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Kuder-Richardson Formula 20 (KR-20) With a Confidence Interval

Source: R/reliability_kr20.R
reliability_kr20.Rd

Estimates Kuder-Richardson formula 20 (Kuder & Richardson, 1937) for a homogeneous composite scored on dichotomous (0/1) items, and returns a confidence interval for the population coefficient.

Usage

reliability_kr20(
  data,
  ci_method = c("feldt", "bonett", "fisher", "hakstian_whalen", "ml", "ml_logistic",
    "adf", "adf_logistic", "bootstrap_se", "bootstrap_se_logistic", "percentile", "bca",
    "none"),
  conf_level = 0.95,
  B = 10000,
  seed = NULL
)

Arguments

data

A numeric matrix or data frame of 0/1 item scores (rows are respondents, columns are items). Rows with any missing values are listwise-deleted. Non-binary values trigger an error.

ci_method

Method for constructing the confidence interval; see reliability_alpha for the full list. Defaults to "feldt".

conf_level

Confidence level for the interval (1 - Type I error rate). Defaults to 0.95.

B

Number of bootstrap replications when a bootstrap method is selected. Defaults to 10000.

seed

Random number seed used for bootstrap reproducibility. Defaults to NULL, which leaves the user's current RNG state intact; supply an integer for reproducibility.

Value

A data.frame with columns term and value and rows "estimate" (sample KR-20), "se" (standard error on the coefficient scale, NA for methods that do not produce one; for the transformation-based intervals "fisher", "bonett", and "hakstian_whalen" it is the delta method back-transform of the transformation-scale standard error), "se_transformed" (only for those transformation-based intervals: the standard error on the transformation scale, with the scale named in the attribute se_transform_scale: "fisher_z", "log(1-alpha)", or "cube_root"), "lower_limit" and "upper_limit" (clamped to [0, 1]), "conf_level", "N" (effective sample size after listwise deletion), "N_complete" (the complete cases; equal to "N" here, and carried so the whole reliability family returns one shape), and "J" (number of items). Attributes coefficient ("kr20") and ci_method record the computation; bootstrap calls also record B.

Details

Kuder and Richardson's (1937) formula 20 for a \(J\)-item composite of binary items is $$KR_{20} = \frac{J}{J-1}\left(1 - \frac{\sum_{j} p_{j} q_{j}}{s_{Y}^{2}}\right),$$ where \(p_{j}\) is the proportion of respondents endorsing item j (i.e., scoring 1), \(q_{j} = 1 - p_{j}\), and \(s_{Y}^{2}\) is the variance of the composite score. For dichotomous items \(p_{j} q_{j}\) is the item variance, so KR-20 is algebraically identical to coefficient \(\alpha\) (Guttman, 1945; Cronbach, 1951) computed from the item covariance matrix. KR-20 predates coefficient \(\alpha\) by 14 years and Feldt's (1965) F-distribution interval was derived specifically for the sampling distribution of KR-20.

Because KR-20 is a special case of coefficient \(\alpha\) (Guttman, 1945; Cronbach, 1951), the same considerations apply: KR-20 equals the population reliability of the composite under essential \(\tau\)-equivalence (i.e., equal factor loadings) and serves as a lower bound otherwise. For dichotomous items see also reliability_omega_categorical (categorical omega), which treats the items via a probit-link single-factor model and does not assume equal loadings.

Available confidence interval methods (set via ci_method) are the same as for reliability_alpha; see that function's Details for full descriptions. The default for KR-20 is "feldt", the F-distribution interval originally developed for KR-20.

The menu also includes the nonparametric bootstrap intervals "percentile", "bca", "bootstrap_se", and "bootstrap_se_logistic", which resample the rows of data with replacement B times and recompute KR-20 on each replication (Efron & Tibshirani, 1993). They are worth the cost when the closed forms are least trustworthy, which for dichotomous items means highly unbalanced item difficulties or a sample size too small for the normal-theory derivations behind "feldt" and "bonett". No bootstrap runs unless ci_method asks for one; when it does, the default is B = 10000 replications, and supplying seed makes the interval reproducible.

Comparison with other packages. The psych package computes the same quantity via alpha (since \(\alpha\) on 0/1 data is KR-20). reliability_kr20 restricts input to raw 0/1 data so the dichotomous-items assumption cannot be quietly violated, presents the historical formula in the documentation, and accompanies the point estimate with a confidence interval drawn from the methods compared in Kelley and Pornprasertmanit (2016).

References

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297–334.

Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. New York, NY: Chapman & Hall/CRC.

Feldt, L. S. (1965). The approximate sampling distribution of Kuder-Richardson reliability coefficient twenty. Psychometrika, 30, 357–370.

Guttman, L. (1945). A basis for analyzing test-retest reliability. Psychometrika, 10(4), 255–282.

Kelley, K., & Cheng, Y. (2012). Estimation of and confidence interval formation for reliability coefficients of homogeneous measurement instruments. Methodology, 8, 39–50. doi:10.1027/1614-2241/a000036

Kelley, K., & Pornprasertmanit, S. (2016). Confidence intervals for population reliability coefficients: Evaluation of methods, recommendations, and software for composite measures. Psychological Methods, 21, 69–92. doi:10.1037/a0040086

Kuder, G. F., & Richardson, M. W. (1937). The theory of the estimation of test reliability. Psychometrika, 2, 151–160.

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge.

Terry, L. J., & Kelley, K. (2012). Sample size planning for composite reliability coefficients: Accuracy in parameter estimation via narrow confidence intervals. British Journal of Mathematical and Statistical Psychology, 65, 371–401. doi:10.1111/j.2044-8317.2011.02030.x

See also

reliability (general wrapper), reliability_alpha, reliability_omega_categorical, alpha.

Other reliability: cohen_kappa(), diagnosis_agreement, fleiss_kappa(), icc(), reliability(), reliability_H(), reliability_alpha(), reliability_omega(), reliability_omega_categorical()

Author

Ken Kelley kkelley@nd.edu

Examples

set.seed(113)
# Ten dichotomous items with a single underlying ability.
N <- 300
J <- 10
ability <- rnorm(N)
loadings <- rep(0.6, J)
latent <- outer(ability, loadings) +
          matrix(rnorm(N * J, sd = sqrt(1 - 0.6^2)), N, J)
items <- (latent > 0) * 1
colnames(items) <- paste0("y", seq_len(J))

reliability_kr20(data = items)
#>  term        value
#>  estimate    0.774
#>  se          <NA> 
#>  lower_limit 0.734
#>  upper_limit 0.81 
#>  conf_level  0.95 
#>  N           300  
#>  N_complete  300  
#>  J           10   
reliability_kr20(data = items, ci_method = "bonett")
#>  term           value 
#>  estimate       0.774 
#>  se             0.0195
#>  se_transformed 0.0864
#>  lower_limit    0.732 
#>  upper_limit    0.809 
#>  conf_level     0.95  
#>  N              300   
#>  N_complete     300   
#>  J              10    

# A bootstrap interval recomputes KR-20 on each of B resamples of the
# rows, so it is shown rather than run; the call is
#   reliability_kr20(data = items, ci_method = "percentile",
#                    B = 10000, seed = 113)
# and the default B = 10000 is what a reported interval deserves.