Description Usage Arguments Details Value Note Author(s) References See Also Examples

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).

1 2 3 | ```
headrick.sheng.lalpha(x, ...)
lalpha(x, ...)
``` |

`x` |
An |

`...` |
Additional arguments to pass. |

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 *i*th examinee on the *j*th 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.

An **R** `list`

is returned.

`alpha` |
The |

`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”. |

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`

.

W.H. Asquith

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | ```
# 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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