# SMHS Cronbachs

## Scientific Methods for Health Sciences - Instrument Performance Evaluation: Cronbach's α

### Overview:

Cronbach’s alpha $\alpha$ is a coefficient of internal consistency and is commonly used as an estimate of the reliability of a psychometric test. Internal consistency is typically a measure based on the correlations between different items on the same test and measures whether several items that propose to measure the same general construct and produce similar scores. Cronbach’s alpha is widely used in the social science, nursing, business and other disciplines. Here we present a general introduction to Cronbach’s alpha, how is it calculated, how to apply it in research and what are some common problems when using Cronbach’s alpha.

### Motivation:

We have discussed about internal and external consistency and their importance in researches and studies. How do we measure internal consistency? For example, suppose we are interested in measuring the extent of handicap of patients suffering from certain disease. The dataset contains 10records measuring the degree of difficulty experienced in carrying out daily activities. Each item is recorded from 1 (no difficulty) to 4 (can’t do). When those data is used to form a scale they need to have internal consistency. All items should measure the same thing, so they could be correlated with one another. Cronbach’s alpha generally increases when correlations between items increase.

### Theory

Cronbach’s Alpha: a measure of internal consistency or reliability of a psychometric instrument and measures how well a set of items measure a single, one-dimensional latent aspect of individuals.

• Suppose we measure a quantity X, which is a sum of K components: $X=Y_{1}+ Y_{2}+⋯+Y_{k}$

then Cronbach’s alpha is defined as $\alpha =\frac{K}{K-1}$ $\choose 1-\frac{\sum_{i=1}^{K}\sigma_{{Y}_{i}^{2}}} {\sigma_{X}^{2}}$, where $\sigma_{X}^{2}$ is the variance of the observed total test scores, and $\sigma_{{Y}_{i}^{2}}$ is the variance of component $i$ for the current sample.

Include the following table in the Methods section!!!

 Subjects Items/Questions Part of the Assessment Instrument Total Score per Subject $Q_1$ $Q_2$ ... $Q_k$ $S_1$ $Y_{1,1}$ $Y_{1,2}$ … $Y_{1,k}$ $X_1=\sum_{j=1}^k{Y_{1,j}}$ $S_2$ $Y_{2,1}$ $Y_{2,2}$ … $Y_{2,k}$ $X_2=\sum_{j=1}^k{Y_{2,j}}$ ... ... ... ... ... ... $S_n$ $Y_{n,1}$ $Y_{n,2}$ … $Y_{n,k}$ $X_n=\sum_{j=1}^k{Y_{n,j}}$ Variance per Item $\sigma_{Y_{.,1}}^2=\frac{1}{n-1}\sum_{i=1}^n{(Y_{i,1}-\bar{Y}_{.,1})^2}$ $$\sigma_{Y_{.,2}}^2=\frac{1}{n-1}\sum_{i=1}^n{(Y_{i,2}-\bar{Y}_{.,2})^2}$$ … $$\sigma_{Y_{.,k}}^2=\frac{1}{n-1}\sum_{i=1}^n{(Y_{i,k}-\bar{Y}_{.,k})^2}$$ $$\sigma_X^2=\frac{1}{n-1}\sum_{i=1}^n{(X_i-\bar{X})^2}$$