cloglog link function to transform the (population) hazard probability into a continuous

Transcription

1 Supplementary material. Discrete time event history analysis Hazard model details. In our discrete time event history analysis, we used the asymmetric cloglog link function to transform the (population) hazard probability into a continuous scale. All observations were censored at 1200 ms after target onset. In order to fit a discrete time hazard model using standard logistic regression software, the original person-by-trial oriented dataset was transformed into a person-by-trial-by-time-bin oriented dataset (i.e., each trial provides as many rows as there are time bins that are at risk for event occurrence). Next, a censoring variable EVENT was added to indicate whether a response occurred in each time bin or not, and a variable TIME was added that indicates the rank c of each time bin. We chose to center time around time bin ]450,500], such that the intercept can be more easily interpreted. As all observations with responses faster than 300 ms were removed from the data analysis and time bins of 50 ms were used, all time bins with a rank below 7 were excluded from the person-by-trial-by-time-bin oriented dataset before fitting the discrete time hazard model. As a consequence, the starting point (time point 0) for the analysis was virtually shifted to 300 ms after target onset. We used the following equation to describe the cloglog discrete time hazard model: cloglog[h(t)] = log(-log[1-h(t)]) = [β 0 + β 1 *(TIME-c) + β 2 *(TIME-c) 2 ] + β 3 *(Trial-k) + β 4 *X 1 + β 5 *X 1 *(TIME-c)+ β 6 *X 1 *(TIME-c) 2. The above equation indicates that the cloglog transformed population hazard, or cloglog[h(t)], is modeled as a linear combination of predictor values, in which TIME is the predictor indicating the rank of the time bin, Trial refers to the trial number, and X 1 is a dichotomous variable indicating the first (X 1 =0) or second (X 1 =1) experimental condition to which a trial can belong. The three terms between brackets model the shape of the baseline

2 hazard function, which is the population hazard function when all predictors (i.e., Trial-k and X 1 ) are set to 0, using a flexible (here: second-order) polynomial specification. The intercept β 0 is the predicted cloglog hazard in time bin c of trial k in the first experimental condition. The parameter for the time-invariant effect of Trial-k indicates how the predicted cloglog hazard value changes (i.e., increases or decreases) with each additional trial in each time bin. The parameter for X 1 indicates how the predicted cloglog hazard value in time bin c changes when going from condition 1 to 2 in a certain trial, while the last two interaction terms allow the effect of the experimental manipulation X 1 on cloglog hazard to change over (discrete) time in a quadratic fashion. Following Allison (2010), we used the surveylogistic procedure in SAS 9.2 to estimate the hazard model parameters, which allows for the specification of the cloglog link and a cluster variable (i.e., subjects) to automatically correct for the otherwise biased standard errors and test statistics due to the repeated measures for each subject. The resulting parameter estimates can be directly interpreted as population-averaged values (in contrast to estimates from a random effects model). Note, however, that unobserved heterogeneity between trials and/or subjects can still lead to artifactually declining hazard functions and parameter estimates that are attenuated toward zero (Allison, 2010). With the parameter estimates at hand, the predicted model-based discrete time cloglog[h(t)] functions can easily be calculated for each time bin (between 300 and 1200 ms after stimulus onset) using the simple operations of multiplication and addition. These can then be transformed back into hazard or conditional probability functions (i.e., h(t) = 1-exp[- exp(cloglog[h(t)])]. From the model-based estimates of h(t) estimates of the discrete time survivor function or S(t) = P(T>t) and the (unconditional) subprobability function or P(t) = P(T=t) can be obtained (see footnote in main text). Due to the presence of censored

