Hierarchical Bayes Models for Response Time Data

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					    Hierarchical Bayes Models for Response Time Data

         Peter Craigmile1      Mario Peruggia1                 Trisha Van Zandt2

                              1 Department of Statistics

                              The Ohio State University
                             2 Departmentof Psychology
                              The Ohio State University


                               SAMSI, July 2009


Supported by:
NSF SES-0214574, SES-0437251, DMS-0604963, and DMS-0605052

   Peruggia (Ohio State)    Hierarchical Bayes Models for RT Data          SAMSI 2009   1 / 32
                                      Outline


Outline


1   RT Data Experiments

2   Modeling Issues
     Removing Trends
     Modeling Detrended Series
     Modeling Trends

3   Hierarchical Bayes Models
     Model Specification
     Model Fits

4   Discussion



     Peruggia (Ohio State)   Hierarchical Bayes Models for RT Data   SAMSI 2009   2 / 32
                           RT Data Experiments


Response Time Data




   RT sequence: Times taken by an individual to respond to a sequence
   of stimuli
   Can be analyzed as time series data
   RT data show interesting features related to changes over time at
   different scale levels and to the behavior of the tails of the marginal
   distributions.




   Peruggia (Ohio State)      Hierarchical Bayes Models for RT Data   SAMSI 2009   3 / 32
                        RT Data Experiments




                4




Peruggia (Ohio State)      Hierarchical Bayes Models for RT Data   SAMSI 2009   4 / 32
                                RT Data Experiments




                        Blank




                4
                                      RT1




Peruggia (Ohio State)              Hierarchical Bayes Models for RT Data   SAMSI 2009   4 / 32
                                RT Data Experiments




                        Blank                  7



                                                         RSI1


                4
                                      RT1




Peruggia (Ohio State)              Hierarchical Bayes Models for RT Data   SAMSI 2009   4 / 32
                                RT Data Experiments




                                                      Blank




                        Blank                  7
                                                                   RT2


                                                         RSI1


                4
                                      RT1




Peruggia (Ohio State)              Hierarchical Bayes Models for RT Data   SAMSI 2009   4 / 32
                                RT Data Experiments




                                                                               Blank




                                                      Blank                6
                                                                                            RT3


                                                                                  RSI2


                        Blank                  7
                                                                   RT2


                                                         RSI1


                4
                                      RT1




Peruggia (Ohio State)              Hierarchical Bayes Models for RT Data                 SAMSI 2009   4 / 32
                           RT Data Experiments


Data from Wagenmakers et al. (2004)



   Subset of RT sequences collected to look at sequential dependencies
   Six subjects
   Stimuli: digits 1 through 9
   Practice phase of 24 stimuli followed by an experimental phase of
   1024 stimuli
   Simple RT: press a key as soon as a digit is detected
   Short vs. Long RSI (Response-Stimulus Interval) distributions
           Short RSI ∼ Uniform[550, 950] (ms)
           Long RSI ∼ Uniform[1150, 1550] (ms)




   Peruggia (Ohio State)      Hierarchical Bayes Models for RT Data   SAMSI 2009   5 / 32
                                        RT Data Experiments


Ln RTs for Subj. 1.S and Subj. 2.S

           ln(RT) for Subj. 1

                                6
                                4
                                2




                                    0      200          400            600         800   1000

                                                              Trial No.
                                7
           ln(RT) for Subj. 2

                                6
                                5
                                4
                                3




                                    0      200          400            600         800   1000

                                                              Trial No.




   Peruggia (Ohio State)                   Hierarchical Bayes Models for RT Data         SAMSI 2009   6 / 32
                                          RT Data Experiments


Ln RTs for Subj. 1.S and Subj. 2.S


                                6.2
           ln(RT) for Subj. 1

                                5.8
                                5.4




                                      0      200          400            600         800   1000

                                                                Trial No.
                                6.2
           ln(RT) for Subj. 2

                                5.8
                                5.4




                                      0      200          400            600         800   1000

                                                                Trial No.




