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The authors have declared that no competing interests exist.

Conceived and designed the experiments: LvM RdJ HvR. Performed the experiments: LvM. Analyzed the data: LvM. Contributed reagents/materials/analysis tools: LvM RdJ HvR. Wrote the paper: LvM RdJ HvR.

When multiple strategies can be used to solve a type of problem, the observed response time distributions are often mixtures of multiple underlying base distributions each representing one of these strategies. For the case of two possible strategies, the observed response time distributions obey the fixed-point property. That is, there exists one reaction time that has the same probability of being observed irrespective of the actual mixture proportion of each strategy. In this paper we discuss how to compute this fixed-point, and how to statistically assess the probability that indeed the observed response times are generated by two competing strategies. Accompanying this paper is a free R package that can be used to compute and test the presence or absence of the fixed-point property in response time data, allowing for easy to use tests of strategic behavior.

Almost all intentional behavior is the result of applying strategies to problems. Theorizing in cognitive psychology thus often involves the assertion that humans have access to a number of alternative strategies to solve a particular task, and that the observed behavior on a particular trial is the result of the execution of one particular strategy. On other trials an alternative strategy might have been selected, which may result in differences in the observed behavior. For example, idiomatic or fixed-phrase language processing is thought to be a dual-route process

Many theoretical paradigms assume that behavior is the result of similar mixtures of processes (e.g., visual word recognition and reading aloud

An important property of a set of mixture distributions that are all based on the combination of two identical base distributions is the so-called fixed-point property

Panel A shows density plots for two base distributions. The blue line reflects an RT distribution for Strategy 1 (d1), with a mean of 500 ms and a standard deviation of 100, the red line an RT distribution (d2) for Strategy 2 (mean = 600, SD = 150). Panel B shows three mixtures of the two base distributions with mixture proportions as indicated in the legend. The vertical line shown in both panels is drawn at the common coordinate or fixed-point at ∼590 ms.

Despite its wide applicability, there are only a few studies that discuss the fixed-point property (e.g,

The second reason that might have withheld researchers to use the fixed-point property is that performing a statistical test to support the presence of the fixed-point property requires supporting the null hypothesis (i.e., the frequency does

The fixed-point property is a mathematical property of binary mixture distributions: The density function of a binary mixture distribution (_{1}(t)_{2}(t)_{p = 0}(t)_{2}(t)_{p = 1}(t)_{1}(t)_{o}_{o}

Thus, the density of the mixture at _{0}_{0})_{0}

To infer the fixed-point property in experimental data, for example if one wants to assess whether two conditions just differ in terms of the relative proportions of Strategy 1 and Strategy 2 usage, the first step is to estimate the continuous density functions of the data. That is, for each condition – reflecting a mixture proportion – the empirical probability density has to be computed. Computing the density instead of a histogram solves in part the issue of the bin size discussed previously. A straightforward method for computing the continuous density function is kernel density estimation (e.g.,

Density estimation can be used to compute the fixed-point property. _{1} = 1_{2} = 3

Probability density (A, C) and density difference (B, D) for data with (A, B) and without (C, D) a fixed-point. The densities in A correspond to binary mixture distributions with mixture proportions of .1 (black line), .3 (red line), and .9 (green line), respectively. The densities in C correspond to shifted distributions with mean _{1} = 1.2_{2} = 1.6_{3} = 2.8

_{1} = 1.2_{2} = 1.6_{3} = 2.8

While a graphical demonstration of the fixed-point property may be convincing, inferences from data should ideally be based on the results of sound statistical tests. In our approach, such tests are concerned with assessing the degree to which the binary-mixture hypothesis is supported by the distribution of estimated between-conditions crossing points. In the case of the fixed-point property in RT data, we want to find support for either the hypothesis that the data comes from binary mixture distributions with different mixture proportions (that is, the fixed-point should be observed), or not. Thus, for the fixed-point property to hold, there should be evidence

Typically in an experiment, we want to infer whether a certain property exists for the population, based on the sample of participants that were tested. In the current discussion, this means that we want to test whether the fixed-point property holds for the sample of participants in a study. This is the case if we find support for the hypothesis that the crossing points for the various pairs of mixture proportions do not differ. Once the distributions of crossing points per pair of mixture proportion conditions for each of the participants are known, Bayes factors for a regular analysis of variance can be computed

To validate the method for computing and testing the fixed-point property, we ran a series of Monte Carlo simulations. Simulation 1 illustrates that our approach produces reasonable results for non-Gaussian distribution functions, as are typically observed in RT data (e.g.,

In Simulation 1, we assume observations are sampled from one of two inverse Gaussian base distributions, with scale _{1}_{2}_{01} = 0.098, which means it is 10.2 times more likely that there is a fixed point in the data than that there is no fixed point (Not surprisingly, standard frequentist statistics show no support for the alternative hypothesis, F(2,49) = 0.40, p = .67). This means that there is reason to accept the null hypothesis that there is a fixed point.

Averaged density (A), density difference curves (B), and boxplots for the distributions of crossing points (C) for the data from Simulation 1.

