# Cronbach's alpha

Cronbach's alpha (Cronbach's ${\displaystyle \alpha }$), also known as rho-equivalent reliability (${\displaystyle \rho _{T}}$) or coefficient alpha (coefficient ${\displaystyle \alpha }$), is a reliability coefficient that provides a method of measuring internal consistency of tests and measures.[1][2][3] Numerous studies warn against using it unconditionally, and note that reliability coefficients based on structural equation modeling (SEM) or generalizability theory are in many cases a suitable alternative in certain situations.[4][5][6][7][8][9]

## History

### Cronbach (1951)

As with previous studies,[10][11][12][13] Cronbach (1951) published an additional method to derive Cronbach's alpha.[14] His interpretation was more intuitively attractive than those of previous studies and became quite popular.[15]

### After 1951

Novick and Lewis (1967)[16] proved the necessary and sufficient condition for ${\displaystyle \rho _{T}}$ to be equal to reliability and named it the condition of being essentially tau-equivalent.

Cronbach (1978)[2]: 263  mentioned that the reason Cronbach (1951) received a lot of citations was "mostly because [he] put a brand name on a common-place coefficient".[3] He explained that he had originally planned to name other types of reliability coefficients (e.g., inter-rater reliability or test-retest reliability) after consecutive Greek letters (e.g., ${\displaystyle \beta }$, ${\displaystyle \gamma }$, ${\displaystyle \ldots }$), but later changed his mind.

Cronbach and Shavelson (2004)[9] encouraged readers to use generalizability theory rather than ${\displaystyle \rho _{T}}$. Cronbach opposed the use of the name Cronbach's alpha, and explicitly denied the existence of studies that had published the general formula of KR-20 prior to Cronbach (1951).

## Prerequisites for using Cronbach's alpha

In order to use Cronbach’s alpha as a reliability coefficient, the data from the measure must satisfy the following conditions:[17][18]

1. Normality distributed and linear
2. Tau-equivalence (essential)
3. Independence between errors

## Formula and calculation

Cronbach’s alpha is calculated by taking the score from each scale item and correlating them with the total score for each observation and then comparing that with the variance for all individual item scores. Cronbach’s alpha is best understood as a function of the number of questions or items in a measure, the between pairs of items average covariance and the overall variance of the total measured score.[19]

${\displaystyle \alpha ={k \over k-1}\left(1-{\sum _{i=1}^{k}\sigma _{y}^{2} \over \sigma _{x}^{2}}\right)}$
${\displaystyle k}$ the number of items in the measure

${\displaystyle \sigma _{y}^{2}}$ variance associated with each

${\displaystyle \sigma _{x}^{2}}$ variance associated of the total scores

## Common misconceptions[7]

### The value of Cronbach's alpha ranges between zero and one

By definition, reliability cannot be less than zero and cannot be greater than one. Many textbooks mistakenly equate ${\displaystyle \rho _{T}}$ with reliability and give an inaccurate explanation of its range. ${\displaystyle \rho _{T}}$ can be less than reliability when applied to data that are not tau-equivalent. Suppose that ${\displaystyle X_{2}}$ copied the value of ${\displaystyle X_{1}}$ as it is, and ${\displaystyle X_{3}}$ copied by multiplying the value of ${\displaystyle X_{1}}$ by -1. The covariance matrix between items is as follows, ${\displaystyle \rho _{T}=-3}$.

Observed covariance matrix
${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{3}}$
${\displaystyle X_{1}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle -1}$
${\displaystyle X_{2}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle -1}$
${\displaystyle X_{3}}$ ${\displaystyle -1}$ ${\displaystyle -1}$ ${\displaystyle 1}$

Negative ${\displaystyle \rho _{T}}$ can occur for reasons such as negative discrimination or mistakes in processing reversely scored items.

Unlike ${\displaystyle \rho _{T}}$, SEM-based reliability coefficients (e.g., ${\displaystyle \rho _{C}}$) are always greater than or equal to zero.

