# stratified.cronbach.alpha: Stratified Cronbach's Alpha In sirt: Supplementary Item Response Theory Models

## Description

This function computes the stratified Cronbach's Alpha for composite scales (Cronbach, Schoenemann & McKie, 1965; Meyer, 2010).

## Usage

 1 stratified.cronbach.alpha(data, itemstrata=NULL) 

## Arguments

 data An N \times I data frame itemstrata A matrix with two columns defining the item stratification. The first column contains the item names, the second column the item stratification label (these can be integers). The default NULL does only compute Cronbach's Alpha for the whole scale.

## Author(s)

Alexander Robitzsch

## References

Cronbach, L.J., Schoenemann, P., & McKie, D. (1965). Alpha coefficient for stratified-parallel tests. Educational and Psychological Measurement, 25, 291-312.

Meyer, P. (2010). Reliability. Cambridge: Oxford University Press.

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 ############################################################################# # EXAMPLE 1: data.read ############################################################################# data( data.read ) dat <- data.read I <- ncol(dat) # apply function without defining item strata stratified.cronbach.alpha( data.read ) # define item strata itemstrata <- cbind( colnames(dat) , substring( colnames(dat) , 1 ,1 ) ) stratified.cronbach.alpha( data.read , itemstrata=itemstrata ) ## scale I alpha mean.tot var.tot alpha.stratified ## 1 total 12 0.677 8.680 5.668 0.703 ## 2 A 4 0.545 2.616 1.381 NA ## 3 B 4 0.381 2.811 1.059 NA ## 4 C 4 0.640 3.253 1.107 NA ## Not run: #************************** # reliability analysis in psych package library(psych) # Cronbach's alpha and item discriminations psych::alpha( dat ) # McDonald's omega psych::omega(dat , nfactors=1) # 1 factor ## Alpha: 0.69 ## Omega Total 0.69 ## => Note that alpha in this function is the standardized Cronbach's ## alpha, i.e. alpha computed for standardized variables. psych::omega(dat , nfactors=2) # 2 factors ## Omega Total 0.72 psych::omega(dat , nfactors=3) # 3 factors ## Omega Total 0.74 ## End(Not run) 

### Example output

- sirt 2.0-25 (2017-05-11 13:52:31)
scale  I alpha mean.tot var.tot alpha.stratified
1 total 12 0.677     8.68   5.668            0.677
scale  I alpha mean.tot var.tot alpha.stratified
1 total 12 0.677    8.680   5.668            0.703
2     A  4 0.545    2.616   1.381               NA
3     B  4 0.381    2.811   1.059               NA
4     C  4 0.640    3.253   1.107               NA

Reliability analysis
Call: psych::alpha(x = dat)

raw_alpha std.alpha G6(smc) average_r S/N   ase mean  sd
0.68      0.69     0.7      0.15 2.2 0.026 0.72 0.2

lower alpha upper     95% confidence boundaries
0.63 0.68 0.73

Reliability if an item is dropped:
raw_alpha std.alpha G6(smc) average_r S/N alpha se
A1      0.66      0.68    0.69      0.16 2.1    0.027
A2      0.64      0.66    0.67      0.15 1.9    0.029
A3      0.65      0.66    0.68      0.15 2.0    0.029
A4      0.66      0.67    0.69      0.16 2.1    0.028
B1      0.67      0.68    0.70      0.16 2.2    0.026
B2      0.67      0.68    0.69      0.16 2.1    0.027
B3      0.67      0.68    0.69      0.16 2.1    0.027
B4      0.65      0.66    0.68      0.15 2.0    0.028
C1      0.66      0.66    0.66      0.15 2.0    0.028
C2      0.64      0.65    0.67      0.15 1.9    0.029
C3      0.66      0.66    0.66      0.15 2.0    0.028
C4      0.66      0.67    0.68      0.15 2.0    0.027