   Peruggia (Ohio State)                     Hierarchical Bayes Models for RT Data         SAMSI 2009   7 / 32
                            Modeling Issues


Trends




   RT sequences exhibit trends over long and medium scales, having
   possibly different interpretations
   Trends over long scales may be associated with learning effects
   (decreasing trends) and tiring (increasing trends)
   Trends over medium scales may be associated with transitory causes
   (persistent distractions, changes in response strategies)




   Peruggia (Ohio State)   Hierarchical Bayes Models for RT Data   SAMSI 2009   8 / 32
                              Modeling Issues


Local Dependencies




   Finer scale, local dependencies may be linked to carry over effects
   from neighboring trials and to errors
   Asymmetric tails in the marginal distribution of the data linked to
           rapid guessing, accidental keypresses (lower tail)
           momentary distractions, noises, itches (upper tail)




   Peruggia (Ohio State)     Hierarchical Bayes Models for RT Data   SAMSI 2009   9 / 32
                            Modeling Issues     Removing Trends


Detrending

   First approach: Construct a realistic model for the data after
   removing the large and medium scales trends
   Detrending performed on the log-RT series after conversion to normal
   scores (Craigmile, Peruggia, and Van Zandt, 2009)
   Detrending uses a procedure that fits cubic smoothing splines to
   autocorrelated Gaussian data (R package ASSIST; Wang 1998)
   Estimated trends are then transformed back to the original scale
   Note: Standard smoothing techniques, which depend on the
   assumption of independence, will result in series that are
   oversmoothed
   Note: How much trend is removed at this stage will have an impact
   on all subsequent modeling, so it is important to establish well
   defined detrending criteria

   Peruggia (Ohio State)   Hierarchical Bayes Models for RT Data   SAMSI 2009   10 / 32
                                              Modeling Issues     Removing Trends


Cubic SS Trends

                       6.2
                                       short RSIs                                    long RSIs




                                                                  6.2
           Subject 1
                       5.8




                                                                  5.8
                       5.4




                                                                  5.4
                             0   200   400   600    800    1000         0    200     400   600   800   1000
                       6.2




                                                                  6.2
           Subject 2
                       5.8




                                                                  5.8
                       5.4




                                                                  5.4
                             0   200   400   600    800    1000         0    200     400   600   800   1000
                       6.2




                                                                  6.2
           Subject 3
                       5.8




                                                                  5.8
                       5.4




                                                                  5.4




                             0   200   400   600    800    1000         0    200     400   600   800   1000
                       6.2




                                                                  6.2




 Wavelet
           Subject 4
                       5.8




                                                                  5.8
                       5.4




                                                                  5.4




    Peruggia (Ohio State)                    Hierarchical Bayes Models for RT Data                 SAMSI 2009   11 / 32
                                              Modeling Issues     Removing Trends




                       6.2




                                                                  6.2
           Subject 3
Cubic SS Trends

                       5.8




                                                                  5.8
                       5.4




                                                                  5.4
                             0   200   400   600    800    1000         0     200    400   600   800   1000
                       6.2             short RSIs                                    long RSIs




                                                                  6.2
           Subject 4
                       5.8




                                                                  5.8
                       5.4




                                                                  5.4
                             0   200   400   600    800    1000         0     200    400   600   800   1000
                       6.2




                                                                  6.2
           Subject 5
                       5.8




                                                                  5.8
                       5.4




                                                                  5.4
                             0   200   400   600    800    1000         0     200    400   600   800   1000
                       6.2




                                                                  6.2
           Subject 6
                       5.8




                                                                  5.8
                       5.4




                                                                  5.4




                             0   200   400   600    800    1000         0     200    400   600   800   1000

 Wavelet




    Peruggia (Ohio State)                    Hierarchical Bayes Models for RT Data                SAMSI 2009   12 / 32
                            Modeling Issues     Modeling Detrended Series


Basic Model for a Single RT Sequence




   Account for sequential dependencies at the likelihood stage
   Model explicitly the long tails of the marginal RT distribution
   Conditional on certain parameters, assume that the Log RTs are
   either Gaussian of ex-Gaussian RVs