This anomaly was first pointed out by Cronbach (1943)[20] to criticize ${\displaystyle \rho _{T}}$, but Cronbach (1951)[14] did not comment on this problem in his article, which discussed all conceivable issues related ${\displaystyle \rho _{T}}$.[9]: 396

### If there is no measurement error, the value of Cronbach's alpha is one

This anomaly also originates from the fact that ${\displaystyle \rho _{T}}$ underestimates reliability. Suppose that ${\displaystyle X_{2}}$ copied the value of ${\displaystyle X_{1}}$ as it is, and ${\displaystyle X_{3}}$ copied by multiplying the value of ${\displaystyle X_{1}}$ by two. The covariance matrix between items is as follows, ${\displaystyle \rho _{T}=0.9375}$.

Observed covariance matrix
${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{3}}$
${\displaystyle X_{1}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 2}$
${\displaystyle X_{2}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 2}$
${\displaystyle X_{3}}$ ${\displaystyle 2}$ ${\displaystyle 2}$ ${\displaystyle 4}$

For the above data, both ${\displaystyle \rho _{P}}$ and ${\displaystyle \rho _{C}}$ have a value of one.

The above example is presented by Cho and Kim (2015).[7]

### A high value of Cronbach's alpha indicates homogeneity between the items

Many textbooks refer to ${\displaystyle \rho _{T}}$ as an indicator of homogeneity[21] between items. This misconception stems from the inaccurate explanation of Cronbach (1951)[14] that high ${\displaystyle \rho _{T}}$ values show homogeneity between the items. Homogeneity is a term that is rarely used in the modern literature, and related studies interpret the term as referring to uni-dimensionality. Several studies have provided proofs or counterexamples that high ${\displaystyle \rho _{T}}$ values do not indicate uni-dimensionality.[22][7][23][24][25][26] See counterexamples below.

Unidimensional data
${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{3}}$ ${\displaystyle X_{4}}$ ${\displaystyle X_{5}}$ ${\displaystyle X_{6}}$
${\displaystyle X_{1}}$ ${\displaystyle 10}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$
${\displaystyle X_{2}}$ ${\displaystyle 3}$ ${\displaystyle 10}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$
${\displaystyle X_{3}}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 10}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$
${\displaystyle X_{4}}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 10}$ ${\displaystyle 3}$ ${\displaystyle 3}$
${\displaystyle X_{5}}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 10}$ ${\displaystyle 3}$
${\displaystyle X_{6}}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 10}$

${\displaystyle \rho _{T}=0.72}$ in the unidimensional data above.

Multidimensional data
${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{3}}$ ${\displaystyle X_{4}}$ ${\displaystyle X_{5}}$ ${\displaystyle X_{6}}$
${\displaystyle X_{1}}$ ${\displaystyle 10}$ ${\displaystyle 6}$ ${\displaystyle 6}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
${\displaystyle X_{2}}$ ${\displaystyle 6}$ ${\displaystyle 10}$ ${\displaystyle 6}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
${\displaystyle X_{3}}$ ${\displaystyle 6}$ ${\displaystyle 6}$ ${\displaystyle 10}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
${\displaystyle X_{4}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 10}$ ${\displaystyle 6}$ ${\displaystyle 6}$
${\displaystyle X_{5}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 6}$ ${\displaystyle 10}$ ${\displaystyle 6}$
${\displaystyle X_{6}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 6}$ ${\displaystyle 6}$ ${\displaystyle 10}$

${\displaystyle \rho _{T}=0.72}$ in the multidimensional data above.

Multidimensional data with extremely high reliability
${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{3}}$ ${\displaystyle X_{4}}$ ${\displaystyle X_{5}}$ ${\displaystyle X_{6}}$
${\displaystyle X_{1}}$ ${\displaystyle 10}$ ${\displaystyle 9}$ ${\displaystyle 9}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle 8}$
${\displaystyle X_{2}}$ ${\displaystyle 9}$ ${\displaystyle 10}$ ${\displaystyle 9}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle 8}$
${\displaystyle X_{3}}$ ${\displaystyle 9}$ ${\displaystyle 9}$ ${\displaystyle 10}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle 8}$
${\displaystyle X_{4}}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle 10}$ ${\displaystyle 9}$ ${\displaystyle 9}$
${\displaystyle X_{5}}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle 9}$ ${\displaystyle 10}$ ${\displaystyle 9}$
${\displaystyle X_{6}}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle 9}$ ${\displaystyle 9}$ ${\displaystyle 10}$

The above data have ${\displaystyle \rho _{T}=0.9692}$, but are multidimensional.