Item statistics
n raw.r std.r r.cor r.drop mean   sd
A1 328  0.43  0.42  0.33   0.29 0.85 0.36
A2 328  0.55  0.54  0.49   0.40 0.74 0.44
A3 328  0.54  0.50  0.43   0.37 0.57 0.50
A4 328  0.49  0.45  0.35   0.31 0.46 0.50
B1 328  0.40  0.38  0.26   0.22 0.71 0.45
B2 328  0.44  0.41  0.30   0.25 0.51 0.50
B3 328  0.37  0.42  0.32   0.26 0.91 0.29
B4 328  0.53  0.51  0.43   0.37 0.68 0.47
C1 328  0.44  0.51  0.48   0.34 0.93 0.25
C2 328  0.55  0.55  0.49   0.40 0.71 0.45
C3 328  0.44  0.51  0.47   0.32 0.87 0.33
C4 328  0.47  0.48  0.41   0.30 0.73 0.44

Non missing response frequency for each item
0    1 miss
A1 0.15 0.85    0
A2 0.26 0.74    0
A3 0.43 0.57    0
A4 0.54 0.46    0
B1 0.29 0.71    0
B2 0.49 0.51    0
B3 0.09 0.91    0
B4 0.32 0.68    0
C1 0.07 0.93    0
C2 0.29 0.71    0
C3 0.13 0.87    0
C4 0.27 0.73    0
Omega_h for 1 factor is not meaningful, just omega_t
Omega
Call: psych::omega(m = dat, nfactors = 1)
Alpha:                 0.69
G.6:                   0.7
Omega Hierarchical:    0.69
Omega H asymptotic:    0.99
Omega Total            0.69

g  F1*   h2   u2 p2
A1 0.30      0.09 0.91  1
A2 0.47      0.22 0.78  1
A3 0.41      0.17 0.83  1
A4 0.33      0.11 0.89  1
B1 0.24      0.06 0.94  1
B2 0.28      0.08 0.92  1
B3 0.32      0.10 0.90  1
B4 0.42      0.17 0.83  1
C1 0.50      0.25 0.75  1
C2 0.52      0.27 0.73  1
C3 0.49      0.24 0.76  1
C4 0.43      0.19 0.81  1

With eigenvalues of:
g F1*
1.9 0.0

general/max  1.652911e+16   max/min =   1
mean percent general =  1    with sd =  0 and cv of  0
Explained Common Variance of the general factor =  1

The degrees of freedom are 54  and the fit is  0.6
The number of observations was  328  with Chi Square =  192.67  with prob <  1.9e-17
The root mean square of the residuals is  0.08
The df corrected root mean square of the residuals is  0.09
RMSEA index =  0.09  and the 10 % confidence intervals are  0.075 0.102
BIC =  -120.15

Compare this with the adequacy of just a general factor and no group factors
The degrees of freedom for just the general factor are 54  and the fit is  0.6
The number of observations was  328  with Chi Square =  192.67  with prob <  1.9e-17
The root mean square of the residuals is  0.08
The df corrected root mean square of the residuals is  0.09

RMSEA index =  0.09  and the 10 % confidence intervals are  0.075 0.102
BIC =  -120.15

g F1*
Correlation of scores with factors            0.84   0
Multiple R square of scores with factors      0.71   0
Minimum correlation of factor score estimates 0.42  -1

Total, General and Subset omega for each subset
g  F1*
Omega total for total scores and subscales    0.69 0.69
Omega general for total scores and subscales  0.69 0.69
Omega group for total scores and subscales    0.00 0.00
Warning message:
In schmid(m, nfactors, fm, digits, rotate = rotate, n.obs = n.obs,  :
Omega_h and Omega_asymptotic are not meaningful with one factor

Three factors are required for identification -- general factor loadings set to be equal.
Proceed with caution.
Think about redoing the analysis with alternative values of the 'option' setting.

Omega
Call: psych::omega(m = dat, nfactors = 2)
Alpha:                 0.69
G.6:                   0.7
Omega Hierarchical:    0.3
Omega H asymptotic:    0.41
Omega Total            0.72

g   F1*   F2*   h2   u2   p2
A1              0.43 0.24 0.76 0.15
A2  0.30        0.45 0.30 0.70 0.30
A3  0.26        0.42 0.24 0.76 0.27
A4  0.21        0.36 0.17 0.83 0.25
B1              0.24 0.08 0.92 0.27
B2              0.26 0.10 0.90 0.30
B3  0.20             0.10 0.90 0.42
B4  0.26        0.41 0.24 0.76 0.29
C1  0.39  0.69       0.64 0.36 0.24
C2  0.32  0.29  0.25 0.25 0.75 0.41
C3  0.36  0.58       0.47 0.53 0.27
C4  0.29  0.38       0.23 0.77 0.35

