doi: 10.1037/cep0000118 safe or out: ralph s. redden

DOI: 10.1037/cep0000118

Safe or Out:

Does the Location of Attention Affect Judgments at First Base in Baseball?

Ralph S. Redden, Ghislain d’Entremont & Raymond M. Klein

Dalhousie University

Copyright 2016 Canadian Psychological Association

This is the author-accepted manuscript (AAM). Please refer to any applicable terms of use of the

publisher. This article may not exactly replicate the authoritative document published in the APA

journal. It is not the copy of record.

Author Note:

The present work was supported by a Natural Sciences and Engineering Research

Council operating grant awarded to Raymond Klein.

Correspondence for this article should be addressed to Ralph Redden, Department of Psychology & Neuroscience, Dalhousie University, 1355 Oxford st., 3rd Floor LSC, Halifax, NS, Canada, B3H 4R2. E-mail: [email protected]

ENDOGENOUS VISUAL PRIOR ENTRY 2

ENDOGENOUS VISUAL PRIOR ENTRY 2 Abstract

Titchener’s law of prior entry states that attended stimuli are perceived prior to unattended

stimuli. Prior entry effects measured with visual stimuli have been generated with both

endogenous and exogenous attentional deployment (e.g., by Shore, Spence & Klein, 2001). In

theory, the endogenous form of prior entry may have implications for baseball umpire

judgments. Conventionally, umpires are instructed to first attend to the ball when it is hit into

play, however where they attend at the imperative instant of the play at first base can vary

between individuals and across scenarios. If the law of prior entry holds in the baseball context,

umpires may be biased to make judgments in favour of the imperative event nearest the locus of

attention. We tested this hypothesis by having non-umpires make ‘Safe’ or ‘Out’ judgments in

response to first base baseball plays wherein the relative arrival times of the runner and baseball

were varied. A novel colour wheel method was implemented in an orthogonal task to bias

attention endogenously and to measure the effectiveness of this manipulation. Attention was

confirmed to be successfully biased to the glove or base by way of improved identification at the

likely probe location. However, there was no evidence that prior entry was affecting ‘Safe’ or

‘Out’ judgments.

Keywords: endogenous visual prior entry, temporal order judgments, baseball, colour

wheel mixture model, Bayesian hierarchical modelling



A question that has been long pursued in the field of cognitive psychology is what is the

effect of attention on how we process information about our environment? One of the earliest

theories on this matter is Titchener’s law of prior entry (Titchener, 1908). The law of prior entry

states that attended stimuli are perceived prior to unattended stimuli (Spence, Shore & Klein,

2001). Prior entry has been studied using a variety of tightly-controlled experimental methods

and with many psychological and neuroscientific tools (for a review, see Spence & Parise, 2010),

however whether this phenomenon extends to real-world contexts is an unanswered question.

Before exploring the implications for the external validity of prior entry, a prototypical task will

be described.

Generally, prior entry is studied using a temporal order judgment (TOJ) task. Participants

are presented with multiple stimuli that vary in their onset time (stimulus onset asynchrony -

SOA). Through some experimental manipulation (e.g., a transient flash, or some experimental

contingency), attention is presumed to be more aligned with one of these two stimuli. The

participant must then report which stimulus appeared first. The data are summarized as

psychometric functions in which proportion of responses to one of the two stimuli are plotted as

a function of the temporal asynchronies of the stimuli. The key parameter of interest is the point

of subjective simultaneity (PSS), which is the SOA at which the proportion of selecting each

stimulus is .50. This represents the amount of temporal asynchrony required between the

presentations of the stimuli for the participant to perceive them as having appeared

simultaneously. The prior entry effect is operationally defined as half the difference between the

PSSs of the attentional conditions (Stimulus A attended vs Stimulus B attended). The prior entry

effect in essence quantifies the effect of attention on TOJs and therefore the rate of information


ENDOGENOUS VISUAL PRIOR ENTRY 4 processing. Another parameter that is often considered is the just-noticeable difference (JND).

The JND reflects the participant’s sensitivity to the temporal order of the presentation of the

stimuli (Spence & Parise, 2010). The JND is defined as the delay in presentation between two

stimuli that is required for participants to perceive the correct order of presentation at a given

level - often calculated as half the SOA interval between the 25 % and 75 % points of the

psychometric function. While JND is typically reported in studies investigating prior entry, it is

important to note that JND is not necessarily about the locus of attention, but might reflect the

amount of attention allocated to the TOJ task itself.

Attention can be modulated by many factors (Klein, 2004). Attention can be driven by

volitional means - commonly referred to as endogenous attention. As well, attention can be

modulated by more reflexive means - commonly referred to as exogenous attention. There is

debate over whether prior entry is driven by either of these attentional processes. Shore, Spence

and Klein (2001) demonstrated prior entry for both forms of attentional control in the visual

modality - albeit reporting a smaller prior entry effect for their endogenous attention condition.

Contrarily, Schneider and Bavelier (2003) argued that visuospatial prior entry effects are

primarily caused by attention-independent factors (i.e., sensory facilitation and attentional

modifications to decision making mechanisms), and concluded from their research that there

exists little to no endogenous prior entry. The issue with this claim however, as argued by Klein -

reported by Spence and Parise (2010 - Footnote 3) - and mentioned by Schneider and Bavelier in

their discussion, is that they did not confirm that their effort to manipulate attention

endogenously had in fact succeeded. In other words, to support their claim regarding the near

non-existence of prior entry caused by endogenous orienting Schneider and Bavelier had to

assume that endogenous orienting had occurred.


