Calculating Reliability.pdf

1
Calculating Reliability of
Quantitative Measures
Dr. K. A. Korb
University of Jos
Reliability Overview
• Reliability is defined as the
consistency of results from a test.
• Theoretically, each test contains
some error – the portion of the score
on the test that is not relevant to the
construct that you hope to measure.
– Error could be the result of poor test
construction, distractions from when
the participant took the measure, or
how the results from the assessment
were marked.
• Reliability indexes thus try to
determine the proportion of the test
score that is due to error.
Unreliable
Reliable
Dr. K. A. Korb
University of Jos

2
Reliability
• There are four methods of evaluating the reliability of an instrument:
– Split-Half Reliability: Determines how much error in a test score is due to poor
test construction.
• To calculate: Administer one test once and then calculate the reliability index by
coefficient alpha, Kuder-Richardson formula 20 (KR-20) or the Spearman-Brown
formula.
– Test-Retest Reliability: Determines how much error in a test score is due to
problems with test administration (e.g. too much noise distracted the
participant).
• To calculate: Administer the same test to the same participants on two different
occasions. Correlate the test scores of the two administrations of the same test.
– Parallel Forms Reliability: Determines how comparable are two different
versions of the same measure.
• To calculate: Administer the two tests to the same participants within a short period
of time. Correlate the test scores of the two tests.
– Inter-Rater Reliability: Determines how consistent are two separate raters of
the instrument.
• To calculate: Give the results from one test administration to two evaluators and
correlate the two markings from the different raters.
Dr. K. A. Korb
University of Jos
Split-Half Reliability
• When you are validating a measure, you will most likely be
interested in evaluating the split-half reliability of your
instrument.
– This method will tell you how consistently your measure assesses the
construct of interest.
• If your measure assesses multiple constructs, split-half reliability will be
considerably lower. Therefore, separate the constructs that you are
measuring into different parts of the questionnaire and calculate the
reliability separately for each construct.
• Likewise, if you get a low reliability coefficient, then your measure is
probably measuring more constructs than it is designed to measure.
Revise your measure to focus more directly on the construct of interest.
– If you have dichotomous items (e.g., right-wrong answers) as you
would with multiple choice exams, the KR-20 formula is the best
accepted statistic.
– If you have a Likert scale or other types of items, use the Spearman-
Brown formula.
Dr. K. A. Korb
University of Jos

3
KR-20
• NOTE: Only use the KR-20 if each item has a right
answer. Do NOT use with a Likert scale.
• Formula:
– rKR20 is the Kuder-Richardson formula 20
– k is the total number of test items
– Σ indicates to sum
– p is the proportion of the test takers who pass an item
– q is the proportion of test takers who fail an item
– σ2 is the variation of the entire test
rKR20 = ( )( )
k
k - 1
1 –
Σpq
σ2
Dr. K. A. Korb
University of Jos
• I administered a 10-item spelling test to 15
children.
• To calculate the KR-20, I entered data in an
Excel Spreadsheet.
KR-20
Dr. K. A. Korb
University of Jos

4
This column lists each
student.
In these columns, I marked a 1 if
the student answered the item
correctly and a 0 if the student
answered incorrectly.
Student Math Problem
Name 1. 5+3 2. 7+2 3. 6+3 4. 9+1 5. 8+6 6. 7+5 7. 4+7 8. 9+2 9. 8+4 10. 5+6
Sunday 1 1 1 1 1 1 1 1 1 1
Monday 1 0 0 1 0 0 1 1 0 1
Linda 1 0 1 0 0 1 1 1 1 0
Lois 1 0 1 1 1 0 0 1 0 0
Ayuba 0 0 0 0 0 1 1 0 1 1
Andrea 0 1 1 1 1 1 1 1 1 1
Thomas 0 1 1 1 1 1 1 1 1 1
Anna 0 0 1 1 0 1 1 0 1 0
Amos 0 1 1 1 1 1 1 1 1 1
Martha 0 0 1 1 0 1 0 1 1 1
Sabina 0 0 1 1 0 0 0 0 0 1
Augustine 1 1 0 0 0 1 0 0 1 1
Priscilla 1 1 1 1 1 1 1 1 1 1
Tunde 0 1 1 1 0 0 0 0 1 0
Daniel 0 1 1 1 1 1 1 1 1 1
Dr. K. A. Korb
University of Jos
• The first value is k, the number of items. My
test had 10 items, so k = 10.
• Next we need to calculate p for each item, the
proportion of the sample who answered each
item correctly.
rKR20 = ( )( )
k
k - 1
1 –
Σpq
σ2
k = 10
Dr. K. A. Korb
University of Jos

