Cronbach’s Alpha: A Brief Overview and Application
Measurement Error and True Score Theory
In the realm of psychological measurement, understanding the relationship between observed scores and true underlying capabilities is crucial. True score theory provides a fundamental framework for conceptualizing measurement reliability.
Theoretical Foundations.
The fundamental equation of true score theory posits:
This model represents a core principle in psychometrics: any measurement contains inherent imperfections. Consider an example where a student knows 80% of test material but scores 85% due to partial guessing. The additional 5% represents measurement error.
Core Assumptions of Reliability Measurement
Reliability measurement in true score theory relies on several critical assumptions. First and foremost, error scores must be randomly distributed, uncorrelated with each other, and centered around zero. This means that the measurement errors should not exhibit any systematic patterns or biases that could skew the observed scores in a particular direction.
Additionally, the test items should demonstrate a property known as tau-equivalence. Tau-equivalence implies that the true scores for different items remain relatively constant within a meaningful range. In other words, the items should be measuring the same underlying construct with a similar level of precision and accuracy.
Implications of Measurement Error
Measurement error introduces significant complexity and challenges in psychological assessment. Perfect measurement is a theoretical ideal that cannot be fully achieved in practice, as every measurement inherently contains some degree of error. These errors, particularly systematic ones, can substantially distort test results and lead to inaccurate or biased interpretations. Reliability coefficients derived from measurement data provide estimates of the consistency and precision of the scores, but they should not be mistaken for absolute certainties.
Practical Considerations for Researchers
Given the pervasive nature of measurement error, researchers must critically evaluate potential sources of error in their studies. Respondent variability, such as fluctuations in fatigue or motivation levels, can introduce unwanted noise into the measurement process. Inconsistencies in test administration procedures, ambiguous or poorly constructed test items, and context-specific response patterns can also contribute to measurement error. By carefully considering and addressing these factors, researchers can minimize the impact of error on their findings and enhance the validity of their conclusions.
Advanced Methodological Insights
To quantify and account for measurement error, researchers often employ advanced methodological techniques. Cronbach’s Alpha, for example, serves as a lower-bound estimate of reliability, providing a conservative assessment of the internal consistency of a measurement instrument. However, violations of the core assumptions underlying reliability measurement can lead to problematic outcomes. Overestimation of reliability, underestimation of measurement precision, and potential misinterpretation of test results are all possible consequences of assumption violations. Researchers must be vigilant in assessing the appropriateness of their measurement models and take steps to mitigate the impact of assumption violations on their findings.
Historical Development
The development of Cronbach’s Alpha as a reliability measure emerged from early 20th-century psychometric research. Charles Spearman established the foundational framework by introducing the split-half method for test consistency evaluation in 1910. This methodological innovation served as a catalyst for subsequent reliability assessment techniques.
In 1951, Lee J. Cronbach introduced the Alpha coefficient, which revolutionized reliability testing by quantifying internal consistency among test items. This statistical advancement provided researchers with a more robust and interpretable measure of scale reliability. Cronbach further expanded the theoretical framework through his Generalizability Theory, which addressed the limitations of generalizing test findings beyond their intended contexts.
A Comprehensive Approach to Assessing Scale Reliability Using Cronbach’s Alpha
Cronbach’s alpha remains a fundamental statistical tool for assessing scale reliability in applied research, particularly in educational and psychological measurements. This paper presents a comprehensive set of R functions for advanced reliability analysis using The alpha coefficient, addressing several critical methodological challenges in psychometric research. When developing and validating psychometric instruments, it is essential to follow a systematic approach to ensure reliable measurement of psychological constructs.
The following steps outline the process of assessing scale reliability using Cronbach’s alpha, one of the most widely used measures of internal consistency.
Verifying item coding to avoid reducing internal consistency.
Determining scale interpretation (interval vs ordinal) and selecting the appropriate correlation coefficient (Pearson or polychoric).
Assessing unidimensionality to confirm items measure the same construct, which is an assumption of Cronbach’s alpha.
Checking for tau-equivalence (equal true score variances), though this is often violated in practice.
calculating coefficient alpha and estimating confidence intervals, such as through bootstrapping, to assess reliability variability.
Compute “alpha if item deleted” values for each individual scale item.
Calculate “item-rest correlations”, i.e., the correlations between each individual item and the sum score of the remaining items (the scale without that item).
Additional Metric. Beyond Cronbach’s alpha, it is beneficial to consider McDonald’s Omega coefficient as an alternative measure of internal consistency. A comprehensive analysis of these indicators provides deeper insight into the constructed scale’s quality and helps determine whether adjustments to its formulation or structure are necessary.
Definition and Key Points
Cronbach’s Alpha (α) - Internal Consistency Reliability Coefficient.
Cronbach’s alpha measures how closely related a set of items are as a group.
It represents internal consistency reliability of measurements in a test or scale.
Higher values indicate greater reliability.
When calculating Cronbach’s Alpha, all items must be coded in the same direction. This means that higher scores on all items should indicate either a higher or lower level of the construct being measured, but not a mix of both. If some items are worded negatively (reverse-coded), they must be recoded before calculating Cronbach’s Alpha to ensure consistent scoring direction across all items.
Formula
α = (k/(k-1)) * (1 - Σσᵢ²/σₜ²)
The Formula Components
k = number of items in the scale/test
σᵢ² = variance of each individual item
σₜ² = total variance of the whole test/scale
Σσᵢ² = sum of variances for all items
Cronbach’s Alpha typically ranges from 0 to 1. A higher alpha value indicates greater internal consistency.
While theoretically Cronbach’s Alpha should fall between 0 and 1, negative values can occur in practice. These negative values indicate serious problems with the data or scale construction.
The Cronbach’s Alpha can be interpreted as the proportion of observed score variance that is attributable to true score variance. For example:
If α = 0.80, then approximately 80% of the observed score variance represents true score variance The remaining 20% represents error variance.
However, it’s important to note several caveats:
This interpretation assumes the test meets the assumptions of:
Alpha typically underestimates the true reliability.
The relationship between Alpha and true score variance becomes less precise.
In practice:
α = 0.70 suggests roughly 70% of observed variance reflects true score variance.
α = 0.90 suggests about 90% of variance is attributable to true scores.
α > 0.95 might indicate item redundancy rather than optimal measurement.
This relationship helps explain why higher Alpha values generally indicate better reliability - they represent a larger proportion of true score variance relative to error variance in the observed scores.