Unidimensional data with unacceptably low reliability
${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{3}}$ ${\displaystyle X_{4}}$ ${\displaystyle X_{5}}$ ${\displaystyle X_{6}}$
${\displaystyle X_{1}}$ ${\displaystyle 10}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
${\displaystyle X_{2}}$ ${\displaystyle 1}$ ${\displaystyle 10}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
${\displaystyle X_{3}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 10}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
${\displaystyle X_{4}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 10}$ ${\displaystyle 1}$ ${\displaystyle 1}$
${\displaystyle X_{5}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 10}$ ${\displaystyle 1}$
${\displaystyle X_{6}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 10}$

The above data have ${\displaystyle \rho _{T}=0.4}$, but are unidimensional.

Uni-dimensionality is a prerequisite for ${\displaystyle \rho _{T}}$. You should check uni-dimensionality before calculating ${\displaystyle \rho _{T}}$, rather than calculating ${\displaystyle \rho _{T}}$ to check uni-dimensionality.[3]

### A high value of Cronbach's alpha indicates internal consistency

The term internal consistency is commonly used in the reliability literature, but its meaning is not clearly defined. The term is sometimes used to refer to a certain kind of reliability (e.g., internal consistency reliability), but it is unclear exactly which reliability coefficients are included here, in addition to ${\displaystyle \rho _{T}}$. Cronbach (1951)[14] used the term in several senses without an explicit definition. Cho and Kim (2015)[7] showed that is ${\displaystyle \rho _{T}}$ is not an indicator of any of these.

### Removing items using "alpha if item deleted" always increases reliability

Removing an item using "alpha if item deleted"[clarification needed] may result in 'alpha inflation,' where sample-level reliability is reported to be higher than population-level reliability.[27] It may also reduce population-level reliability.[28] The elimination of less-reliable items should be based not only on a statistical basis but also on a theoretical and logical basis. It is also recommended that the whole sample be divided into two and cross-validated.[27]

## Ideal reliability level and how to increase reliability

### Nunnally's recommendations for the level of reliability

The most frequently cited source of how high reliability coefficients should be is Nunnally's book.[29][30][31] However, his recommendations are cited contrary to his intentions. What he meant was to apply different criteria depending on the purpose or stage of the study. However, regardless of the nature of the research, such as exploratory research, applied research, and scale development research, a criterion of 0.7 is universally used.[32] 0.7 is the criterion he recommended for the early stages of a study, which most studies published in the journal are not. Rather than 0.7, the criterion of 0.8 referred to applied research by Nunnally is more appropriate for most empirical studies.[32]

Nunnally's recommendations on the level of reliability
1st edition[29] 2nd[30] & 3rd[31] edition
Early stage of research 0.5 or 0.6 0.7
Applied research 0.8 0.8
When making important decisions 0.95 (minimum 0.9) 0.95 (minimum 0.9)

His recommendation level did not imply a cutoff point. If a criterion means a cutoff point, it is important whether or not it is met, but it is unimportant how much it is over or under. He did not mean that it should be strictly 0.8 when referring to the criteria of 0.8. If the reliability has a value near 0.8 (e.g., 0.78), it can be considered that his recommendation has been met.[33]

### Cost to obtain a high level of reliability

Nunnally's idea was that there is a cost to increasing reliability, so there is no need to try to obtain maximum reliability in every situation.

Measurements with perfect reliability lack validity.[7] For example, a person who take the test with the reliability of one will get a perfect score or a zero score, because the examinee who gives the correct answer or incorrect answer on one item will give the correct answer or incorrect answer on all other items. The phenomenon in which validity is sacrificed to increase reliability is called attenuation paradox.[34][35]

A high value of reliability can be in conflict with content validity. For high content validity, each item should be constructed to be able to comprehensively represent the content to be measured. However, a strategy of repeatedly measuring essentially the same question in different ways is often used only for the purpose of increasing reliability.[36][37]

When the other conditions are equal, reliability increases as the number of items increases. However, the increase in the number of items hinders the efficiency of measurements.

