivity  The conditional probability of a positive decision provided that the condition is TRUE
.  sens
= hi
/cond.true

 3.  mirt
 miss rate  The conditional probability of a negative decision provided that the condition is TRUE
.  mirt
= mi
/cond.true

 4.  spec
 specificity  The conditional probability of a negative decision provided that the condition is FALSE
.  spec
= cr
/cond.false

 5.  fart
 false alarm rate  The conditional probability of a positive decision provided that the condition is FALSE
.  fart
= fa
/cond.false

 6.  ppod
 proportion of positive decisions  The proportion (baseline probability or rate) of the decision being positive (but not necessarily TRUE
).  ppod
= dec.pos
/N

 7.  PPV
 positive predictive value  The conditional probability of the condition being TRUE
provided that the decision is positive.  PPV
= hi
/dec.pos

 8.  FDR
 false detection rate  The conditional probability of the condition being FALSE
provided that the decision is positive.  FDR
= fa
/dec.pos

 9.  NPV
 negative predictive value  The conditional probability of the condition being FALSE
provided that the decision is negative.  NPV
= cr
/dec.neg

 10.  FOR
 false omission rate  The conditional probability of the condition being TRUE
provided that the decision is negative.  FOR
= mi
/dec.neg

Note that the ratios of frequencies are straightforward consequences of the probabilities' definitions:
The two unconditional probabilities (1. and 6.) are proportions of the entire population:
prev
= cond.true
/N
ppod
= dec.pos
/N
All conditional probabilities (2.5. and 7.10.) can be computed as a proportion of the reference group on which they are conditional. More specifically, if we schematically read each definition as "The conditional probability of $X$ provided that $Y$", then the ratio of the corresponding frequencies is X & Y
/Y
. More explicitly,
the ratio's numerator is the frequency of the joint occurrence (i.e., both X & Y
) being the case;
Y
) being the case.The following network diagram is based on the following inputs:
prev = .50
);sens = .80
);spec = .60
);N = 10
);and illustrates the relationship between frequencies and probabilities:
plot_fnet(prev = .50, sens = .80, spec = .60, # 3 essential probabilities N = 10, # 1 frequency area = "no", # all boxes have the same size p.lbl = "num", # show numeric probability values on edges title.lbl = "A simple example")
Verify that the probabilities (shown as numeric values on the edges) match the ratios of the corresponding frequencies (shown in the boxes). What are the names of these probabilities?
What is the frequency of dec.cor
and dec.err
cases? Where do these cases appear in the diagram?
Gigerenzer, G. (2002). Reckoning with risk: Learning to live with uncertainty. London, UK: Penguin.
Gigerenzer, G. (2014). Risk savvy: How to make good decisions. New York, NY: Penguin.
Gigerenzer, G., & Hoffrage, U. (1999). Overcoming difficulties in Bayesian reasoning: A reply to Lewis and Keren (1999) and Mellers and McGraw (1999). Psychological Review, 106, 425430.
Hájek, A (2012) Interpretations of Probability. In Edward N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. URL: https://plato.stanford.edu/entries/probabilityinterpret/ 2012 Archive
Hoffrage, U., Gigerenzer, G., Krauss, S., & Martignon, L. (2002). Representation facilitates reasoning: What natural frequencies are and what they are not. Cognition, 84, 343352.
We appreciate your feedback, comments, or questions.
Please report any riskyr
related issues at https://github.com/hneth/riskyr/issues.
For general inquiries, please email us at contact.riskyr@gmail.com.
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 
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