Calculating Cronbach’s Alpha Using Inter-Item Correlations
Cronbach’s alpha can be calculated using only the correlations between items (inter-item correlations). This approach yields standardized Cronbach’s Alpha. There is a formula based on the average correlation between items:
α = (k × r̄) / (1 + (k-1) × r̄)
where:
k = number of items r̄ = mean correlation between all pairs of items
This formula is mathematically equivalent to the original variance-based formula. It is particularly useful when:
You only have access to the correlation matrix
You need a quick reliability estimate knowing only the average correlation between items Let’s do a sample calculation:
If you have a scale with 7 items (k=7) and the average correlation between items is r̄=0.43:
This simplified formula proves especially valuable in research contexts where raw data might be unavailable but correlation matrices are provided in published studies or technical reports.
Note: While this method is very practical, having access to the full dataset is still preferable as it allows for more comprehensive reliability analyses, including item-level diagnostics.
Recommended minimum α depends on the stakes of assessment:
* High-stakes testing: α > 0.9
* Research scales: α > 0.7
* Exploratory work: α > 0.6 might be acceptable
Scale Types and Correlation Coefficients in Cronbach’s Alpha
Cronbach’s Alpha can be calculated for different types of measurement scales by selecting the appropriate correlation coefficient for each case. When working with interval scales, which assume that the data are continuous, the Pearson correlation coefficient is used. In contrast, for ordinal scales — especially those involving Likert-type items — the polychoric correlation coefficient is more appropriate to accurately capture the underlying relationships between the items.
Polychoric correlation estimates the correlation between two theoretically continuous latent variables that are measured on ordinal scales. It assumes:
Underlying bivariate normal distribution.
Ordinal variables are discretized versions of continuous latent variables.
For ordinal data with 5+ categories, Pearson correlations often provide similar results to polychoric correlation. With fewer categories or highly skewed data, polychoric correlations are more appropriate.
Checking Dimensionality
Before calculating Cronbach’s alpha, it is essential to verify the scale’s unidimensionality, as this coefficient assumes items measure a single underlying construct. The dimensionality assessment should begin with parallel analysis to determine the optimal number of factors to retain. This can be followed by parallel analysis.
Parallel analysis is a powerful technique used to empirically determine the optimal number of factors or principal components to retain in exploratory factor analysis (EFA). This method compares the eigenvalues of the observed correlation matrix to those obtained from randomly generated data with the same number of variables and observations.
The process involves the following steps:
Generate a large number of random datasets with the same dimensions as the original data. Conduct EFA on each simulated dataset and calculate the eigenvalues.
Compute the average eigenvalues and specific percentiles (e.g., 95th) across the simulated datasets.
Compare the eigenvalues from the observed data to the average and percentile eigenvalues from the simulated data.
Retain factors with eigenvalues greater than the corresponding average or percentile values from the simulated data.
The results of parallel analysis provide two key pieces of information:
The number of factors determined by the mean eigenvalue criterion, which indicates how many factors have eigenvalues exceeding the average eigenvalues from the simulated data.
The number of factors determined by the eigenvalue percentile criteria, typically focusing on the 95th percentile. This suggests retaining factors with eigenvalues greater than the 95th percentile of eigenvalues from the simulated data.
By considering both the mean and percentile criteria, researchers can make informed decisions about the appropriate number of factors to retain in their EFA models. Parallel analysis offers an objective, data-driven approach to determining factor retention, which helps to avoid over- or under-extraction of factors.
If multiple dimensions emerge, researchers should consider calculating alpha coefficients separately for each subscale or employing alternative reliability measures better suited for multidimensional instruments, such as omega coefficients.
Factors Affecting Alpha Values
Factors Affecting Cronbach’s Alpha Values Sample Size Considerations Larger samples (n > 300) typically provide more stable alpha estimates. Small samples can lead to imprecise estimates with wider confidence intervals. A minimum sample size of 30 is recommended for preliminary analyses, though 50+ is preferred for more reliable results.
Impact of Scale Length Adding more items generally increases alpha values, even without improving true reliability. This occurs due to the mathematical properties of the coefficient rather than actual improvement in measurement quality. Researchers should balance comprehensive coverage against unnecessary redundancy.
Response Variability Effects Limited response variability (e.g., when most participants choose similar answers) can artificially deflate alpha values. This commonly occurs with ceiling or floor effects, where items are too easy or too difficult. Ensuring appropriate item difficulty levels and response distributions helps achieve more accurate reliability estimates. Cultural and Linguistic Factors Cross-cultural applications require careful consideration:
Translation quality affects item understanding and response patterns Cultural response styles (e.g., tendency toward extreme or middle responses) can impact alpha values Cultural relevance of items may vary, affecting overall scale consistency
An extremely high Cronbach’s alpha (above 0.95) can signal problems in scale construction. Such values often indicate that items are mere paraphrases of each other rather than distinct aspects measuring the full scope of the construct. While this creates strong internal consistency, it reduces content validity and construct coverage. The scale becomes inefficient, burdening respondents with redundant questions without gaining additional measurement value. This demonstrates why reliability must be balanced against other key psychometric properties when developing measurement instruments.
Improving Low Alpha Values
Review item quality. Check for unclear or ambiguous wording Verify translation accuracy if applicable Ensure items target appropriate difficulty level
Examine item statistics. Remove items with low item-total correlations Identify items that substantially increase alpha when deleted
Consider scale modifications. Add items that better represent the construct Revise response options if variability is limited Address potential cultural or contextual misalignment
Cronbach’s Alpha for Two Items
While it’s technically possible to calculate Cronbach’s alpha for two items, it’s generally not recommended. Here’s why:
Mathematical Equivalence
For two items, Cronbach’s alpha is mathematically equivalent to the Spearman-Brown coefficient and simply equals the Pearson correlation between the items. Specifically:
For k=2, the formula reduces to: α = (2r)/(1+r), where r is the correlation between the two items
Reliability Assessment Issues Alpha was designed to assess internal consistency across multiple items.
With only two items, you can’t properly assess the consistency pattern across the scale.
Two items provide very limited information about the construct being measured.
Better Alternatives for Two Items For two-item scenarios, it’s recommended to:
Simply report the correlation coefficient between the items.
Use the Spearman-Brown coefficient.
Consider adding more items to your scale for better reliability assessment.
At least 3 items for calculating Cronbach’s alpha.
Preferably 4+ items for a more stable and meaningful reliability estimate.
Note: If you must work with only two items, be transparent about the limitations in your methodology section and consider reporting both the correlation and Cronbach’s alpha for completeness.
