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Combining reaction-time distributions to conserve shape

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Abstract

To improve the estimate of the shape of a reaction-time distribution, it is sometimes desirable to combine several samples, drawn from different sessions or different subjects. How should these samples be combined? This paper provides an evaluation of four combination methods, two that are currently in use (the bin-means histogram, often called "Vincentizing", and quantile averaging) and two that are new (linear-transform pooling and shape averaging). The evaluation makes use of a modern method for describing the shape of a distribution, based on L-moments, rather than the traditional method, based on central moments. Also provided is an introduction to shape descriptors based on L-moments, whose advantages over central moments—less biased and less sensitive to outliers—are demonstrated. Whether traditional or modern shape descriptions are employed, the combination methods currently in use, especially bin-means histograms, based on averaged bin means, prove to be substantially inferior to the new methods. Averaged bin-means themselves are less deficient when estimating differences between distribution shapes, as in delta plots, but are nonetheless inferior to linear-transform pooling.

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Code availability

R code for the simulations will be made available by the author to anyone who plans to pursue these investigations further.

Notes

  1. The meaning of "kurtosis", when it is defined, traditionally, as the standardized fourth central moment, is unclear (Westfall, 2014). The interpretation is clearer when it is defined in terms of L-moments, discussed below and in Appendix A. The corresponding density functions in Panels A and B of Fig. D1 have the same values of L-skewness, but differ in L-kurtosis, with values of L-kurtosis smaller in Panel B. The light curve in Panel A has about the same L-kurtosis as the broken curve in Panel C (0.175 vs. 0.172), but differs in L-skewness (.100 vs .172).

  2. Such speech detection variations can be as great as 100 ms, larger than the word- frequency effect often of primary interest (See Table 2 in Andrews & Heathcote, 2001, and Table 4 in Spieler & Balota, 1997). Detection delays can be reduced by using a speech detector that applies appropriately different but low thresholds to high and low audio frequencies, along with filtering that rejects the brief non-speech lip noises that exceed the low thresholds. Alternatively, estimates of such effects can be used to correct the RTs before combining data from different words. Without either suitable instrumentation or such correction, the shapes of distributions of combined RTs may be misleading.

  3. Thomas and Ross (1980) and Jiang et al. (2004) showed that quantile averaging has desirable properties when the parent distributions of the components belong to the same LSF, and Thomas and Ross (1980) recommended the use of Q-Q plots to test this. However, I have seen no evidence in the psychological literature of this test having been applied as a prelude to quantile averaging. Furthermore, the evaluations described below show that with realistic sample sizes, quantile averaging is inadequate even when the samples are from members of the same LSF.

  4. For an example of applying L-moments in the analysis of the shapes of RT distributions, see Sternberg & Backus, 2015.

  5. Because of the standardization by m2, β1 and β2 are dimensionless quantities: The same (β1, β2) point represents distributions that can differ in location and/or scale.

  6. Traditionally, β1 is defined as \({\mu }_{3}^{2}/{\mu }_{2}^{3}\), and the skewness measure as \(\sqrt{{\beta }_{1}}\) (Kendall & Stuart, 1969, Chapter 3). For the purpose of this paper, β1 will be defined as \({\mu }_{3}/{\mu }_{2}^{3/2}\).

  7. I use the term "family of distributions", or just "family" to indicate a theoretical distribution with the values of its parameters unspecified, and the term "distribution" to denote a particular member of that family, with specified parameter values.

  8. The variability of these measures, emphasized for C-moments of RT data by Ratcliff (1979) and Heathcote et al. (1991), increases with the variance of the underlying distribution. Thus, when their use is anticipated, experimental methods are called for that reduce this variance, such as extensive practice, rest periods, adequate warning signals, and performance incentives and feedback.

  9. L-skewness and L-kurtosis have often been represented by τ1 and τ2, respectively, by analogy to β1 and β2. To avoid their being confused with τ, often used for the exponential parameter of the ex-Gaussian distribution, popular in psychology, this paper will use λs3 and λs4, etc., instead, where the "s" indicates the standardization.

