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Cronbach’s Alpha and Maximized Lambda4
The Data Download the KJ.dat data file from my StatData page. These are the data from the research that was reported in: Wuensch, K. L., Jenkins, K. W., & Poteat, G. M. (2002). Misanthropy, idealism, and attitudes towards animals. Anthrozoös , 15 , 139-149. A summary of the research can be found at Misanthropy, Idealism, and Attitudes About Animals. SAS To illustrate the computation of Cronbach’s alpha with SAS, I shall use the data set we used back in PSYC 6430 when learning to do correlation/regression analysis on SAS, KJ.dat. Columns 1 through 10 contain the participants’ responses to the first ten items, which constitute the idealism scale. Obtain the program file, Alpha.sas , from my SAS Programs page. The simple way to get Cronbach’s alpha is to use the NOMISS and ALPHA options with PROC CORR. I have also included an illustration of how Cronbach’s alpha can be computed from the item variances and an illustration of how to compute maximized 4 (lambda4) and estimated maximum 4. SPSS Boot up SPSS and click File, Read Text Data. Point the wizard to the KJ.dat file. On step one, indicate that there is no predefined format. On step two, indicate that the data file is of FIXED WIDTH, and that there are no variable names at the top. On step three indicate that the data start on line one, there is one line per case, and all cases should be read. On step four you will see a screen like this: (^) Copyright 2007, Karl L. Wuensch - All rights reserved. Alpha.doc
To indicate that the scores for item one are in column one, item two in column two, etc., just click between columns one and two, two and three……… ten and eleven.
Click Continue, OK. The Questionnaire and Results of the Analysis Each item had a 5-point Likert-type response scale (strongly disagree to strongly agree). Here are the items:
- People should make certain that their actions never intentionally harm others even to a small degree.
- Risks to another should never be tolerated, irrespective of how small the risks might be.
- The existence of potential harm to others is always wrong, irrespective of the benefits to be gained.
- One should never psychologically or physically harm another person.
- One should not perform an action which might in any way threaten the dignity and welfare of another individual.
- If an action could harm an innocent other, then it should not be done.
- Deciding whether or not to perform an act by balancing the positive consequences of the act against the negative consequences of the act is immoral.
- The dignity and welfare of people should be the most important concern in any society.
- It is never necessary to sacrifice the welfare of others.
- Moral actions are those which closely match ideals of the most "perfect" action. Look at the output. With SAS, use the statistics for the raw (rather than standardized) variables. The value of alpha, .744, is acceptable. There are two items, numbers 7 and 10, which have rather low item-total correlations, and the alpha would go up if they were deleted, but not much, so I retained them. It is disturbing that item 7 did not perform better, since failure to do ethical cost/benefit analysis is an important part of the concept of ethical idealism. Perhaps the problem is that this item does not make it clear that we are talking about ethical cost/benefit analysis rather than other cost/benefit analysis. For example, a person might think it just fine to do a personal, financial cost/benefit analysis to decide whether to lease a car or buy a car, but immoral to weigh morally good consequences against morally bad consequences when deciding whether it is proper to keep horses for entertainment purposes (riding them). Somehow I need to find the time to do some more work on improving measurement of the ethical cost/benefit component of ethical idealism.
Computing Alpha from Item Variances Coefficient alpha can be computed from the item variances with this formula: 1
( 1 2 /^2 )
n n I T , where the ratio of variances is the sum of the n item variances divided by the total test variance. The second part of the SAS program illustrates the application of this method for computing coefficient alpha. Maximized 4 (Lambda4)Lambda4) H. G. Osburn (Coefficient alpha and related internal consistency reliability coefficients, Psychological Methods , 2000, 5 , 343-355) noted that coefficient alpha is a lower bound to the true reliability of a measuring instrument, and that it may seriously underestimate the true reliability. Osburn used Monte Carlo techniques to study a variety of alternative methods of estimating reliability from internal consistency. Their conclusion was that maximized lambda4 was the most consistently accurate of the techniques. 4 is computed as coefficient alpha, but on only one pair of split-halves of the instrument. To obtain maximized 4 , one simply computes 4 for all possible split-halves and then selects the largest obtained value of 4. The problem is that the number of possible split halves is (^2) (! )
. 5 ( 2 )! n n for a test with 2 n items. If there are only four or five items, this is not so bad. For example, suppose we decide to use only the first four items in the idealism instrument. Look at the third part of the SAS program. In the data step I create the three possible split-halves and the total test score for a scale comprised of only the first four items of the idealism measure. Proc Corr is used to obtain an alpha coefficient of .717. Then I use Proc Corr three more times, obtaining the correlations for each pair of split halves. Those correlations are .49581, .66198, and .53911. When I apply the Spearman-Brown correction (taking into account that each split half has only half as many items as does the total instrument), r r 1 2 4 , the values of 4 are .6629, .7966, and .7005. The largest of these, .7966, is maximized 4. The average of these, .72, is Cronbach’s alpha.
If you have an even number of items,
2 2 4
T A B
, where the three
variances are for half A, half B, and the total test variance. My program computes variances for each half and the total variance, and then 4 is computed for each split- half using these variances. The maximized 4 is .796. I also computed coefficient alpha as the mean of the three possible pairs of split-halves. Note that value obtained is the same as reported by Proc Corr. If you have an odd number of items, use the method which actually computes r for each split-half. For example, I had data from a 5-item misanthropy scale. I created 10 split-halves (in each, one set had only 2 items, the other had 3 items). The correlations, with the Spearman-Brown correction,