Best Practices for Reporting Cronbach’s Alpha
Confidence Intervals for Alpha Always report 95% confidence intervals alongside point estimates of alpha. These intervals provide crucial information about estimate precision and sampling error. Wider intervals suggest less precise estimates and may indicate a need for larger samples. Bootstrapping is a nonparametric procedure that involves repeatedly resampling your data with replacement to create numerous “bootstrap samples” by calculating Cronbach’s Alpha within each resampled dataset, you obtain a distribution of Alpha values that can be used to construct confidence intervals.
Item-Total Correlations Report corrected item-total correlations for each scale item. These correlations show how well individual items relate to the overall scale score. Corrected item-total correlation above 0.30 was considered satisfactory.
Alpha-if-Item-Deleted Values include alpha coefficients calculated with each item removed. This information helps identify problematic items that may reduce scale reliability. Items whose removal substantially increases alpha should be carefully reviewed for potential elimination or revision. Sample Characteristics
Report relevant sample characteristics that could influence reliability:
Sample size and demographic composition Testing conditions and administration method Missing data patterns and handling procedures any unusual response patterns or outliers population-specific factors that might affect interpretation.
Example of Reporting:
“Internal consistency reliability was assessed using Cronbach’s alpha. The scale demonstrated good reliability (α = .83, 95% CI [.79, .86]). All item-total correlations exceeded .30, ranging from .49 to .68.”
Testing Tau-Equivalence for Cronbach’s Alpha
Tau-equivalence is a crucial assumption for the proper application of Cronbach’s alpha. It assumes that all scale items measure the same latent construct with approximately equal “weight” or strength of association.
Testing for tau-equivalence can be accomplished through several complementary methodological approaches. The process typically begins with basic statistical analysis, where researchers examine descriptive statistics by comparing means and variances across items. Significant differences in these basic statistics may signal potential violations of tau-equivalence, making it essential to visualize the distributions for a more comprehensive understanding.
Correlation analysis serves as another fundamental approach, focusing on the examination of the inter-item correlation matrix. In cases of tau-equivalence, correlations between items should demonstrate approximate equality, with any substantial deviations potentially indicating violation of the tau-equivalence assumption. This analysis extends to item-rest correlations, where researchers should not observe substantial differences between these correlations if tau-equivalence holds.
Exploratory Factor Analysis (EFA) provides a more sophisticated method for assessing tau-equivalence. When conducting EFA with a single factor, researchers carefully analyze the factor loadings, which should be approximately equal under conditions of tau-equivalence. A practical guideline suggests that the difference between the highest and lowest loading should not exceed 0.2. Additionally, the percentage of variance explained by the first factor becomes crucial, with a high percentage (exceeding 50%) lending support to construct unidimensionality.
Confirmatory Factor Analysis (CFA) offers perhaps the most rigorous approach through model comparison. This method involves testing two competing models: a congeneric model that allows different factor loadings, serving as a less restrictive baseline, and an equal factor loadings model that imposes the tau-equivalence assumption by fixing all loadings to be equal. The assessment of these models relies on various fit indices, including CFI (which should exceed 0.95), RMSEA (which should be less than 0.08), and SRMR (which should be less than 0.08).
The comparison between these CFA models involves multiple criteria. Researchers typically employ a chi-square difference test to assess statistical significance. The difference in CFI values between models provides another crucial metric, with a difference less than 0.01 suggesting that the more restrictive tau-equivalent model might be acceptable.
While tau-equivalence is a fundamental assumption underlying Cronbach’s alpha, empirical evidence consistently shows that strict tau-equivalence is rarely achieved in psychological measurement, with most scales exhibiting varying degrees of congeneric properties across their items.
Limitations and Criticisms
Despite its utility, Cronbach’s Alpha has faced scrutiny due to its assumptions of tau-equivalence and unidimensionality, which can lead to misleading results in certain contexts. Furthermore, its value can be artificially inflated with an increasing number of items, prompting researchers to consider additional measures of reliability.
Alternatives to Cronbach’s Alpha
However, methodological debates persist regarding the unconditional reliance on Cronbach’s Alpha, with alternative coefficients and techniques - such as McDonald’s Omega and test-retest reliability - being suggested to provide a more nuanced understanding of reliability in diverse research contexts.
An Illustrative Example
All examples are based on data from an online survey “Student youth of Kyiv under the corona crisis”. Period: February 24 - April 18, 2021. Participants: 253 students of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”. The survey was conducted via Google Forms.
This study attempted a Ukrainian-language adaptation of the the Fear of COVID-19 Scale (FCV-19S) (Бова, 2023).
The FCV-19S includes such items (Ahorsu et al., 2020).
I am most afraid of Corona (Afraid).
It makes me uncomfortable to think about Corona (Umcomf).
My hands become clammy when I think about Corona (ClammyHands).
I am afraid of losing my life because of Corona (AfraidDie).
When I watch news and stories about Corona on social media, I become nervous or anxious (AnxNews).
I cannot sleep because I’m worrying about getting Corona (Insomnia).
My heart races or palpitates when I think about getting Corona (HeartRaces).
Respondents used the five-point Likert scale to answer. Level of Agreement. 1 - strongly disagree, 2 - disagree, 3 - neutral, 4 - agree, 5 - strongly agree.
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library(haven) # Read SPSS filelibrary(labelled) # Work with labelled datalibrary(fmsb) # Calculate Cronbach's alphalibrary(misty) # Item analysis and reliabilitylibrary(psych) # Reliability and item statisticslibrary(psychometric) # Confidence intervals for alphalibrary(sjPlot) # Create item-scale tableslibrary(jmv) # Calculate Cronbach's alpha library(EFA.MRFA) # Parallel аnalysislibrary(ufs) # Ordinal reliability analysislibrary(Cronbach) # CI for summary alpha datalibrary(ltm) # Bootstrapped alpha confidence intervalslibrary(coefficientalpha) # Tau-equivalence testinglibrary(cocron) # Compare alpha coefficientslibrary(presize) # Sample Size for Cronbach's alphalibrary(caret) # For creating cross-validation folds
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# Setting the working directorysetwd("D:/Lichnoe/2025/Cronbach")# Loading the necessary library for reading SPSS fileslibrary(haven)# Reading the data from the SPSS filedf <-read_sav("Stud2021.sav")# Selecting the items for Cronbach's Alpha calculationitems <- df[, 42:48]# Removing variable labels from the items data framevar_label(items) <-NULL
Evaluating Unidimensionality
Before calculating Cronbach’s alpha, it is crucial to evaluate whether the scale is unidimensional. Parallel analysis and exploratory factor analysis are commonly employed at the initial stage. If these methods indicate that respondents’ answers can be explained by a single latent variable, and if the correlations between items align with this assumption, then researchers can proceed to interpret Cronbach’s alpha as a general measure of the scale’s internal consistency.