  10. When considering the sums of stochastically independent random variables, it should be noted that (functions of) L-moments do not share the very useful additivity property of the cumulants of a distribution (κr), which are functions of the C-moments. For example, when x and y are stochastically independent, then, for their sum, x + y, the cumulants are additive: κr (x + y) = κr (x) + κr (y), (r ≥ 2). It follows, for example, that the variance, third moment, and higher cumulants of the duration of a sequence of stochastically independent iterated processes all increase linearly with the number of iterations.

  11. These two distributions are among those that have been applied to reaction-time data; see, e.g., Luce (1986), Anders et al. (2016).

  12. These levels of bias for L-skewness and L-kurtosis can be further reduced by adopting the estimates developed by Withers & Nadarajah (2011).

  13. Examples of distributions that are represented by regions, rather than points or curves, are the four-parameter Kappa distribution (Hosking & Wallis, 1997; Kjeldsen et al., 2017) and the generalized lambda distribution (Asquith, 2007).

  14. But see, e.g., Asquith, 2014; Hosking, 2007; Karvanen & Nuutinen, 2008.

  15. To derive the density function of the distribution of combined samples when using the shape averaging combination method, or to generate random samples from the combined distribution, Headrick’s procedure (or an equivalent) would be necessary.

  16. I have not included the method devised by Cousineau et al. (2016), because it requires estimating the minimum RT, which depends on assuming a feature of the shape of the distribution.

  17. Referring to these two methods, Jiang et al. (2004, p. 187), who distinguish them, claimed that "In practice, the two variants yield highly similar results." In what follows we shall see that one can be substantially inferior to the other, especially if the number of bins in bin-means histograms is as small as has been recommended. See footnote 21.

  18. The R function I used to obtain bin means is provided in Appendix G.

  19. The midpoint of the jth bin is ( pj + pj+1)/2. It has sometimes been claimed that the jth bin mean is the quantile associated with the jth bin midpoint. This is (approximately) true if and only if the distribution of observations within the jth bin is (approximately) symmetric. Thus, bin means for different samples are quantiles whose associated probabilities are likely to differ.

  20. The bin means have sometimes been called "Vincentiles" (Van Zandt, 2000). They are also quantiles, of course, as are any values within the bounds of the sample, but these quantiles are not defined in terms of the proportions to which they correspond.

  21. According to Ratcliff (1979, p. 459), whose examples included nine bins (Fig. 2) and 24 bins (Fig. 4), the procedure can be used with "as few as ten observations per subject". According to Heathcote (1996), who recommends between 4 and 19 bins, and whose example (Fig. 7) has 19 bins (p. 435), "For good ex-Gaussian fits, Vincentizing requires as few as 20 observations per condition". He also suggests (p. 436) that "Vincent averaging combines data across subjects or conditions without distorting the shape of the underlying distribution." According to Andrews & Heathcote (2001, p. 517), "Average vincentiles approximately preserve the shape of underlying distributions for each participant, as long as the individual participants’ distributions are unimodal and smooth." In what follows, the validity of these claims is tested.

  22. I used Hyndman & Fan’s (1996) type-8 quantile estimator. This method might be improved by using a better quantile estimator, such as the one proposed by Navruz & Ozdemir (2020).

  23. This method (which they call "Vincentizing") has been thoroughly discussed by Jiang et al. (2004), Rouder & Speckman (2004), and by Thomas & Ross (1980), who also discuss variants that make use of transformed observations. The differences between quantile histograms and bin-means histograms (sometimes called "Vincent histograms") are discussed by Van Zandt (2000). See also footnote 2 in Andrews & Heathcote (2001).

  24. This last measure was used by Wolfe et al. (2010), who called it the "x- score transform".

  25. Qn is given by the .25 quantile of the distances {|xi − xj|; i < j}

  26. The linear-transform method was used by Roberts and Sternberg in their "summation test" (1993; Sternberg & Backus, 2015, Appendix), but using the median and the interquartile range as measures of location and scale, respectively, which are not the choices that minimize the mean estimation error in my simulations (Appendix E).

  27. A contrasting view is expressed by Ratcliff and Smith (2010, p. 90), according to whom "Invariance of distribution shape is one of the most powerful constraints on models of RT distributions.That the diffusion model predicts this invariance is a strong argument in support of its use in performing process decomposition of RT data." See Sternberg & Backus (2015) for a discussion of arguments for (and evidence against) shape invariance.

  28. See Footnote 10, and, for a recent validation of their use for inferring distributional shape, Section 6.3 in Sternberg (2016).