   Peruggia (Ohio State)   Hierarchical Bayes Models for RT Data            SAMSI 2009   13 / 32
                            Modeling Issues     Modeling Detrended Series


Basic Model for a Single RT Sequence


Wk = detrended log(RTk ),         k = 1, . . . , 1024




   Peruggia (Ohio State)   Hierarchical Bayes Models for RT Data            SAMSI 2009   14 / 32
                                 Modeling Issues     Modeling Detrended Series


Basic Model for a Single RT Sequence


Wk = detrended log(RTk ),              k = 1, . . . , 1024
     
      Xk                  with prob. pX
Wk =   X + Yk              with prob. pY
      k
       Xk − Zk             with prob. pZ




   Peruggia (Ohio State)        Hierarchical Bayes Models for RT Data            SAMSI 2009   14 / 32
                                  Modeling Issues     Modeling Detrended Series


Basic Model for a Single RT Sequence


Wk = detrended log(RTk ),               k = 1, . . . , 1024
     
      Xk                   with prob. pX
Wk =   X + Yk               with prob. pY
      k
       Xk − Zk              with prob. pZ

Xk ∼ Gaussian AR(1) (φ, µ, σ 2 )




    Peruggia (Ohio State)        Hierarchical Bayes Models for RT Data            SAMSI 2009   14 / 32
                                    Modeling Issues     Modeling Detrended Series


Basic Model for a Single RT Sequence


Wk = detrended log(RTk ),                 k = 1, . . . , 1024
     
      Xk                     with prob. pX
Wk =   X + Yk                 with prob. pY
      k
       Xk − Zk                with prob. pZ

Xk ∼ Gaussian AR(1) (φ, µ, σ 2 )

Yk ∼ Exp(λY ),              Zk ∼ Exp(λZ )




    Peruggia (Ohio State)          Hierarchical Bayes Models for RT Data            SAMSI 2009   14 / 32
                                    Modeling Issues     Modeling Detrended Series


Basic Model for a Single RT Sequence


Wk = detrended log(RTk ),                 k = 1, . . . , 1024
     
      Xk                     with prob. pX
Wk =   X + Yk                 with prob. pY
      k
       Xk − Zk                with prob. pZ

Xk ∼ Gaussian AR(1) (φ, µ, σ 2 )

Yk ∼ Exp(λY ),              Zk ∼ Exp(λZ )

Priors: Normal for φ and µ, Inverse Gamma for σ 2 ,
        Gamma for λY and λZ , Dirichlet for p



    Peruggia (Ohio State)          Hierarchical Bayes Models for RT Data            SAMSI 2009   14 / 32
                              Modeling Issues     Modeling Detrended Series


Hidden AR(1) Process and Tails

   {Xk } is a hidden AR(1) process, to capture serial dependence in the
   detrended RTs
   {Xk } is defined by

                   X1 − µ = U1 ,
                   Xk − µ = φ(Xk−1 − µ) + Uk , k = 2, . . . , 1024,
                         2
   where U1 is a N(0, σ1 ) RV and
        1024 is an IID sequence of N(0, σ 2 ) RVs
   {Ut }t=2
   The possible occurrence of extreme observations is modeled by the
   two independent sequences of exponentially-distributed RVs:
   {Yt } with, mean µY = 1/λY (upper tails), and
   {Zt }, with mean µZ = 1/λZ (lower tails)


   Peruggia (Ohio State)     Hierarchical Bayes Models for RT Data            SAMSI 2009   15 / 32
                                                  Modeling Issues        Modeling Detrended Series


Time Series Parameters



                                mu                                   phi                                 sigma




                                                     1.0




                                                                                        0.14
                5.70




                                                     0.8




                                                                                        0.12
                                                     0.6
                5.60




                                                     0.4




                                                                                        0.10
                5.50




                                                     0.2




                                                                                        0.08
                5.40




                                                     0.0




                       1   2    3   4    5   6              1   2    3    4   5   6            1     2    3   4    5   6
                               subject                              subject                              subject

