Kappa is a statistical measure of inter-rater agreement, often used in the fields of psychology, sociology, and medicine. It is used to determine the degree of agreement between two or more raters or observers, and is particularly useful when dealing with qualitative data. The expected agreement in kappa is a key factor in determining the reliability and validity of the measure, and is an important consideration when interpreting kappa values.

The formula for kappa is based on the observed agreement and the expected agreement between raters. The observed agreement is the proportion of cases in which the raters agreed, while the expected agreement is the proportion of cases in which the raters would be expected to agree if they were making ratings by chance alone. The expected agreement is calculated by multiplying the marginal proportions of the categories for each rater and adding them together.

To calculate the expected agreement in kappa, follow these steps:

1. Create a contingency table of your data. The contingency table should include the ratings of each rater for each category.

2. Calculate the marginal proportions for each rater. The marginal proportion is the total number of cases in each row or column divided by the total number of cases in the table.

3. Multiply the marginal proportions for each rater for each category and add them together. This will give you the expected frequency of agreement for each category.

4. Calculate the sum of the expected frequencies of agreement for each category. This will give you the overall expected frequency of agreement.

5. Divide the overall expected frequency of agreement by the total number of cases in the table.

6. Subtract the result from 1. This will give you the expected agreement in kappa.

For example, let`s say you have two raters who each rated 50 cases for the categories of "agree" and "disagree". The contingency table looks like this:

Rater 1

Agree: 30

Disagree: 20

Rater 2

Agree: 40

Disagree: 10

The marginal proportions for Rater 1 are:

Agree: 30 / 50 = 0.6

Disagree: 20 / 50 = 0.4

The marginal proportions for Rater 2 are:

Agree: 40 / 50 = 0.8

Disagree: 10 / 50 = 0.2

To calculate the expected frequency of agreement for Agree-Agree, you would multiply the marginal proportions for Rater 1 and Rater 2 for Agree:

0.6 * 0.8 = 0.48

To calculate the expected frequency of agreement for Agree-Disagree, you would multiply the marginal proportions for Rater 1 for Agree and for Rater 2 for Disagree:

0.6 * 0.2 = 0.12

To calculate the expected frequency of agreement for Disagree-Agree, you would multiply the marginal proportions for Rater 1 for Disagree and for Rater 2 for Agree:

0.4 * 0.8 = 0.32

To calculate the expected frequency of agreement for Disagree-Disagree, you would multiply the marginal proportions for Rater 1 and Rater 2 for Disagree:

0.4 * 0.2 = 0.08

The overall expected frequency of agreement is the sum of the expected frequencies for each category:

0.48 + 0.12 + 0.32 + 0.08 = 1

The expected agreement in kappa is the overall expected frequency of agreement divided by the total number of cases in the table:

1 / 100 = 0.01

To calculate the kappa coefficient, you would divide the observed agreement by the expected agreement:

Kappa = (Observed agreement - Expected agreement) / (1 - Expected agreement)

It is important to note that kappa values can range from -1 to 1, with 0 indicating no agreement above chance and 1 indicating perfect agreement. A kappa value of 0.01 indicates very poor agreement between the raters. Therefore, if the expected agreement in kappa is low, you may need to reevaluate your data collection process or look for other causes of disagreement between the raters.