1 Introduction
The science of human judgment provides numerous examples of people “discovering” patterns where none exist (Reference BaronBaron, 2008; Reference GilovichGilovich, 1991; Reference VyseVyse, 2013). Famous examples include illusory associations in judgments about people (Reference Chapman and ChapmanChapman & Chapman, 1969; Reference Hamilton and RoseHamilton & Rose, 1980), locations (Reference ClarkeClarke, 1946), or other stimuli (Reference ChapmanChapman, 1967). Here, we consider the potential for illusory correlation in judgment of a common type of graph — an interrupted time-series graph — where viewers judge whether an interrupting event is associated with a change in the subsequent height of the data points. Such graphs are common in many parts of life, for example, in news articles claiming that a stock market “tumbles on” some major incident or prominent figure’s comments. Relatedly, at the time of this writing, these graphs are being judged to infer whether health policy implementation was followed by a change in disease infection rate. These graphs are also common in single-subject (n=1) A-B designs, where the comparison is between data points before versus after some intervention on a single participant. In addition to being common, these graphs can be useful in the study of covariation detection because they are relatively simple. Unlike stimuli that are slowly revealed over the course of an experiment (e.g., Reference Chapman and ChapmanChapman & Chapman, 1969), graphs provide a static display of all relevant information at once, avoiding complications of forgetting and memory biases. Another simplification is that only one of the relevant variables is continuous (data height); the other is binary (before-versus-after the interruption). Hence, judgment of these graphs can be viewed as point-biserial correlation detection. Despite the simplicity of these graphs, we worry that they are especially prone to misjudgment because of a common attribute of time-series data: serial dependence.
One example of serial dependence is that warm days tend to be followed by warm days, and cold by cold. This pattern can be seen, for instance, when examining daily high temperatures of New York City’s Central Park for each day in January 2020 (National Centers for Environmental Information, 2020). Perhaps the simplest measure of serial dependence is the lag-1 autocorrelation, and those temperatures show an autocorrelation of r = .54. This autocorrelation is just the correlation between the temperature on each day and the day that follows it (Jan. 1 paired with 2, 2 with 3, and so on). Non-zero autocorrelation is also common in time-series of behavior. An analysis of psychology and education journals from 2008 found that single case design datasets had a mean autocorrelation of +.20 (bias corrected; Reference Shadish and SullivanShadish & Sullivan, 2011). Serial dependence also varied across studies and types of designs. For example, the simplest designs (e.g., 10 trials of condition A followed by 10 trials of condition B) had relatively high serial dependence with a mean autocorrelation of .75, whereas alternating treatment designs — in which the condition can change on every trial (e.g., ABABBA) — had a mean autocorrelation of -.01 on average.
To visualize serial dependence, consider a simplified scenario where some outcome is measured over 6 days, with an intervention occurring between the 3rd and 4th days. Even if the intervention has no effect, one could expect some fluctuation in the outcome due to random error. The top panel of Figure 1 illustrates a situation where random error is independent across days. The gray dashed horizontal line at a height of 10 indicates the baseline (error-free) outcome. Relative to this baseline, error may increase or decrease the outcome measure, as indicated by arrows and the e corresponding to each day. Without serial dependence, error on one day affects the outcome only on that day. In other words, error on each day is independent of error on other days. A large random error immediately after the intervention (4) will alter the outcome, but just for a single measurement. So, without serial dependence, it is apparent that the intervention is not associated with any meaningful change. The temporary increase in the outcome on day 4 would likely be judged correctly as random noise rather than systematic change produced by the intervention.
Figure 1: Illustration of the impact of random error and serial dependence in a situation where the intervention has no effect (i.e., the Null Hypothesis is true). When serial dependence is absent, error (colored arrows labelled with e) at each time point has no effect on other time points. In contrast, when serial dependence is present, error on each day persists with diminishing effect on the days that follow. The horizontal dashed line indicates an error-free baseline.
As illustrated in the bottom panel of Figure 1, serial dependence can cause error on one day to persist for the following days, decreasing in its influence over time. One can see, for example, that the large error on day 4 (e4) impacts the outcomes in the days that follow it. For illustration purposes, Figure 1 was generated with an extreme serial dependence, with a 1-lag autoregression parameter set to .8. With this setting, error on day 4 is still 80% (i.e., .81) as impactful one day later, 64% (.82) as impactful two days later, and so on. There are a wide variety of time series models, but in the simple one considered here, autoregression parameters typically range from −1 to +1, with values closer to 0 indicating less serial dependence, and a value of exactly 0 indicating its absence. The lag-1 autocorrelation will be identical to the lag-1 autoregression parameter, at least in the long-run.