### Methods to increase reliability

Despite the costs associated with increasing reliability discussed above, a high level of reliability may be required. The following methods can be considered to increase reliability.

Before data collection:

• Eliminate the ambiguity of the measurement item.
• Do not measure what the respondents do not know.[38]
• Increase the number of items. However, care should be taken not to excessively inhibit the efficiency of the measurement.
• Use a scale that is known to be highly reliable.[39]
• Conduct a pretest - discover in advance the problem of reliability.
• Exclude or modify items that are different in content or form from other items (e.g., reverse-scored items).

After data collection:

• Remove the problematic items using "alpha if item deleted". However, this deletion should be accompanied by a theoretical rationale.
• Use a more accurate reliability coefficient than ${\displaystyle \rho _{T}}$. For example, ${\displaystyle \rho _{C}}$ is 0.02 larger than ${\displaystyle \rho _{T}}$ on average.[40]

## Which reliability coefficient to use

${\displaystyle \rho _{T}}$ is used in an overwhelming proportion. A study estimates that approximately 97% of studies use ${\displaystyle \rho _{T}}$ as a reliability coefficient.[3]

However, simulation studies comparing the accuracy of several reliability coefficients have led to the common result that ${\displaystyle \rho _{T}}$ is an inaccurate reliability coefficient.[41][42][6][43][44]

Methodological studies are critical of the use of ${\displaystyle \rho _{T}}$. Simplifying and classifying the conclusions of existing studies are as follows.

1. Conditional use: Use ${\displaystyle \rho _{T}}$ only when certain conditions are met.[3][7][8]
2. Opposition to use: ${\displaystyle \rho _{T}}$ is inferior and should not be used.[45][5][46][6][4][47]

### Alternatives to Cronbach's alpha

Existing studies are practically unanimous in that they oppose the widespread practice of using ${\displaystyle \rho _{T}}$ unconditionally for all data. However, different opinions are given on which reliability coefficient should be used instead of ${\displaystyle \rho _{T}}$.

Different reliability coefficients ranked first in each simulation study[41][42][6][43][44] comparing the accuracy of several reliability coefficients.[7]

The majority opinion is to use SEM-based reliability coefficients as an alternative to ${\displaystyle \rho _{T}}$.[3][7][45][5][46][8][6][47]

However, there is no consensus on which of the several SEM-based reliability coefficients (e.g., unidimensional or multidimensional models) is the best to use.

Some people suggest ${\displaystyle \omega _{H}}$[6] as an alternative, but ${\displaystyle \omega _{H}}$ shows information that is completely different from reliability. ${\displaystyle \omega _{H}}$ is a type of coefficient comparable to Revelle's ${\displaystyle \beta }$.[48][6] They do not substitute, but complement reliability.[3]

Among SEM-based reliability coefficients, multidimensional reliability coefficients are rarely used, and the most commonly used is ${\displaystyle \rho _{C}}$,[3] also known as composite or congeneric reliability.

#### Software for SEM-based reliability coefficients

General-purpose statistical software such as SPSS and SAS include a function to calculate ${\displaystyle \rho _{T}}$. Users who don't know the formula of ${\displaystyle \rho _{T}}$ have no problem in obtaining the estimates with just a few mouse clicks.

SEM software such as AMOS, LISREL, and MPLUS does not have a function to calculate SEM-based reliability coefficients. Users need to calculate the result by inputting it to the formula. To avoid this inconvenience and possible error, even studies reporting the use of SEM rely on ${\displaystyle \rho _{T}}$ instead of SEM-based reliability coefficients.[3] There are a few alternatives to automatically calculate SEM-based reliability coefficients.

1. R (free): The psych package[49] calculates various reliability coefficients.
2. EQS (paid):[50] This SEM software has a function to calculate reliability coefficients.
3. RelCalc (free):[3] Available with Microsoft Excel. ${\displaystyle \rho _{C}}$ can be obtained without the need for SEM software. Various multidimensional SEM reliability coefficients and various types of ${\displaystyle \omega _{H}}$ can be calculated based on the results of SEM software.

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