Parallel analysis is often considered one of the most accurate criteria for determining the number of factors to retain. If parallel analysis confirms the unidimensionality of a test, and Cronbach’s alpha is also high, one can conclude that the test has high reliability and internal consistency in measuring a single construct.
However, for highly correlated factor structures, the correct number of factors may be underestimated.
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par<-parallelMRFA(items[1:7], display=F)
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# Output the number of factors based on the mean eigenvalue criterioncat("Number of factors based on mean eigenvalue criterion:", par$N_factors_mean, "\n")
Number of factors based on mean eigenvalue criterion: 1
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# Output the number of factors based on the eigenvalue percentile criteriacat("Number of factors based on the 95th percentile of eigenvalues:", par$N_factors_percentiles, "\n")
Number of factors based on the 95th percentile of eigenvalues: 1
In R, the psych package is one of the most popular tools for computing Cronbach’s alpha and other related metrics. Below, we will walk through the key outputs from the alpha() function and interpret their meaning.
Cronbach’s alpha coefficient (raw_alpha), raw_alpha = 0.83. This is the primary measure of internal consistency. A value of 0.83 indicates good reliability , as it exceeds the commonly accepted threshold of 0.7 for psychological scales.
95% confidence boundaries. These boundaries provide an estimate of the precision of the alpha coefficient. The narrow confidence interval (0.79–0.86) suggests that the reliability estimate is robust.
Alpha-if-item-deleted (alpha.drop). This table shows how Cronbach’s alpha would change if each item were removed from the scale. For example, removing “AnxNews” would reduce alpha to 0.78, suggesting that this item contributes significantly to the overall reliability.
Item-total correlations (r.drop). These are the corrected item-total correlations, which indicate how strongly each item correlates with the total score (excluding itself). Higher values (e.g., > 0.5) suggest that the item is well-aligned with the construct being measured.
Average inter-item correlation (average_r), average_r = 0.43. This is the mean correlation between all pairs of items. A value of 0.43 suggests moderate inter-item consistency, which is desirable for a reliable scale.
Standardized alpha (std.alpha), std.alpha = 0.84. This is Cronbach’s alpha computed after standardizing the items (i.e., converting them to z-scores). It is slightly higher than raw_alpha, indicating that the items have have similar variances.
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# Calculating Cronbach's alpha and item statistics using the psych packageitem_alpha_psych <- psych::alpha(items)# Displaying the resultscat("Cronbach's Alpha and Item Statistics:\n")
# Item analysis of a scale or index and Cronbach's alpha: sjPlot::tab_itemscalesjPlot::tab_itemscale(items)
Component 1
Row
Missings
Mean
SD
Skew
Item Difficulty
Item Discrimination
α if deleted
Afraid
0.00 %
2.17
0.92
0.57
0.43
0.49
0.81
Umcomf
0.00 %
2.79
1.18
0.07
0.56
0.51
0.82
ClammyHands
0.00 %
1.42
0.69
2.26
0.28
0.60
0.80
AfraidDie
0.00 %
2.42
1.26
0.53
0.48
0.59
0.80
AnxNews
0.00 %
2.19
1.09
0.65
0.44
0.67
0.78
Insomnia
0.00 %
1.38
0.65
2.25
0.28
0.59
0.81
HeartRaces
0.00 %
1.73
0.9
1.22
0.35
0.68
0.78
Mean inter-item-correlation=0.432 · Cronbach's α=0.825
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options(digits =3) # Set the number of significant digits to display# Perform reliability analysis on items 1 to 7# Display the correlation plot and Cronbach's alpha for each item if deletedjmv::reliability(items[1:7], corPlot=T, alphaItems=T)
Warning: The `size` argument of `element_line()` is deprecated as of ggplot2 3.4.0.
ℹ Please use the `linewidth` argument instead.
ℹ The deprecated feature was likely used in the jmvcore package.
Please report the issue at <https://github.com/jamovi/jmvcore/issues>.
Estimating Cronbach’s Alpha with Confidence Intervals
Coefficient alpha applies Cronbach’s formula directly, whereas alpha-cfa constructs a factor model with a single latent factor and imposes the condition of equal loadings (tau-equivalence). This makes it methodologically different from the basic formula and can lead to a slightly different result.
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# Calculate Cronbach's alphaalpha1 <- MBESS::ci.reliability(data = items[1:7], type ="alpha", interval.type="bca")# Display the results for Cronbach's alphacat("Cronbach's Alpha (Classical):", round(alpha1$est, 3), "\n")
# Calculate alpha using CFAalpha2 <- MBESS::ci.reliability(data = items[1:7], type ="alpha-cfa")# Display the results for CFA-based alphacat("Cronbach's Alpha (CFA-based):", round(alpha2$est, 3), "\n")
Cronbach's Alpha (CFA-based): 0.786
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# Calculating Cronbach's alpha using the CronbachAlpha() functionalpha_result_base <-CronbachAlpha(items)# Displaying the result with clear formattingcat("Cronbach's Alpha:", round(alpha_result_base, 3), "\n")
Cronbach's Alpha: 0.825
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# Calculating Cronbach's alpha and confidence intervals using Cronbach::cronbach()alpha_result_cronbach <- Cronbach::cronbach(as.matrix(items))# Confidence intervals using cronfree.ci()ci_result_free <-cronfree.ci( alpha_result_cronbach,p =7, # Number of itemsn =nrow(items), # Sample sizeconf =0.95,type ="kf")# Displaying the resultscat("Cronbach's Alpha:", round(alpha_result_cronbach, 3), "\n")
Cronbach's Alpha: 0.825
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cat("Confidence Interval:\n")
Confidence Interval:
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print(ci_result_free)
0.025% 0.975%
0.808 0.841
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# Confidence intervals using Cronbach::cron.ci()ci_result_cron <-cron.ci(as.matrix(items), conf =0.95, type ="logit")# Displaying the resultcat("Confidence Interval:\n")
Confidence Interval:
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print(round(ci_result_cron, 3))
0.025% 0.975%
0.791 0.855
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# Calculating Cronbach's alpha and confidence interval using ltm::cronbach.alpha()alpha_result_ltm <- ltm::cronbach.alpha(items, CI =TRUE, probs =c(0.025, 0.975), B =1000)# Displaying the resultscat("Cronbach's Alpha:", round(alpha_result_ltm$alpha, 3), "\n")
# Calculating Cronbach's alpha using psychometric::alpha()alpha_result_psychometric <- psychometric::alpha(items)# Displaying the Cronbach's alpha resultcat("Cronbach's Alpha:", round(alpha_result_psychometric, 3), "\n")
Cronbach's Alpha: 0.825
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# Confidence intervals using psychometric::alpha.CI()ci_result_psychometric <- psychometric::alpha.CI(alpha_result_psychometric, k =7, N =nrow(items))# Displaying the confidence interval resultcat("Confidence Interval:\n")
Confidence Interval:
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print(ci_result_psychometric)
LCL ALPHA UCL
1 0.796 0.825 0.852
Sample Size for Cronbach’s Alpha: presize::prec_cronb
The function allows you to determine the number of study participants (sample size) needed to achieve a desired level of precision in estimating Cronbach’s alpha (a measure of scale reliability), or conversely, what level of precision can be expected with a given sample size.