 Hierarchical




    Peruggia (Ohio State)                        Hierarchical Bayes Models for RT Data                            SAMSI 2009   16 / 32
                                         Modeling Issues     Modeling Detrended Series


Tail Distribution Parameters

                                     p_Y                                         p_Z
                 0.8




                                                            0.8
                 0.4




                                                            0.4
                 0.0




                                                            0.0
                            1   2   3     4    5    6               1     2     3    4    5   6
                                    subject                                     subject


                                    mu_Y                                        mu_Z
                 0.35




                                                            2.5
                 0.25




                                                            1.5
                 0.15




                                                            0.5




                            1   2   3     4    5    6               1     2     3    4    5   6
                                    subject                                     subject

 Hierarchical


    Peruggia (Ohio State)               Hierarchical Bayes Models for RT Data                     SAMSI 2009   17 / 32
                            Modeling Issues     Modeling Detrended Series


Analysis and Model Validation



   Within subjects, there is substantial agreement between the posterior
   parameter estimates for long and short RSIs
   Further analysis showed no evidence of dependence in the transitions
   from one mixture component to another
   RT series simulated from the posterior predictive distributions showed
   satisfactory agreement with the statistical properties of the original
   RT sequences, both in terms of marginal properties and first order
   serial dependence properties.




   Peruggia (Ohio State)   Hierarchical Bayes Models for RT Data            SAMSI 2009   18 / 32
                            Modeling Issues     Modeling Trends


Wavelet Model for Trends




   Second approach: Model trends rather than removing them, i.e.,
   introduce explicitly a stochastic process that generates a time varying
   mean for the AR mixture component
   Model for trends based on a wavelet decomposition
   Linear regression model in which the various predictors account for
   changes at different scale levels and different locations in time




   Peruggia (Ohio State)   Hierarchical Bayes Models for RT Data   SAMSI 2009   19 / 32
                            Modeling Issues     Modeling Trends


Wavelet Model for Trends


   We have control over the number of detail levels included in the
   model, with higher detail levels corresponding to changes at finer
   scales
   Normal priors on the regression coefficient
   Variances at different detail levels are proportional to a common term,
   ω, so as to learn about the smoothness of the trend across wavelet
   scales
   Coefficients corresponding to smoother changes have larger variance
   (akin to Vidakovic 1999)
   Include enough detail levels to account for large and medium scale
   changes, letting the AR structure account for the fine scale changes



   Peruggia (Ohio State)   Hierarchical Bayes Models for RT Data   SAMSI 2009   20 / 32
                                           Modeling Issues     Modeling Trends


Wavelet Trends for Subj. 1.S and Subj. 2.S

           ln(RT) for Subj. 1

                                6.2
                                5.8
                                5.4




                                      0   200           400           600         800   1000

                                                              Trial No.
           ln(RT) for Subj. 2

                                6.2
                                5.8
                                5.4




                                      0   200           400           600         800   1000

                                                              Trial No.




   Peruggia (Ohio State)                  Hierarchical Bayes Models for RT Data         SAMSI 2009   21 / 32
                            Modeling Issues     Modeling Trends


Individual Wavelet Fits




    Comparison between the estimated wavelet trends and the cubic
    smoothing splines trends removed in the 1-st analysis
    Comparison of the corresponding parameter estimates




   Peruggia (Ohio State)   Hierarchical Bayes Models for RT Data   SAMSI 2009   22 / 32
                                            Modeling Issues      Modeling Trends


Wavelet and Cubic SS Fits for Sbj. 1.S

           6.2
           6.0
           5.8
           5.6
           5.4




                     0             200                 400                600              800       1000




                 phi              sigma                p_Y                 p_Z             mu_Y           mu_Z
                          0.120




                 q
                 q                 q                   q
                                                       q                        q
                                                                                q                q             q