A crucial question is to what extend the method to detect mixture distributions described here depends on the nature of the base distributions. Clearly, if the base distributions have a large difference in means relative to their standard deviations, the mixtures will show signs of bimodality. In contrast, when the means of the base distributions are very similar, it might not be possible to distinguish binary mixture distributions from non-mixture distributions. In Simulation 2, we generated data from two inverse Gaussian distributions with different means and scale parameters. One base distribution was always fixed with with _{1}_{1}_{2}_{2}_{1}_{2}_{1}

In addition to the mixture data, we also simulated data in which the three observed distributions were shifted relative to each other (cf. _{1}_{1}_{2}_{2}_{1}_{1}_{2}_{2}

(A) Bayes factors for mixture data (solid black line) and shifted data (dashed red line). (B) F-values for mixture data (solid black line) and shifted data (dashed red line). (C) The average differences between the crossing points for mixture data (solid black lines) and shifted data (dashed red lines). (D) Base distributions of Simulation 2. Solid line represents Process 1, dashed lines represent alternatives of Process 2. In particular, the dashed lines represent the smallest

As an illustration of the size of the effects that the fp method detects, consider the example base distributions in

Because both the power of a study as well as the type I error rate depend on the sample size, we explored the impact of sample size on the probability of finding the fixed-point property. To achieve this, we simulated data from varying numbers of participants (Simulation 3), as well as from varying numbers of observations (Simulation 4). This way, both the sample size (the number of participants) and the precision of the fixed point estimate (based on the number of observations) can be considered. In the next sections, we varied the width of the smoothing kernel used for density estimation to study how this impacts the test statistics.

We simulated data for either 10, 50, or 100 participants, with 200 observations per condition. In this simulation we assumed two normally distributed base distributions, with _{1} = 0_{2} = 1.5

Bayes factors (A) and F statistics (B) differ with sample size (lines) and the standard deviation of the Gaussian kernel. (C) The precision of the estimated crossing points does not vary with sample size. samp: sample size (i.e., the number of participants).

Simulation 4 was set up in a similar way as Simulation 3. That is, again 10,000 simulations were performed, while generating data from distributions with the same properties. The difference lies in the ratio between the number of samples and the number of observations. The number of samples in Simulation 4 was kept constant at 50, while the number of observations per condition varied from 100, to 200, to 500.

Bayes factors (A) and F statistics (B) differ with the number of observations (lines) and the standard deviation of the Gaussian kernel. (C) The precision of the estimated crossing points varies with the number of observations. obs: the number of observations (i.e., repeated measures).

As an illustration of how the fp package can easily be applied to test the prediction that a binary mixture distribution underlies the data, we studied the “failure-to-engage” hypothesis of task switching (FTE,

The FTE hypothesis explains residual switch costs by proposing that task preparation only occurs on a subset of trials. That is, on some trials participants fail to prepare for the new task, leading to additional time costs when executing the task. Formally, the FTE hypothesis thus proposes that the RT distribution of switch trials is

Here, _{engaged} and _{not engaged} refer to the RT distributions of prepared and not prepared trials, respectively.

De Jong (

For each participant, we first estimated density functions for each RSI condition, with a smoothing kernel SD of 0.1 s. Next, the difference between these densities was computed as well as the crossing points. (Bayesian) mixed-design ANOVAs are used to infer the presence or absence of the fixed-point property. In particular, a mixed-design ANOVA model was fit to the data with block as a between-subject factor and RSI as a within-subject factor. Next, the fit of this model against a model that omits each factor separately results in a Bayes factor indicating the likelihood that a particular factor is required to explain the data

The FTE hypothesis predicts that there exists a fixed-point in the data. In particular, the RT distributions of the three different RSI conditions that we compared should have a common fixed point, as well as the RSI conditions across the between-subject block duration manipulation. _{RSI} = 0.29 (the data is 3.4 times more likely under the null hypothesis than under a model that includes RSI as a factor) and BF_{RSI × block} = 0.27 (the data is is 3.7 times more likely under a model without the interaction – but with main effects – than under the full model with RSI, block and the interaction). A classical mixed-effect ANOVA with block length as between-subjects factor and RSI as within-subjects factor indeed does not find support for the alternative hypothesis (F_{RSI}(2,36) = 0.78, p = 0.47, F_{RSI × block}(2,36) = 0.28, p = 0.76). In addition, there was no clear effect of the block duration (BF_{block} = 0.51), suggesting that the data is only 2.0 times more likely to come from a model without block duration than with block duration (A standard frequentist test yields F_{block}(1,18) = 2.8, p = 0.11).

Averaged density (A, D) density difference curves (B, E), and boxplots for the distributions of crossing points (C, F) for the data from De Jong (2000), Experiment 2. The top row (A, B, C) shows the short blocks, the bottom row (D, E, F) shows the long blocks.

The average

The results of the fixed-point analysis on the data of Experiment 2 of De Jong

The fixed-point property in binary mixture data is an interesting prediction for many theories in cognitive psychology that assume mixtures of processes. If the mixture proportions are experimentally manipulated, then it can be easily verified whether the fixed-point property holds in the data. This paper has outlined how this can be achieved. Accompanying this paper is an R package called

In a series of simulations, we tested the method proposed here as well as the R package, and found that it can successfully distinguish between data sets from binary mixture distributions and data sets with other but comparable differences in RT. In particular, we tested the method on a data set in which three distributions were shifted relative to each other (instead of mixed), and found that for large enough

Furthermore, the test is robust against variations in the Gaussian kernel standard deviation, which determines the smoothness of the estimates density functions. When the standard deviation of the kernel was set at a suitably high value exceeding one standard deviation, the results remained comparable. However, there is a practical limit on increasing the kernel SD. If the SD is too large, the density estimate oversmoothes important properties of the RT distribution related to bimodality. The test is also reasonably robust against low number of observations and small sample sizes such that it can be applied to relatively small data sets.

Finally, to show the applicability of the fixed-point property test, we analyzed data from De Jong

These simulations and analysis of an existing data set demonstrate that the fixed-point property, and the

(GZ)