Chapter 5, Problem 11RQ

Summary Introduction

Introduction: The Kappa Statistic is a measure of agreement in between the two observers. It provides an idea on how well an agreement between the observer’s readings of a diagnostic test, than would be expected by chance alone.

Answer to Problem 11RQ

Correct answer: “The value of kappa is: 0.4.”

Explanation of Solution

As given in the problem statement, two independent physicians (physician 1 and physician 2) classified 100 chest X-rays as normal or abnormal as given in the table.

Given,

a = 40

b = 20

c = 10

d = 30

The kappa statistic is calculated using the below formula:

$\text{Kappa}\text{=}\frac{\left(\begin{array}{c}\text{Percent}\text{agreement}\\ \text{observed}\end{array}\right)-\left(\begin{array}{c}\text{Percent}\text{agreement}\\ \text{expected}\text{by}\text{chance}\text{alone}\end{array}\right)}{\text{100\%}-\left(\begin{array}{c}\text{Percent}\text{agreement}\\ \text{expected}\text{by}\text{chance}\text{alone}\end{array}\right)}$

 To compute Kappa, first calculate the percentage of observed agreement.

$\begin{array}{l}\text{Percent}\text{agreement}\text{=}\frac{\text{a}\text{+}\text{d}}{\text{a}\text{+}\text{b}\text{+}\text{c}\text{+}\text{d}}\text{×}\text{100}\\ \text{=}\frac{\text{40}+30}{40+20+10+30}\text{×}\text{100}\\ =\frac{70}{100}\text{×}\text{100}\\ =70\%\end{array}$

Now, calculate the percentage of agreement expected by chance alone.

For each cell, we need to find it by calculating,

$\begin{array}{l}\text{Percentage of agreement expected by chance alone}=\frac{\text{column}\text{total}\text{×}\text{row}\text{total}}{\text{Grand}\text{total}}\\ =\frac{\text{(60 ×}50\text{)}}{100}+\frac{\text{(40 ×}50\text{)}}{100}\\ \text{=}\frac{3000}{\text{100}}+\frac{2000}{\text{100}}\\ \text{=}50\text{\%}\end{array}$

Therefore, the kappa statistic is calculated as below:

$\begin{array}{l}\text{Kappa(κ)}\text{=}\frac{\text{70\%}-\text{50\%}}{\text{100\%}-\text{50\%}}\\ \text{=}\frac{\text{20}}{\text{50}}\\ \text{=}\text{0}\text{.4}\end{array}$

Thus, the value of kappa is 0.4