For example, if a scale has 7 items and the expected Cronbach’s alpha is 0.7, then a sample of 326 respondents is needed. With 95% confidence, the true population Cronbach’s alpha would fall within the range of [0.622, 0.822], if estimated from a sample of this size.
Test of Tau-Equivalence via Confirmatory Factor Analysis
Model Comparison Framework
To assess tau-equivalence, we conducted a comparative analysis of two CFA models. The first model implemented tau-equivalent constraints with equal factor loadings (Model A), while the second allowed for congeneric relationships with freely estimated factor loadings (Model B).
Results Analysis
Factor Loading Patterns
The tau-equivalent model demonstrated uniform standardized factor loadings of 0.561 across all items, reflecting the imposed equality constraints inherent to tau-equivalence assumptions. In contrast, the congeneric model revealed substantial variability in factor loadings, ranging from 0.463 (Item 6) to 0.791 (Item 5). This considerable range of 0.328 suggests meaningful item-level differences in factor relationships.
Model Fit Assessment.
The tau-equivalent model (Model A) exhibited suboptimal fit across multiple indices:
The Comparative Fit Index (CFI) reached 0.848, falling short of the 0.90 acceptability threshold.
The Tucker-Lewis Index (TLI) similarly underperformed at 0.840.
The Root Mean Square Error of Approximation (RMSEA) indicated poor fit at 0.136.
The chi-square test yielded χ² = 205.587 (df = 20, p < 0.001).
The congeneric model (Model B) demonstrated superior fit characteristics:
CFI improved to 0.928, exceeding the acceptable threshold.
TLI approached acceptability at 0.892.
RMSEA showed improvement at 0.112, though still indicating some fit concerns.
Chi-square results improved to χ² = 101.727 (df = 14, p < 0.001).
Comparative Analysis
Direct model comparison revealed substantial differences:
A significant chi-square difference (Δχ² = 103.86, df = 6, p < 0.001).
The CFI difference (ΔCFI = 0.08) exceeded the standard 0.01 threshold for model invariance.
Theta parameters represented the residual covariance matrix for observed variables.
Key Findings
The analysis provides strong evidence for the violation of tau-equivalence assumptions. This conclusion is supported by both the significant chi-square difference and the substantial CFI change between models. While neither model achieved optimal fit across all indices, the congeneric model demonstrated markedly superior fit characteristics compared to the tau-equivalent specification. These findings suggest that assuming tau-equivalence for this measure may be inappropriate, and a congeneric measurement model may better represent the underlying factor structure.
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#' Unidimensional confirmatory factor analysis#'#' @param cov observed covariances#' @param what e.g., "est", "std", "fit"#' @param sample_size number of sample observations#' @param nonneg_loading if TRUE, constraint loadings to nonnegative values#' @param nonneg_error if TRUE, constraint loadings to positive values#' @param taueq if TRUE, a tau-equivalent model is estimated#' @param parallel if TRUE, a parallel model is estimated#' @examples uni_cfa(Graham1)#' @import lavaan#' @export uni_cfa#' @return parameter estimates of unidimensional cfa modeluni_cfa <-function(cov, what ="est", sample_size =500, nonneg_loading =FALSE,nonneg_error =TRUE, taueq =FALSE, parallel =FALSE) {stopifnot(requireNamespace("lavaan")) k <-nrow(cov)rownames(cov) <-character(length = k)for (i in1:k) {rownames(cov)[i] <-paste0("V", i)if (i ==1) { model_str <-paste("F =~ NA*V1") } elseif (taueq | parallel) { # tau-equivalent or parallel model_str <-paste0(model_str, " + equal('F=~V1')*V", i) } else {# congeneric model_str <-paste0(model_str, " + l", i, "*V", i) } }colnames(cov) <-rownames(cov) model_str <-paste0(model_str, " \n F ~~ 1*F", collapse ="\n")if (parallel) {for (i in1:k) { # all errors are constained to be equal model_str <-paste0(model_str, "\n V", i, " ~~ e*V", i) } } elseif (!taueq) { # congenericfor (i in1:k) { # to prevent negative errorsif (nonneg_error) { model_str <-paste0(model_str, "\n V", i, " ~~ e", i, "*V", i, "\n e", i,"> 0") }if (i >1& nonneg_loading) { # prevent negative loadings model_str <-paste0(model_str, "\n l", i, "> .0") } } } fit <- lavaan::cfa(model_str, sample.cov = cov, sample.nobs = sample_size)if (lavaan::inspect(fit, what ="converged")) { out <- lavaan::inspect(fit, what = what) } else { out <-NA }return(out)}##########################################################################################################' Obtain the covariance matrix#'#' If the input data is a square matrix, it is converted into a matrix,#' otherwise the covariance matrix is obtained.#'#' @param x A dataframe or a matrix#' @param cor if TRUE, return correlation matrix. if FALSE, return covariance matrix#' @return The covariance or correlation matrixget_cov <-function(x, cor =FALSE) {if (nrow(x) ==ncol(x)) {if (cor) { out <- stats::cov2cor(as.matrix(x)) } else { out <-as.matrix(x) } } else {if (cor) { out <- stats::cor(x, use ="pairwise.complete.obs") } else { out <- stats::cov(x, use ="pairwise.complete.obs") } }return(out)}##########################################################################################################' Test the essential tau-equivalence of the data#' #' The goodness-of-fit indices are compared with the parameter estimates of the #' essential tau-equivalence model and the cognate model. It can also be used for #' the purpose of investigating the unidimensionality of the data.#' @export test.tauequivalence#' @author Eunseong Cho, \email{bene@kw.ac.kr}#' @param data a dataframe or a matrix (unidimensional)#' @return taueq_cfi#' @return taueq_tli#' @return taueq_rmsea#' @return taueq_df#' @return taueq_pvalue#' @return taueq_chisq#' @return conge_cfi#' @return conge_tli#' @return conge_rmsea#' @return conge_df#' @return conge_pvalue#' @return conge_chisq#' @return diff_df#' @return diff_chisq#' @return diff_pvalue#' @examples test.tauequivalence(Graham1)#' @references Graham, J. M. (2006). Congeneric and (essentially) tau-equivalent #' estimates of score reliability what they are and how to use them. Educational #' and Psychological Measurement, 66(6), 930-944.