MathPsyc 2009 International Conference.

A Poisson Race Model Analysis of the Implicit Association Test

M. Vianello L. Stefanutti P. Anselmi E. Robusto University of Padua Department of General Psychology

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A Poisson Race Model Analysis of the Implicit Association Test

The IAT The Implicit Association Test (IAT, Greenwald, McGhee & Schwartz, 1998) is at present the only implicit technique which possesses all the characteristics of a real psychological test. Many studies investigated its psychometric properties, while some publications serve as a real manual (e.g. Lane, Banaji, Nosek & Greenwald, 2007) and a normative sample (Nosek, Smyth, Hansen, Devos, Lindner, Ranganath, Smith, Olson, Chugh, Greenwald & Banaji, 2007). The materials used in an IAT can be grouped in four categories, defined by category labels (e.g. Flowers, Insects, Good Words, Bad Words) and stimuli that serve as examples of these categories (e.g. images of flowers and insects and words with good and bad meanings). In most IATs, the four categories represent opposing couples, sometimes separated in targets (e.g. Flowers-Insects) and attributes (e.g. Good-Bad). These two dimensions generally define the nominal categories of interest, which generate the combined identification tasks (Greenwald, Nosek, Banaji & Klauer, 2005). In these tasks, participants rapidly classify individual stimuli that represent category and attribute into one of four distinct categories with only two responses. Labels representing categories are provided on the top left and top right corners of the screen. Stimuli appears individually in the center of the screen. The underlying assumption is that responses will be facilitated — and thus will be faster and more accurate— when categories that are closely associated share a response (compatible task, which provides, e.g., the ―White and Good‖ label on the left and the ―Black and Bad‖ label on the right), as compared to when they do not (incompatible task, which provides, e.g., the ―Black and Good‖ label on the left and the ―White and Bad‖ label on the right ). The IAT effect is a relative measure of the association between these two nominal categories obtained comparing individual performances at the two critical blocks (compatible and incompatible). In our example, the score represents an implicit preference for flowers over insects. The Poisson Race Model A Poisson process is a stochastic process that can be formulated as a set {X (t ): t ≥ 0} of random variables where X(t ) is the number of events that occurred independently in time t (Townsend & Ashby, 1983). In a homogeneous Poisson process, λ represents the expected number of events that occur in each unit of time. The probabilities that a given number of events K will occur in a unit of time is given by:

(1)

Given a decision task where two stimuli (SA and SB) are repeatedly provided in order to obtain respectively responses RA and RB from the participant, a Poisson race model is given assuming that responses are controlled by two parallel independent processes XA (t ) and XB (t ) which represent the total number of evidences occurred toward respectively RA and RB. Each process has an event rate for each stimulus category (λAA, λAB, λBA, λBB,), which represents the number of evidences occurred in each time unit, and a threshold (KA or KB), which represents the number of events needed to quit the process and give the answer. Given the minimum time needed for a process to reach its threshold, TA = inf {t : XA (t ) ≥ KA} and TB = inf {t : XB (t ) ≥ KB }, participants’ response time at the decision task is given by T = min (TA, TB), from which the ―Race‖ diction derives. A Two-process Poisson Race Model of the IAT: Theoretical Formulation Let T1 and T2 be the two sets of target stimuli T and A3 and A4 be the two sets of Attribute stimuli A, such that S=T ∪ A. Let C = {C1, C2,C3,C4} be the set of categories used as labels in the IAT (e.g. Flowers, Insects, Good, Bad). We assume that each element of S might activate each element of C simultaneously and independently at different levels of intensity (0<λ<∞). These intensities determine participants’ speed (λcs ) at the IAT, such that 16 different rates can be identified (L=C X S). It is assumed that two different competing processes are involved in the IAT (X1,X0), which lead respectively to a correct or incorrect answer. In each block, these processes are determined by different rates, according to the scheme provided in table 1, where the first subscript refer to categories. We also

assumed that each trial within a block (practice, compatible, incompatible) in the IAT has the same difficulty, hence we defined six different thresholds (K0P, K1P, K0C, K1P, K0I , K1I), one for each block and each process (correct, incorrect). Lastly, we added to the model three more parameters to account for residual latencies that are not involved in the lexical decision task (e.g. word recognition). Table 1 Lambda parameters involved in each task. Stimuli Categories Processes T1 T2 A3 Practice trials X1P(t) λ11 λ22 λ33 X0P(t) λ21 λ12 λ43 Compatible Trials X1C(t) λ11 + λ31 λ22 + λ42 λ13 + λ33 X0C(t) λ21 + λ41 λ12 + λ32 λ23 + λ43 Incompatible trials X1I(t) λ11 + λ41 λ22 + λ32 λ23 + λ33 X0I(t) λ21 + λ31 λ12 + λ42 λ13 + λ43