ENDOGENOUS VISUAL PRIOR ENTRY 5 The present investigation is foremost concerned with this conclusion from Schneider and

Bavelier (2003). There is evidence for prior entry effects resulting from the endogenous orienting

of attention within the visual modality (Shore, Spence & Klein, 2001) and manipulations

employing cross-modal stimuli (Spence, Shore & Klein, 2001; Vibell, Klinge, Zampini, Spence

& Nobre, 2007; Zampini, Shore & Spence, 2005) and as such, there are numerous real-world

scenarios in which prior entry resulting from endogenous orienting could have important

consequences. Drivers approaching a four-way stop simultaneously with vehicles in other

directions could have their appraisal of right-of-way influenced by such an attentional

consequence. Less gravely important - but critical nonetheless - officials in many sporting

contexts must make judgments about the temporal sequence of events in order to appropriately

rule on a given circumstance (e.g., ‘offside’ in ice hockey requires judging whether the puck

crosses the blue line before or after an attacking player crosses the same blue line).

The present investigation will look at one example of these real-world sporting scenarios.

Prior entry effects could have significant consequences in a sport such as baseball. Umpires must

judge whether a player running to first base is "Safe" or "Out" based on their perception of the

runner’s and the baseball's arrival at the base and glove, respectively. The potential applications

of this research are emphasized by the fact that umpires go through extensive training programs

in order to reach the professional level, however there is debate about how umpires precisely

approach this specific scenario (Rodgers, 2015; MiLB, 20161), and even anecdotal reports of

1 We would like to thank Joel Rodgers of the Baseball Nova Scotia Umpires Division, and Darren Spagnardi of the MiLB Umpire Training Academy for their correspondence. 2 This is also how trainees are instructed in the WPBU 105: Plate and Base Umpire Positioning (Field Work) course at the Wendelstedt School for Umpires, according to personal communication with the president of the school - Hunter Wendelstedt. However, no training manuals were available for direct citation. The Wendelstedt School for Umpires claims to be the most successful US academy for producing MLB officials (http://www.umpireschool.com/).


ENDOGENOUS VISUAL PRIOR ENTRY 6 individual differences in strategy for these judgments (Weber, 2011). For instance, the Northwest

Association of Umpires recommends umpires to attend the ball in play until it is thrown, shift

their gaze to the base while listening for the sound of the ball impacting the first baseman’s

glove, and then finally reorienting visually to the glove to affirm the catch (Evans, 2016). In

contrast, the Baseball Nova Scotia Umpires Division instructs officials to “assume standard set to

see entire play at first base” and makes no reference to the use of auditory information in their

training manual, but rather to visually attend the entire play (Rodgers, 2015, p.14). The Minor

League Baseball umpire training protocol advises trainees to attend the base visually while

listening for the sound of the ball impacting the glove, but emphasizes proper field positioning so

as to also simultaneously attend the ball arriving in the glove using peripheral vision (MiLB,

2016; Wendelstedt School for Umpires, personal communication, November 6, 20162).

Individual reports (Weber, 2011) also suggest umpires are trained to use all available

information, but must always be prepared for instances where auditory information is not readily

available (e.g., noisy environment; lightly thrown ball) or is unreliable (e.g., ball ‘short-hops’,

creating two auditory signals; ball impacts at the webbing of the glove rather than the pocket).

Importantly, previous research by Zampini, Shore and Spence (2005) has shown that TOJ

performance is poorer when making cross-modal judgments, and that visual signals need to

precede auditory signals in order for them to appear simultaneous. Accordingly, because

Zampini and colleagues’ basic research suggests that umpires ought not judge close plays at first

2 This is also how trainees are instructed in the WPBU 105: Plate and Base Umpire Positioning (Field Work) course at the Wendelstedt School for Umpires, according to personal communication with the president of the school - Hunter Wendelstedt. However, no training manuals were available for direct citation. The Wendelstedt School for Umpires claims to be the most successful US academy for producing MLB officials (http://www.umpireschool.com/).


ENDOGENOUS VISUAL PRIOR ENTRY 7 base bimodally, and because umpire training suggests they should always be prepared for a

unimodal visual assessment of the play, we will focus our investigation on purely visual TOJs.

Assuming the law of prior entry does apply to this real-world attentional scenario,

allocating visual attention asymmetrically could result in a systematic bias in these umpire

judgments. Let’s presume, for example, an umpire systematically prioritizes attending the foot

arriving on the base. Should the prior entry effect to be equivalent to what was observed by

Shore, Spence, & Klein (2001), and taking the average speed at which an MLB player traverses

the path to first base (6.33m/s; Coleman, 2007), we infer that umpires could be making “Safe”

calls in situations in which the runner is up to 11 cm away from the base at the time that the ball

arrives at the glove. The only other study (to our knowledge) that has empirically evaluated

umpire judgments has shown, by way of a computer-animated ‘close play’, that umpires are

more likely than untrained observers to call a runner out when the play is close (Larsen &

Rainey, 1991), which the authors attribute to umpires having a more stringent criterion for ‘safe’

judgments than untrained observers. The objective of our research is to determine to what extent

an observer’s allocation of attention actually influences his or her "Safe" or "Out" judgments. If

we should discover that the safe/out decisions of non-umpire observers are biased by the locus of

attention, then a dedicated study of professional umpires would be a logical next step.

The current study is an extension of Redden and Klein (2012). In their study, a TOJ task

was used in which participants had to make “Safe” or “Out” judgments when shown actual MLB

footage of first base plays. Observers were assigned to one of three attention conditions: attend

ball/glove, attend runner/base, or no instruction. Although their findings showed no effect of

prior entry (‘Safe’ or ‘Out’ judgments were not influenced by attentional instruction), their

methods are challenged by several shortcomings, most notably the one attributed above to


ENDOGENOUS VISUAL PRIOR ENTRY 8 Schneider and Bavelier (2003): No manipulation check was conducted to ensure attention was

actually deployed to the instructed location. Furthermore, the videos shown were cropped from

actual game footage, so there was little control over the visual sequence of events - leading to

unequal display times for the imperative to-be-attended stimuli3. Finally, although the umpire

(when visible) was digitally blurred out of the video so as to not influence the observer’s

judgment, this digital blurring also briefly occluded the trajectory of the ball on a subset of trials.