With eigenvalues of:
g  F1*  F2*
0.86 1.10 1.10

general/max  0.78   max/min =   1
mean percent general =  0.29    with sd =  0.07 and cv of  0.25
Explained Common Variance of the general factor =  0.28

The degrees of freedom are 43  and the fit is  0.15
The number of observations was  328  with Chi Square =  47.06  with prob <  0.31
The root mean square of the residuals is  0.04
The df corrected root mean square of the residuals is  0.04
RMSEA index =  0.019  and the 10 % confidence intervals are  0 0.042
BIC =  -202.04

Compare this with the adequacy of just a general factor and no group factors
The degrees of freedom for just the general factor are 54  and the fit is  0.78
The number of observations was  328  with Chi Square =  251.83  with prob <  2.6e-27
The root mean square of the residuals is  0.12
The df corrected root mean square of the residuals is  0.14

RMSEA index =  0.107  and the 10 % confidence intervals are  0.093 0.119
BIC =  -60.99

g  F1*  F2*
Correlation of scores with factors             0.56 0.76 0.71
Multiple R square of scores with factors       0.32 0.58 0.50
Minimum correlation of factor score estimates -0.36 0.15 0.00

Total, General and Subset omega for each subset
g  F1*  F2*
Omega total for total scores and subscales    0.72 0.66 0.62
Omega general for total scores and subscales  0.30 0.23 0.16
Omega group for total scores and subscales    0.34 0.43 0.46
Omega
Call: psych::omega(m = dat, nfactors = 3)
Alpha:                 0.69
G.6:                   0.7
Omega Hierarchical:    0.32
Omega H asymptotic:    0.43
Omega Total            0.75

g   F1*   F2*   F3*   h2   u2   p2
A1              0.50       0.28 0.72 0.06
A2  0.28        0.45       0.29 0.71 0.28
A3  0.22        0.44       0.25 0.75 0.20
A4              0.39       0.18 0.82 0.16
B1              0.21       0.09 0.91 0.28
B2              0.26       0.10 0.90 0.28
B3  0.43              0.75 0.75 0.25 0.25
B4  0.25        0.40       0.23 0.77 0.26
C1  0.42  0.69             0.66 0.34 0.27
C2  0.31  0.30  0.26       0.26 0.74 0.39
C3  0.38  0.56             0.46 0.54 0.32
C4  0.29  0.38             0.24 0.76 0.35

With eigenvalues of:
g  F1*  F2*  F3*
0.99 1.02 1.16 0.62

general/max  0.85   max/min =   1.88
mean percent general =  0.26    with sd =  0.09 and cv of  0.34
Explained Common Variance of the general factor =  0.26

The degrees of freedom are 33  and the fit is  0.1
The number of observations was  328  with Chi Square =  32.03  with prob <  0.52
The root mean square of the residuals is  0.03
The df corrected root mean square of the residuals is  0.04
RMSEA index =  0  and the 10 % confidence intervals are  0 0.039
BIC =  -159.14

Compare this with the adequacy of just a general factor and no group factors
The degrees of freedom for just the general factor are 54  and the fit is  0.79
The number of observations was  328  with Chi Square =  255.24  with prob <  6.6e-28
The root mean square of the residuals is  0.12
The df corrected root mean square of the residuals is  0.14

RMSEA index =  0.108  and the 10 % confidence intervals are  0.094 0.12
BIC =  -57.58

g  F1*  F2*  F3*
Correlation of scores with factors             0.63 0.75 0.74 0.78
Multiple R square of scores with factors       0.40 0.57 0.55 0.60
Minimum correlation of factor score estimates -0.20 0.14 0.10 0.20

Total, General and Subset omega for each subset
g  F1*  F2*  F3*
Omega total for total scores and subscales    0.75 0.69 0.61 0.75
Omega general for total scores and subscales  0.32 0.24 0.13 0.19
Omega group for total scores and subscales    0.35 0.45 0.48 0.57


sirt documentation built on May 29, 2017, 11:32 p.m.

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