  29. The order statistics of a sample are the sample values in ascending order. One approach to computing the nth L-moment, λn, of a sample would be to determine the mean of each of the mth order statistics (m = 1, 2,.., n) of all possible combinations of n values of the sample, and then to combine these means as in the equation for λn below. See Wang (1996). For example, suppose a sample of size n = 5 with values 1, 2, 3, 4, and 5, and we want to calculate λ3 for that sample. The set of all possible combinations of size 3 is: [ (1,2,3), (1,2,4), (1,2,5), (1,3,4), (1,3,5), (1,4,5), (2,3,4), (2,3,5), (2,4,5), and (3,4,5)]. (These would appear with equal probability in a random sample.) The second order-statistics of this set are 2, 2, 2, 3, 3, 4, 3, 3, 4, and 4. Their mean, E(X2:3) = 3. 0. Similarly, E(X1:3) = 1. 5, and E(X3:3) = 4. 5. Thus, λ3 = [1. 5 −(2 × 3. 0) + 4. 5] = 0, as expected, given that the sample, thought of as a distribution, is symmetric.

  30. Thus, unlike the measure of kurtosis, β2, based on the standardized fourth central moment, the meaning of L-kurtosis is clear.

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Acknowledgements

I thank Vincent Hurtubise and Mark Davidson for computer support.

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  1. University of Pennsylvania, Philadelphia, PA, USA

    Saul Sternberg

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The computations on which this report is based were conducted in R (R Core Team, 2022). Packages used include lmom (Hosking, 2019), lmomco (Asquith, 2021), robustbase (Maechler et al., 2022), SuppDists (Wheeler, 2022), and gamlss.dist (Stasinopoulos & Rigby, 2022).

Appendices

Appendix A: definitions and properties of L-moments

The first six L-moments are defined as follows: Let Xm:n be the mth order-statistic in a sample of size n, and E denote expectation.Footnote 29

$$\begin{array}{l}\lambda_1=E\left(X\right)\\\lambda_2 = E(X_{2:2}-X_{1:2})/2\\\lambda_3 = E(X_{3:3}-{{2X}_{2:3}+X}_{1:3})/3\\\lambda_4 = E(X_{4:4}-{{3X}_{3:4}+{3X}_{2:4}-X}_{1:4})/4\\\lambda_5 = E(X_{5:5}-{{4X}_{4:5}+{6X}_{3:5}-4_{2:5}+X}_{1:5})/5\\\lambda_6 = E(X_{6:6}-{{5X}_{5:6}+{10X}_{4:6}-10X_{3:6}+5X}_{2:6}-X_{1:6})/6\end{array}$$

To understand these L-moments, it helps to rewrite the expressions inside the parentheses as follows:

For λ3

$$\left(X_{3:3}-X_{2:3}\right)-(X_{2:3}-X_{1:3}).$$

Thus, λ3 will be zero if the distribution is symmetric, such that the mean separation between the middle and third order-statistic is equal to the that between the middle and first order-statistic; if it is larger (smaller), it will be positive (negative).

For λ4

$$\left(X_{4:4}-X_{1:4}\right)-3(X_{3:4}-X_{2:4}).$$

If the means of the four order-statistics are equally spaced, the separation between the means of the two middle ones will be one third of the that between the first and fourth, and we will have λ4 = 0. If the separation is less (more) than one third, e.g., if either or both tails are heavy (light) relative to the middle of the distribution, λ4 will be positive (negative).

For λ5

Footnote 30

$$\left[\left(X_{5:5}-X_{4:5}\right)-(X_{2:5}-X_{1:5})\right]-3\left[\left(X_{4:5}-X_{3:5}\right)-(X_{3:5}-X_{2:5})\right].$$

The value of λ5 indicates how much of the asymmetry measured by λ3 is due to the tails (large first term) rather than the middle region of the distribution (large second term).

For λ6

$$\left[\left(X_{6:6}-X_{1:6}\right)-5(X_{4:6}-X_{3:6})\right]-5\left[\left(X_{5:6}-X_{4:6}\right)+5(X_{3:6}-X_{2:6})\right].$$

This will be large if either tail and/or the peak (first term) is large relative to the "shoulders" (second term) of the distribution. Thus, unlike λ4, and supplementing it, λ6 distinguishes the shoulders from other parts of the distribution.