3 observations, the unconditional probabilities do not sum to 1, hence the prefix sub (Chechile, 2003). In Experiment 1 we started with a hazard model with 12 effect parameters. The shape of the cloglog hazard function in the long flanker condition (which was taken as the reference condition) was modeled using a fourth-order polynomial (5 parameters). The effect of trial was assumed to be linear and time-invariant (i.e., the baseline hazard function was assumed not to change its shape during the experiment, resulting in a single parameter for Trial). In contrast, the effects of small and short flankers were allowed to vary over time within a trial using a second-order polynomial specification (6 predictors). This full hazard model was iteratively and automatically reduced to the final hazard model with 8 parameters (reported in Table 1A) by eliminating in each step the effect with the highest p- value that was not part of any higher-order interaction, and refitting the model. To model the CAFs we used the genmod procedure in SAS, and selected the logit link and the repeated statement. As for the hazard model, we started with 12 parameters and applied the same backward procedure to select a final conditional accuracy model. This procedure reduced the full conditional accuracy model with 12 parameters (same effects as in the hazard model) to a final conditional accuracy model with 8 parameters (reported in Table 1B). A similar approach was used in Experiment 2. Continuous and discrete time methods. Depending on how the passage of time is measured, a discrete time or continuous time method can be used in event history analysis. Although continuous time methods (e.g., Cox regression) would seem best for RTs, there are a few advantages to using discrete time methods instead (Allison, 2010). First, statistical models for discrete time hazard can be estimated using widely available standard logistic regression

4 software packages. Second, discrete time hazard analysis is easier to comprehend for novices and can help in understanding and appreciating continuous time methods. Third, when a complementary log-log (cloglog) link is used instead of a logit link, the same underlying parameters are estimated when one fits a discrete time hazard model with an unconstrained specification of the shape of h(t) in the reference condition, as in a continuous time Cox regression model which ignores the shape of λ(t) in the reference condition (Allison, 2010). In other words, the parameter estimates from a discrete time cloglog[h(t)] model can be directly interpreted as ratios of continuous time hazard rates after exponentiating them. Finally, discrete time methods always provide an estimate of the shape of the baseline hazard function (in contrast to Cox regression). Discrete time methods also have a few disadvantages compared to continuous time methods, such as the subjective choice of the number and width of the time bins, and a lower temporal resolution. However, it can be shown that, within limits, the exact assumptions about the time bins do not affect the outcomes of the analysis much (Allison, 2010). Time bins of 50 ms provide sufficiently stable estimates and a sufficient temporal resolution to answer our research questions. Modeling more than 1 type of event. In a classical event history analysis there is only one event of interest (e.g., death) for each experimental unit. When there are two or more events of interest, one speaks of a competing risks situation in event history analysis parlance. As our RT analysis must deal with correct and incorrect responses, the competing risk situation applies and one of two approaches may be applied (Allison, 2010). In the conditional processes approach (Allison, 2010, p ), one process is assumed to govern the occurrence and timing of responses (studied by fitting a discrete time hazard

5 model with a single event, i.e., any response, to model the speed of the overt decision to respond), and a second conditional process is assumed to govern the latency-dependent accuracy of an observed response. In contrast to estimating separate hazard functions for correct and error responses (the second possible approach), the advantage of the conditional processes approach is that it does not require that the process generating correct responses must be independent from the process generating error responses, which we know is not the case because response channels influence each other through competition below threshold (Eriksen, Coles, Morris, & O Hara, 1985). In other words, one should not estimate the h(t) of correct response occurrence because then the random censoring introduced by error responses is not noninformative for the mechanism of the occurrence of correct responses; when the hazard of an error response is high at time t and the error occurs, the hazard of a correct response will be on average low (due to response competition) and trials with an error at time t are therefore not representative of all trials that survive beyond t. Thus, whenever uncontrollable error responses occur, the estimate of h(t) of correct response occurrence will be biased. It is therefore more appropriate to estimate h(t) of (any) response occurrence and to subsequently estimate the corresponding conditional accuracy functions (CAFs; plotting accuracy of the responses per time bin). Furthermore, assuming that errors are produced by the same dynamics that produce correct responses is also conservative (Holden, Van Orden, & Turvey, 2009).