In the example at the bottom of Figure 1, serial dependence could create the illusion of a durable change in the outcome. That is, because the effect of error persists over time, error could be misattributed to an effect of the interrupting event, even when the Null hypothesis (no effect) is true. In other words, a non-zero autocorrelation could create the illusion of a nonzero-correlation between data height and phase (before-versus-after the intervention). Incorrectly judging the data height as being associated with the intervention could be described as a Type-I error. The phrase “Type I error” often refers to false positives from mechanical decision rules, most commonly formal statistical hypothesis tests. Here, we use the phrase more broadly to include false positives in informal intuitive judgment (e.g., Bishara et al., 2021). The informal judgment of interest here involves visually inspecting a graph to judge the association between the intervention and data point height.
Type I errors in informal judgment could plausibly result from serial dependence. Serial dependence allows random error to temporarily steer measurements upward or downward, giving the appearance of a stable change in the height of data points. This effect could increase the chances of large absolute differences between phases of an intervention. If viewers relied on a simple heuristic, comparing the average before versus after, they would be more likely to make Type-I errors. Such a heuristic has been found in related judgments where people compare two sets of numbers (e.g., ratings of two products). For such judgments, people tend to weigh the mean difference much more than they weigh other relevant cues, such as sample size or standard deviation (Reference Obrecht, Chapman and GelmanObrecht et al., 2007).
Even if people were aware that serial dependence could influence their judgments, it is not clear how they would informally estimate serial dependence and adjust for it when visually inspecting graphical displays of data. For example, positive autocorrelation often leads to visually smoother times-series plots, but so does low noise (i.e., small residuals), and so it is not clear how serial dependence and noise would be distinguished. Furthermore, even if people were able to somehow estimate serial dependence by visually inspecting a graph, they would still have to adjust their judgments properly. Unfortunately, people can have inaccurate lay intuitions about serial dependence and evidentiary value. For example, people sometimes believe that dependent information provides stronger evidence than independent information even when the reverse is true (Reference Xie and HayesXie & Hayes, 2020). This belief would lead to adjustments in the wrong direction.
Some work has already hinted that serial dependence increases Type-I errors in interrupted time-series graph judgments. Reference Harrington and VelicerHarrington and Velicer (2015) compared expert graph judgment to statistical analysis of actual data published in the Journal of Applied Behavior Analysis. This journal was of particular interest because some applied behavior analysts prefer informal judgments of graphs in place of formal statistical tests (also see Smith, 2012), perhaps due to B. F. Skinner’s (1956) skepticism of the latter. When informal judgments of the articles’ authors clashed with statistical test results, serial dependence was usually present (also see Jones et al., 1978). Furthermore, in these clashing cases, graph judgment often resulted in a determination of significant treatment effects even when statistical analysis did not. Such results could suggest that graph judgment amidst serial dependence leads to Type I errors, but only if one assumes that the statistical analysis was accurate and graph judgment was not. Unfortunately, with real samples of data, one does not know whether the treatment effect exists in the population or not. So, these inconsistencies could indicate Type I errors via graph judgment, but they could also indicate Type II errors (missing a true effect) via statistical analysis.
To objectively identify errors, it is necessary to know the population from which the graph data were generated. If, in the population, there is no difference before versus after the intervention, then deciding that there is a treatment effect is truly a Type I error. Reference Matyas and GreenwoodMatyas and Greenwood (1990) took this approach, using populations with and without treatment effects to generate line graphs of A-B designs. When participants judged these graphs, they frequently made Type I errors, and especially so amidst serial dependence. When the serial dependence was created with an autoregressive coefficient of just .3, the Type I error rate was as high as 84%. To put this number in context, it is useful to compare it to another practice known to inflate Type I errors: p-hacking. P-hacking refers to a variety of questionable statistical practices, for example, repeatedly analyzing data and adding participants until a p-value happens to stray below .05, at which point the experiment is stopped (Reference John, Loewenstein and PrelecJohn et al., 2012; Reference Simmons, Nelson and SimonsohnSimmons et al., 2011). One classic simulation study showed that a simultaneous combination of four different p-hacking techniques inflated Type I error rates from the intended 5% up to 61% (Reference Simmons, Nelson and SimonsohnSimmons et al., 2011), a dangerously high number, though still not as high as the number in Matyas and Greenwood.
Even setting aside the unusually high Type I error rate, the Reference Matyas and GreenwoodMatyas and Greenwood (1990) experiment attracted skepticism for several methodological reasons. First, in that experiment, the response options did not allow for different degrees of certainty that a treatment effect had occurred (Reference Brossart, Parker, Olson and MahadevanBrossart et al., 2006; Reference Parsonson, Baer, Kratochwill and LevinParsonson & Baer, 1992). Instead, participants had five response options: A. no intervention effect, B. a level change, C. a trend change, D. combined level and trend change, and E. other type of systematic change during intervention. Thus, participants had to choose between no-change (A), and change (B-E), but could not express varying degrees of certainty about whether change happened. So, it is unclear whether the Type I errors occurred with high confidence, or merely when participants were guessing.
