Callender and Osburn (A Method for Maximizing Split-Half Reliability Coefficients, Educational

1

Estimating Maximized Lambda4

Callender and Osburn (A method for maximizing split-half reliability coefficients, Educational and Psychological Measurement, 1977, 37, 819-825) described a method of estimating the maximized 4 . Although tedious, it is not completely unreasonable when the number of items is about 10. The trick is to find the particular split-half which is most likely to maximize 4. Look at my program Lambda4.sas. This program uses the idealism scale data discussed in my handout Cronbach's Alpha and Maximized Lambda4. Proc Corr is used to obtain the item covariances. I then use these covariances to create a 10 x 10 matrix of my 10 items, arranged such that the Spearman-Brown corrected correlation between the sum of the scores on the items in the first five rows and the sum of the scores on the remaining five items will be a good estimate of 4. This is rather tedious, as you will see from my outline of the solution below.

Step 1: Put x,y into row 1, col 10, where x,y is the pair of questions that has the greatest covariance. For the idealism instrument, that was questions 4,5.

Step 2: Put x,y into row 2, col 9, where x,y is the pair of questions that maximizes the covariances in the shaded cells -- that is, 5,x + x,y + 4,y

C1
q4 / C2 / C3 / C4 / C5 / C6 / C7 / C8 / C9 / C10
q5
R1 q4 / 4,y / 4,5
R2 / x,y / x,5

There are 56 possibilities (ouch). Here are the summed covariances:

x,y = . / cov x,5 / covx,y / cov 4,y / Sum
1,2 / 286 / 368 / 230 / 884
1,3 / 286 / 287 / 416 / 989
1,6 / 286 / 137 / 098 / 521
1,7 / 286 / 183 / 212 / 681
1,8 / 286 / 182 / 075 / 543
1,9 / 286 / 128 / 280 / 694
1,10 / 286 / 017 / 268 / 571
2,1 / 335 / 368 / 246 / 949
2,3 / 335 / 474 / 416 / 1225
2,6 / 335 / 222 / 098 / 655
2,7 / 335 / 027 / 212 / 574
2,8 / 335 / 154 / 075 / 569
2,9 / 335 / 139 / 280 / 754
2,10 / 335 / 129 / 268 / 732
3,1 / 374 / 289 / 246 / 909
3,2 / 374 / 287 / 230 / 891
3,6 / 374 / 307 / 098 / 779
3,7 / 374 / 221 / 212 / 807
3,8 / 374 / 199 / 075 / 648
3,9 / 374 / 413 / 280 / 1067
3,10 / 374 / 149 / 268 / 791
6,1 / 202 / 137 / 246 / 585
6,2 / 202 / 222 / 230 / 654
6,3 / 202 / 307 / 416 / 925
6,7 / 202 / 042 / 212 / 456
6,8 / 202 / 090 / 075 / 367
6,9 / 202 / 134 / 280 / 616
6,10 / 202 / 014 / 268 / 484
7,1 / 119 / 183 / 246 / 548
7,2 / 119 / 027 / 230 / 376
7,3 / 119 / 221 / 416 / 756
7,6 / 119 / 042 / 098 / 259
7.8 / 119 / 075 / 075 / 264
7,9 / 119 / 296 / 280 / 645
7,10 / 119 / 112 / 268 / 399
8,1 / 209 / 181 / 246 / 636
8,2 / 209 / 158 / 230 / 597
8,3 / 209 / 199 / 416 / 824
8,6 / 209 / 090 / 098 / 397
8,7 / 209 / 075 / 212 / 496
8,9 / 209 / 343 / 280 / 832
8,10 / 209 / 027 / 268 / 504
9,1 / 301 / 128 / 246 / 675
9,2 / 301 / 139 / 230 / 670
9.3 / 301 / 413 / 416 / 1130
9,6 / 301 / 134 / 098 / 535
9,7 / 301 / 296 / 212 / 809
9,8 / 301 / 343 / 075 / 719
9,10 / 301 / 183 / 268 / 752
10,1 / 147 / 018 / 246 / 191
10,2 / 147 / 129 / 230 / 506
10,3 / 147 / 149 / 416 / 712
10,6 / 147 / 014 / 098 / 259
10,7 / 147 / 112 / 212 / 471
10,8 / 147 / 027 / 075 / 249
10,9 / 147 / 183 / 280 / 610

x,y = 2,3

Step3: Find x,y such that cov x,5 + x,3 + x,y + 2,y + 4,y is maximized. There are 30 possibilities