                                                                                                     8
                 q
                 q                 q                   q
                                                       q                        q
                                                                                q                q             q
                                                       q                                         q
                                                       q                                         q
                                                                                                 q             q
                                       q               q                        q                              q
                                       q
                                       q               q                        q                q
                                                       q     q                  q                q
                                                                                                 q
                                                 0.5




                                       q               q                        q            q   q
                                                                                                 q             q
           0.4




                                       q                                   q    q                q
                                                                                                 q             q



                                                                  0.020
                                       q               q                   q    q                q
                                                                                                 q
                                                       q     q             q    q
                                                                                q                q




                                                                                                     6
                                                       q
                                                       q                   q    q            q                 q
                                                       q                   q    q




                                                                                    0.25
                                                       q                   q
                                                                           q    q
                                                                                q            q                 q
                                                       q     q
                                                             q             q    q
                                                                                q            q
                                                             q                  q            q
                                                                                             q                 q
                          0.105




                                                             q
                                                             q                                                 q
                                                                                                               q
                                                             q
                                                             q                                                 q
                                                                                                               q
                     q
                     q                                       q
                                                             q                                                 q
                                                                                                               q
                     q                                       q
                                                             q                                                 q
                                                                                                               q
                     q
                     q                                       q                                                 q




                                                                                                     4
                                                                                                               q
                                                                                                               q
                                                 0.3




                                                                                                               q
           0.2




                 q                                                                                             q
                                                                                                               q
                 q                                                                                             q
                                                                                                               q
                                                                                                               q
                                                                                                               q
                                                                                                           q
                                                                                                           q   q
                                                                                                               q
                                                                                                           q
                                                                                                           q
                                                                                                           q
                                                                                                           q
                                                                  0.005

                                                                                                           q




                                                                                                     2
                                                                                    0.15
                          0.090




                                       q
                                       q
                                   q
                                   q   q
                                       q
                                   q   q
                                                 0.1
           0.0




                     q
                     q             q
                     q                 q                     q




  Peruggia (Ohio State)                    Hierarchical Bayes Models for RT Data                         SAMSI 2009   23 / 32
                                           Modeling Issues      Modeling Trends


Wavelet and Cubic SS Fits for Sbj. 3.S

           6.2
           6.0
           5.8
           5.6
           5.4




                     0            200                 400               600                             800         1000




                 phi             sigma                p_Y                p_Z                            mu_Y             mu_Z




                                                                                  0.22 0.26 0.30 0.34




                                                                                                                  1.6
                                                0.9
           0.8




                 q
                 q                    q                     q            q                                    q               q
                 q                                          q
                                                            q            q                                    q
                                                                                                              q
                                                                                                              q
                 q                                          q                                                 q
                                                                                                          q   q
                                                                                                              q
                                  q   q                     q
                                                            q                                             q
                                                                                                          q   q
                                                                                                              q               q
                                                                                                                              q
                          0.12




                     q            q   q
                                      q                     q
                                                            q
                                                            q                                                 q
                                                                                                              q               q
                     q
                     q            q   q
                                      q                     q            q
                                                                         q                                                    q
                                                                                                                              q
                                                                                                                              q
                     q                                      q            q
                                                                         q                                                    q




                                                                                                                  1.2
                                                            q
                                                            q            q
                                                                         q    q                                               q
                                                      q
                                                      q     q
                                                            q            q                                                q   q
                                                      q     q
                                                            q            q
                                                                         q    q                                               q
                                                                                                                              q
                                                      q                  q    q                                               q

                                                                 0.06
                                                                              q                                               q
                                                                                                                              q
                                                                                                                              q
                                                0.7




                                                                              q
                                                                              q                                               q
                                                                                                                              q
           0.6




                                                                              q
                                                                              q                                           q   q
                                                                                                                              q
                                                                              q
                                                                              q
                                                                              q
                                                                              q                                           q
                                                                                                                          q
                                                                                                                          q
                                                                                                                          q
                          0.10




                                                                                                                          q




                                                                                                                  0.8
                                                                                                                          q
                                                                                                                          q
                                                                                                                          q
                                                                                                                          q
                                                                                                                          q


                 q
                 q
                 q
                                                0.5
           0.4