#' @references Cho, E. (2016). Making reliability reliable: A systematic #' approach to reliability coefficients. Organizational Research Methods, 19(4), #' 651-682. #' @references Cho, E., & Kim, S. (2015). Cronbach’s coefficient alpha: Well #' known but poorly understood. Organizational Research Methods, 18(2), 207-230. #'test.tauequivalence <-function(data) { cov <-get_cov(data) taueq_fit <-uni_cfa(cov, what ="fit", taueq =TRUE) conge_fit <-uni_cfa(cov, what ="fit", taueq =FALSE) ############################################################# Converting lavaan's result to variables############################################################ taueq_cfi = taueq_fit[9] taueq_tli = taueq_fit[10] taueq_rmsea = taueq_fit[23] taueq_df = taueq_fit[4] taueq_pvalue = taueq_fit[5] taueq_chisq = taueq_fit[3] conge_cfi = conge_fit[9] conge_tli = conge_fit[10] conge_rmsea = conge_fit[23] conge_df = conge_fit[4] conge_pvalue = conge_fit[5] conge_chisq = conge_fit[3] diff_df = taueq_df - conge_df diff_chisq = taueq_chisq - conge_chisq diff_pvalue =1- stats::pchisq(diff_chisq, diff_df)############################################################# print############################################################cat("Parameter estimates of the tau-equivalent model\n")print(uni_cfa(cov, what ="est", taueq =TRUE))cat("Parameter estimates of the congeneric model\n")print(uni_cfa(cov, what ="est", taueq =FALSE))cat(" CFI TLI RMSEA df chisquare pvalue\n")cat(paste("tau-equivalent (A) ", round(taueq_cfi, 3), round(taueq_tli, 3), round(taueq_rmsea, 3),round(taueq_df,3), " ",round(taueq_chisq, 3),round(taueq_pvalue, 3), "\n"))cat(paste("congeneric (B) ", round(conge_cfi, 3), round(conge_tli, 3), round(conge_rmsea, 3),round(conge_df,3), " ",round(conge_chisq, 3),round(conge_pvalue, 3), "\n"))cat(paste("Difference (A - B) ", round(diff_df, 3), " ",round(diff_chisq, 3),round(diff_pvalue, 3)))############################################################# invisible return############################################################ test_taueq <-list(taueq_cfi = taueq_cfi,taueq_tli = taueq_tli,taueq_rmsea = taueq_rmsea,taueq_df = taueq_df,taueq_pvalue = taueq_pvalue,taueq_chisq = taueq_chisq,conge_cfi = conge_cfi,conge_tli = conge_tli,conge_rmsea = conge_rmsea,conge_df = conge_df,conge_pvalue = conge_pvalue,conge_chisq = conge_chisq,diff_df = diff_df,diff_chisq = diff_chisq,diff_pvalue = diff_pvalue )invisible(test_taueq)}
Tau-Equivalence: What is it?
Tau-equivalence posits that all items on a scale contribute equally to the measurement of the underlying latent variable, or the true score. In essence, this implies that each item is related to the true score to the same extent. To assess this assumption, a Confirmatory Factor Analysis (CFA) is employed. Tau-equivalence posits that all items on a scale contribute equally to the measurement of the underlying latent variable, or the true score. In essence, this implies that each item is related to the true score to the same extent. To assess this assumption, a CFA is employed.
How the Function Tests Tau-Equivalence
Building the CFA Model for Tau-Equivalence: A CFA model is constructed positing a single latent factor that accounts for all items in the scale. Then, the factor loadings of each item on this latent factor are constrained to be equal to one. This constraint implies that each item in the model is assumed to have an equivalent contribution to the underlying latent construct.
Assessing Model Fit: The fit of this constrained model to the observed data is evaluated. This evaluation uses a robust F-statistic, which quantifies how well the model represents the sample data. The associated p-value from this statistic is then used to determine the probability of observing the data given the model is true, that is, given equal factor loadings.
Interpreting the Results: If the obtained p-value is small (e.g., less than 0.05), it suggests that the model assuming equal item contributions does not adequately explain the observed data. In this instance, the appropriateness of utilizing Cronbach’s alpha may be questioned, or its interpretation should be approached with caution.
Item Homogeneity: What is it?
Item homogeneity means that all the items in the scale measure the same underlying construct, and that they all “speak to the same thing”. If a scale is homogeneous, it supports its unidimensionality (measuring a single construct). The opposite of homogeneity is heterogeneity, where the items in a scale measure different things, or different aspects of the same construct.
How the Function Tests Item Homogeneity
Building the CFA Model for Homogeneity: A CFA model is created where all the scale items load onto a single latent factor (the underlying construct). However, the item loadings on the latent factor are NOT fixed to be equal. Instead, they are freely estimated by the CFA model. This allows each item to have a different degree of association with the latent factor.
Assessing Model Fit: As with the tau-equivalence test, the fit of this CFA model (with freely estimated item loadings) to the data is assessed. The robust F-statistic and its associated p-value are calculated.
Interpreting the Results: If the p-value is high (e.g., greater than 0.05), it indicates that the model with freely estimated item loadings fits the data well. This suggests that the items measure the same underlying construct, though potentially with varying degrees of association. If the p-value is low (e.g., less than 0.05), it suggests that the model with a single latent factor does not describe the data well, and that the items may measure different things. In this case, item homogeneity is called into question.
Key Differences:
Tau-equivalence requires all items to have the same strength of association with the latent factor (item loadings fixed to 1).