Second, each condition was represented by just one graph, so if that one graph happened to have sampling error in the shape of an intervention effect, then the Type I error rate would be artificially inflated for that condition (Reference Fisher, Kelley and LomasFisher et al., 2003). In fact, the one graph that produced the most extreme Type I error rate (84%) by informal judgments also produces a Type I error when analyzed by a common statistical model (Reference Fisher, Kelley and LomasFisher et al., 2003). So, that extreme Type I error rate could simply be an artifact of sampling error in graph generation.
Third, when examining the equation used to generate the graphs (Reference Matyas and GreenwoodMatyas & Greenwood, 1990, p. 343), we noticed that there is an alternative explanation of the increased Type I errors amidst serial dependence. Even when there is no treatment effect in that equation, it creates on-average differences between baseline and intervention periods. Those differences could cause participants to choose something other than “no intervention effect,” thereby inflating the Type I error rate. Importantly, this problem arises only when there is serial dependence, and so this problem could account for the elevated Type I error rates in that condition (see Appendix for a proof that the expected difference was non-zero even under the Null Hypothesis).
Fourth, in the Matyas and Greenwood experiment, 50% of graphs had no effect in them, but only 20% of options (option A) indicated no effect. If participants inferred that they should spread their responses equally among options A through E, that incorrect inference could have inflated Type I errors generally, for all conditions. Although this concern cannot account for the relatively high Type I errors amidst serial dependence, at first glance, it could account for the high absolute error rate.
There have been no published attempts to address reasons for skepticism of Reference Matyas and GreenwoodMatyas and Greenwood (1990), despite the surprisingly high Type I error rate observed in that study. We conducted two experiments to do so, one with a general adult sample, and one with a sample of graduate students who already had training relevant to the task. Because results were nearly identical across experiments, findings are presented in a single Results section.
The primary question of interest was whether Type I errors were more common when serial dependence was present in the graphs than when it was absent, and if so, would this pattern merely reflect guesses, or also more confident responses. Several secondary questions were also considered. If serial dependence increases Type I errors, could this be due to a heuristic where people compare the average before versus after the intervention? Does serial dependence create a bias to decide “treatment”, or decrease people’s ability to distinguish between treatments and non-treatments, or both? Additionally, do the absolute rates of Type I errors reach the extremes indicated by Matyas and Greenwood (84%); are they elevated even relative to the customary statistical standard (5%)? Finally, how well do people’s predictions about their performance align with reality?
2 Experiment 1 Method
2.1 Participants
Initially, 54 adult participants were recruited through Amazon Mechanical Turk, with the restrictions that they were in the United States and had a worker rating above 90%. Prior to unblinding to condition, we decided to exclude any participants performing below chance (.50 correct) overall, having a median response time under 1 second, or an experiment completion time of over two hours, as these could demonstrate a lack of attention to the task. These rules resulted in the exclusion of 4 participants for a final sample size of n=50 (32 male). Age ranged from 21 to 56 (M = 34.0, SD = 7.37). Participants were paid $5 plus a bonus that ranged between $0 and $7.20 depending on performance.
2.2 Design and Materials
A 2 (Serial Dependence Absent vs. Present) x 2 (Treatment Effect vs. No Treatment Effect) repeated-measures factorial design was used. Both factors were manipulated through the graph generating equations.
Specifically, graphs were generated using a 1-lag autoregressive interrupted time series model:
where y t was the observed score at time t (see Figure 1). Each graph had 20 data points (t ranged from 1 to 20). Reviews of actual single-case designs have found a similar number of datapoints, with medians of 19 to 20 data points (Reference Harrington and VelicerHarrington & Velicer, 2015; Reference Shadish and SullivanShadish & Sullivan, 2011). The constant b was set to 20 to avoid the complication of negative values of y t. In the No Treatment Effect condition, d t was equal to 0. In the Treatment Effect condition, d t = 0 if t ≤ 10, and d t = 5 if t > 10. This created an intervention period 5 points higher than the baseline period, on average. The v t term was a function of the preceding trial (v t−1) and random error on the current trial (e t):
with a representing the autoregression parameter. Specifically, a = 0 in the Serial Dependence Absent condition, and a = .3 in the Serial Dependence Present condition. The .3 value was chosen based on the extreme Type I errors that it appeared to produce in Reference Matyas and GreenwoodMatyas and Greenwood (1990). Random error (e t) was generated independently for each trial using a normal distribution with mean of 0 and standard deviation of 5. Thus, when there was a treatment effect (when d t=5), the treatment had an effect size of 1 standard deviation relative to the random error term. To initiate the data stream, v 1 was defined as e 1.Footnote 1
Equations 1–2 are similar to ones previously used (Reference Matyas and GreenwoodMatyas & Greenwood, 1990; also see Ximenes et al., 2009) with a 1-lag autoregressive model that can approximate other time series models (Reference Harrop and VelicerHarrop & Velicer, 1985). However, previous equations multiplied the autoregression parameter times t-1 (the prior data value). In contrast, our equations multiplied the autoregression parameter times v t−1, thus avoiding the interaction between a and b described in the Appendix. Equations 1-2 here correctly result in zero difference (on average) between baseline and intervention phases in the No Treatment Effect condition.
