Jim Clark is correct: mathematically, there is nothing preventing a Cronbach's
alpha or KR-20 (which is equivalent to alpha for dichotomous items) from being
negative if the item intercorrelations are negative. I've occasionally seen
this in my own questionnaire data for just the reason Jim posits - when we had
inadvertently forgotten to flip the coding of reverse-scored items (or more
often, flipped them and then inadverently unflipped them - a good reason,
incidentally, to recode reverse-scored variables into new variables with little
Rs after their labels).
...Scott
________________________________________
From: Jim Clark [j.cl...@uwinnipeg.ca]
Sent: Sunday, March 04, 2012 5:04 PM
To: Teaching in the Psychological Sciences (TIPS)
Subject: Re: [tips] statistical oddity: please help
Hi
As shown below, the data in its original form has a negative average
intercorrelation of items, hence a negative alpha (and KR20). Is it possible
that some of the items on your measure need to be reversed (i.e., they were
negatively worded)? The summary statistics show possible candidates for
reversing.
Another possibility (less likely) is that rows and columns are reversed. When
I FLIPped the file (i.e., switched rows and columns), the intercorrelation (and
alpha) become positive. This could again reflect failure to reverse items
(i.e., subjects responded similar to one another to pos and neg items).
You may need to convert following to fixed font (e.g., courier) to view
properly.
Take care
Jim
data list free / Examinee A B C D E F G H I J.
begin data
1 1 1 0 1 1 0 1 0 1 0
2 1 0 0 0 0 1 0 0 0
1
3 1 1 1 1 1 0 1 0 0 0
4 1 1 1 0 0 1 0 1 0 0
5 1 0 1 1 0 0 0 0 0 0
6 1 1 1 0 1 1 1 0 0 0
7 1 0 1 1 0 0 1 1 0
1
8 1 1 1 0 1 1 0 0 1 0
9 1 1 0 1 1 1 0 1 0 0
10 1 1 1 1 1 0 0 0 1 0
end data.
reliab /vari = a to j / summ = all.
Reliability
Warnings
No SCALE subcommand was found. Scaling will be done on
all specified variables.
Each of the following component variables has
zero variance and is removed from the scale:
A
The determinant of the covariance matrix is zero
or approximately zero. Statistics based on its
inverse matrix cannot be computed and they are
displayed as system missing values.
Scale: ALL VARIABLES
Case Processing Summary
N %
Cases Valid 10 100.0
Excluded(a) 0 .0
Total 10 100.0
a Listwise deletion based on all variables in the procedure.
Reliability Statistics
Cronbach's Cronbach's N of Items
Alpha(a) Alpha Based on
Standardized
Items(a)
-.447 -.544 9
a The value is negative due to a negative average covariance among items. This
violates reliability model assumptions. You may want to check item codings.
Summary Item Statistics
Mean Minimum Maximum Range Maximum / Variance N of Items
Minimum
Item Means .478 .200 .700 .500 3.500 .034 9
Item Variances .243 .178 .278 .100 1.563 .001 9
Inter-Item -.009 -.222 .200 .422 -.900 .006 9
Covariances
Inter-Item -.041 -.816 .802 1.618 -.982 .107 9
Correlations
Item-Total Statistics
Scale Mean if Scale Variance Corrected Squared Cronbach's
Item Deleted if Item Deleted Item-Total Multiple Alpha if Item
Correlation Correlation Deleted
B 3.600000 .933 .429 . -1.252(a)
C 3.600000 1.600 -.218 . -.254(a)
D 3.700000 1.567 -.206 . -.259(a)
E 3.700000 .900 .408 . -1.298(a)
F 3.800000 1.956 -.452 . .026
G 3.900000 1.211 .078 . -.671(a)
H 4.000000 1.556 -.184 . -.294(a)
I 4.000000 1.333 .000 . -.533(a)
J 4.100000 2.100 -.582 . .048
a The value is negative due to a negative average covariance among items. This
violates reliability model assumptions. You may want to check item codings.
flip /newnames = examinee.
FLIP performed on 10 cases and 11 variables, creating 10 cases
and 11 variables. The working file has been replaced.
Variable examinee has been used to name the new variables. It has
not been transformed into a case.
A new variable has been created called CASE_LBL. Its
contents are the old variable names.
New variable names:
CASE_LBL K_1.0000 K_2.0000 K_3.0000 K_4.0000 K_5.0000 K_6.0000 K_7.0000
K_8.0000 K_9.0000 K_10.000
list.