5
rKR20 = ( )( )
k
k - 1
1 –
Σpq
σ2
To calculate the proportion of the sample
who answered the item correctly, I first
counted the number of 1’s for each item.
This gives the total number of students who
answered the item correctly.
Second, I divided the number of students
who answered the item correctly by the
number of students who took the test, 15 in
this case.
Name 1. 5+3 2. 7+2 3. 6+3 4. 9+1 5. 8+6 6. 7+5 7. 4+7 8. 9+2 9. 8+4 10. 5+6
Sunday 1 1 1 1 1 1 1 1 1 1
Monday 1 0 0 1 0 0 1 1 0 1
Linda 1 0 1 0 0 1 1 1 1 0
Lois 1 0 1 1 1 0 0 1 0 0
Ayuba 0 0 0 0 0 1 1 0 1 1
Andrea 0 1 1 1 1 1 1 1 1 1
Thomas 0 1 1 1 1 1 1 1 1 1
Anna 0 0 1 1 0 1 1 0 1 0
Amos 0 1 1 1 1 1 1 1 1 1
Martha 0 0 1 1 0 1 0 1 1 1
Sabina 0 0 1 1 0 0 0 0 0 1
Augustine 1 1 0 0 0 1 0 0 1 1
Priscilla 1 1 1 1 1 1 1 1 1 1
Tunde 0 1 1 1 0 0 0 0 1 0
Daniel 0 1 1 1 1 1 1 1 1 1
Number of 1's 6 8 12 12 7 11 10 10 12 11
Proportion Passed (p) 0.40 0.53 0.80 0.80 0.47 0.73 0.67 0.67 0.80 0.73
Dr. K. A. Korb
University of Jos
• Next we need to calculate q for each item, the
proportion of the sample who answered each
item incorrectly.
• Since students either passed or failed each
item, the sum p + q = 1.
– The proportion of a whole sample is always 1.
– Since the whole sample either passed or failed an
item, p + q will always equal 1.
rKR20 = ( )( )
k
k - 1
1 –
Σpq
σ2
Dr. K. A. Korb
University of Jos

6
rKR20 = ( )( )
k
k - 1
1 –
Σpq
σ2
I calculated the percentage who failed by
the formula 1 – p, or 1 minus the proportion
who passed the item.
You will get the same answer if you count
up the number of 0’s for each item and then
divide by 15.
Name 1. 5+3 2. 7+2 3. 6+3 4. 9+1 5. 8+6 6. 7+5 7. 4+7 8. 9+2 9. 8+4 10. 5+6
Number of 1's 6 8 12 12 7 11 10 10 12 11
Proportion Failed (q) 0.60 0.47 0.20 0.20 0.53 0.27 0.33 0.33 0.20 0.27
Dr. K. A. Korb
University of Jos
• Now that we have p and q for each item, the
formula says that we need to multiply p by q
for each item.
• Once we multiply p by q, we need to add up
these values for all of the items (the Σ symbol
means to add up across all values).
rKR20 = ( )( )
k
k - 1
1 –
Σpq
σ2
Dr. K. A. Korb
University of Jos

7
rKR20 = ( )( )
k
k - 1
1 –
Σpq
σ2
In this column, I took p times q.
For example, (0.40 * 0.60) = 0.24
• Once we have p x q for every item, we sum up
these values.
.24 + .25 + .16 + … + .20 = 2.05
Σpq = 2.05
Name 1. 5+3 2. 7+2 3. 6+3 4. 9+1 5. 8+6 6. 7+5 7. 4+7 8. 9+2 9. 8+4 10. 5+6
Number of 1's 6 8 12 12 7 11 10 10 12 11
Proportion Failed (q) 0.60 0.47 0.20 0.20 0.53 0.27 0.33 0.33 0.20 0.27
p x q 0.24 0.25 0.16 0.16 0.25 0.20 0.22 0.22 0.16 0.20
Dr. K. A. Korb
University of Jos
• Finally, we have to calculate σ2, or the
variance of the total test scores.
rKR20 = ( )( )
k
k - 1
1 –
Σpq
σ2
Dr. K. A. Korb
University of Jos