We used Equations 1–2 to generate 124 graphs, 31 for each of the 4 conditions. In each set of 31, the first 10 were used for practice trials, the next 20 for critical trials, and the last graph was reserved for examples in the instructions. We generated graphs in R (R Core Team, 2018) and executed the experiment in Qualtrics.
2.3 Procedure
Participants answered demographic questions about race, gender, ethnicity, and the amount of schooling completed. Next, to provide context, participants were asked to imagine a scenario:
A kindergarten student sometimes engages in aggressive behavior in the classroom. The teacher collected data for 10 days (BASELINE period), observing how often aggression occurred. Then, the child was placed into a new classroom and observed for 10 more days (INTERVENTION period). Did the intervention (classroom change) have any effect on aggression? Either an increase or decrease would count as an effect.
Participants then saw three flat line graphs in which there was no random error (e t fixed at 0) to illustrate instances in which the intervention produced an increase in aggression, a decrease in aggression, or no change in aggression. Participants were then told that, in real-life, judgments about effects are more difficult due to the presence of random variance. Then they saw example graphs from the reserve set with error (e t) generated as described earlier, as would be the case in the experiment.
The participants were instructed to make decisions for each graph on a 6-point Likert scale ranging from “Definitely No Effect (Wager 3)” to “Definitely An Effect (Wager 3)” (see Figure 2). A zero-wager option was not included to avoid missing data in accuracy calculations. Participants were instructed to use all six buttons. Participants earned points exchangeable for money based on their responses. The “Wager” number in each option corresponded directly to the number of points a participant could win or lose by selecting that option. Every point was worth 2 cents. Participants began with 250 points ($5.00). Participants were instructed about the scoring and bonus payment, and then took a four-question instruction quiz and received feedback about each question.
Figure 2: An example trial that participants were shown during instructions. Critical trials in the experiment had the same layout, but without the current score shown.
Next, to further assure understanding of the task and point system, participants did 40 practice trials. During practice trials, the current number of points (i.e., current score) was always visible. The starting score was 250 and points could be earned or lost based on participants’ wagers. Participants made decisions by selecting a button on the Likert scale with the mouse, and then selecting the “next” button. On practice trials, the subsequent screen continued to show the graph they had just judged, but the correct answer (e.g., “Effect Present”) was shown across the bottom of the screen in place of the Likert scale. The current score was shown above the graph. Immediately above the current score, they were shown how the decision had affected their score (raised or lowered and by how much). There were an equal number of graphs (10) from each condition during practice trials.
Next, participants completed 80 critical trials, with an equal number of graphs (20) for each condition. Participants were not provided information as to the number of critical trials, or the distribution of the types of graphs. Order of trials was random for each participant. To mimic most real-world circumstances, no feedback on accuracy, points earned, or point totals appeared during the critical trials. Participants were informed that they would still gain and lose points despite the absence of feedback. At the end of the experiment, the screen revealed their final score.
3 Experiment 2 Method
3.1 Participants
Initially, 47 graduate students were recruited from the Teaching Behavior Analysis mailing list at University of Houston, Downtown, with 46 from U.S. institutions and one from a Northern Ireland institution. These graduate students were primarily from programs in behavior analysis, including both the experimental analysis of behavior and applied behavior analysis. We chose this cohort because single-subject designs are prominent in this area of psychology. One participant was excluded using the same exclusion rules as before, leading to a final sample size of n = 46 (12 male). Nearly all participants (95.7%) had completed a course with an emphasis on single-subject design, and the majority (71.7%) had used single-subject design in their own research or projects. At the time of the experiment, 41% of participants had a Master’s degree, and 7% a Ph.D. Age ranged from 21 to 38 (M = 27.2, SD = 3.9). Participants were paid 5 cents (rather than 2 cents) per point, for a total payment between $12.50 and $30.50. Graduate student status was verified via online department directories when available, and when not available, respondents were verified to have a university or college email address at the institution with the graduate program that they reported to be attending.