C1
q4 / C2
q2 / C3 / C4 / C5 / C6 / C7 / C8 / C9
q3 / C10
q5
R1 q4 / 4,y / 4,3 / 4,5
R2 q2 / 2,y / 2,3 / 2,5
R3 / x,y / x,3 / x,5

There are 30 possibilities (ouch). Here are the summed covariances:

x,y / cov x,5 / cov x,3 / covx,y / cov 2,y / cov 4,y / Sum
1,6 / 286 / 287 / 137 / 222 / 098 / 1030
1,7 / 286 / 287 / 183 / 027 / 212 / 995
1,8 / 286 / 287 / 182 / 159 / 075 / 989
1,9 / 286 / 287 / 128 / 139 / 280 / 1120
1,10 / 286 / 287 / 017 / 129 / 268 / 987
6,1 / 202 / 307 / 137 / 368 / 246 / 1260
6,7 / 202 / 307 / 042 / 027 / 212 / 790
6,8 / 202 / 307 / 090 / 159 / 075 / 833
6,9 / 202 / 307 / 134 / 139 / 280 / 1062
6,10 / 202 / 307 / 014 / 129 / 268 / 920
7,1 / 119 / 221 / 183 / 368 / 246 / 1137
7,6 / 119 / 221 / 042 / 222 / 098 / 702
7,8 / 119 / 221 / 075 / 159 / 075 / 649
7,9 / 119 / 221 / 296 / 139 / 280 / 1055
7,10 / 119 / 221 / 112 / 129 / 268 / 849
8,1 / 209 / 199 / 182 / 368 / 246 / 1204
8,6 / 209 / 199 / 090 / 222 / 098 / 818
8,7 / 209 / 199 / 075 / 027 / 212 / 722
8,9 / 209 / 199 / 343 / 139 / 280 / 1170
8,10 / 209 / 199 / 027 / 129 / 268 / 832
9,1 / 301 / 413 / 128 / 368 / 246 / 1456
9,6 / 301 / 413 / 134 / 222 / 098 / 1168
9,7 / 301 / 413 / 296 / 027 / 212 / 1249
9,8 / 301 / 413 / 343 / 159 / 075 / 1291
9,10 / 301 / 413 / 183 / 129 / 268 / 1294
10,1 / 147 / 149 / 017 / 368 / 246 / 927
10,6 / 147 / 149 / 014 / 222 / 098 / 630
10,7 / 147 / 149 / 112 / 027 / 212 / 647
10,8 / 147 / 149 / 027 / 159 / 075 / 557
10,9 / 147 / 149 / 183 / 139 / 280 / 898

x,y is 9,1

Step 4: Find x,y such that cov x,1 + x,3 + x,5 + x,y + 2,y + 4,y + 9,y is maximized

C1
q4 / C2
q2 / C3 / C4 / C5 / C6 / C7 / C8
q1 / C9
q3 / C10
q5
R1 q4 / 4,y / 4,1 / 4,3 / 4,5
R2 q2 / 2,y / 2,1 / 2,3 / 2,5
R3 q9 / 9,y / 9,1 / 9,3 / 9,5
R4 / x,y / x,1 / x,3 / x,5

There are 12 possibilities Here are the summed covariances:

x,y = / cov x,1 / cov x,3 / cov x,5 / covx,y / cov 2,y / cov 4,y / cov 9,y / Sum
6,7 / 137 / 307 / 202 / 042 / 027 / 212 / 296 / 1223
6,8 / 137 / 307 / 202 / 090 / 159 / 075 / 343 / 1313
6,10 / 137 / 307 / 202 / 014 / 129 / 268 / 183 / 1240
7,6 / 183 / 221 / 119 / 042 / 222 / 098 / 134 / 1019
7,8 / 183 / 221 / 119 / 075 / 159 / 075 / 343 / 1175
7,10 / 183 / 221 / 119 / 112 / 129 / 268 / 183 / 1215
8,6 / 182 / 199 / 209 / 090 / 222 / 098 / 134 / 1134
8,7 / 182 / 199 / 209 / 075 / 027 / 212 / 296 / 1200
8,10 / 182 / 199 / 209 / 027 / 129 / 268 / 183 / 1197
10,6 / 017 / 149 / 147 / 014 / 222 / 098 / 134 / 781
10,7 / 017 / 149 / 147 / 112 / 027 / 212 / 296 / 960
10,8 / 017 / 149 / 147 / 027 / 159 / 075 / 343 / 917

x,y is 6,8

C1
q4 / C2
q2 / C3
q9 / C4
q6 / C5 / C6 / C7
q8 / C8
q1 / C9
q3 / C10
q5
R1 q4 / 4,y / 4,8 / 4,1 / 4,3 / 4,5
R2 q2 / 2,y / 2,8 / 2,1 / 2,3 / 2,5
R3 q9 / 9,y / 9,8 / 9,1 / 9,3 / 9,5
R4 q6 / 6,y / 6,8 / 6,1 / 6,3 / 6,5
R5 / x,y / x,8 / x,1 / x,3 / x,5

