# Intraclass Correlation

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Intraclass correlation(ICC) refers to the correlation within the groups or classes. It tells us how the members of a group are correlated with respect to a particular characteristic. The correlation between the heights of members of a family is an example of intraclass correlation.

We compare two different variables when calculating the usual correlation coefficient, whereas we use the same characteristic for the two variables when calculating intraclass correlation. For example, when calculating the intraclass correlation between the heights of family members, we take both of the variables X and Y to be equal to the height.

### Intraclass Correlation Coefficient Formula:

Suppose that our data has ‘n’ families/groups each of size ‘k’. Then the intraclass correlation coefficient can be calculated using the formula,

\text{ICC } = \frac{k\sigma_m^2 - \sigma^2}{(k-1)\sigma^2}

where \sigma^2 denotes the variance of all data values and \sigma_m^2 denotes the variance of the means of the families.

### Interpretation of Intraclass Coefficient:

The value of the ICC always lies within the following range:

\frac{-1}{k-1} \leq \text{ICC}\leq 1

So we see the values of the intraclass coeffiecient are skewed to the right. Values closer to +1 indicate higher positive intraclass correlation. If the value of ICC is close to zero then it implies that the data points within a group are independent of each other.

### What is a good intraclass correlation coefficient?

Values of ICC lying between 0.75 and 1 are considered to be good. It indicates that the data has high reliability. If the value of ICC is less than 0.5 then we conclude that our data has less reliability. Thus values less than 0.5 are considered to be bad.

### Calculating ICC using statistical software:

1. We can calculate ICC using R software by using the command icc(ratings, model, type, unit).
2. We calculate the ICC in SPSS by clicking on the following commands: Analyze > Scale > Reliability Analysis > Intraclass Correlation Coefficient.

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