3.2 Procedure
There were three changes relative to the previous experiment. First, there were extended demographic questions pertaining to experience with single-subject designs. Second, after the instructions and examples, but prior to the practice trials, participants were asked, “What percentage of graphs do you expect to judge correctly (50% would be random guessing, and 100% would be completely accurate)?” Third, after the experiment was complete, they were asked to estimate their accuracy again.
4 Results (Experiments 1 and 2)
Analyses of graph judgment focus on the 80 critical graphs. First, for a simplified overview, accuracy was analyzed by collapsing across confidence levels. For example, if there was a treatment effect in the graph, any of the three “…an Effect” responses were considered accurate. As shown in Figure 3, accuracy declined when serial dependence was present, but only if there was no treatment effect. In other words, serial dependence led to Type I errors, but not Type II errors. In Experiment 1, a repeated measures ANOVA showed a significant main effect of treatment (F(1,49) = 33.1, p <. 001, 
=.40), a significant main effect of serial dependence (F(1,49) = 39.5, p < .001, =.45), and a significant interaction effect between treatment effect and serial dependence (F(1,49) = 56.7, p < .001, =.54). When there was a treatment effect, there was no significant effect of serial dependence, as shown by Tukey’s HSD (p = .81). In contrast, when there was no treatment effect, the presence of serial dependence led to a significant decline in accuracy (p < .001).
Figure 3: Accuracy in each condition collapsed across confidence levels. Dots show individual participants. Brackets show 95% CIs of the mean.
In Experiment 2, despite recruiting participants with relevant training, the results were very similar (see Figure 3, right). Both main effects and the interaction were significant (all ps < .001, = .34, .49, and .50, respectively). When there was a treatment effect, there was no significant effect of serial dependence (p = .59). In contrast, when there was no treatment effect, the presence of serial dependence led to a significant decline in accuracy (p < .001).
Next, Type I errors (decisions that a treatment effect existed even when it did not) were analyzed separately at three levels of confidence. In Experiment 1 (see Figure 4, left panel), Type I errors increased when serial dependence was present, and this pattern occurred regardless of the level of confidence. When confidence was “Guess An Effect” or higher, Type I error rates significantly increased from the Serial Dependence Absent (M = .33) to Present (M = .49) conditions (t(49) = 9.19, p < .001, d = 1.30). This pattern still held when counting only “Probably An Effect” or higher (Ms = .20 and .34; t(49) = 8.38, p < .001, d = 1.19), and when counting only the highest level of confidence, “Definitely An Effect” (Ms = .09 and .20; t(49) = 7.64, p < .001, d = 1.08). Experiment 2 showed similar results (all ps <.001), with d = 1.20, 1.38, and 1.04, moving from left to right in the right panel of Figure 4.
Figure 4: Type I Error rates as a function of confidence level (upper boxes) and serial dependence. Dots show individual participants. Brackets show 95% CIs of the mean. Dashed lines show the customarily desired Type I error rate of .05.
Random number generators will sometimes produce graphs that coincidentally have a significant treatment effect even where none exists in the population (see Reference Fisher, Kelley and LomasFisher et al., 2003). To determine the prevalence of such coincidences, we applied a formal hypothesis test known to be robust to serial dependence even in small samples of data. Specifically, we analyzed graphs with a parametric bootstrap of the null hypothesis, with the sample autocorrelation used for the simulation of 10,000 bootstrap replicates per graph (see Borckardt et al., 2008, for details and evidence of robustness). This analysis yielded a Type I error rate of .10 regardless of serial dependence being present or absent (2 graphs out of 20 for each), suggesting that the different judgment results for those conditions was not due simply to a coincidence of graph generation. To verify that the population generating equations behaved properly, we also conducted this analysis with 10,000 runs of Equations 1–2 with no treatment effect, once with serial dependence present and once with it absent. This resulted in estimated Type I error rates below .05 regardless of serial dependence (.046 when present, .043 when absent), providing further verification that informal judgment Type I error rates were not artifacts of the generating equations.
Next, we considered the possibility that inflated Type I errors amidst serial dependence were consistent with a simple heuristic: comparing the absolute difference between the average baseline and average intervention data points. To consider this in the long-run, we used Equations 1–2 to simulate 100,000 time-series for each of the serial dependence conditions, always with the null hypothesis true (no treatment effect). As expected, the average raw difference between baseline and intervention periods was .00, but importantly, the result was more variable when serial dependence was present (SD=3.0) than when it was absent (SD=2.2). This heightened variability led to significantly higher absolute differences. Specifically, when serial dependence was present, the intervention period was on average 2.4 points higher or lower than baseline, as compared to only 1.8 points when serial dependence was absent, Welch’s t(184,282)=85.7, p < .001. When considering only the 20 critical stimuli in each condition, there was a similar trend, with an absolute difference of 2.6 when serial dependence was present versus 1.8 when it was absent, though this was not significant in this smaller sample, Welch’s t(34.8) =1.54, p=.13. Overall, inflated Type I errors amidst serial dependence would be expected if participants were relying on an absolute difference heuristic.
