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 essentially 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)^[21] to criticize ${\displaystyle \rho _{T}}$, but Cronbach (1951)^[10] did not comment on this problem in his article that otherwise discussed potentially problematic 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^[22] between items. This misconception stems from the inaccurate explanation of Cronbach (1951)^[10] that high ${\displaystyle \rho _{T}}$ values show homogeneity between the items. Homogeneity is a term that is rarely used in 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.^{[23][7][24][25][26][27]} See counterexamples below.

Uni-dimensional 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 uni-dimensional 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.

Uni-dimensional 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 uni-dimensional.

Uni-dimensionality is a prerequisite for ${\displaystyle \rho _{T}}$. One 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)^[10] 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" may result in 'alpha inflation,' where sample-level reliability is reported to be higher than population-level reliability.^[28] It may also reduce population-level reliability.^[29] 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.^[28]

Nunnally's recommendations for the level of reliability

Nunnally's book^[30][31] is often mentioned as the primary source for determining the appropriate level of dependability coefficients. However, his proposals contradict his aims as he suggests that different criteria should be used depending on the goal or stage of the investigation. Regardless of the type of study, whether it is exploratory research, applied research, or scale development research, a criterion of 0.7 is universally employed.^[32] He advocated 0.7 as a criterion for early stages of a study, most studies published in the journal do not fall under that category. Rather than 0.7, Nunnally's applied research criterion of 0.8 is more suited for most empirical studies.^[32]

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.

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]

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., uni-dimensional 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 Reveille'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.