The Sample Headrick and Sheng L-alpha

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Description

Compute the sample Headrick and Sheng “L-alpha” (Headrick and Sheng, 2013) by

α_L = \frac{d}{d-1} \biggl(1 - \frac{∑_j λ^{(j)}_2}{∑_j λ^{(j)}_2 + ∑∑_{j\ne j'} λ_2^{(jj')}} \biggr)\mbox{,}

where j = 1,…,d for dimensions d, the ∑_j λ^{(j)}_2 is the summation of all the 2nd order (univariate) L-moments (L-scales, λ^{(j)}_2), and the double summation is the summation of all the 2nd order L-comoments (λ_2^{(jj')}). In other words, the double summation is the sum of all entries in both the lower and upper triangles (not the primary diagonal) of the L-comoment matrix (the L-scale and L-coscale [L-covariance] matrix).

Usage

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Arguments

x

An R data.frame of the random observations for the d random variables X, which must be suitable for internal dispatch to the Lcomoment.matrix function for the k=2 order L-comoment. Alternatively, x can be a precomputed 2nd order L-comoment matrix (L-scale and L-coscale matrix).

...

Additional arguments to pass.

Details

Headrick and Sheng (2013) propose α_L to be an alternative estimator of reliability based on L-comoments. They describe its context as follows: “Consider [a statistic] alpha (α) in terms of a model that decomposes an observed score into the sum of two independent components: a true unobservable score t_i and a random error component ε_{ij}.” And the authors continue “The model can be summarized as X_{ij} = t_i + ε_{ij}\mbox{,} where X_{ij} is the observed score associated with the ith examinee on the jth test item, and where i = 1,...,n [for sample size n]; j = 1,…,d; and the error terms (ε_{ij}) are independent with a mean of zero.” The authors go on to observe that “inspection of [this model] indicates that this particular model restricts the true score t_i to be the same across all d test items.”

Headrick and Sheng (2013) show empirical results for a simulation study, which indicate that α_L can be “substantially superior” to [a different formulation of α (Cronbach's Alpha) based on product moments (the variance-covariance matrix)] in “terms of relative bias and relative standard error when distributions are heavy-tailed and sample sizes are small.”

The authors remind the reader that the second L-moments associated with X_j and X_{j'} can alternatively be expressed as λ_2(X_j) = 2\mathrm{Cov}(X_j,F(X_j)) and λ_2(X_{j'}) = 2\mathrm{Cov}(X_{j'},F(X_{j'})). And that the second L-comoments of X_j toward (with respect to) X_{j'} and X_{j'} toward (with respect to) X_j are λ_2^{(jj')} = 2\mathrm{Cov}(X_j,F(X_{j'})) and λ_2^{(j'j)} = 2\mathrm{Cov}(X_{j'},F(X_j)). The respective cumulative distribution functions are denoted F(x_j). Evidently the authors present the L-moments and L-comoments this way because their first example (thanks for detailed numerics!) already contain nonexceedance probabilities. Thus the function headrick.sheng.lalpha is prepared for two different contents of the x argument. One for a situation in which only the value for the random variables are available, and one for a situation in which the nonexceedances are already available. The numerically the two α_L will not be identical as the example shows.

Value

An R list is returned.

alpha

The α_L statistic.

title

The formal name “Headrick and Sheng L-alpha”.

source

An attribute identifying the computational source of the Headrick and Sheng L-alpha: “headrick.sheng.lalpha”.

Note

Headrick and Sheng (2013) use k to represent d as used here. The change is made because k is an L-comoment order argument already in use by Lcomoment.matrix.

Author(s)

W.H. Asquith

References

Headrick, T.C. and Sheng, Y., 2013, An alternative to Cronbach's Alpha—A L-moment based measure of internal-consistency reliability: Book Chapters, Paper 1, http://opensiuc.lib.siu.edu/epse_books/1

See Also

Lcomoment.matrix

Examples

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# Table 1 in Headrick and Sheng (2013)
TV1 <- # Observations in cols 1:3, estimated nonexceedance probabilities in cols 4:6
c(2, 4, 3, 0.15, 0.45, 0.15,       5, 7, 7, 0.75, 0.95, 1.00,
  3, 5, 5, 0.35, 0.65, 0.40,       6, 6, 6, 0.90, 0.80, 0.75,
  7, 7, 6, 1.00, 0.95, 0.75,       5, 2, 6, 0.75, 0.10, 0.75,
  2, 3, 3, 0.15, 0.25, 0.15,       4, 3, 6, 0.55, 0.25, 0.75,
  3, 5, 5, 0.35, 0.65, 0.40,       4, 4, 5, 0.55, 0.45, 0.40)
T1 <- matrix(ncol=6, nrow=10)
for(r in seq(1,length(TV1), by=6)) T1[(r/6)+1, ] <- TV1[r:(r+5)]
colnames(T1) <- c("X1", "X2", "X3", "FX1", "FX2", "FX3"); T1 <- as.data.frame(T1)

lco2 <- matrix(nrow=3, ncol=3)
lco2[1,1] <- lmoms(T1$X1)$lambdas[2]
lco2[2,2] <- lmoms(T1$X2)$lambdas[2]
lco2[3,3] <- lmoms(T1$X3)$lambdas[2]
lco2[1,2] <- 2*cov(T1$X1, T1$FX2); lco2[1,3] <- 2*cov(T1$X1, T1$FX3)
lco2[2,1] <- 2*cov(T1$X2, T1$FX1); lco2[2,3] <- 2*cov(T1$X2, T1$FX3)
lco2[3,1] <- 2*cov(T1$X3, T1$FX1); lco2[3,2] <- 2*cov(T1$X3, T1$FX2)
headrick.sheng.lalpha(lco2)$alpha     # Headrick and Sheng (2013): alpha = 0.807
# 0.8074766
headrick.sheng.lalpha(T1[,1:3])$alpha #              FXs not used: alpha = 0.781
# 0.7805825

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