lmomco (version 2.3.7)

headrick.sheng.lalpha: The Sample Headrick and Sheng L-alpha

Description

Compute the sample Headrick and Sheng “L-alpha” (Headrick and Sheng, 2013) by $$\alpha_L = \frac{d}{d-1} \biggl(1 - \frac{\sum_j \lambda^{(j)}_2}{\sum_j \lambda^{(j)}_2 + \sum\sum_{j\ne j'} \lambda_2^{(jj')}} \biggr)\mbox{,}$$ where \(j = 1,\ldots,d\) for dimensions \(d\), the \(\sum_j \lambda^{(j)}_2\) is the summation of all the 2nd order (univariate) L-moments (L-scales, \(\lambda^{(j)}_2\)), and the double summation is the summation of all the 2nd order L-comoments (\(\lambda_2^{(jj')}\)). 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

headrick.sheng.lalpha(x, ...)
lalpha(x, ...)

Value

An R

list is returned.

alpha

The \(\alpha_L\) 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”.

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.

Author

W.H. Asquith

Details

Headrick and Sheng (2013) propose \(\alpha_L\) to be an alternative estimator of reliability based on L-comoments. They describe its context as follows: “Consider [a statistic] alpha (\(\alpha\)) 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 \(\epsilon_{ij}\).” And the authors continue “The model can be summarized as \(X_{ij} = t_i + \epsilon_{ij}\mbox{,}\) where \(X_{ij}\) is the observed score associated with the \(i\)th examinee on the \(j\)th test item, and where \(i = 1,...,n\) [for sample size \(n\)]; \(j = 1,\ldots,d\); and the error terms (\(\epsilon_{ij}\)) 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 \(\alpha_L\) can be “substantially superior” to [a different formulation of \(\alpha\) (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 \(\lambda_2(X_j) = 2\mathrm{Cov}(X_j,F(X_j))\) and \(\lambda_2(X_{j'}) = 2\mathrm{Cov}(X_{j'},F(X_{j'}))\). 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 \(\lambda_2^{(jj')} = 2\mathrm{Cov}(X_j,F(X_{j'}))\) and \(\lambda_2^{(j'j)} = 2\mathrm{Cov}(X_{j'},F(X_j))\). 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 \(\alpha_L\) will not be identical as the example shows.

References

Headrick, T.C., and Sheng, Y., 2013, An alternative to Cronbach's Alpha---An L-moment-based measure of internal-consistency reliability: in Millsap, R.E., van der Ark, L.A., Bolt, D.M., Woods, C.M. (eds) New Developments in Quantitative Psychology, Springer Proceedings in Mathematics and Statistics, v. 66, tools:::Rd_expr_doi("10.1007/978-1-4614-9348-8_2").

See Also

Lcomoment.matrix

Examples

# 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