A HOW TO GUIDE ON THE QUAD MODEL

Thomas J. Allen and Jeffrey W. ShermanDepartment of PsychologyUniversity 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

AC

Black Face

1 - AC

1 - D

D

G

1 - G

Compatible

-

+

+

+Correct = [AC +

(1-AC)*D +

(1-AC)(1-D)(1-G)]

Error =(1-AC)(1-D)(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)*(1-OB) + (ACbb)*(1-D) + (1-ACbb)(1-D)(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

AC

Black Face

1 - AC

1 - D

D

G

1 - G

Incompatible

-

+

+

+D

1 - D

OB

1 - OB -

-

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)*(1-D)*(G)

Pleasant error = ACwg + (1-ACwg)(1-D)(1-G)

Unpleasant correct = (1-ACbb)*D + (1-ACbb)*(1-D)*(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.

Compatibleactual frequency

observed proportion

predicted frequency

predicted proportion X^2

white correct 558 0.96 558.65 0.96 0.00error 22 0.04 21.35 0.04 0.02

These are the raw correct and incorrect responses that the participants make.

These are the probabilities estimated by the parameters.

These are the chi-squares that compare the actual and predicted counts.

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:

AC Black-Bad 0.13AC White-Good 0.38OB 0.87G 0.50D 0.88

Parameters

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, 469-487.

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.