The present experiment instead will use video from a single live first base play generated by our

own means. We created a video of a player running to first base with a defender on first base.

We then animated the trajectory of a baseball such that the arrival time of the baseball going into

the fielder’s glove relative to the runner's foot touching the base could be precisely manipulated.

Our experiment provided an unchanging observer viewpoint to avoid the bias across conditions

suggested by Redden and Klein. Also, as in Redden and Klein (2012), we will explore this

question using observers with no prior umpiring experience as it is highly conceivable that the

orthogonal attentional manipulation would be less (or not) efficacious for individuals with prior

training in the baseball scenario, as training may in fact preclude the individual’s amenability to

any experimental attention instructions. Furthermore, it is common in most amateur baseball

settings (where instant replay is non-existent) for umpires to adopt the heuristic that a tie (or

indistinguishable difference in relative arrivals) shall be awarded in favour of the runner. To that

end, we will probe observers post-test to ascertain their knowledge of the ‘Tie Goes to Runner’

convention. With this information in hand we will be able to determine the extent to which this

heuristic may have affected behaviour in the present paradigm.

3 In a typical major network broadcast of an MLB game, the camera primarily follows the ball during the sequence of play.


ENDOGENOUS VISUAL PRIOR ENTRY 9 To encourage and verify an endogenous shift of attention, the present study employed a

colour discrimination task using the colour wheel method (Prinzmetal, Amiri, Allen, & Edwards,

1998; Zhang & Luck, 2008). This method entails presenting a randomly selected colour probe at

either the location of the base or the glove on a random subset of trials with the observer required

to indicate at the end of the trial, using the color wheel and mouse, the colour of a probe. The

endogenous attention manipulation was generated by systematically biasing the probable

location of the colour probe within a block of trials and explicitly instructing observers about this

probability manipulation. As such, if the manipulation successfully encourages a shift in

attention then colour wheel response accuracy ought to be greater when assessing probes that

appeared at the more likely location. As demonstrated by Zhang and Luck (2008; see also, Brady

et al., 2013; Fawcett, Lawrence, & Taylor, 2016) the use of a continuous reporting dimension

like color allows the researcher to decompose accuracy into two potentially independent

parameters: the probability that the probe was encoded and when it was, the fidelity of this

encoding.

Should endogenous attention elicit a prior entry effect, we expected a greater proportion

of ‘Safe’ judgments when the colour probe is more likely to occur at the base, and a greater

proportion of ‘Out’ judgments when the colour probe is more likely to occur at the glove.

Methods

Participants. Fifty seven undergraduate students (37 male; 2 unreported) participated in

the study. The median participant age was 20. Participants received credits towards an

undergraduate class for their participation. Participants provided informed consent approved by

the Research Ethics Board, Dalhousie University. After testing, participants were asked to

answer yes/no to a question presented on screen: “Are you familiar with the baseball convention


ENDOGENOUS VISUAL PRIOR ENTRY 10 which states that a tie goes to the runner?”. This convention states that if the temporal difference

between the runner arriving at the base and the ball arriving at the glove is imperceptible, then

the runner is judged to be “Safe”. If observers responded ‘yes’, they were asked a second

question: “Did you implement that convention?”

Apparatus. Stimuli were presented on iMacs running OS X 10.8.4. Images were

displayed on 27” or 24” monitors with a resolutions of 2560 x 1440 or 1920 x 1200 pixels,

respectively. Participants sat approximately 57 cm from the screen. The colour probes had a

diameter of approximately 0.5° of visual angle. The colour wheel (see below) had a diameter of

18.4 , with an annulus width of 1.8°.

Trial preparation. A video (27°x 24°) of a runner (female) running towards first base

with a fielder standing on first base positioned to catch a baseball was recorded at a baseball

field. The video angle is unchanging and aligned with the first base line such that the runner is

running toward the camera (see Figure 1). A computer-generated baseball was animated into the

video such that the baseball travelled horizontally from the right side of the screen to the fielder’s

glove. The relative arrival times of runner’s foot at the base and the baseball and glove (SOAs)

were varied across trials (see Table 1). The first imperative event (ball or runner arrival) occurred

at a random interval (1300-2300 ms) from trial initiation in order to reduce any temporal

predictiveness that may come from trial onset.



Figure 1. Screenshot of the video of the first base baseball play. In this trial, the colour probe (pink/purple) appeared at the glove. The ball is seen in the top right quadrant of the image, approaching the fielder’s glove.

Procedure. Each participant completed a total of four blocks each consisting of 120

trials. Two different tasks were randomly intermixed in each block. Eighty trials required

participants to make an unspeeded judgment as to whether the runner was “Safe” or “Out”-

whether the runner’s foot arrived before or after the baseball reached the fielder’s glove,

respectively. Participants indicated their safe/out responses by pressing either the “s” key for

“Safe” or the “o” key for “Out” on a USB QWERTY keyboard. On the remaining 40 trials a

coloured probe was presented for 350 ms at the location of the glove or base 175ms prior to the

time of the first of the two possible TOJ task events. At the end of a trial with a colour probe

trial, a colour wheel was displayed (see Figure 2) and participants indicated, by way of a USB


ENDOGENOUS VISUAL PRIOR ENTRY 12 mouse click, the colour of the probe. The colour wheel was randomly rotated from trial to trial to

avoid response biases to the position of certain colours with respect to the computer screen

and/or the mouse cursor. Colour wheel responses were recorded as angle deviations from the

actual probe colour. The colour wheel trials in each block were designed to incentivize an

endogenous shift of attention by presenting probes (when they occurred) at one of two locations

(glove or base) 80% of the time. This attentional manipulation alternated between blocks, and the

order of alternation was counterbalanced across participants. The number of trials for each SOA

and trial type was fixed (Table 1). However, the order of trial SOAs for each task was

randomized for each block.