                                                                 0.02


                 q   q
                     q




                                                                                                                  0.4
                     q
                          0.08




                     q                q
                     q
                     q            q   q
                                      q                                                                       q
                                                                                                              q
                     q            q
                     q            q                                                                           q
                     q                q                     q




  Peruggia (Ohio State)                   Hierarchical Bayes Models for RT Data                                         SAMSI 2009   24 / 32
                                           Modeling Issues      Modeling Trends


Wavelet and Cubic SS Fits for Sbj. 6.S

           6.2
           6.0
           5.8
           5.6
           5.4




                     0            200                 400               600              800         1000




                 phi             sigma                p_Y                p_Z             mu_Y             mu_Z
                                                0.9




                                                                                  0.30
                     q            q                                           q                q               q
                                  q                                           q            q                   q
           0.8




                                                                                               q




                                                                                                   1.2
                                  q                                           q                q
                                                                                               q
                                                                                               q




                                                                 0.16
                                  q   q                                       q            q   q
                                                                                               q               q
                                                                                                               q
                                  q
                                  q                                           q                q           q   q
                                                                                                               q
                                  q
                                  q                                      q    q                q
                                                                                               q           q
                                                                                                           q   q
                                      q
                                      q                                       q            q
                                                                                           q               q   q
                                                                                                               q
                                      q
                                      q                                  q    q
                                                                              q                            q   q
                                      q
                                      q                                  q    q                                q
                                                                                                               q
                                      q                                  q    q
                          0.12




                                      q
                                      q                                  q
                                      q
                                      q
                                      q




                                                                                                   1.0
                                                0.7




                                                                                  0.26
                                                                 0.12
           0.4




                     q
                     q
                     q
                     q
                     q
                     q
                     q
                     q




                                                                                                   0.8
                     q
                     q
                     q
                     q
                     q
                     q
                          0.08




                     q
                     q
                     q
                     q
                     q
                     q                                q
                                                                 0.08
                                                0.5




                     q                                q




                                                                                  0.22
                     q                                q
                                                      q
                                                      q     q                 q
                                                                              q                q
           0.0




                                                                         q




                                                                                                   0.6
                 q                                                       q                     q
                                                                                               q
                                                                                           q   q
                 q                    q                     q                 q            q   q           q




  Peruggia (Ohio State)                   Hierarchical Bayes Models for RT Data                          SAMSI 2009   25 / 32
                           Hierarchical Bayes Models   Model Specification


Hierarchical Bayes Model


   Tie together some of the model coefficients for different subjects and
   RSI conditions through a hierarchical Bayes structure, so as to learn
   across subjects and RSI condition
   The following analysis is based on a hierarchical structure for:
       1   the constant in the wavelet expansion
           (i.e., the intercept in the absence of the extremes)
       2   the logs of the means of the exponential tails
   Besides subject specific effects, we included fixed effects for RSI
   condition in the models describing the parameters in (1) and (2)
   Routine specification of mildly informative hierarchical priors
   Used JAGS to fit the model



   Peruggia (Ohio State)           Hierarchical Bayes Models for RT Data    SAMSI 2009   26 / 32
                                    Hierarchical Bayes Models    Model Fits


Wavelet Trends

                                     short RSIs                                     long RSIs
            5.4 5.8 6.2




                                                                 6.2
              Subject 1




                                                                 5.8
                                                                 5.4
                          0   200    400    600    800    1000         0      200   400   600   800     1000
            5.4 5.8 6.2




                                                                 6.2
              Subject 2




                                                                 5.8
                                                                 5.4
                          0   200    400    600    800    1000         0      200   400   600   800     1000
            5.4 5.8 6.2




                                                                 6.2
              Subject 3




                                                                 5.8
                                                                 5.4




                          0   200    400    600    800    1000         0      200   400   600   800     1000


 Cubic SS




    Peruggia (Ohio State)                   Hierarchical Bayes Models for RT Data                     SAMSI 2009   27 / 32
                                    Hierarchical Bayes Models    Model Fits