Performance across all possible confidence criteria was analyzed via Receiver Operating Characteristics (ROCs; Reference Peterson and BirdsallPeterson & Birdsall, 1953; Reference Swets, Dawes and MonahanSwets et al., 2000). ROCs also allow estimation of both the ability to discriminate between treatment and no-treatment effects, and the bias toward deciding that there was a treatment effect. As shown in Figure 5, ROCs indicated better performance (closer to the upper left corner) when serial dependence was absent than when it was present. Each dot represents a different confidence criterion. For example, on the dashed line, the left-most dot represents only the highest confidence criterion: “Definitely an Effect.” The second dot from the left represents a more relaxed confidence criterion: “Definitely an Effect” or “Probably an Effect”. The more relaxed confidence criterion produces more power, but also more Type I errors. Exactly five criteria (dots) are possible here, one for each location in-between the six response options.
Figure 5: Receiver Operating Characteristics (ROCs) closer to the upper left corner indicate better discriminability. Specifically, ROCs here show participants’ mean probabilities of correctly declaring a treatment effect (Power) and incorrectly declaring a treatment effect (Type I Error) across all confidence criteria.
Participants’ discrimination between treatment effect and no treatment effect graphs was measured by the area under the curve (AUC; specifically, A g in MacMillan & Creelman, 2005). In Experiment 1, AUC was significantly larger when serial dependence was absent (M = .75) than when it was present (.67), suggesting that serial dependence made it harder to discriminate treatment effects from no treatment effects (t(49) = 7.34, p < .001, d = 1.04). Similar results occurred in Experiment 2 (Ms = .77 and .69; t(45)=6.85, p < .001, d = 1.01). Bias to decide that there was an effect was measured by Kornbrot’s (2006) nonparametric ln(
) on the threshold between “Guess No Effect” and “Guess An Effect.” The higher the ln(), the more bias there is to decide that there is an effect in the graph. In Experiment 1, participants were significantly more biased toward selecting a button with “…An Effect” when serial dependence was present (M=.75) than when it was absent (.19; t(49) = 7.39, p < .001, d = 1.04). Experiment 2 produced a similar pattern (Ms = .68 and .14; t(45) = 6.71, p < .001, d = .99). Overall, ROCs suggested that serial dependence affected performance in two ways, first by reducing discriminability between treatment and no treatment effects, and second by increasing bias toward deciding that there was a treatment effect.
The effect of serial dependence on discriminability can also be seen in the proportion of responses to each individual confidence category, as shown in Figure 6. When a treatment effect was in the graph, participants rarely chose “…No Effect” confidence categories, and more often chose “…An Effect” categories, as expected. The reverse pattern appeared when there was no treatment effect in the graph, or at least when serial dependence was absent (black dashed lines on bottom panels). However, when there was no treatment effect and serial dependence was present (red lines on bottom panels), responses were spread more evenly across all categories, consistent with the impaired discriminability in that condition.
Figure 6: Proportion of responses in each confidence category as a function of treatment effect and serial dependence. Brackets show 95% CIs of the mean.
In Experiment 2, participants also made metacognitive judgments. Immediately after the instructions and examples, participants significantly overestimated the accuracy that they would achieve on the task, predicting that they would achieve on average .80 proportion accurate while achieving only .69 (t(45)=8.84, p < .001, d = 1.30). After the experiment was finished, participants significantly underestimated their actual accuracy, estimating on average .65 accuracy (t(45)=2.79, p = .008, d = .41). Actual accuracy was positively correlated with estimated accuracy, both at the beginning (r = .43, p = .003) and end (r = .37, p = .011) of the experiment.
5 General Discussion
In both experiments, serial dependence increased the Type I error rate of informal judgments of graphs, as measured by decisions that there was an effect even when no effect existed in the population. This pattern replicates the general finding of Reference Matyas and GreenwoodMatyas and Greenwood (1990), and it is difficult to discount these findings using criticisms leveled at the earlier work. The inflated Type I error rate occurred for most participants, and in both guesses and highly confident responses. Because Type I error inflation occurred even with high confidence responses, it is unlikely that offering a zero-wager or “pass” option would eliminate this pattern (see Reference Dhar and SimonsonDhar & Simonson, 2003). Furthermore, Type I error inflation occurred even though each no-treatment condition was represented by 20 random graphs rather than just 1. It occurred even with a corrected data-generating equation, and even though the proportion of response options for no treatment effect (.5) was equal to the proportion of trials with no treatment effect (.5).
