CMC (version 1.0)

alpha.curve: Step-by-step Cronbach-Mesbah Curve

Description

The function calculates and plots the Cronbach-Mesbah Curve for a given data set.

Usage

alpha.curve(x)

Arguments

x

an object of class data.frame or matrix with $n$ subjects in the rows and $k$ items in the columns.

Value

• The functions returns: 1) an object of class data.frame with 3 columns. The first column N.Item contains the number of item used for computing the Cronbach $\alpha$ coefficients. It contains the values between 2 and $k$ corresponding, respectively, to the case when only 2 items or all the items are used. The second column, Alpha.Max, refers to the maximums of the Cronbach coefficients calculated at each step of the procedure, that is $(\max\tilde\alpha^{k-2},\ldots,\max\tilde\alpha^1,\max\tilde\alpha^0)$. Finally, the last column, Removed.Item, reports the name of the item removed at each step, that is $(arg\max\tilde\alpha^{k-2},\ldots,arg\max\tilde\alpha^1,arg\max\tilde\alpha^0)$. 2) The corresponding Cronbach-Mesbah curve plot created using the first 2 columns of the data.frame described above. Note that also the names of the removed items are reported in the graph.

Warning

No missing values are admitted.

Details

There is a direct connection between the Cronbach alpha coefficient $\alpha$ (see alpha.cronbach) and the percentage of variance explained by the first component in the Principal Component Analysis (PCA) on $k$ items. The PCA is usually based on the analysis of the roots of the correlation matrix R of $k$ variables which, under the hypothesis of a parallel model (see Lord and Novick, 1968) is:

lllll{ 1 $\rho$ ... $\rho$ $\rho$ 1 ... $\rho$ R = ... ... ... ... $\rho$ $\rho$ ... 1 }

This matrix has only two different roots. The greater root is $\lambda_1=1+\rho(k-1)$ and the other roots are $\lambda_2=\ldots=\lambda_k=1-\rho=\frac{k-\lambda_1}{k-1}$. Thus, using the Spearman-Brown formula, we can express the reliability of the sum of the $k$ items as follows:

$$\tilde \rho=\frac{k}{k-1}\left[1-\frac{1}{\lambda_1}\right].$$

This indicates that there is a monotonic relationship between $\tilde\rho$, estimated by $\alpha$, and the first root $\lambda_1$, which in practice is estimated using the observed correlation matrix and thus gives the percentage of variance of the first principal component. Then, $\alpha$ is considered as a measure of unidimensionality.

In particular, to assess the unidimensionality of a set of items, it is possible to plot a curve, called step-by-step Cronbach-Mesbah curve, which reports the number of items (from 2 to $k$) on the x-axis and the corresponding maximum $\alpha$ coefficient on the y-axis obtained through the following steps:

1. first of all the Cronbach coefficient$\alpha=\tilde\alpha^0$is computed using all the$k$items.
2. One at a time, the$i$-th item ($i=1,\ldots,k$) is left out and the Cronbach coefficient, denoted by$\alpha_{-i}$, is computed using the remaining$(k-1)$items. All the coefficients are collected in a set given by$$\tilde\alpha^1=\left(\alpha_{-1},\ldots,\alpha_{-j},\ldots,\alpha_{-k}\right)$$where the apex refers to the number of item removed at each time. Then, the maximum of$\tilde \alpha^1$is detected and the corresponding item is taken out. For example, if$\alpha_{-j}$is the maximum of$\tilde \alpha^1$, the$j$-th item is removed definitely from the scale.
3. The procedure of step 2 is repeated conditionally on the item removed previously. Supposing that item$j$was removed, the remaining items are left out one at a time and the corresponding Cronbach coefficient is calculated. This gives rise to the following set of$(k-1)$coefficients$$\tilde\alpha^2=\left(\alpha_{-(1,j)},\ldots,\alpha_{-(j-1,j)},\alpha_{-(j+1,j)},\ldots,\alpha_{-(k,j)}\right).$$The item corresponding to the maximum of$\tilde\alpha^2$is then removed definitely. For example, if$\alpha_{-1}$is the maximum of$\tilde \alpha^2$, the first item is removed definitely from the scale together with the$j$-th item removed at step 2.

This procedure is repeated until only 2 items remain. Note that at each step the removed item is the one which leaves the scale with its maximum $\alpha$ value. If we remove a poor item, the $\alpha$ coefficient will increase, whereas if we remove a good item $\alpha$ must decrease. More precisely, the Spearman-Brown formula shows that increasing the number of items leads to increase the reliability of the total score. Thus, a decrease of the Cronbach-Mesbah curve, after adding a variables, would suggest that the added variable do not constitute an unidimensional set with the other variables. On the other hand, if the step-by-step Cronbach-Mesbah curve increases monotonically, then all the items contribute to measure the same latent trait and the bank of items is characterized by unidimensionality.

References

Curt, F., Mesbah, M., Lellouch, J. and Dellatolas, G. (1997) Handedness scale how many and which items? Laterality, 2, 137--154.

Hamon, A. and Mesbah, M. (2002) Questionnaire reliability under the Rasch model. Statistical Methods for Quality of Life Studies: Design, Measurement and Analysis. Mesbah, M., Cole, B.F. and Lee, M.L.T. (Eds.), Kluwer Academic Publishing, Boston, 155--168.

Mesbah, M. (2010) Statistical quality of life. Method and Applications of Statistics in the Life and Health Sciences. Balakrishnan, N. (Editor), Wiley, 839--864.

Nordmann, J., Mesbah, M., Berdeaux, G. (2005) Scoring of Visual Field Measured through Humphrey Perimetry: Principal Component Varimax Rotation Followed by Validated Cluster Analysis. Investigative Ophthalmology & Visual Science 46, 3169--3176.

See Also

See Also alpha.cronbach and cain

Examples

data(cain)

out = alpha.curve(cain)
out