K_1.00 K_2.00 K_3.00 K_4.00 K_5.00 K_6.00 K_7.00 K_8.00 K_9.00
CASE_LBL 00 00 00 00 00 00 00 00 00 K_10.000
A 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
B 1.0000 .0000 1.0000 1.0000 .0000 1.0000 .0000 1.0000 1.0000 1.0000
C .0000 .0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .0000 1.0000
D 1.0000 .0000 1.0000 .0000 1.0000 .0000 1.0000 .0000 1.0000 1.0000
E 1.0000 .0000 1.0000 .0000 .0000 1.0000 .0000 1.0000 1.0000 1.0000
F .0000 1.0000 .0000 1.0000 .0000 1.0000 .0000 1.0000 1.0000 .0000
G 1.0000 .0000 1.0000 .0000 .0000 1.0000 1.0000 .0000 .0000 .0000
H .0000 .0000 .0000 1.0000 .0000 .0000 1.0000 .0000 1.0000 .0000
I 1.0000 .0000 .0000 .0000 .0000 .0000 .0000 1.0000 .0000 1.0000
J .0000 1.0000 .0000 .0000 .0000 .0000 1.0000 .0000 .0000 .0000
Number of cases read: 10 Number of cases listed: 10
reliab /vari = K_1.0000 to K_10.000 / summ = all.
Warnings
No SCALE subcommand was found. Scaling will be done on
all specified variables.
The determinant of the covariance matrix is zero
or approximately zero. Statistics based on its
inverse matrix cannot be computed and they are
displayed as system missing values.
Scale: ALL VARIABLES
Case Processing Summary
N %
Cases Valid 10 100.0
Excluded(a) 0 .0
Total 10 100.0
a Listwise deletion based on all variables in the procedure.
Reliability Statistics
Cronbach's Cronbach's N of Items
Alpha Alpha Based on
Standardized
Items
.610 .608 10
Summary Item Statistics
Mean Minimum Maximum Range Maximum / Variance N of Items
Minimum
Item Means .530 .300 .600 .300 2.000 .016 10
Item Variances .261 .233 .278 .044 1.190 .000 10
Inter-Item .035 -.178 .156 .333 -.875 .007 10
Covariances
Inter-Item .134 -.667 .583 1.250 -.875 .096 10
Correlations
Item-Total Statistics
Scale Mean if Scale Variance Corrected Squared
Cronbach's
Item Deleted if Item Deleted Item-Total Multiple Alpha
if Item
Correlation Correlation
Deleted
K_1.0000 4.700000 5.122 .171 . .610
K_2.0000 5.000000 5.778 -.096 . .662
K_3.0000 4.700000 4.233 .607 . .502
K_4.0000 4.800000 4.844 .287 . .583
K_5.0000 5.000000 4.444 .546 . .523
K_6.0000 4.700000 4.456 .489 . .533
K_7.0000 4.700000 6.011 -.193 . .686
K_8.0000 4.700000 4.678 .378 . .561
K_9.0000 4.700000 4.900 .272 . .587
K_10.000 4.700000 4.456 .489 . .533
James M. Clark
Professor of Psychology
204-786-9757
204-774-4134 Fax
j.cl...@uwinnipeg.ca
>>> Julie Osland <osla...@wju.edu> 04-Mar-12 2:36 PM >>>
Hi all--
In my tests and measurements course, we just covered reliability, including
different ways to compute reliability (test-retest, Cronbach's, KR-20, etc).
Part of their homework assignment was compute the KR-20 for a given set of
data. After giving the assignment, I performed all of the assigned
reliability. In so doing, I arrived at a negative KR-20 value, which all
of my T & M books say should *not* happen. I have double and triple
checked the math both by hand and via MS Exel and I get the same result.
When I collected the assignment, my students obtained the same outcome.
Prior to the assignment, we worked two problems in class and nothing
'strange' happened.
I've attached as an excel file the KR 20 computational problem data, in
hopes someone would be willing to verify what we (my class and I) obtained.
If a negative value is possible for reliability, what does it mean or
communicate about the data? I'm having difficulty wrapping my mind around
this right now.
Any help and/or insight would be greatly appreciated. Feel free to e-mail
me off list.
Thank you,
Julie
Dr. Julie A. Osland, M.A., Ph.D.
Assistant Professor of Psychology
Wheeling Jesuit University
316 Washington Avenue
Wheeling, WV 26003
Office: (304) 243-2329
e-mail: osla...@wju.edu
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