Homogeneity only requires that all items are associated with the same latent factor, but with potentially different strengths of association (item loadings are freely estimated). Homogeneity is a weaker assumption than tau-equivalence.
By first examining the stronger assumption of tau-equivalence and then the weaker assumption of homogeneity, the function provides a comprehensive assessment of the scale’s underlying structure, guiding researchers in the appropriate use and interpretation of Cronbach’s alpha. This approach helps researchers to determine whether their scale fits the assumption of unidimensionality and understand the nature of relationships between the scale’s items and the measured construct.
Testing Tau-Equivalence and Item Homogeneity
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res <- coefficientalpha::tau.test(items[1:7])
Test of tau equivalent
The robust F statistic is 4.58
with a p-value 0
Test of homogeneous items
The robust F statistic is 2.77
with a p-value 8e-04
Model of the tau equivalent model (fixed factor loadings)
Robust Cronbach’s alpha is a version of the traditional reliability coefficient designed to assess internal consistency while reducing the effects of atypical or extreme data. In this approach, the data are evaluated using a weighting mechanism that assigns full weight to observations that fit well and lower weight to those that do not. This adjustment helps ensure that the resulting reliability measure reflects the overall consistency of the items without being overly influenced by outliers.
A central component of this method is the “downweight rate,” often controlled by a parameter called varphi. Varphi determines how strongly the algorithm reduces the influence of observations that deviate from the norm. In other words, if some responses are unusual or do not follow the general pattern, they are downweighted, meaning their impact on the final reliability estimate is lessened. This results in a more stable and realistic evaluation of the internal consistency of the scale or test.
In sum, robust Cronbach’s alpha provides a way to measure reliability that is less sensitive to irregular data by effectively managing the influence of outlying observations through a controlled weighting process.
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# Calculating robust coefficient alpha using coefficientalpha::alpha()cat("Robust coefficient alpha")
Test of tau-equavilence
The robust F statistic is 4.58
with a p-value 0
**The F test rejected tau-equavilence**
The alpha is 0.801.
About 20.9% of cases were downweighted.
Estimation of the Confidence Interval for Robust Cronbach’s Alpha
The estimated alpha is 0.805
Its bootstrap se is 0.018
Its bootstrap confidence interval is [ 0.768 , 0.833 ]
Cross-Validation of Cronbach’s Alpha
Cross-validation is a statistical technique used to assess the stability and generalizability of a measurement’s reliability, such as Cronbach’s alpha. Instead of calculating the reliability coefficient on a single sample, the dataset is partitioned into multiple folds, and Cronbach’s alpha is computed for each subsample. This approach minimizes the impact of sample-specific peculiarities and reduces the likelihood of overfitting, thereby providing a more robust estimate of internal consistency.
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# Load required packages# Function to compute Cronbach's alpha using cross-validation# Parameters:# data: A dataframe that contains at least 7 columns representing items# n_folds: Number of cross-validation folds (default: 5)# seed: Seed for reproducibility (default: 123)calculate_cronbach_cv <-function(data, n_folds =5, seed =123) {# Set the seed for reproducibilityset.seed(seed)# Create cross-validation folds based on row indices fold_indices <-createFolds(1:nrow(data), k = n_folds)# Initialize a vector to store Cronbach's alpha values for each fold alpha_values <-numeric(length(fold_indices))# Loop over each fold to compute Cronbach's alphafor (i inseq_along(fold_indices)) {# Subset the data for the current fold using the first 7 columns (items) fold_data <- data[fold_indices[[i]], 1:7]# Compute Cronbach's alpha using psych::alpha (ensuring to check keys) alpha_out <- psych::alpha(fold_data, check.keys =TRUE)# Store the obtained Cronbach's alpha value for the current fold alpha_values[i] <- alpha_out$total$raw_alpha# Print the result for the current foldcat("Fold", i, "- Cronbach's alpha:", alpha_out$total$raw_alpha, "\n") }# Calculate the average Cronbach's alpha across all folds mean_alpha <-mean(alpha_values)cat("Mean Cronbach's alpha across", n_folds, "folds:", mean_alpha, "\n")# Return a list with individual fold alphas and the mean alphareturn(list(fold_alphas = alpha_values, mean_alpha = mean_alpha))}# Example usage:# Assuming 'items' is your dataframe with at least 253 rows and 7 (or more) columns.result <-calculate_cronbach_cv(items, n_folds =5)
By employing cross-validation, any variability in the reliability measure across different subsamples becomes apparent. For instance, when evaluating the results — Fold 1 (0.862), Fold 2 (0.834), Fold 3 (0.728), Fold 4 (0.735), and Fold 5 (0.872) — we observe some differences among the folds. Such variation suggests that while the scale consistently demonstrates acceptable reliability overall, individual subsamples may exhibit slight discrepancies in internal consistency. The mean Cronbach’s alpha across the folds is 0.806, which generally indicates a reliable scale.
Overall, cross-validation improves the reliability analysis by providing insights into the consistency of the measurement across various segments of the data. This method not only yields a more generalizable estimate by averaging the alpha values across folds but also highlights potential variability that could be critical for improving the instrument’s performance.
Sensitivity Analysis of Cronbach’s Alpha via Data Perturbation
Measurement error is a perennial concern in psychometric research. Even small fluctuations in item responses (e.g., due to respondent fatigue or ambiguity) may affect reliability estimates. The following function introduces controlled perturbations (via additive Gaussian noise) to your dataset, recalculating Cronbach’s alpha over many iterations. The output is a distribution of alpha values from which you can derive a “sensitivity index”—a measure of how robust your reliability estimate is to minor data fluctuations.
The ‘sensitivity_index’ (the range of collected alpha values) provides a numerical assessment of the reliability estimate’s robustness.