Table 1

Distribution of trials in a given block across all SOAs and both tasks (TOJ and Colour Wheel). Negative SOAs indicate that the ball arrived at the glove before the runner’s foot arrived at the base.

Trial Type

SOA (ms)

-250 -150 -100 -50 -17 17 50 100 150 250

Colour Wheel 2 4 4 4 8 8 4 4 4 2

TOJ 4 8 8 8 12 12 8 8 8 4

Before starting the experiment, participants received verbal instructions (Appendix A)

about all possible endogenous attention orienting conditions. Further, at the beginning of each

block participants received on-screen message describing the endogenous attention orienting

condition of said block.



Figure 2. Constant luminance colour wheel. Participants used a cursor to indicate the perceived colour of the colour probe.

Analysis. A mixture model consisting of a uniform and a Von Mises (circular normal)

distribution was used to model participant colour wheel data. The model was fitted with respect

to two parameters: the probability and the fidelity of memory encoding (for review, see

Lawrence, 2010). Probability of memory encoding ( ) is the probability that the participant

encoded the colour of the probe into memory (i.e., that the participant is not guessing). Fidelity

of memory encoding ( ) is a reciprocal measure of the variability with which the participant

reports the colour, given that they have encoded it.

Probability of encoding contributes to the relative proportion of Von Mises distribution

over uniform distribution, whereas fidelity of encoding contributes to the narrowness of the Von

Mises distribution. An intuitive description of the model is as follows: the uniform distribution is

representative of the participant guessing the colour of the probe, whereas the Von Mises

distribution, an approximation of the circular analog of the normal distribution, is representative


ENDOGENOUS VISUAL PRIOR ENTRY 14 of the fidelity with which the participant encoded the colour of the probe.

Participant TOJ data, summarized as psychometric functions, were modelled as

cumulative normal distributions. The PSS was defined as the mean of the distribution, whereas

the JND was defined as the standard deviation of the distribution. It is worth noting that the

current JND computation, which is equivalent to taking half the SOA interval between the 16 %

and 84 % points of the psychometric function, differs slightly from what is often used (i.e., using

25 % and 75 % points). Data from both tasks (TOJ and colour wheel) were evaluated in a joint

model (introduced below) which allowed for the estimation of the correlation of parameters

between tasks.

As with Fawcett et al. (2016), who similarly ran a repeated measures design using the

colour wheel method, we used a Bayesian hierarchical model as our method of statistical

analysis. A hierarchical (or multilevel) approach was used for a priori reasons relating to its

ability to incorporate information trial-by-trial data from repeated measures experiments (for

discussion, see McElreath, 2015). In particular, the hierarchical model provided value in its

capacity for appropriately estimating the attention condition effect parameters (i.e., probability

and fidelity effects) as these were estimated from uneven repeated measurement samples for each

participant. Bayesian data analysis was used because we believe it to be a more powerful and

appropriate method of analysis than the frequentist approach (for discussions, see Dienes, 2011;

Gizenger, 2004; Kruschke, 2014; McElreath, 2015). Also, a benefit afforded by the Bayesian

approach not provided by confidence intervals and other frequentist outcome measures is the

computation of probability distributions regarding the estimates of our parameters of interest

(i.e., posterior distributions). Finally, the Bayesian approach facilitates the incorporation of non-

normal distributions (e.g., mixture model), and the analysis of data from unequal sample sizes


ENDOGENOUS VISUAL PRIOR ENTRY 15 (Kruschke, 2012).

We report highest density intervals (HDIs) along with their respective posterior

distributions (in figures) for our parameters of interest (Fawcett et al., 2016; Kruschke, 2014).

Given that Bayesian analysis is the reallocation of credibility over the set a possible values for a

given parameter, posteriors indicate the relative credibility of parameter values along a

continuum of possible values, having accounted for one’s empirically inspired prior beliefs and

the present data (Kruschke, 2014). We refer to the probability (not credibility) of parameter

estimates. The distinction is nuanced. It arises from the fact that we derive posterior distributions

from the combination of the data (empirical) and our prior beliefs (empirically inspired). Since

the latter is not purely empirical, the posterior is not technically an explicit probability density

distribution. However, the posterior has all the properties of a probability density distribution and

can be thought of as an updated representation of our beliefs in the relative probability of

parameter values for a given parameter. From the parameter estimates (posteriors), we extract

95% and 50% HDIs, which represent our belief that the probability that the true population

parameter value lies within those HDIs is 95% and 50%, respectively. Ultimately, our model

provided posterior distributions for each population parameter mean (e.g., PSS intercept mean, or

JND effect of attention mean). We took the median value of these posteriors to be point

estimates for each parameter.

We present posteriors and their respective HDIs graphically using violin plots, which

simply consist of posterior distributions mirrored across the vertical axis. The width of these

‘violins’ represents the relative probability of values for the parameter in question as calculated

by the model. In other words, values for which the plot is widest are most probable. We depict

the 95% and 50% HDIs as thin and thick bars within these violin plots, respectively, which


ENDOGENOUS VISUAL PRIOR ENTRY 16 facilitates the reader's ability to make statements about the specific probability with which we

believe certain values will be found within a given range. Ultimately, violin plots are

comprehensive depictions of where we believe our model parameters exist in the real world.

Fortunately, the nature of Bayesian parameter estimation is that we can assign probabilities to the

truth of certain ranges of values for our parameters of interests.

To ensure consistency in our evaluation of the strength of the effects presented herein, we

present broad criteria for how we made qualitative interpretations of our posteriors. If the 50%

HDI of an effect included zero, we reported that we believed that effect is null. If the 50% HDI

of an effect excluded zero, but the 95% HDI captured it, then we tentatively made statements

about the size of the effect. These statements were graded depending on how close either bound

of the 95% HDI were to zero. Finally, if the 95% HDI of an effect excluded zero, then we

expressed strong beliefs in the presence of the effect. Overall, we commented on the spread

(uncertainty) and position (magnitude) of our parameter estimates (posteriors and HDIs) to the

extent that these details were pertinent to our hypotheses in order to provide the reader with a

rich and relevant description of our results.

