Editing Bayes' theorem (section)

===Drug testing===
Suppose, a particular test for whether someone has been using cannabis is 90% [[Sensitivity (tests)|sensitive]], meaning the [[true positive rate]] (TPR) = 0.90. Therefore, it leads to 90% true positive results (correct identification of drug use) for cannabis users.

The test is also 80% [[Specificity (tests)|specific]], meaning [[true negative rate]] (TNR) = 0.80. Therefore, the test correctly identifies 80% of non-use for non-users, but also generates 20% false positives, or [[false positive rate]] (FPR) = 0.20, for non-users.

Assuming 0.05 [[prevalence]], meaning 5% of people use cannabis, what is the [[probability]] that a random person who tests positive is really a cannabis user?

The [[Positive predictive value]] (PPV) of a test is the proportion of persons who are actually positive out of all those testing positive, and can be calculated from a sample as:
:PPV = True positive / Tested positive

If sensitivity, specificity, and prevalence are known, PPV can be calculated using Bayes' theorem. Let <math>P(\text{User}\vert \text{Positive}) </math> mean "the probability that someone is a cannabis user given that they test positive", which is what PPV means. We can write:

:<math>
\begin{align}
P(\text{User}\vert \text{Positive}) &= \frac{P(\text{Positive}\vert \text{User}) P(\text{User})}{P(\text{Positive})} \\
 &= \frac{P(\text{Positive}\vert\text{User}) P(\text{User})}{P(\text{Positive}\vert\text{User}) P(\text{User}) + P(\text{Positive}\vert\text{Non-user}) P(\text{Non-user})} \\[8pt] 
&= \frac{0.90 \times 0.05}{0.90 \times 0.05 + 0.20 \times 0.95}
= \frac{0.045}{0.045 + 0.19} \approx 19\%
\end{align}</math>

The denominator <math>
P(\text{Positive}) = P(\text{Positive}\vert\text{User}) P(\text{User}) + P(\text{Positive}\vert\text{Non-user}) P(\text{Non-user})
</math> is a direct application of the [[Law of Total Probability]]. In this case, it says that the probability that someone tests positive is the probability that a user tests positive times the probability of being a user, plus the probability that a non-user tests positive, times the probability of being a non-user. This is true because the classifications user and non-user form a [[partition of a set]], namely the set of people who take the drug test. This combined with the definition of [[conditional probability]] results in the above statement.

In other words, if someone tests positive, the probability that they are a cannabis user is only 19%—because in this group, only 5% of people are users, and most positives are false positives coming from the remaining 95%.

[[File:Bayes-rule3.png|thumb|right|Using a frequency box to show <math>P(\text{User}\vert \text{Positive}) </math> visually by comparison of shaded areas. Note how small the pink area of true positives is compared to the blue area of false positives.]]
If 1,000 people were tested:
* 950 are non-users and 190 of them give false positive (0.20 &times; 950) 
* 50 of them are users and 45 of them give true positive (0.90 &times; 50)

The 1,000 people thus have 235 positive tests, of which only 45 are genuine, about 19%.
====Sensitivity or specificity====
The importance of [[Specificity (tests)|specificity]] can be seen by showing that even if sensitivity is raised to 100% and specificity remains at 80%, the probability that someone who tests positive is a cannabis user rises only from 19% to 21%, but if the sensitivity is held at 90% and the specificity is increased to 95%, the probability rises to 49%.

{| class="wikitable" style="display:inline-table;"
! {{diagonal split header|Actual|Test}}
! style="width:5ex;"|Positive
! style="width:5ex;"|Negative
! rowspan="5" style="padding:0;"|
! Total
|-
! User
| style="text-align:right" | '''45'''
| style="text-align:right" | 5
| style="text-align:right" | 50
|-
! Non-user
| style="text-align:right" | 190
| style="text-align:right" | 760
| style="text-align:right" | 950
|-
| colspan="5" style="padding:0;"|
|-
! Total
| style="text-align:right" | '''235'''
| style="text-align:right" | 765
| style="text-align:right" | 1000
|-
| colspan="5" style="padding:0 0 1ex 0;border:none;background:transparent;"| 90% sensitive, 80% specific, PPV=45/235 &asymp; 19%
|}
{| class="wikitable" style="display:inline-table;"
! {{diagonal split header|Actual|Test}}
! style="width:5ex;"|Positive
! style="width:5ex;"|Negative
! rowspan="5" style="padding:0;"|
! Total
|-
! User
| style="text-align:right" | '''50'''
| style="text-align:right" | 0
| style="text-align:right" | 50
|-
! Non-user
| style="text-align:right" | 190
| style="text-align:right" | 760
| style="text-align:right" | 950
|-
| colspan="5" style="padding:0;"|
|-
! Total
| style="text-align:right" | '''240'''
| style="text-align:right" | 760
| style="text-align:right" | 1000
|-
| colspan="5" style="padding:0 0 1ex 0;border:none;background:transparent;"| 100% sensitive, 80% specific, PPV=50/240 &asymp; 21%
|}
{| class="wikitable" style="display:inline-table;"
! {{diagonal split header|Actual|Test}}
! style="width:5ex;"|Positive
! style="width:5ex;"|Negative
! rowspan="5" style="padding:0;"|
! Total
|-
! User
| style="text-align:right" | '''45'''
| style="text-align:right" | 5
| style="text-align:right" | 50
|-
! Non-user
| style="text-align:right" | 47
| style="text-align:right" | 903
| style="text-align:right" | 950
|-
| colspan="5" style="padding:0;"|
|-
! Total
| style="text-align:right" | '''92'''
| style="text-align:right" | 908
| style="text-align:right" | 1000
|-
| colspan="5" style="padding:0 0 1ex 0;border:none;background:transparent;"| 90% sensitive, 95% specific, PPV=45/92 &asymp; 49%
|}