For k ≥ 3, the L-moments are usually standardized: λsk = λk / λ2, which defines L-skewness (λs3) and L-kurtosis (λs4).

The standardized L-moments are constrained by several inequalities, including |λsk| < 1, (k ≥ 3); λs4 ≥ -1/4; λs6 ≥ – 1/6; and 4λs4 ≥ 5(λs3)2 – 1 (Hosking, 1996; Jones, 2004). Given the limited ranges of λsk when k ≥ 3, small differences in their values matter greatly.

Appendix B: bias of shape measures based on L-moments and C-moments

Among the advantages of measures of distribution shape based on L-moments over those based on C-moments is that the former are less biased, especially for small samples. This difference is demonstrated by the findings pictured below (Fig. B1 and B2).

Fig. B1
figure 9

Bias versus Sample Size of Two Pairs of Shape Measures for Seven Ex - Gaussian Distributions. Plotted values are based on means of 5000 replications. Estimates of the same shape based on different sample sizes are connected by line segments. Panel A: Shape measures β1 and β2, based on central moments. Panel B: Shape measures λs3 and λs4 based on L-moments. Centers of the large open circles represent the "true values" of these measures (based on samples of size 10,000). Panel A shows that even for sample sizes as large as 100 and 200 (triangles), the estimated shape values based on central moments are far from the true values, especially for more skewed distributions, whereas the corresponding estimates in Panel B, based on L-moments, have relatively little bias

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Fig. B2
figure 10

Bias versus Sample Size of Two Pairs of Shape Measures for Seven Inverse-Gaussian (Wald) Distributions. Results are similar to those for the ex-Gaussian distributions in Fig. B1

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Appendix C: outlier effects on shape measures based on L-moments and C-moments

Among the advantages of measures of distribution shape based on L-moments over those based on C-moments is that the former are less influenced by outliers. This difference is demonstrated by the findings pictured below, which show the effects on mean shape measures of introducing one high outlier among 200 RTs, for seven ex-Gaussian distributions (Fig. C1) and seven inverse-Gaussian (Wald) distributions (Fig. C2).

Fig. C1
figure 11

Effects on mean shape measures of introducing one high outlier among 200 RTs, for seven ex-Gaussian distributions. Plotted values are based on means of 10,000 replications. For each distribution and each replication, the outlier was introduced by adding 3 × SD to the largest observation among 200 sampled observations. Shape measures based on L-moments (λs3 and λs4) and based on C-moments (β1 and β2) were determined for each sample with and without the outlier. Panel A: Means of the two skewness measures with and without the outlier, after linear normalization such that without the outlier, measures for seven different distributions increase linearly from zero to one. Small filled circles and large open circles represent values of λs3 and β1, respectively. Panel B: Means of the two kurtosis measures with and without the outlier, represented similarly

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Fig. C2
figure 12

The same analysis as in Fig. C1, but with samples drawn from seven inverse-Gaussian (Wald) distributions

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Appendix D: density functions of nine basic distributions

Fig. D1
figure 13

Density Functions of the Nine Basic Distributions. Panel A: Generalized logistic distributions. From the distribution with highest to lowest peak, values of L-skewness are 0.400, 0.250, 0.100, and values of L-kurtosis are 0.300, 0.219, 0.175, respectively. Panel B: Shifted inverse Gaussian (Wald) distributions. From the distribution with highest to lowest peak, values of L-skewness are 0.400, 0.249, and 0.100, and values of L-kurtosis are 0.232, 0.162, and 0.129, respectively. Panel C: Ex-Gaussian distributions. From the distribution with highest to lowest peak, values of L-skewness are 0.264, 0.172, and 0.050, and values of L-kurtosis are 0.181, 0.172, and 0.138, respectively. All distributions were adjusted to have means of 400 units and standard deviations of 45 units

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Appendix E: choice among variants of linear-transform pooling

Tables E1, E2, E3, E4, E5 and E6 show the mean estimation error (Tables E1A and E2A, etc.) and the mean bias (Tables E1B and E2B, etc.) for each of the thirty variants of the linear-transform pooling method, defined by combining five measures of location: the mean, the 20% winsorized mean (winmn), the 20% trimmed mean (trimmn), the mean of the 0.25 and 0.75 quantiles (quartmn), and the median, with six measures of scale: Qn (Rousseeuw & Croux, 1993), the second L-moment (lm2), the square root of the 20% winsorized variance (swv, Wilcox, 2017), the median absolute deviation (mad), the interquartile range (iqr), and the standard deviation (sdev). The values in each table are the means from two independent simulations, each involving 500 replications, and averaged over the nine basic distributions. Sample sizes of 60 and 100 were chosen because they seemed to approximate the sizes most likely to be used.

