A HOW TO GUIDE ON
THE
QUAD MODEL
Thomas J. Allen and Jeffrey W. Sherman
Department of Psychology
University of California, Davis
Why Do We Model?

Performance on implicit tasks can be influenced
by either:
Automatic processes OR
 Controlled processes (Jacoby, 1991)


Thus, implicit measures alone cannot tell us if
bias is strong b/c of
Stronger Associations
 Weak Cognitive Control OR
 Both

What Do We Model?




Association Activation (AC): the degree to which
stimuli spontaneously activate attributes
Discriminability (D): the degree to which the correct
response can be determined
Overcoming Bias (OB): the likelihood that D will be
selected over AC when D & AC conflict
Guessing (G): the likelihood of response bias (e.g.
left vs. right key) when D and AC fail
When Can We Model?

Data can be modeled when the
number of categories in an implicit
measure exceeds the numbers of
parameters estimated

That is when there is at least one
or more degree(s) of freedom
(categories – parameters = df)
When Can We Model?

For example, the Implicit Association Test (IAT)
has 8 categories:
4 stimuli (Black faces, White faces, pleasant
words, unpleasant words)
 2 conditions (Compatible trials: When Black &
Unpleasant share a key and White and Pleasant
share a key versus Incompatible trials: When
Black & Pleasant share a key and White &
Unpleasant share a key)

What Can We Model?


With 8 categories, theoretically, we can estimate up to 7
parameters.
However, because the equations for 2 of the categories of
responses are identical (more on this below), we can only
estimate 5 parameters from the IAT:


1 AC for associations between Black and Bad when viewing
Black faces or unpleasant words (this parameter biases
responses toward the unpleasant key for Black stimuli and the
Black key for unpleasant stimuli)
1 AC for associations between White and Good when viewing
White faces or pleasant words (this parameter biases
responses toward the pleasant key for White stimuli and the
White key for pleasant stimuli)
What Can We Model?
D for determining the correct response to all 4 types
of stimuli
 OB for determining whether AC or D win out when
they conflict (only occurs in incompatible blocks)
 G for response biases toward the right (pleasant)
with G > .50 or left (unpleasant) keys with G < .50

How Do We Model?



Using the parameters we can construct
equations that predict the frequency of correct
responses and errors to each of the 8 stimulus
categories
We can compare these predicted frequencies to
the actual frequencies
When Chi Squares are non-significant that
means there is a good match between predicted
and actual values (i.e. model fit)
How Do We Model?




Equations for each stimulus category are
constructed using a processing tree.
The processing tree contains all of the paths that
can lead to a correct response or an error
Each point in the path is multiplied (i.e.
conditional probability)
All the different paths are added together to
predict the frequencies of correct responses and
errors
Parameter Interpretation



Parameters are probabilities so they are
estimated on a scale from zero to one.
Values closer to zero can be interpreted as
representing less of a process occurring than
values closer to one.
In equations, a parameter is assumed to either
present (AC; D) or absent (1-AC; 1-D)
Interpretation of G




G is usually coded as right key response (or positivity
response if the right key is always the ‘pleasant’ key in
both compatible and incompatible blocks)
1-G is usually coded as left key response
Values below .50 represent response biases in the
direction of the left key; values above .50 represent
response biases in the direction of the right key
Values no different than .50 represent random guessing
(no right or left responses biases)
How Do We Model?

Example: Black Trials in the Compatible block
Proportion of Correct Response =
ACbb + (1-ACbb)*D + (1-ACbb)*(1-D)*(1-G)
Proportion of Errors = (1-ACbb)*(1-D)*(G)
How Do We Model?

Breaking Down the Compatible Black equations:
Correct Proportion
ACbb – Association Activation (Black-Bad)
should lead to making a correct response on
compatible trials because Black and Bad share
the same response key
How Do We Model?

Breaking Down the Compatible Black equations:
Correct Proportion
(1-ACbb)*D – When Associations are not
activated, but the correct response can be
determined (D), the correct response should be
made
How Do We Model?

Breaking Down the Compatible Black equations:
Correct Proportion
(1-ACbb)*(1-D)*(1-G) – When Associations are
not activated AND the correct response cannot
be determined, a guess must be made. When
that guess is made with the left key (1-G), the
correct response will be made.
How Do We Model?

