# 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.

If items are scored from 0 to 1, then $\alpha =\frac{K}{K-1}$ $\choose 1-\frac{\sum_{i=1}^{K}P_{i}Q_{i}} {\sigma_{X}^{2}}$, where $P_{i}$ is the proportion scoring 1 on item $i$ and $Q_{i=1}-P_{i}$, alternatively, Cronbach’s alpha can be defined as $\alpha$=$\frac{K\bar c}{(\bar v +(K-1) \bar c )}$,where K is as above, $\bar v$ is the average variance of each component and $\bar c$ is the average of all covariance between the components across the current sample of persons.

• The standardized Cronbach’s alpha can be defined as $\alpha_{standardized}=\frac{K\bar r} {(1+(K-1)\bar r )}$, $\bar r$ is the mean of $\frac {K(K-1)}{2}$ non redundant correlation coefficients (i.e., the mean of an upper triangular, or lower triangular, correlation matrix).
• The theoretical value of alpha varies from 0 to 1 considering it is ratio of two variance. $\rho_{XX}=\frac{\sigma_{T}^{2}} {\sigma_{X}^{2}}$, reliability of test scores is the ratio of the true score and total score variance.

Internal consistency: a measure of whether several items that proposed to measure the same general construct produce similar score. It is usually measured with Cronbach’s alpha, which is calculated from the pairwise correlation between items. Internal consistency can take values from negative infinity to 1. It is negative when there is greater within subject variability than between-subject variability. Only positive values of Cronbach’s alpha make sense. Cronbach’s alpha will generally increases as the inter-correlations among items tested increase.

 Cronbach's alpha Internal consistency $\alpha$ ≥ 0.9 Excellent (High-Stakes testing) 0.7 ≤ $\alpha$ < 0.9 Good (Low-Stakes testing) 0.6 ≤ $\alpha$ < 0.7 Acceptable 0.5 ≤ $\alpha$ < 0.6 Poor $\alpha$ < 0.5 Unacceptable

Intra-class correlation: Cronbach’s alpha equals to the stepped-up intra-class correlation coefficient, which is commonly used in observational studies if and only if the value of the item variance component equals zero. If this variance component is negative, then alpha will underestimate the stepped-up intra-class correlation coefficient; if it’s positive, alpha will overestimate the stepped-up intra-class correlation.

Generalizability theory: Cronbach’s alpha is an unbiased estimate of the generalizability. It can be viewed as a measure of how well the sum score on the selected items capture the expected score in the entire domain, even if that domain is heterogeneous.

Problems with Cronbach’s alpha: (1) it is dependent not only on the magnitude of the correlations among items, but also on the number of items in the scale. Hence, a scale can be made to look more homogenous simply by increasing the number of items though the average correlation remains the same; (2) if two scales each measuring a distinct aspect are combined to form a long scale, alpha would probably be high though the merged scale is obviously tapping two different attributes; (3) if alpha is too high, then it may suggest a high level of item redundancy.

Split-Half Reliability: The test is split in half (e.g., odd / even) creating “equivalent forms”. The two “forms” are correlated with each other and the correlation coefficient is adjusted to reflect the entire test length, using the Spearman-Brown Prophecy formula. Suppose the $Corr(Even,Odd)=r$ is the raw correlation between the even and odd items. Then the adjusted correlation will be:$r’ = \frac{n r}{n-1 r+1},$

Where n = number of items (in this case n=2).

Example:

 Index Q1 Q2 Q3 Q4 Q5 Q6 Odd Even 1 1 0 0 1 1 0 2 1 2 1 1 0 1 0 1 1 3 3 1 1 1 1 1 0 3 2 4 1 0 0 0 1 0 2 0 5 1 1 1 1 0 0 2 2 6 0 0 0 0 1 0 1 0 mean 1.833333333 1.33333333 SD 0.752772653 1.21106014 corr(Even,Odd) 0.073127242 AdjCorr(Even,Odd)=$\frac{n*r}{(n-1)*(r+1)}$ 0.136288111
• K-R20 is a very reliable internal reliability estimate which simulates calculating split-half reliability for every possible combination of items.

$K-R20=\frac{k} {k-1}$ $\choose{1-\frac {\sum S_{i}^{2}} {S_{t}^{2}}},$ where $S_{i}$ and $S_{t}$ are the per-item and total variances, respectively.

• K-R21 may be slightly less accurate than KR-20, but can be calculated with just descriptive statistics and tends to underestimate reliability

$K-R21=\frac{k}{k-1}$ $\choose{1-\frac{M(k-M)} {kS_{t}^{2}}}$

• Standard Error of Measurement (SEM): The greater the reliability of the test, the smaller the SEM.

$SEM=S\sqrt{1-r_{xx}},$ where $r_{xx’}$ is the correlation between two instances of the measurements under identical conditions, and $S$ is the total standard deviation.

### Cronbach's $\alpha$ calculations

The table below illustrates the setting and core calculations involved in computing the Cronbach's $\alpha$.

 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 $\sigma_{Y_{.,1}}^2=\frac{1}{n-1}\sum_{i=1}^n{(Y_{i,1}-\bar{Y}_{.,1})^2}$ $$\sigma_{Y_{.,2}}^2=\frac{1}{n-1}\sum_{i=1}^n{(Y_{i,2}-\bar{Y}_{.,2})^2}$$ … $$\sigma_{Y_{.,k}}^2=\frac{1}{n-1}\sum_{i=1}^n{(Y_{i,k}-\bar{Y}_{.,k})^2}$$ $$\sigma_X^2=\frac{1}{n-1}\sum_{i=1}^n{(X_i-\bar{X})^2}$$