Table E1 Ten samples, same-shape parents, N = 60
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Table E2 Ten samples, same-shape parents, N = 100
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Table E3 Four samples, same-shape parents, N = 60
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Table E4 Four samples, same-shape parents, N = 100
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Table E5 Four samples, different-shape parents, N = 60
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Table E6 Four samples, different-shape parents, N = 100
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When ten samples from the same parent distributions were combined (Tables E1 and E2) the mean estimation error was smallest for the mean + Qn variant. When four samples from the same parent distributions were combined (Tables E3 and E4) the mean estimation error was smallest for the mean + lm2 variant. However, all six tables show that the mean bias associated with the mean + lm2 variant is considerably greater than with the mean + Qn variant, while their mean estimation errors do not differ greatly. I therefore selected the mean+Qn variant of the linear-transform method for comparison with the other methods.

It is worth noting, however, that relative to the differences in mean estimation error produced by the various methods (Fig. 2), the range of values of the mean estimation error produced by the thirty variants of the linear-transform pooling method is small. For example, the range when four samples are combined and N = 100 (Table E4) is (4.35, 4.55), a difference of 0.20, while the range of means across all methods for this case (Fig. 2B) is (4.45, 8.95), a difference of 4.50.

Appendix F: why are bin-means histograms so bad?

A hint as to the reasons for the large bias of the results of bin means averaging is provided by noting how the bin-means histogram represents the tails of a distribution. Recall that to represent a distribution, a set of m bins between m +1 equally spaced proportions is defined, and the means {Xk•}, (k = 1, 2, . . . , m), of the values in each bin determined. The bin-means histogram contains m equal-area rectangles bounded by the successive pairs of these means. As noted by van Zandt (2000, p. 430), such histograms exclude the data below the smallest and above the greatest of the m means. To demonstrate this, I created a distribution by combining a uniformly distributed center with low and high tails. A standard histogram of this distribution together with its density function are shown in Panel A of Fig. F1. The same data are represented as bin-means histograms with 6 and 15 bins, respectively, in Panels B and C of Fig. F1. Note how much area under the density function is outside the range of the bin-means histograms.

Fig. F1
figure 14

Standard and Bin − Means Histograms. Representations of a distribution by standard and bin-means histograms. Panel A. Standard histogram of a distribution composed of a central uniformly distributed portion together with low and high tails. Also shown is a density function for the distribution. Panel B. Bin-means histogram of the same data with 6 bins. The same density function is shown. The red regions under the x-axis indicate values in the distribution that are excluded from the binmeans histogram. Panel C. Bin-means histogram of the same data with 15 bins. The same density function is shown. The red regions under the x-axis indicate values in the distribution that are excluded from the bin-means histogram

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The L-moments associated with the sample values and with the two bin-means histograms are shown in Table F1. Consistent with the bias shown in Figs. 4, 5, and 6, we see that the values associated with the bin-means histograms are underestimates.

Table F1 L-moments of sample and bin-means histogram representations
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Appendix G: R code for selected functions

The following function starts with a sample, and generates bin means ("Vincentiles") with nbin bins.

figure a

quantiles were generated using the type-8 quantile function:

figure b

Given averaged quantiles, a sample was generated using the following function (a deterministic version of inverse transform sampling), which distributes nobs observations uniformly in each interval, thus creating total obsns = nobs*(length(quantiles)-1).

The same function was used to generate a sample from bin-means averaging, replacing quantiles by bin means.

figure c

Appendix H: R code to implement linear-transform pooling and shape averaging

Each sample of reaction-time data is a vector. The input to the program, called "sample.list" is a list of these vectors. The program outputs are "pool" and "meanshape".

figure d

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Sternberg, S. Combining reaction-time distributions to conserve shape. Behav Res 56, 1164–1191 (2024). https://doi.org/10.3758/s13428-023-02084-7

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