When serial dependence was present, the Type I error rate was inflated relative to the typical statistical standard of 5%, and even relative to a statistical test of the graphs themselves (10%). In contrast to earlier work, though, the Type I error rate never exceeded 49% by any measure, let alone reached the 84% observed in Reference Matyas and GreenwoodMatyas and Greenwood (1990), or the 61% observed in classic simulations of p-hacking (Reference Simmons, Nelson and SimonsohnSimmons et al., 2011). The more tempered Type I error rates observed here might be the result of addressing methodological concerns. Of course, this pattern provides little comfort, as Type I errors rates still exceeded the customary 5% standard.
All participants were paid based on performance, so it is unlikely that the elevated error rates were due to lack of motivation. They are also unlikely due to simple misunderstandings, as all participants received extensive training with feedback before beginning the experiment. Furthermore, most graduate students in Experiment 2 had first-hand experience using single-subject designs. Nevertheless, their performance was not markedly different from that of a general adult sample. Indeed, others have found little or no impact of single-subject design experience on graph judgment (Reference Harbst, Ottenbacher and HarrisHarbst et al., 1991; Reference Richards, Taylor and RamasamyRichards et al., 1997), though such disappointing findings are not universal (see Reference Kratochwill, Levin, Horner, Swoboda, Kratochwill and LevinKratochwill et al., 2014). In our experiments, it is possible that any effects of graduate training were overshadowed by the extensive task-specific practice with feedback that all our participants received. It has been suggested that such practice is crucial, as real-life rarely provides such clear feedback (Reference Parsonson, Baer, Kratochwill and LevinParsonson & Baer, 1992). Indeed, prior to this practice, most graduate students overestimated their own abilities for this task. The practice and the remainder of the experiment were sobering, as by the end of the experiment, participants tended to slightly underestimate their performance.
Overconfidence sometimes disappears when participants make metacognitive judgments about trials in aggregate at the end of an experiment (Reference Gigerenzer, Hoffrage and KleinböltingGigerenzer et al., 1991). One reason that could have happened here is that participants were relying on different cues with different validities at the beginning versus at the end of the experiment. At the beginning of the experiment, graduate students’ metacognition probably relied on cues that had only modest validity. For example, prior to the experiment, they may have noticed when their belief in a treatment effect agreed with the belief expressed by textbook authors, their mentors, or fellow students when informally judging graphs, and used this agreement as a cue to assess their own performance. The experiment, though, provided a cue with perfect validity – feedback about the correct answer – at least for the practice trials. By the end of this experiment, participants could rely on memory for practice trial feedback to assess their own performance.
As indicated by the ROCs, serial dependence affected performance in at least two manners. First, serial dependence made it more difficult for participants to distinguish between treatment and no-treatment conditions. This difficulty might have occurred because the data points on the graph (y t) become more variable as the autoregressive parameter (a) gets farther from 0. Second, serial dependence created a bias toward more “treatment” responses. This bias could have occurred because, when serial dependence is present, one especially large error value (e t) can affect several data points that follow it, producing the appearance of a lasting change (Reference Matyas, Greenwood, Franklin, Allison and GormanMatyas & Greenwood, 1997). Relatedly, serial dependence tends to create larger absolute differences between baseline and intervention periods. So, the pattern of results is consistent with a heuristic whereby people attempt to mentally subtract the mean of the baseline period from the mean of the intervention period, and if the absolute difference is large enough, decide that a treatment effect has occurred. Such a heuristic would be useful for judging intervention effectiveness, but it also makes one vulnerable to illusions of intervention effectiveness, particularly if one uses the same threshold regardless of the degree of dependence. Of course, it is unlikely that this heuristic is the only heuristic that people rely upon. Disentangling heuristics for statistical judgment is challenging because many plausible heuristics make predictions that are strongly correlated (e.g., Obrecht et al., 2007; Reference Soo and RottmanSoo & Rottman, 2018, 2020). For example, participants could adjust for variability in the data, treating absolute differences as more meaningful when variability is smaller. Such an approach would be an informal estimate of the absolute standardized difference (and also an informal estimate of the absolute point-biserial correlation, which is just a multiple of the absolute standardized difference). In our stimuli, the absolute standardized difference is almost perfectly correlated with the absolute raw difference, preventing us from disentangling the two.
The current work differs from classic examples of illusory correlation (e.g., Reference Chapman and ChapmanChapman & Chapman, 1969) where the judged datapoints were typically independent (e.g., judging independent people). The results here show that dependent observations across time can cause changes in judgment, perhaps because an error term that applies to multiple time points can be misattributed to a real change even where none exists. This illusory correlation between the interruption and datapoint height was observed here even in the absence of prior beliefs or expectations (c.f., Reference Chapman and ChapmanChapman & Chapman, 1969). Of course, such alternative causes of illusory correlation could impact time series judgment as well, potentially interacting with serial dependence. For example, a pattern produced by serial dependence might be especially attended to when that pattern also confirms the viewer’s expectations.