The model parameters themselves can be interpreted as standard regression terms (i.e.,

intercept, main effects, and interaction effects). The model computed parameter estimates for

probability in logit space, and those for fidelity and JND in logarithmic space to facilitate the

computation of population intercepts and effects. However, we report back-transformed

estimates for these parameters to facilitate interpretability4. Furthermore, for the reader interested

in seeing how a more traditional frequentist analysis would result, a summary of null-hypothesis

significance tests of all parameters can be found in Appendix D.

4 Transformed parameter estimates are reported in Appendix C


ENDOGENOUS VISUAL PRIOR ENTRY 17 Results

Exclusions. Three participants were excluded from analysis because they did not learn

the response mappings (i.e., they did not learn that ‘Safe’ meant that the runner arrived first, or

‘Out’ meant the ball arrived first). Due to a coding error, nine participants were excluded

because they inadvertently did not receive block-wise unequal distributions of probe locations,

and therefore could not have had their attention endogenously oriented. Finally, one participant

was excluded because her psychometric function indicated that they were likely not sufficiently

engaged in the task. Of the 44 participants that remain, 12 reported that they were familiar with

the ‘tie goes to runner’ convention. Nine of these reported that they followed this convention in

making their TOJs5.

Summary of Results. Analysis of overall accuracy on the colour wheel task

demonstrated that our manipulation of endogenous attention was successful. Separation of global

accuracy into the components described above yielded some interesting findings as will be seen

in the next section. The psychometric functions in Figures 4 and 6 illustrate the common sense

expectation that when the ball arrives at the glove before the runner arrives at the base,

judgments of “out” prevail whereas with the opposite arrival times (runner before the ball) “safe”

judgments prevail. Importantly, the judgment becomes more difficult as the difference in arrival

times decreases. The effect of attention, however, was not at all as expected. The PSS was

approximately veridical when participants were attending the base; but when attending the glove

(which according to the law of prior entry ought to have accelerated the perception of the ball’s

arrival, thereby increasing “out” responses) there was a slight increase in “safe” judgments and a 5 We split based on the “Knowledge of Convention” factor rather than the self-report “Followed Convention” factor for two reasons: 1) because practically, this afforded more data in the smaller group, and 2) theoretically, we believe the awareness of such a heuristic is capable of biasing responding regardless of one’s subjective experience of employing it.


ENDOGENOUS VISUAL PRIOR ENTRY 18 corresponding shift in the PSS (in the negative direction given the legend). Of greatest import to

our hypothesis, no effect of prior entry was found, as measured by PSS. We verify that the locus

of attention was successfully manipulated by way of the colour probe diagnostic - showing

improved colour wheel performance at the more likely probe location. Furthermore, we show

that participants’ knowledge of the ’tie goes to runner’ convention impacts the remaining factors

of interest (probability of encoding, fidelity of encoding, and JND).

Colour wheel responses. Probability of Encoding: Although the overall probability of

encoding was very high (Mdn = .980; HDI95%=.970, .988), we detected a small effect of

attention on probability of encoding (‘Attended’ minus ‘Unattended’: Mdn= .008; HDI95%=-.003,

.020). There did not appear to be an effect of knowledge regarding the ‘tie goes to runner’

convention (‘Don’t Know’ minus ‘Know’) on probability (Mdn=-.002; HDI95%=-.019, .017).

Importantly, there was a relatively large, near 2% interaction (Mdn= .018), between attention and

“Knowledge of Convention” (HDI95%=-.004, .041). The directionality of this interaction is

represented in Figure 3 (Left) and Table 2 (Top). The figure illustrates that the effect of attention

on probability of memory is greater when participants are unaware of the baseball convention

than when they have knowledge of the convention.

Table 2

Point estimates (medians of posterior distributions) for population means of probability ( ) and fidelity ( ) of encoding as a function of all possible pairings of convention knowledge (‘Don’t Know’ vs. ‘Know’) and attention condition (‘Attended’ vs. ‘Unattended’) levels.

Parameter Know Don’t Know

Attended Unattended Mean Attended Unattended Mean

Probability .981 .983 .982 .988 .971 .981

Fidelity 10.59 11.10 10.84 14.42 12.95 13.66


ENDOGENOUS VISUAL PRIOR ENTRY 19 Fidelity of Encoding: The estimate of the mean for fidelity of encoding for all

participants was centered at 12.29 (HDI95%=11.28, 13.34), and there did not appear to be an

effect of attention on fidelity (‘Attended’ minus ‘Unattended’: Mdn=.47; HDI95%=-0.85, 1.77).

However, it is very probable that fidelity was greater for the ‘Don’t Know’ group than it was for

the ‘Know’ group by a substantial amount, with a difference of Mdn=2.83 (HDI95%=0.71, 4.85).

Moreover, the interaction between ‘Knowledge of Convention’ and attention was notable

(Mdn=1.99; HDI95%=-0.53, 4.68). However, in light of the rather strong interaction observed for

probability of memory, we show the effects of attention on fidelity of encoding as a function of

convention knowledge in Figure 3 (Right) and Table 2 (Bottom). Consistent with what was

observed with the probability of memory parameter, the color wheel performance of participants

that do not have knowledge of the baseball convention appear to show a larger effect.

Focusing only on the subgroup of participants who were not aware of the ‘tie goes to

runner’ convention, Figure 3 shows that there were effects on both probability and fidelity of

memory and that therefore the attentional manipulation was successful for that sub-group.

Figure 3. Polygons depict the posterior credibility density distributions of the effects of attention


ENDOGENOUS VISUAL PRIOR ENTRY 20 (‘Attended’ minus ‘Unattended’) for both ‘Knowledge of Convention’ groups (‘Know’ and ‘Don’t’ Know’) on the probability (left panel) and fidelity (right panel) of encoding, with and on the y-axis, respectively. The dotted lines represent effects of zero. The thick lines represent 50 % HDIs, whereas the thin lines represent 95 % HDIs. The large dot depicts the median value of the distribution.

