# Chapter 1 The role of biostatistics in clinical research

Under development

From clinical inquiry book - not sure where this may apply in this text; maybe as a review in an appendix since they should know this (it is entry level material for most clinicians)

## 1.1 Preliminaries

### 1.1.1 Contingency Table

\(D\) | \(\neg D\) | |
---|---|---|

\(S\) | True Positive (TP) (a) | False Positive (FP) (b) |

\(\neg S\) | False Negative (FN) (c) | True Negative (TN) (d) |

### 1.1.2 Sensitivity / Specificity

Sensitivity: \(P(S|D)=\dfrac{a}{a+c}\) (Rate of True Positives)

Specificity: \(P(\neg S|\neg D)=\dfrac{d}{b+d}\) (Rate of True Negatives)

1 - Sensitivity: \(P(\neg S|D)=\dfrac{c}{a+c}\) (Rate of False Negatives)

1 - Specificity: \(P(S|\neg D)=\dfrac{b}{b+d}\) (Rate of False Positives)

### 1.1.3 Positive Likelihood Ratio

\(+LR=\dfrac{Sensitivity}{1-Specificity}\)

\(+LR=\dfrac{P(S|D)}{P(S|\neg D)}\)

\(+LR= \dfrac{P(TP)}{P(FP)}\)

\(+LR= \dfrac{a(b+d)}{b(a+c)}\)

### 1.1.4 Negative Likelihood Ratio

\(-LR=\dfrac{1-Sensitivity}{Specificity}\)

\(-LR=\dfrac{P(\neg S|D)}{P(\neg S|\neg D)}\)

\(-LR= \dfrac{P(FN)}{P(TN)}\)

\(-LR= \dfrac{c(b+d)}{d(a+c)}\)

## 1.2 Bayes Formula

\(P(D|S)=\dfrac{P(S|D)\cdot P(D)}{P(S)}\)

The \(P(D)\) and \(P(S)\) are the “priors” - or “baseline” probabilities of the disease and the sign

### 1.2.1 Alternative format

\(P(\neg D|\neg S)=\dfrac{P(\neg S|\neg D)\cdot P(\neg D)}{P(\neg S)}\)

The \(P(\neg D)\) and \(P(\neg S)\) are the “priors” - or “baseline” probabilities of not having the disease or the sign