Step 5: Find x,y to maximize the sum of covariances (cov (x,y) constant, so left out).

x,y = / x,1 / x,3 / x,5 / x,8 / 2,y / 4,y / 6,y / 9,y / Sum
7,10 / 183 / 221 / 119 / 075 / 129 / 268 / 014 / 183 / 1192
10,7 / 017 / 149 / 147 / 027 / 027 / 212 / 042 / 296 / 917

x,y is 7,10.

C1
q4 / C2
q2 / C3
q9 / C4
q6 / C5
q7 / C6
q10 / C7
q8 / C8
q1 / C9
q3 / C10
q5
R1 q4 / 4,10 / 4,8 / 4,1 / 4,3 / 4,5
R2 q2 / 2,10 / 2,8 / 2,1 / 2,3 / 2,5
R3 q9 / 9,10 / 9,8 / 9,1 / 9,3 / 9,5
R4 q6 / 6,10 / 6,8 / 6,1 / 6,3 / 6,5
R5 q7 / 7,10 / 7,8 / 7,1 / 7,3 / 7,5
R6 q10
R7 q8
R8 q1
R9 q3
R10 q5

The split halves are now (2,4,6,7,9) vs (1,3,5,8,10). The program obtains the r between these split halves is .70639. Applying the Spearman-Brown correction:

Alternatively, using the variances obtained by the program,

A Similar Method That Can Be Used With Scales That Have More Items

H. G. Osburn (Coefficient alpha and related internal consistency reliability coefficients, Psychological Methods, 2000, 5, 343-355) described a procedure "similar to" that of Callender and Osburn (1977). Osburn was dealing with an 8 item instrument, but I have employed it with scales having as many as 28 items. His description of the method was terse. I quote "First, find the two components with the largest covariance. Assign these two components to separate halves. Second, find the two components with the next largest covariance and assign these to components to separate halves, and so on until four pairs of components with the largest covariances are assigned to separate halves."

This method is certainly simpler than that of Callender and Osburn, but I was stumped with respect to which half to assign each member of each pair -- and it does matter. For example, for the ten item measure of idealism, items 2 and 3 had the highest covariance. I assigned item 2 to half A and item 3 to half B. The next highest covariance was between items 4 and 5. Which half receives item 4, half A or half B? I assigned it to B. I ended up with half A being comprised of items 2, 5, 8, 7, and 6, with half B being comprised of items 3, 4, 9, 1, and 10. The resulting estimated maximum 4 was .783, somewhat less than the estimate from the more complicated procedure explained above. This simpler method might be adequate when the number of items is too large for the more complicated method to be feasible. I did email Osburn asking him about how to decide into which half each item into a pair should be assigned, but I never got a response.

I used this less complex method to estimate maximized 4 for the 28-item Animal Rights scale also included in the KJ.dat file. The program I employed is Lambda4.sas, which is available on my SAS programs page. The program computes Cronbach’s alpha, obtains the covariances needed to construct the split-half that is used to estimate maximized 4, and computes the correlation between the halves obtained.

The halves I employed are defined as variables A and B in the data step, but I first needed to obtain the entire 28 x 28 covariance matrix. I got it one column at a time to make it easier for me to sort it in a way that I could find the pair of items with the highest covariance, the next highest pair, etc.

I brought the covariances into Word and removed all of the blanks on the left and then replaced with tabs the blanks between item number and covariance. Then I converted the text to table and sorted by column 2. I then used the sorted table to assign variables to halves, following Osburn’s (2000) method.

The correlation between halves A and B was .87525, which yields an estimated maximized 4 of .93348 after applying the Spearman-Brown correction. The Cronbach alpha was .91.

The sorted table is 22 pages long, so I shall reproduce here only the top several rows of the table.

q34 / 0.6915436242
q39 / 0.6915436242
q26 / 0.6758389262
q37 / 0.6758389262
q26 / 0.6285906040
q57 / 0.6285906040
q22 / 0.5975391499
q30 / 0.5975391499
q22 / 0.5894854586
q27 / 0.5894854586
q28 / 0.5864429530
q31 / 0.5864429530
q37 / 0.5724832215
q57 / 0.5724832215

Copyright 2002, Karl L. Wuensch - All rights reserved.

Copyright 2002, Karl L. Wuensch - All rights reserved.