TOJs. To conceptualize the meaning of the effect of attention on PSS (the primary

outcome) in the context of the TOJ task, psychometric functions were generated for both

attention conditions (‘Attend Base’ and ‘Attend Glove’) and both ‘Knowledge of Convention’

conditions (‘Know’ vs. ‘Don’t Know’) (Figure 4). The curves are cumulative normal

distributions with the median of the posterior of the PSS and JND as the mean and standard

deviation, respectively. The raw data (proportion of responses by condition and by SOA) are

depicted as points to illustrate the degree to which the model fit the data.

Point of Subjective Simultaneity: The mean of the population intercept of the PSS was

centered at -7.3 ms (HDI95%=-16.8 ms, 1.7 ms), indicating that overall judgments favored the

runner. The mean of the population effect of locus of attention on the PSS (i.e., ‘Attend Glove’

minus ‘Attend Base’) was centered at -9.1 ms (HDI95%=-18.5 ms, 0.5 ms). This effect can be

seen in Figure 4 as a leftward shift in the psychometric function when attending the glove.

Therefore, contrary to prediction (which was that there would be more “safe” judgments when

attending the base than the glove), we observed a reversal of prior entry such that it was more

probable for participants to report that the runner appeared first when they were attending the

glove than it was when they were attending the base. The effect of ‘Knowledge of Convention’

on PSS, and its interaction effect with locus of attention, were unclear as both their estimates

were widely spread around 0 ms (Mdn=4.9 ms; HDI95%=-13.5 ms, 23.4 ms; and Mdn=0.0 ms;

HDI95%=-18.8 ms, 19.4 ms; respectively). This null interaction, along with the moderate effects

of attention on PSS (albeit in the reverse direction predicted by prior entry), is illustrated in


ENDOGENOUS VISUAL PRIOR ENTRY 21 Figure 5 (Left) and reported in Table 3 (Top).

Table 3

Point estimates (medians of posterior distributions) for population means of PSS (ms) and JND (ms) as a function of all possible pairings of convention knowledge (‘Don’t Know’ vs. ‘Know’) and attention condition (‘Attend Glove’ vs. ‘Attend Base’) levels. Negative PSS values indicate that the ball had to lead the runner for the participant to perceive the runner and the ball as appearing at the same time.

Parameter Know Don’t Know

Glove Base Mean Glove Base Mean

PSS -14.3 -5.2 -9.7 -9.3 -0.3 -4.8

JND 87.7 86.0 86.9 121.5 106.5 113.8



Figure 4. Cumulative normal distributions for both attention conditions (Attend Glove, red; Attend Base, blue) across both ‘Knowledge of Convention’ conditions (‘Don’t Know’, top; ‘Know’, bottom), with the median of the posterior distribution of the PSS and JND as the mean and standard deviation, respectively. The curves represent the fitted functions, whereas the colored points depict the raw data (proportion of ‘safe’ responses by condition and by SOA). Proportions of ‘Safe’ responses are plotted on the y-axis. Negative SOAs indicate that the runner was in fact ‘Out’.



Figure 5. Polygons depict the posterior credibility density distributions of the effects of attention (‘Attend Glove’ minus ‘Attend Base’) for both ‘Knowledge of Convention’ groups (‘Know’ and ‘Don’t Know’) on the PSS (left panel) and JND (right panel). The dotted lines represent effects of zero. The thick lines represent 50 % HDIs, whereas the thin lines represent 95 % HDIs. The large dot depicts the median value of the distribution.

Just-Noticeable Difference: The mean of the population intercept of the JND was

centered at 100.6 ms (HDI95%=90.9 ms, 111.7 ms). There was an effect of the locus of attention

on the JND (Mdn=8.3 ms; HDI95%=-0.1 ms, 16.3 ms), where JNDs were smaller (ergo observers

were more sensitive to the temporal order) in the ‘Attend Base’ condition. Also, the effect of

“Knowledge of Convention” on the JND was large with high probability (Mdn=27.1 ms;

HDI95%=6.0 ms, 47.5 ms), with the ‘Don’t Know’ group having a much larger JND than the

‘Know’ group (Figure 6; Table 3). Moreover, the interaction between ‘Knowledge of

Convention’ and locus of attention on JND was notable (Mdn=13.3 ms; HDI95%=-2.8 ms, 29.7

ms). Figure 5 (Right) and Table 3 (Bottom) shows that participants that are not aware of the

convention are more sensitive to the effects of the locus of attention on their performance on the

TOJ task as indicated by JND.

We judged none of the correlations to be notable and had no concrete a priori


ENDOGENOUS VISUAL PRIOR ENTRY 24 hypotheses about them. Therefore, they will not be discussed further6.

Figure 6. Cumulative normal distributions for the mean psychometric function for both ‘Knowledge of Convention’ conditions, with the median of the posterior distribution of the PSS and JND as the mean and standard deviation, respectively. Points, with grey fill, depict the raw data (proportion of ‘safe’ responses by condition and by SOA). Proportions of ‘Safe’ responses are plotted on the y-axis. Negative SOAs indicate that the runner was in fact ‘Out’.

Discussion We have shown that the locus of endogenous visual attention, when it was confirmed by

our colour wheel task, did not generate a prior entry effect. We did, however, see an effect on

6The 95% HDIs and medians corresponding to each correlation coefficient are presented in Appendix C.