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alphaSensitivityAnalysis <-function(data, noise_level =0.05, R =500, seed =NULL) {if (!is.null(seed)) set.seed(seed) data <-as.data.frame(data) n_items <-ncol(data)# Simple Cronbach's alpha function (unweighted, standard version) computeAlpha <-function(df) { k <-ncol(df) total_scores <-rowSums(df) item_vars <-apply(df, 2, var, na.rm =TRUE) (k/(k -1)) * (1-sum(item_vars) /var(total_scores, na.rm =TRUE)) } base_alpha <-computeAlpha(data) alpha_vals <-numeric(R)# Apply perturbations across R replicationsfor (i in1:R) { perturbed_data <- datafor (j in1:n_items) {# Add random noise scaled by noise_level * (item SD) sd_item <-sd(data[[j]], na.rm =TRUE) perturbed_data[[j]] <- data[[j]] +rnorm(nrow(data), mean =0, sd = noise_level * sd_item) } alpha_vals[i] <-computeAlpha(perturbed_data) }# Define a sensitivity index as the range of alpha values from the perturbations sensitivity_index <-max(alpha_vals) -min(alpha_vals)list(base_alpha = base_alpha,alpha_distribution = alpha_vals,sensitivity_index = sensitivity_index )}sensitivity <-alphaSensitivityAnalysis(items[1:7], noise_level =0.1, R =1000, seed =101)cat("Base Cronbach's Alpha:", round(sensitivity$base_alpha, 3), "\n")
hist(sensitivity$alpha_distribution,main ="Distribution of Perturbed Cronbach's Alpha",xlab ="Cronbach's Alpha Values",ylab ="Frequency",col ="skyblue",border ="black",las =1# Ensure labels are horizontal)
Interpretation of Base Alpha Value
The Cronbach’s alpha value of 0.825, obtained from the initial data, indicates a high level of internal consistency, signifying a reliable measurement instrument. Typically, in social sciences, values above 0.8 are considered very good, implying that the test items consistently measure the same concept or construct. The sensitivity of 0.01022825 indicates that the variations in Cronbach’s alpha when adding a small level of noise (perturbation) to the data are very stable, meaning that minor fluctuations in the data do not significantly impact the final result.
Comparison of Cronbach’s Alphas in Independent Samples: cocron::cocron.n.coefficients()
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# Adding gender as a factor to the datasetitems$gender <-factor(df$Sex, levels =c(1, 2), labels =c("women", "men"))# Gender Distributioncat("Gender Distribution:\n")
Gender Distribution:
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print(table(items$gender))
women men
131 122
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# Subset data for women# Calculating Cronbach's Alpha for Womenwomen_data <-subset(items, gender =="women")CA_women <- psych::alpha(women_data[, 1:7])# Display results for women# Cronbach's Alpha for Womencat("Cronbach's Alpha for Women:", round(CA_women$total$raw_alpha, 3), "\n")
Cronbach's Alpha for Women: 0.804
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# Print the message indicating the 95% confidence interval for womencat("95% Confidence Interval for Women\n")
95% Confidence Interval for Women
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# Display the lower bound of the confidence intervalcat("Lower bound:")
Lower bound:
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round(CA_women$feldt$lower.ci, 3)
raw_alpha
0.747
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# Display the upper bound of the confidence intervalcat("Upper bound:")
Upper bound:
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round(CA_women$feldt$upper.ci, 3)
raw_alpha
0.851
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# Calculating Cronbach's Alpha for Menmen_data <-subset(items, gender =="men")CA_men <- psych::alpha(men_data[, 1:7])# Cronbach's Alpha for Mencat("Cronbach's Alpha for Men:", round(CA_men$total$raw_alpha, 3), "\n")
Cronbach's Alpha for Men: 0.837
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# 95% Confidence Interval for Men# Print the message indicating the 95% confidence interval for mencat("95% Confidence Interval for Men\n")
95% Confidence Interval for Men
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# Display the lower bound of the confidence interval cat("Lower bound:")
Lower bound:
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round(CA_men$feldt$lower.ci, 3)
raw_alpha
0.788
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# Display the upper bound of the confidence interval cat("Upper bound:")
Upper bound:
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round(CA_men$feldt$upper.ci, 3)
raw_alpha
0.877
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# Comparing Cronbach's Alpha Coefficients Between Women and Mencomparison_result <- cocron::cocron.n.coefficients(alpha =c(CA_women$total$raw_alpha, CA_men$total$raw_alpha),n =c(nrow(women_data), nrow(men_data)),items =c(7, 7),dep =FALSE,los =0.05,conf.level =0.95)# Display comparison resultscat("Comparison of Cronbach's Alpha Coefficients (Women vs. Men):\n")
Comparison of Cronbach's Alpha Coefficients (Women vs. Men):
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print(comparison_result)
Compare n alpha coefficients
Comparison between: a1 = 0.8035, a2 = 0.8365
The coefficients are based on independent groups
95% confidence intervals: CI1 = 0.7475 0.8509, CI2 = 0.7880 0.8773
Group sizes: n1 = 131, n2 = 122
Item count: i1 = 7, i2 = 7
Null hypothesis: a1 and a2 are equal
Alternative hypothesis: a1 and a2 are not equal
Level of significance: 0.05
chisq = 0.7898, df = 1, p-value = 0.3742
Null hypothesis retained
Overall, Cronbach’s Alpha remains a cornerstone in measurement theory, influencing both the design and evaluation of assessments in a variety of disciplines. Its ease of calculation and interpretation has established it as a standard practice among researchers, despite ongoing discussions about its limitations and the need for comprehensive reliability assessment methodologies.
Despite known violations of tau-equivalence in most psychological measures, Cronbach’s alpha continues to be widely reported in research for several reasons. First, it has been the dominant reliability coefficient in psychological research for over half a century, making it a de facto standard for peer review and cross-study comparisons. Second, its widespread use facilitates meta-analytic studies and systematic reviews, as it allows for direct comparison of reliability estimates across different studies and time periods. Third, while alpha may underestimate reliability when tau-equivalence is violated, it is often viewed as providing a conservative lower-bound estimate of reliability. Finally, many journals and reporting guidelines still specifically request Cronbach’s alpha, reflecting its deeply entrenched position in research methodology, even as more sophisticated reliability coefficients become available.
References
Бова, А. А. (2023). Шкала страху перед COVID-19 (FCV-19S): факторна валідність та надійність на вибірці київських студентів. Габітус, 46, 11-23 [in Ukrainian]. https://doi.org/10.32782/2663-5208.2023.46.1
Ahorsu, D. K., Lin, C.-Y., Imani, V., Saffari, M., Griffiths, M. D., & Pakpour, A. H. (2020). The Fear of COVID-19 Scale: Development and Initial Validation. International Journal of Mental Health and Addiction. https://doi.org/10.1007/s11469-020-00270-8
Citation
BibTeX citation:
@online{bova2025,
author = {Bova, Andrii},
title = {Cronbach’s {Alpha} in {R:} {A} {Set} of {Useful} {Functions}},
date = {2025-01-19},
url = {https://www.rpubs.com/abova/alpha-cronbach},
langid = {en}
}