In a related finding, informal judgments for forecasting (extrapolating into future parts of time-series data) are also prone to error, as they are often less accurate than forecasts achieved by formal statistical models (Reference Carbone and GorrCarbone & Gorr, 1985). It is possible that informal forecasting is also hampered by serial dependence. Serial dependence, at least when positive, can make data appear to be smoother and more stable, even while simultaneously creating more variability in the possible realizations of time series patterns. So, ironically, informal forecasts might be most certain when certainty is least warranted.
Unfortunately, serial dependence is a common attribute of time series data (Reference Harrington and VelicerHarrington & Velicer, 2015; Reference Shadish and SullivanShadish & Sullivan, 2011). Serial dependence is what necessitates tools for time series beyond the typical inferential statistics (e.g., t-tests, ANOVAs). Perhaps more troubling, our experiments involved a rather minor form of dependence, a 1-lag autoregressive coefficient of .3. Datasets that show stronger dependencies may be more likely to lead to misattribution of error to real patterns.
Conversely, it is likely that serial dependence is harmless in some situations. If effect sizes are extremely large, judgment should reach perfect performance, eliminating all types of errors. Informal judgment seems sufficient, for example, to determine that the American stock market dropped in the great crash of 1929. Additionally, negative serial dependence produces high scores that are followed by low scores, and vice-versa. It is unclear whether such alternating data patterns would also bias respondents toward treatment effect decisions (see Reference Ximenes, Manolov, Solanas and QueraXimenes et al., 2009). Furthermore, dependencies could involve longer lags, multiple lags, or seasonal effects. Considering the breadth of the standard model for time series (Autoregressive Integrated Moving Average, i.e., ARIMA), we examined just a simple case that is often used as a proxy for potentially more complicated ones (see Reference Harrop and VelicerHarrop & Velicer, 1985).
We hope that our results are not oversimplified so as to malign informal graph judgment in general. After all, we have relied on readers’ informal graph judgment, in addition to formal statistical judgment, to evaluate our data. However, interrupted time-series graphs may justify extra caution, as they often involve serial dependence. Misjudgment of these graphs could have substantial costs, particularly if illusory correlation leads to illusory causation. Those who judge interrupted time-series graphs might infer that certain events affect the stock market, public health, individual patients, or other outcomes, even when they do not.
The risk of misjudgment could be mitigated by corroborating informal graph judgment with formal statistical procedures (Reference Bishara, Li and ConleyBishara et al., 2021). Of course, formal statistical procedures can also fail if they do not accommodate the serial dependence in time-series data. Serial dependence necessitates the use of special statistical models (e.g., ARIMA; Reference Box, Jenkins, Reinsel and LjungBox et al., 2015) rather than more typical ones that assume independent observations (e.g., t-tests). Additionally, in small sample sizes – such as the 20-observation graphs used here — typical time-series models often fail to control Type I errors (Reference Greenwood and MatyasGreenwood & Matyas, 1990) due to noisy estimates of serial dependence. Such small sample situations also require bootstrapping to prevent Type I error inflation (Borckardt et al., 2008; Lin et al., 2016; McKean & Zhang, 2018; McKnight et al., 2000; ). As daunting as these formal statistical options may be, the alternative – informal judgment – can too easily lead to “discovery” of patterns where none exist.
Appendix: Proof that generating equations in previous work created on-average differences between the periods before versus after the interruption
From Matyas and Greenwood (1990, p. 343):
When there is no treatment effect, d = 0. Also, for clarity, relabeling e as e t:
The long-run average is the expectation ( 
). For example, with the first trial (t=1),
There is no trial 0, so the term with y 0 drops out. Also, b is a constant, so [b] = b. Because e t is generated from a population with a mean of 0, [e t] = 0 for all t. This simplifies the average data point of trial 1:
For trial 2,
For trial 3,
More generally, for any trial,
We need only show that the expectations differ before (t ≤ 10) versus after the interruption (t > 10) for at least one pair of trials. For simplicity, consider the average difference between the trials 11 and 1:
The very first trial and the first trial after the interruption differ on average then, so long as a ≠ 0, a ≠ −1, and b ≠ 0. Although b was never explicitly defined in the text, the graphs (Reference Matyas and GreenwoodMatyas & Greenwood, 1990, p. 344) clearly indicate a value much larger than 0 (b ≈ 20). Importantly, a ≠ 0 only in the conditions with serial dependence. So, when there is serial dependence, the generating equation in Matyas and Greenwood creates differences before versus after the interruption even when there is no treatment effect (d = 0). Response options indicating these differences (options B-E) were coded as Type I errors. Thus, this problem could account for their elevated Type I error rates in conditions with serial dependence.
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