8
The variation of the Total Exam Score is the squared standard deviation. I discussed calculating
the standard deviation in the example of a Descriptive Research Study in side 34. The standard
deviation of the Total Exam Score is 2.36. By taking 2.36 * 2.36, we get the variance of 5.57.
For each student, I calculated their
total exam score by counting the
number of 1’s they had.
rKR20 = ( )( )
k
k - 1
1 –
Σpq
σ2
σ2 = 5.57
Student Math Problem Total
Exam
Score
Name 1. 5+3 2. 7+2 3. 6+3 4. 9+1 5. 8+6 6. 7+5 7. 4+7 8. 9+2 9. 8+4 10. 5+6
Sunday 1 1 1 1 1 1 1 1 1 1 10
Monday 1 0 0 1 0 0 1 1 0 1 5
Linda 1 0 1 0 0 1 1 1 1 0 6
Lois 1 0 1 1 1 0 0 1 0 0 5
Ayuba 0 0 0 0 0 1 1 0 1 1 4
Andrea 0 1 1 1 1 1 1 1 1 1 9
Thomas 0 1 1 1 1 1 1 1 1 1 9
Anna 0 0 1 1 0 1 1 0 1 0 5
Amos 0 1 1 1 1 1 1 1 1 1 9
Martha 0 0 1 1 0 1 0 1 1 1 6
Sabina 0 0 1 1 0 0 0 0 0 1 3
Augustine 1 1 0 0 0 1 0 0 1 1 5
Priscilla 1 1 1 1 1 1 1 1 1 1 10
Tunde 0 1 1 1 0 0 0 0 1 0 4
Daniel 0 1 1 1 1 1 1 1 1 1 9
Dr. K. A. Korb
University of Jos
• Now that we know all of the values in the
equation, we can calculate rKR20.
rKR20 = ( )( )
k
k - 1
1 –
Σpq
σ2
k = 10
rKR20 = ( )( )
10
10 - 1
1 –
2.05
5.57
rKR20 = 1.11 * 0.63
Σpq = 2.05
σ2 = 5.57
r
rKR20
KR20 =
= 0.70
0.70
Dr. K. A. Korb
University of Jos

9
Likert Tests
• If you administer a Likert Scale or have another measure
that does not have just one correct answer, the preferable
statistic to calculate the split-half reliability is coefficient
alpha (otherwise called Cronbach’s alpha).
– However, coefficient alpha is difficult to calculate by hand. If you
have access to SPSS, use coefficient alpha to calculate the
reliability.
– However, if you must calculate the reliability by hand, use the
Spearman Brown formula. Spearman Brown is not as accurate, but
is much easier to calculate.
1 + rhh
rSB =
2rhh
where rhh = Pearson correlation
of scores in the two half tests.
Spearman-Brown Formula
where σi
2 = variance of one test
item. Other variables are identical to
the KR-20 formula.
Coefficient Alpha
rα= ( )( )
k
k - 1
1 –
Σσi
2
σ2
Dr. K. A. Korb
University of Jos
• To demonstrate calculating the Spearman Brown formula, I used the
PANAS Questionnaire that was administered in the Descriptive Research
Study.
– See the PowerPoint for the Descriptive Research Study for more information
on the measure.
• The PANAS measures two constructs via Likert Scale: Positive Affect and
Negative Affect.
– When we calculate reliability, we have to calculate it for each separate
construct that we measure.
• The purpose of reliability is to determine how much error is present in the test score.
If we included questions for multiple constructs together, the reliability formula would
assume that the difference in constructs is error, which would give us a very low
reliability estimate.
– Therefore, I first had to separate the items on the questionnaire into essentially
two separate tests: one for positive affect and one for negative affect.
• The following calculations will only focus on the reliability estimate for positive affect.
We would have to do the same process separately for negative affect.
Spearman Brown Formula
Dr. K. A. Korb
University of Jos