ENDOGENOUS VISUAL PRIOR ENTRY 25 various experimental outcomes as a function of whether the participant knew the ‘tie goes to

runner’ convention for decision-making in the baseball TOJ task. Importantly, regardless of

whether participants knew the convention or not, no prior entry was observed. However, that the

remaining factors (JND, probability and fidelity of encoding) each interact with ‘Knowledge of

Convention’ offers reason to believe that individuals in the ‘Know’ group were dividing their

attention between the two tasks differently than those in the ‘Don’t Know’ group. The ‘Know’

group shows a smaller JND than the ‘Don’t Know’ group, suggesting a better sensitivity to the

temporal asynchrony in the baseball task for the ‘Know’ group. Conversely, on the colour wheel

task, the ‘Don’t Know’ group outperformed the ‘Know’ group (see Table 2). If, as we propose,

the ‘Know’ group was investing more resources in the baseball task (a strategy that could be

related to a greater knowledge of baseball), then it is not surprising that this group was not only

worse than the ‘Don’t Know’ group on the colour wheel task but was also insensitive to the

attentional manipulation embedded in this task.

There was an effect of the locus of endogenous attention on JND exclusively in the

‘Don’t Know’ group, whereby these participants were more sensitive to temporal asynchrony in

the ‘Attend Base’ condition. We offer a post-hoc explanation for this pattern: the runner’s path of

motion carries on beyond the base, however the trajectory of the ball is terminated upon its

arrival at the glove. As such, the arrival of the runner at the base must be inferred as a precise

instant within a motion sequence that continues after the imperative arrival has occurred, but the

arrival of the ball at the glove can be inferred as the precise instant a motion sequence is

terminated. Accordingly, this “design” feature (which is inherent in the real-world situation we

are exploring here) creates a context in which the visual cues an observer may use to evaluate the

two events are not balanced. The data suggest that, for the “Don’t Know” group, it is easier to


ENDOGENOUS VISUAL PRIOR ENTRY 26 perform the task in the ‘Attend Base’ condition, as evidenced by a smaller JND, presumably

because of the benefit from allocating attention to the location with the more difficult (attention-

demanding) arrival judgment.

The present experiment revealed no effect of prior entry on TOJ responses as a result of

an endogenous shift in visual attention. Participants were not more likely to judge a runner to be

‘Safe’ when attending the base than when attending the glove, nor more likely to judge the

runner to be ‘Out’ when attending the glove than when attending the base. Furthermore, we have

shown that attention was in fact endogenously oriented as a result of the spatial contingency

assigned to the orthogonal colour probe task, at least for the “Don’t Know” group. This

manipulation check is represented as an effect on the probability of encoding of the colour probe.

These results support the conclusions from Schneider and Bavelier (2003) who found no

evidence for prior entry as a result of endogenous orienting, and is in discord with the findings of

Shore et al. (2001) and Vibell et al. (2007), where significant prior entry under conditions of

endogenous orienting was obtained. Although in agreement with the conclusions of Schneider

and Bavelier, the present investigation is more strongly suited to argue that endogenous attention

does not elicit prior entry as the present design has in fact confirmed that endogenous orienting

had occurred by way of the orthogonal colour probe task. The colour probe task has shown

evidence that endogenous attention was in fact at the more likely probe location as evidenced by

an effect of increased probability and fidelity of encoding.

The present pattern of results offer insight into how attention may operate in a real-world

context. To this end, it is fair to consider the myriad of factors that may also contribute to the

decision-making process for first base judgments. Baseball is a sport with fast, abrupt, and highly

dynamic sequences of play. Accordingly, umpires are trained in a variety of domains beyond


ENDOGENOUS VISUAL PRIOR ENTRY 27 how they should attend, such as their on-field positioning to maximize one’s vantage point for

numerous potential scenarios. Also, the speed at which the ball or the runner approach their

terminal location is highly variable between plays. Lastly, as noted in the Introduction, umpires

often use auditory signals in hopes that this will facilitate their judgments of the rapid sequence

of events at first base. It would be pertinent to extend the current investigation to explore the

degree to which prior entry may influence cross-modal judgments in this real-world scenario.

However, of practical real-world importance, we have shown there was no prior entry effect on

these judgments. Furthermore, since Zampini, Shore and Spence (2005) showed improved

accuracy on TOJs for unimodal judgments relative to bimodal judgments, and since sound is

processed rapidly relative to vision (i.e., the two signal types are not on equal footing temporally)

and may not be readily available due to a variety of reasons (e.g., crowd noise, velocity of the

throw, the impact point of the ball in the glove), it may behoove those who design umpire

training modules to consider emphasizing a unimodal attentional strategy for this scenario. More

research would be required in order to confidently recommend such a prescription. Be that as it

may, it would be naive to assume our investigation is entirely representative of the complex

environment in which the umpire is forced to work. It is in this spirit that we hope that our

investigation may inspire further research into teasing apart whether these on-field actions may

influence the outcome of these temporal decisions.



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Appendix A

Script for Verbal Instructions for Participant Prior to Experiment

There are four blocks. In each block you will be shown a total of 120 first base baseball

plays in which the runner is running towards first base as the baseball is making its way to the

catcher’s glove. At the end of some of these plays you will be asked whether the runner was

“Safe” or “Out”. The runner is “Safe” if her foot touches the base before the ball enters the

glove, and is “Out” if the ball enters the glove before her foot touches the base. There is no tie,

the runner must be either “Safe” or “Out”. You will indicate your judgment, on your own time,

by clicking either the “s” key for “Safe” or the “o” key for “Out”.

At the end of the rest of these plays you will be asked to estimate the colour of a disk

which will have been flashed briefly over either the base near the time that the runner’s foot

reaches it, or the glove near the time that the ball enters it. You will indicate the colour of the

disk using a colour wheel. Importantly, the coloured disk will NOT appear on either location -

the base, or the catcher’s glove - with equal probability. In a given block, on colour wheel plays,

the colour disk will be more likely to appear on one location than the other. You will be

informed at the beginning of each block where the colour disk is likely to appear most often.

There is no particular order with respect to which of the two possible tasks you will be required

to do after any given baseball play - the tasks are randomly distributed throughout the block.



