Severe
Discrepancy Determination by Formula
Kate obtains an IQ score of 90 and an achievement
score of 74. Is this 16point difference large enough to be considered a ‘significant
difference’ between ability and achievement? Below is a table showing a
statistical manipulation of the scores.
Ability Score 
90 
Reliability of Ability Score 
.91 
Achievement Score 
74 
Achievement Reliability 
.91 
Correlation Between Ability
and Achievement Scores*** 
.47 


Predicted Achievement Score 
95 
Difference between Predicted
and Actual Achievement 
21 
Magnitude of Difference
required at .05 level 
17.75 
The 21.3 point difference between
ability and achievement was found to be significant using the
PredictedAchievement method.
Using this approach to
assessment, Kate should be considered to be functioning significantly below what
was expected of her.
Above is an example of the use of
a regression equation to determine a severe discrepancy. Regression is necessary
because of the imperfect correlations between the ability and achievement
measures. Just because someone has a low IQ score, we should not assume that
they would also have a correspondingly low score on other measures. No one, I
hope, would assume that just because a person had a very high IQ that the person
would obviously be able to excel at gymnastics, or be able to hold their breath
longer than ‘normal.’ Just because someone is good or bad at one thing does
not guarantee that the person will also be good or bad at other things. The IQ
test does not measure the same thing as an achievement test. Regressing the
scores on the tests used also allows one to compare individuals to others with
the same IQ. For example, if one child has an IQ of 120, while another an IQ of
80, the expectation of how each might be performing on an achievement test would
be different. If the tests correlated at the .60 level, the first child would be
expected to obtain an achievement score of 112 while the second child would be
expected to achieve a score of 88.
Because most tests do not report
scores in terms of Z, we can use a formula to calculated Z:
Z=
The simplest way to regress a
particular score (either IQ or achievement) is to multiply the correlation
between the two measures by the Z score you wish to regress. The formula would
be:
Regressed score = (Correlation*Z
)
The best way to determine the
correlation between two measures is to look the correlation up in the manual of
the test used. Unfortunately, not all manuals offer that information; the
information is often based on absurdly small samples; and not all tests have
been compared to each other. One way around this shortcoming is to estimate the
correlation between the two tests. If we know the reliability of the ability
test and the reliability of the achievement test, the correlation between the
two tests can be estimated.
The equation for estimating the
correlation between two tests is
**The values rxx and ryy are
the internal consistency reliability coefficients for the aptitude and
achievement tests used.
*** note
that in the example at the beginning of this page, the correlation is listed
as .47.
If the user knows from some source what the correlation really is, that
should override the estimated correlation obtained from a formula. The
formula provides the upper end of what is possible given the two
reliabilities but those are estimates. In the example given, since
the correlation between the IQ and the achievement (.47) was known,
it was used. The example is something like using the PIQ versus a reading
test (correlation .47) versus using the VIQ versus a reading test
(correlation .65).
Use this link for Tables
for the Reliability
Finally, the score must be turned
back into a Standard Score with a mean of 100 and a standard deviation of 15.
This is done by multiplying the "regressed Z" by the standard
deviation and adding 100 to the result.
Predicted achievement score =
or
15*(Correlation*((Ability100)/15))+100
This regressed score provides the examiner with a
different point for comparison. Comparing the expected score with the actual
score gives a better idea of the magnitude of the difference between the scores
obtained. When Kate obtained an IQ score of 90 and an achievement score of 74
her 16point difference was considered nonsignificant. However, she was
expected (using the regression formula) to obtain an achievement score of 95,
not 90. This increases the simple difference from 16 points to 21 points. We
still don’t know if this new 21point difference is large enough to be
considered a ‘severe discrepancy.’ For that determination, a different
formula is needed.
Magnitude of Difference required at .05 level =
1.96*15*(SQRT(1(Correlation^2)))1.65*((15*SQRT(1(Correlation^2))*
SQRT(1((RAbility)+(RAchievement*(Correlation^2))(2*(Correlation^2)))/(1(Correlation^2)))))
Thank heavens for computers!!
The first part of this monstrous equation:
1.96*15*(SQRT(1(Correlation^2))
sets the level of our decision
making to the 95% confidence (z=1.96) and then determines the Standard Error of
Estimate for the obtained score. The second part of the equation:
1.65*((15*SQRT(1(Correlation^2))*SQRT(1((RAbility)+(RAchievement*(Correlation^2))
(2*(Correlation^2)))/(1(Correlation^2))))
reduces the final cutoff score
by subtracting the standard error of the relevant difference score. The end
result is a statistically justifiable ‘severe discrepancy’ upon which to
make clinical decisions.
When a psychologist doesn’t
bother to struggle with the true identification problems, the result can be an
overidentification of children as disabled when they are not. This increases
the caseloads of the school and the individual staff, as well as making the
whole label of "Learning Disability" into a meaningless catchall. It
must also be noted that even when a child does have a severe discrepancy between
ability and achievement, this by itself does not constitute a diagnosis. It is a
necessary, but insufficient factor, to be used in the determination.
For a wonderful explanation of
this and other issues see:
Reynolds, C. R. Conceptual and technical problems in learning disability diagnosis, (Chapter 24) in
Handbook of Psychological and Educational Assessment of
Children: Intelligence and Achievement (Reynolds & Kamphaus) (1990) Guilford Press
A copy of the template for determining severe
discrepancy using this method is available.
To download template press here.
Please send me
an email if you download the template. I will be happy to answer
any questions you may have. Please do not distribute the template to
anyone else.