A4 λ44 λ34 λ44 + λ24 λ14 + λ34 λ14 + λ44 λ24 + λ34

The PRM-IAT model provides 16 rates and 6 thresholds, which measure many different associations strengths involved in the IAT effect. λ11, λ22, λ33, λ44 measure stimuli correct discrimination; λ21, λ12, λ43, λ34 ; measure stimuli incorrect discrimination; λ31, λ32, λ41, λ42 measure stimulus-driven association strengths; λ13, λ14, λ23, λ24 measure category-driven association strengths; K0P, K1P, K0C, K1P, K0I , K1I measure blocks’ difficulty. Figure 1 provides an example of the association explained by rate parameters involved in the compatible task. Identification of the PRM-IAT The PRM-IAT arises as a special case of a model which is the composition of six separate Poisson race models operating independently one another. As a consequence of this, if each of these separate PRMs is identifiable, so is the PRM-IAT. The building block in the compound model is a PRM consisting of two independent and competing processes X i t  and X j t  , activated in parallel by two stimulus categories p and q. This elementary PRM contains four rate parameters ip , iq ,  jp ,  jq and two stopping criteria, one for each response category, K1 and K 0 . The compound model embeds two such elementary PRMs for each of the three blocks in the IAT: one for the pair ( T1 , T2 ) of target categories and one for the pair ( A3 , A4 ) of attribute categories. Looking at Table 1 it is not difficult to recognize the six elementary PRMs. To give an example, the portion of the table corresponding to rows X 1C t  and X 0C t  and columns T1 and T2 is the elementary PRM that, in the compatible block, is involved when either of the two target categories T1 , T2 appears on the screen. The PRM-IAT model can then be derived by introducing the following equality and inequality constraints to the compound model described above:

1Ck  1Pk , 0Ck  0 Pk , 1Ik  1Pk , 0 Ik  0 Pk , and

1Ck  1Pk  0 Ik  0 Pk , 0Ck  0 Pk  1Ik  1Pk , where, for each lambda parameter the leftmost index refers to the observed response (1 = correct, 0 = wrong); the second index refers to the block (P = practice, C = compatible, I = incompatible); the rightmost index specifies the stimulus category that appears on the screen (k = 1, 2, 3, 4).

Empirical test of the model A Conscientiousness-IAT (Steffens, 2004) was developed and presented to 101 psychology students. Maximum likelihood estimates of PRM-IAT parameters were obtained with Matlab’s (v. 2007b) optimization tool separately for each participant. 68 models were estimated successfully. Better estimation algorithm are currently under investigation (suggestions are welcome). As expected, repeated parameter estimations gave rise to the same individual values, regardless of the starting point. Residuals of each model were analyzed to obtain a chi-square test of data-model fit, in a way similar to, e.g., Van Zandt, Colonius and Proctor (2000) and Klauer, Voss, Schmitz and Teige-Mocigemba (2007). Less then half of the models without parameters accounting for residual latencies satisfactorily fit the data. Models with residual terms seem to achieve a better fit. Yet, they are currently under investigation.

References Greenwald, A. G., McGhee, D. E., & Schwartz, J. L. K. (1998). Measuring individual differences in implicit cognition: The implicit association test. Journal of Personality and Social Psychology, 74, 14641480. Lane, K. A., Banaji, M. R., Nosek, B. A., & Greenwald, A. G. (2007). Understanding and using the Implicit Association Test: IV: Procedures and validity. In B. Wittenbrink & N. Schwarz (Eds.), Implicit measures of attitudes: Procedures and controversies (pp. 59-102). New York: Guilford Press. Nosek, B. A., Smyth, F. L., Hansen, J. J., Devos, T., Lindner, N. M., Ranganath, K. A., Smith, C. T., Olson, K. R., Chugh, D., Greenwald, A. G., & Banaji, M. R. (2007). Pervasiveness and correlates of implicit attitudes and stereotypes. European Review of Social Psychology, 18, 36-88. Greenwald, A. G., Nosek, B. A., Banaji, M. R., & Klauer, K. C. (2005). Validity of the salience asymmetry interpretation of the IAT: Comment on Rothermund and Wentura, 2004. Journal of Experimental Psychology: General, 134, 420–425. Townsend, J. T., & Ashby, F. G. (1983). Stochastic modelling of elementary psychological processes. Cambridge, England: Cambridge University Press. Steffens, M. C. (2004). Is the Implicit Association Test Immune to Faking? Experimental Psychology, 51, 165-179. Van Zandt, T., Colonius, H. and Proctor, R., 2000. A comparison of two reaction time models applied to perceptual matching. Psychonomic Bulletin & Review, 7, 208–256. Klauer, K.C., Voss, A., Schmitz, F., & Teige-Mocigemba, S. (2007). Process components of the Implicit Association Test: A diffusion-model analysis. Journal of Personality and Social Psychology, 93, 353–368.

Figure 1 Trials in the compatible block involving four different categories of stimuli and PRM-IAT processes and estimates of Stimulus-Category association

X1C ( t )

X0C ( t )

X0C ( t )

X1C ( t )

X1C ( t )

X0C ( t )

Flowers (λ11) + Good (λ31)

Insects (λ21) + Bad (λ41)

Flowers (λ12) + Good (λ32)

Insects (λ22) + Bad (λ42)

Flowers (λ13) + Good (λ33)

Insects (λ23) + Bad (λ43)

Heaven

Note: Text in parentheses was, obviously, not provided.

X0C ( t )

X1C ( t )

Flowers (λ14) + Good (λ34)

Insects (λ24) + Bad (λ44)

Nasty