Use \posclass and \negclass instead of + and \neg + in Bayes' theorem diagnostic test example

_subfiles/_sol-base-rate.qmd

@@ -1,16 +1,16 @@
 ::: {#sol-base-rate}
 
-Write $S$ for having the disease and $+$ for a positive test.
+Write $S$ for having the disease and $\posclass$ for a positive test.
 The three given numbers are
 
-$$\Pr(+ \mid S) = 0.99, \qquad \Pr(+ \mid \text{not } S) = 0.01, \qquad \Pr(S) = 0.001$$
+$$\Pr(\posclass \mid S) = 0.99, \qquad \Pr(\posclass \mid \text{not } S) = 0.01, \qquad \Pr(S) = 0.001$$
 
-@eq-bayes needs $\Pr(+)$, the overall chance of a positive result.
+@eq-bayes needs $\Pr(\posclass)$, the overall chance of a positive result.
 Assemble it with @thm-total-prob, splitting on whether the person is
 sick:
 
 $$\begin{aligned}
-\Pr(+) &= \Pr(+ \mid S)\Pr(S) + \Pr(+ \mid \text{not } S)\Pr(\text{not } S) \\
+\Pr(\posclass) &= \Pr(\posclass \mid S)\Pr(S) + \Pr(\posclass \mid \text{not } S)\Pr(\text{not } S) \\
 &= (0.99)(0.001) + (0.01)(0.999) \\
 &= 0.00099 + 0.00999 \\
 &= 0.01098
@@ -20,8 +20,8 @@ Then
 
 $$
 \begin{aligned}
-\Pr(S \mid +)
-&= \frac{\Pr(+ \mid S)\,\Pr(S)}{\Pr(+)} \\
+\Pr(S \mid \posclass)
+&= \frac{\Pr(\posclass \mid S)\,\Pr(S)}{\Pr(\posclass)} \\
 &= \frac{0.00099}{0.01098} \\
 &\approx 0.0902
 \end{aligned}

_subfiles/_thm-bayes.qmd

@@ -35,16 +35,16 @@ Suppose a disease test has 99% sensitivity and 99% specificity,
 and the prevalence of the disease in the population is 7%.
 
 Let $D$ be the event "person has the disease"
-and $+$ be the event "test is positive".
+and $\posclass$ be the event "test is positive".
 Then:
 
-- $\Pr(+ \mid D) = 0.99$ (sensitivity)
-- $\Pr(\neg + \mid \neg D) = 0.99$ (specificity),
+- $\Pr(\posclass \mid D) = 0.99$ (sensitivity)
+- $\Pr(\negclass \mid \neg D) = 0.99$ (specificity),
   so the false positive rate is
 
   $$
   \begin{aligned}
-  \Pr(+ \mid \neg D) &= 1 - 0.99 \\
+  \Pr(\posclass \mid \neg D) &= 1 - 0.99 \\
   &= 0.01
   \end{aligned}
   $$
@@ -56,10 +56,10 @@ By @thm-bayes, with the denominator expanded by the
 
 $$
 \ba
-\Pr(D \mid +)
-&= \frac{\Pr(+ \mid D) \cd \Pr(D)}{\Pr(+)}
+\Pr(D \mid \posclass)
+&= \frac{\Pr(\posclass \mid D) \cd \Pr(D)}{\Pr(\posclass)}
 && \text{(Bayes' theorem)} \\
-&= \frac{\Pr(+ \mid D) \cd \Pr(D)}{\Pr(+ \mid D) \cd \Pr(D) + \Pr(+ \mid \neg D) \cd \Pr(\neg D)}
+&= \frac{\Pr(\posclass \mid D) \cd \Pr(D)}{\Pr(\posclass \mid D) \cd \Pr(D) + \Pr(\posclass \mid \neg D) \cd \Pr(\neg D)}
 && \text{(law of total probability)} \\
 &= \frac{0.99 \cd 0.07}{0.99 \cd 0.07 + 0.01 \cd 0.93}
 && \text{(substitute the given values)} \\

inst/WORDLIST

@@ -45,6 +45,8 @@ glitchy
 homoskedasticity
 lintr
 navbar
+negclass
+posclass
 probabilitybook
 qwt
 renormalizing