Survey reliability (round 2)

The original formulation for Cronbach’s alpha takes this form:

$$\alpha =\frac{N}{N-1}(1-\frac{\sum _{i=1}^N{\sigma }_{y_i}^2}{{\sigma }_T^2})$$

, where $N$ represents the number of items in the measure, ${\sigma }_{y_i}^2$ is the variance associated with each item $i$, and ${\sigma }_T^2$ is the variance associated with the total scores.

It’s often written in this form:

$$\alpha =\frac{N\overline{c}}{\overline{v}+(N-1)\overline{c}}$$

, where $\overline{c}$ is the average of the inter-item covariance, and $\overline{v}$ is the average variance.

To get the above, note that:

$$\overline{c}=\frac{\sum _{i<j}COV(y_i,y_j)}{(\genfrac{}{}{0ex}{}{N}{2})}=\frac{2\sum _{i<j}{\sigma }_{y_i{y}_j}}{N(N-1)}$$

Tune in next time for Generalizability theory and SEM as alternatives to Cronbach alpha.