Wavelet Trends

                                     short RSIs                                     long RSIs
            5.4 5.8 6.2




                                                                 6.2
              Subject 4




                                                                 5.8
                                                                 5.4
                          0   200    400    600    800    1000         0      200   400   600   800     1000
            5.4 5.8 6.2




                                                                 6.2
              Subject 5




                                                                 5.8
                                                                 5.4
                          0   200    400    600    800    1000         0      200   400   600   800     1000
            5.4 5.8 6.2




                                                                 6.2
              Subject 6




                                                                 5.8
                                                                 5.4




                          0   200    400    600    800    1000         0      200   400   600   800     1000


 Cubic SS




    Peruggia (Ohio State)                   Hierarchical Bayes Models for RT Data                     SAMSI 2009   28 / 32
                                    Hierarchical Bayes Models                          Model Fits


Time Series Parameters



                                µ                                                      φ                                  σ




                                                     0.2 0.4 0.6 0.8 1.0
                    q




                                                                                                       0.14
             5.70




                            q
                                                                                   q
             5.60




                                                                                                                  q




                                                                                                       0.10
                                                                                                                                  q
                                                                                                              q
                        q                   q                                                                         q
                                                                                           q                                          q
             5.50




                                                                           q                       q                          q
                                    q
                                                                               q               q




                                                                                                       0.06
                                        q
                                                     −0.2
             5.40




                    1   2    3   4      5   6                              1   2    3   4      5   6          1   2    3   4      5   6
                            subject                                                subject                            subject




 Detrended




    Peruggia (Ohio State)                       Hierarchical Bayes Models for RT Data                                             SAMSI 2009   29 / 32
                                               Hierarchical Bayes Models      Model Fits


Tail Distribution Parameters
                                                     pY                                                         pZ




                                                                              0.15
             0.0 0.2 0.4 0.6 0.8 1.0

                                                                                                                              q




                                                                              0.10
                                                                     q
                                                 q
                                                                                                                     q




                                                                              0.05
                                                          q                                             q
                                       q   q                                                                q            q

                                                               q
                                                                                                    q




                                                                              0.00
                                       1   2      3       4     5     6                             1   2   3       4    5    6
                                                   subject                                                   subject

                                                     µY                                                         µZ




                                                                              0.5 1.0 1.5 2.0 2.5
             0.40
             0.30




                                                               q
                                                 q                   q
                                                                                                    q
             0.20




                                           q                                                                                  q
                                       q
                                                                                                        q                q
                                                                                                            q
                                                          q                                                          q
             0.10




                                       1   2      3       4     5     6                             1   2   3       4    5    6
                                                   subject                                                   subject




 Detrended


    Peruggia (Ohio State)                                 Hierarchical Bayes Models for RT Data                              SAMSI 2009   30 / 32
                                       Hierarchical Bayes Models            Model Fits


Fixed Effects



                                  γµ                                        γY                             γZ
             4




                                                           2.5




                                                                                            1.5
                                                     1.0 1.5 2.0
             3




                                                                                            1.0
          Density




                                                       Density




                                                                                         Density
            2




                                                                                            0.5
             1




                                                           0.5
                                                           0.0




                                                                                            0.0
             0




                    −0.6   −0.2    0.2      0.6                    −0.5   0.0    0.5               −1.0   0.0   0.5    1.0   1.5




   Peruggia (Ohio State)                          Hierarchical Bayes Models for RT Data                               SAMSI 2009   31 / 32
                                 Discussion


Discussion and Ongoing Work



   Important distinction between changes at various scale levels
   Design trials in which manipulation of experimental conditions affects
   changes at various scales (e.g., vary the experimental task difficulty
   over time, change the reward system, etc.)
   Can these changes be related to specific changes in the distributions
   of the wavelet decomposition parameters?
   What is the impact on the AR parameters and the other model
   parameters?




   Peruggia (Ohio State)   Hierarchical Bayes Models for RT Data   SAMSI 2009   32 / 32