10
Fourteen participants took
the test.
These are the 10 items that
measured positive affect on
the PANAS.
S/
No
Questionnaire Item Number
1 3 5 9 10 12 14 16 17 19
1 5 4 4 5 1 4 4 5 3 4
2 5 3 3 4 5 2 3 4 5 4
3 5 3 3 4 5 2 3 4 5 4
4 5 3 3 4 3 4 3 5 4 3
5 5 3 3 4 3 4 3 5 4 3
6 5 3 5 3 1 3 2 3 3 1
7 5 4 4 4 3 3 3 4 4 3
8 5 3 5 4 3 4 4 4 5 3
9 5 5 3 4 2 5 3 5 5 4
10 5 4 3 4 3 4 4 4 4 4
11 4 4 4 3 3 4 4 4 5 4
12 4 3 3 3 1 3 3 5 4 3
13 5 5 3 3 2 3 4 4 3 3
14 3 2 2 3 1 4 3 4 3 4
1 + rhh
rSB =
2rhh
The data for each participant
is the code for what they
selected for each item: 1 is
slightly or not at all, 2 is a little,
3 is moderately, 4 is quite a
bit, and 5 is extremely.
• The first step is to split the questions into half. The recommended
procedure is to assign every other item to one half of the test.
– If you simply take the first half of the items, the participants may have become
tired at the end of the questionnaire and the reliability estimate will be
artificially lower.
Dr. K. A. Korb
University of Jos
The first half total was
calculated by adding up the
scores for items 1, 5, 10,
14, and 17.
1 + rhh
rSB =
2rhh
S/
No
Questionnaire Item Number
1 Half
Total
2 Half
Total
1 3 5 9 10 12 14 16 17 19
1 5 4 4 5 1 4 4 5 3 4 17 22
2 5 3 3 4 5 2 3 4 5 4 21 17
3 5 3 3 4 5 2 3 4 5 4 21 17
4 5 3 3 4 3 4 3 5 4 3 18 19
5 5 3 3 4 3 4 3 5 4 3 18 19
6 5 3 5 3 1 3 2 3 3 1 16 13
7 5 4 4 4 3 3 3 4 4 3 19 18
8 5 3 5 4 3 4 4 4 5 3 22 18
9 5 5 3 4 2 5 3 5 5 4 18 23
10 5 4 3 4 3 4 4 4 4 4 19 20
11 4 4 4 3 3 4 4 4 5 4 20 19
12 4 3 3 3 1 3 3 5 4 3 15 17
13 5 5 3 3 2 3 4 4 3 3 17 18
14 3 2 2 3 1 4 3 4 3 4 12 17
The second half total was
calculated by adding up the
scores for items 3, 9, 12,
16, and 19
Dr. K. A. Korb
University of Jos

11
• Now that we have our two halves of the test,
we have to calculate the Pearson Product-
Moment Correlation between them.
Spearman Brown Formula
√[Σ (X – X)2] [ΣY – Y)2]
rxy=
Σ(X – X) (Y – Y)
In our case, X = one person’s score on the first half of items, X = the mean
score on the first half of items, Y = one person’s score on the second half of
items, Y = the mean score on the second half of items.
Dr. K. A. Korb
University of Jos
We first have to calculate
the mean for both halves.
The first half total is X.
The second half total is Y.
S/No
1 Half
Total
2 Half
Total
1 17 22
2 21 17
3 21 17
4 18 19
5 18 19
6 16 13
7 19 18
8 22 18
9 18 23
10 19 20
11 20 19
12 15 17
13 17 18
14 12 17
Mean 18.1 18.4
This is X. This is Y
√[Σ(X – X)2] [Σ(Y – Y)2]
rxy=
Σ(X – X) (Y – Y)
Dr. K. A. Korb
University of Jos