Breaking Down the Compatible Black equations:
Error Proportion
(1-ACbb)*(1-D)*(G) – When Associations are not
activated AND the correct response cannot be
determined, a guess must be made. When that
guess is made with the right key (G), an error
will be made.
All Paths for Compatible
Black
Compatible
Correct = [AC +
+
AC
(1-AC)*D +
Black Face
D
+
(1-AC)(1-D)(1-G)]
G
1 - AC
1-D
+
Error =(1-AC)(1-D)(G)
1-G
-
Another Example

How would the equations for Black
stimuli differ in the Incompatible
Block?
Correct Proportion = ACbb*D*OB +
(1-ACbb)*D + (1-ACbb)*(1-D)*(G)
Error Proportion = (ACbb)*(D)*(1OB) + (ACbb)*(1-D) + (1-ACbb)(1D)(1-G)
Another Example

Breaking Down the Incompatible Black
equations:
Correct Proportion
(ACbb)*(D)*(OB) – When Associations are
activated (bias toward the unpleasant left key)
they conflict with the correct response (right
key). Presence (OB) or absence of OB (1-OB)
must then determine the response. In this case,
presence (OB) favors D over AC.
Another Example

Breaking Down the Incompatible Black
equations:
Correct Proportion
(1-ACbb)*(D) – When Associations are not
activated (bias toward the unpleasant left key)
the correct response can be made if it has been
determined.
Another Example

Breaking Down the Incompatible Black
equations:
Correct Proportion
(1-ACbb)*(1-D)*(G) – When Associations are not
activated (bias toward the unpleasant left key)
and the correct response has not been
determined, a guess can be made. B/c the
Black key is on the right now, a right key (G)
guess must be made to get the correct
response.
Another Example

Breaking Down the Incompatible Black
equations:
Error Proportion
(ACbb)*(D)*(1-OB) – When Associations are
activated (bias toward the unpleasant left key)
they conflict with the correct response (right
key). Presence (OB) or absence of OB (1-OB)
must then determine the response. In this case,
absence (1-OB) favors AC over D, producing an
incorrect response.
Another Example

Breaking Down the Incompatible Black
equations:
Correct Proportion
(ACbb)*(1-D) – When Associations are activated
(bias toward the unpleasant left key) and the
correct response cannot be determined, AC will
produce an incorrect response.
Another Example

Breaking Down the Incompatible Black
equations:
Correct Proportion
(1-ACbb)*(1-D)*(1-G) – When Associations are
not activated (bias toward the unpleasant left
key) and the correct response has not been
determined, a guess can be made. B/c the
Black key is on the right now, a left key (1-G)
guess will produce an incorrect response.
All Paths for Incompatible Black
Incompatible
OB
+
1 - OB
-
D
AC
-
1-D
Black Face
+
D
1 - AC
G
+
1-D
1-G
-
Equations

All of the other equations can be
viewed on the accompanying
model template. Look at the
predicted frequencies column

On the next slide are all the
possible pathways for Black and
White stimuli in the compatible and
incompatible conditions
Quad Model Processing Tree
What about the
Attributes?

In the compatible blocks, the
processes that predict correct and
incorrect responses for Black and
White stimuli are assumed to be
the same for Unpleasant and
Pleasant stimuli, respectively.
What about the
Attributes?



In older versions of the Quad Model, a separate
OB was estimated for Attributes (pleasant and
unpleasant stimuli). Often, this parameter was
no different from zero (indicating no need for it).
Also, theoretically, it makes more sense that
individuals attempt to overcome bias on the
racial categories rather than the attributes
That is, the associations are not bi-directional:
Black may activate bad things, but bad things do
not activate Black people
What about the
Attributes?

