# 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