12
To get X – X, we take each second
person’s first half total minus the
average, which is 18.1. For example,
21 – 18.1 = 2.9.
To get Y – Y, we take each
second half total minus the
average, which is 18.4. For
example,
18 – 18.4 = -0.4
S/No
1 Half
Total
2 Half
Total X - X Y - Y
1 17 22 -1.1 3.6
2 21 17 2.9 -1.4
3 21 17 2.9 -1.4
4 18 19 -0.1 0.6
5 18 19 -0.1 0.6
6 16 13 -2.1 -5.4
7 19 18 0.9 -0.4
8 22 18 3.9 -0.4
9 18 23 -0.1 4.6
10 19 20 0.9 1.6
11 20 19 1.9 0.6
12 15 17 -3.1 -1.4
13 17 18 -1.1 -0.4
14 12 17 -6.1 -1.4
Mean 18.1 18.4
√[Σ(X – X)2] [Σ(Y – Y)2]
rxy=
Σ(X – X) (Y – Y)
Dr. K. A. Korb
University of Jos
Next we multiply (X – X) times (Y – Y).
S/No
1 Half
Total
2 Half
Total X - X Y - Y (X - X)(Y - Y)
1 17 22 -1.1 3.6 -3.96
2 21 17 2.9 -1.4 -4.06
3 21 17 2.9 -1.4 -4.06
4 18 19 -0.1 0.6 -0.06
5 18 19 -0.1 0.6 -0.06
6 16 13 -2.1 -5.4 11.34
7 19 18 0.9 -0.4 -0.36
8 22 18 3.9 -0.4 -1.56
9 18 23 -0.1 4.6 -0.46
10 19 20 0.9 1.6 1.44
11 20 19 1.9 0.6 1.14
12 15 17 -3.1 -1.4 4.34
13 17 18 -1.1 -0.4 0.44
14 12 17 -6.1 -1.4 8.54
Mean 18.1 18.4 Sum 12.66
After we have multiplied, we
sum up the products.
Σ(X – X) (Y – Y) = 12.66
√[Σ(X – X)2] [Σ(Y – Y)2]
rxy=
Σ(X – X) (Y – Y)
Dr. K. A. Korb
University of Jos

13
To calculate the
denominator, we have to
square (X – X) and (Y – Y)
S/N
o
1 Half
Total
2 Half
Total X - X Y – Y (X - X)2 (Y - Y)2
1 17 22 -1.1 3.6 1.21 12.96
2 21 17 2.9 -1.4 8.41 1.96
3 21 17 2.9 -1.4 8.41 1.96
4 18 19 -0.1 0.6 0.01 0.36
5 18 19 -0.1 0.6 0.01 0.36
6 16 13 -2.1 -5.4 4.41 29.16
7 19 18 0.9 -0.4 0.81 0.16
8 22 18 3.9 -0.4 15.21 0.16
9 18 23 -0.1 4.6 0.01 21.16
10 19 20 0.9 1.6 0.81 2.56
11 20 19 1.9 0.6 3.61 0.36
12 15 17 -3.1 -1.4 9.61 1.96
13 17 18 -1.1 -0.4 1.21 0.16
14 12 17 -6.1 -1.4 37.21 1.96
Sum 90.94 75.24
√[Σ(X – X)2] [Σ(Y – Y)2]
rxy=
Σ(X – X) (Y – Y)
Next we sum the squares across the
participants. Then we multiply the
sums.
90.94 * 75.24 = 6842.33.
Finally, √6842.33 = 82.72.
√[Σ(X – X)2] [Σ(Y – Y)2] = 82.72
Dr. K. A. Korb
University of Jos
• Now that we have calculated the numerator
and denominator, we can calculate rxy
r
rxy
xy =
= 0.15
0.15
√[Σ(X – X)2] [Σ(Y – Y)2]
rxy=
Σ(X – X) (Y – Y)
√[Σ(X – X)2] [Σ(Y – Y)2] = 82.72
Σ(X – X) (Y – Y) = 12.66
82.72
rxy=
12.66
Dr. K. A. Korb
University of Jos

14
• Now that we have calculated the Pearson
correlation between our two halves (rxy = 0.15),
we substitute this value for rhh and we can
calculate rSB
r
rxy
xy =
= 0.26
0.26
1 + 0.15
rSB=
2 * 0.15
1 + rhh
rSB =
2rhh
1.15
rSB=
0.3
The measure did not
have good reliability in
my sample!
Dr. K. A. Korb
University of Jos

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