Thus, attributes have the following equations in the
incompatible block:
Pleasant correct = (1-ACwg)*D + (1-ACwg)*(1D)*(G)
Pleasant error = ACwg + (1-ACwg)(1-D)(1-G)
Unpleasant correct = (1-ACbb)*D + (1-ACbb)*(1D)*(1-G)
Unpleasant error = ACbb + (1-ACbb)(1-D)(G)
Using the Excel Template

If the solver is not already under the Tools drop
down menu, you can go to Tools, then Add-Ins.
Another window will open, then click “Solver
Add-In” and then Ok. Excel will then load the
program and it should thereafter be visible in the
Tools drop down menu.
Using the Excel Template


The Solver add-on in Excel is needed to perform
maximum likelihood estimation (MLE) to solve all
the equations of the model simultaneously and
produce parameter estimates.
The solver (using MLE) will attempt to find
parameter estimates that minimize the
differences between predicted and actual
frequencies as much as possible. This will
produce the smallest chi square value possible
(hopefully non-significant)
Using the Excel Template

First, the raw correct responses and errors need to be
calculated for each of the 8 categories and inserted in
the Actual Frequency column.
These are the raw correct and
incorrect responses that the
participants make.
Compatible
white
correct
error
These are the chi-squares that
compare the actual and
predicted counts.
actual
observed predicted
predicted
frequency proportion frequency proportion X^2
558
0.96
558.65
0.96 0.00
22
0.04
21.35
0.04 0.02
These are the probabilities estimated by the parameters.
Using the Excel Template

To begin parameter estimation, go to Tools, then Solver.
The following window will open:

The cells where parameters are located (Column b) are
in the changing cell box
Using the Excel Template

To begin parameter estimation, go to Tools, then Solver.
The following window will open:

In the constraints box, each parameter is constrained to be a
value between .000001 and .999999; This can be changed
by clicking “change” or additional cells can be included by
clicking “add”
Using the Excel Template

To begin parameter estimation, go to Tools, then
Solver. The following window will open:

In the target cell box, the cell where the overall
fit (chi square) will be calculated is selected
Using the Excel Template

The solver will produce parameter values that
will produce the smallest possible chi square
value (best fit). It is best to initiate the solver 2-3
times to obtain optimal values. The parameters
appear in column b:
Parameters
AC Black-Bad
AC White-Good
OB
G
D
0.13
0.38
0.87
0.50
0.88
Making Comparisons




After obtaining the initial model fit, parameters in
different experimental treatments can be tested
for differences
To do this, open up the solver and click “add” to
put in another constraint
Set the parameters you want to compare (e.g.
ACbb treatment vs. ACbb control) equal to each
other.
If the difference between the new chi square
value and the old chi square value is significant,
then it can inferred that the parameter values for
control versus treatment are genuinely different
Making Comparisons

The change in chi square can be tested using 1
degree of freedom on any online chi square
calculator.
What Stats to Report?

In results sections, the critical
numbers to report are the
parameter values, overall fit of the
model (chi square and p value),
and the chi square changes (and
their p values) for each parameter
comparison
Other Issues



Sometimes, the research question we
have requires that we have parameter
estimates at the individual level.
The previous slides and the template are
limited to aggregate comparisons
between conditions
Individual parameter estimates can be
more efficiently calculated using the
HMMTree program, which can be
downloaded for free at HMMTree
Other Issues

Model identifiability is another
issue that can be addressed by
setting the initial parameter values
to different values. If the model is
identifiable, the solver should settle
on the same parameter estimates
regardless of the start values.
Other Issues


When aggregate data is used to estimate
parameters and make comparisons,
homogeneity of variance is being assumed.
This assumption can be tested several ways
including using the latent class function on
HMMTree. More about latent class analysis can
be found in this article, Latent Class modeling
using HMMTree
Good Luck Modeling!

Further references
Conrey, Sherman, Gawronski, Hugenberg, and Groom (2005). Separating
multiple processes in implicit social cognition: The Quad-Model of implicit
task performance. Journal of Personality and Social Psychology, 89, 469487.
Sherman (2005). Automatic and Controlled components of implicit stereotyping
and prejudice. Psychological Science Agenda, 19 (3). link
Sherman (2006). On building a better process model: It’s not only how many,
but which ones and by which means. Psychological Inquiry, 17, 173-184.
Sherman, Gawronski, Gonsalkorale, Hugenberg, Allen, and Groom (2008).
The self-regulation of automatic associations and behavioral impulses.
Psychological Review, 115, 314-335.