Appendix B

Bayesian Hierarchical Model Details

The model was implemented using RStan (Guo et al., 2016), the R interface to the Stan

modelling language (Stan Development Team, 2015). Here, we will simply give a high-level

summary of the model. The outcome variable for TOJ trials was a binary judgment (‘Safe’ or

‘Out’) which was modelled with a bernoulli distribution. The probability of a particular judgment

followed a cumulative normal distribution with parameters JND (standard deviation) and PSS

(mean). Each parameter was modelled as a function of participant-wise intercept and condition

effect parameters. These parameters followed population distributions, which adaptively pooled

information across participants. Relatively wide (i.e., less informed) priors were placed on the

parameters of these population distributions. The outcome variable for colour wheel trials was

response accuracy measured as angle deviation, which was modelled with the mixture model

described in the methods section. The primary parameters, (probability) and (fidelity), were

also modelled hierarchically. Finally, the overall model incorporated a correlation matrix to

quantify the linear relationship between population parameters. Ultimately, posterior credibility

distributions were obtained for the PSS, JND, , and population intercept and attention effect

means, and a between-subject effect of knowledge of ‘tie goes to runner’ convention along with

its interaction with the within-subject factor of attention7.

Model samples were drawn with a No-U-Turn-Sampler which is an extension of

Hamiltonian Monte Carlo, a Markov Chain Monte Carlo algorithm (Hoffman & Gelman, 2014).

7 All code relating to the Bayesian analysis, including an explicit declaration of the model (e.g., priors), can be found online (http://or.psychology.dal.ca/~klein/dentremont/toj_color.stan.zip).


ENDOGENOUS VISUAL PRIOR ENTRY 33 Eight chains were run, and each chain had 20,000 iterations and a ‘burn-in’ period of 10,000.

Therefore, we were left with 80,000. Model convergence was verified via a couple key

diagnostics parameters (for all parameters, R-hat 1, NEffective>1000). Model validity was verified

via posterior predictive checks whereby one evaluates whether predictions from the posterior

distribution of the model capture, within reason, the raw data from which the model was derived.



Appendix C

95% HDIs and medians corresponding to posteriors for transformed values and all correlation

coefficients.

Median and 95% HDI for the logarithm of Fidelity of Encoding

Population Parameter (Means) Median 95% HDI

Intercept 2.50 2.41, 2.59

Effect of Attention 0.03 -0.08, 0.14

Effect of Convention Knowledge 0.23 0.06, 0.40

Interaction Effect 0.15 -0.06, 0.38

Median and 95% HDI for the log-odds of Probability of Encoding


Intercept 3.97 3.53, 4.46


Effect of Convention Knowledge -0.05 -0.97, 0.84


Median and 95% HDI for the logarithm of JND


Intercept -0.92 -1.03, -0.81


Effect of Convention Knowledge 0.27 0.06, 0.49


Note. The logarithm of normalized JNDs (divided by 250) is presented here. All back-transformations reported in this paper account for this normalization, which was meant to facilitate the sampling of the posterior distributions relating to the JND outcome.



Appendix C - Figure 1. Caterpillar plot of all possible correlation coefficients (r). Each of four outcome variables (JND, PSS, probability, and fidelity) have a intercept and a locus of attention effect parameter (means), meaning there are eight parameters and therefore 28 unique, non-trivial, correlations. The parameters that were computed in a transformed space are presented in that space (i.e., all but PSS intercept and effect).The dotted line represents effects of zero. The thick lines represent 50 % HDIs, whereas the thin lines represent 95 % HDIs. The large dots depicts the median value of the distribution.



Appendix D

To facilitate comparisons between the Bayesian analysis and a more traditional analytic

approach, we have subject the same data set (N=448) to frequentist analyses. Analyses were

performed on transformed data to meet assumptions of frequentist analysis, however for

interpretability, the non-transformed data are reported in all summaries and tables.

Absolute Color Wheel Error A paired t-test on the log transform of absolute colour wheel error showed a main effect of

attention on performance [t(43) = -2.43, p = 0.019], where observers were more accurate on

colour wheel judgments when the probe appeared at the attended location (15.4°) than at the

unattended location (17.0°). Analysis of variance (ANOVA) on attention (‘attended’ vs.

‘unattended’) and ‘Knowledge of Convention’ (‘Know’ vs. ‘Don’t Know’) was performed on the

same dependent variable, and means are shown in Table D1. This analysis showed a main effect

of attention [F(1, 42) = 6.42, p = 0.016, ges = 0.022], but no main effect of ‘Knowledge’ [F(1,

42) = 1.89, p = 0.176, ges = 0.037] nor an interaction [F(1, 42) = 3.50, p = 0.068, ges = 0.012].

8 The full data set includes 5 outliers which are not well-handled by traditional analyses (but are in the Bayesian approach. We recomputed all the analyses reported here excluding these 5 participants. There were two effects that are marginal in the full analysis that become significant when outliers were removed: the effect of knowledge on fidelity and the effect of locus of attention on the PSS. Importantly the interesting interactions between knowledge and attention that were apparent in the main Bayesian analyses are not significant in these less powerful ANOVAs.


ENDOGENOUS VISUAL PRIOR ENTRY 37 Table D1. Absolute colour wheel error data shown relative to Attention condition and Knowledge of ‘Tie Goes to Runner’ Convention (standard deviations are shown in parentheses).

Attention Convention Knowledge

Absolute Color Error (degrees)

Unattended Don’t Know 16.6 (6.4)

Attended Don’t Know 14.1 (4.6)

Unattended Know 18.0(9.9)

Attended Know 18.6 (11.7)

Probability of Encoding A paired t-test on the logit transform of rho (probability of encoding) parameter showed a main

effect of attention on performance [t(43) = 5.02, p = 9.39e-06], where observers were more likely

to encode the probe when it appeared at the attended location (0.964) than when it appeared at

the unattended location (0.942). Analysis of variance (ANOVA) on attention (‘attended’ vs.

‘unattended’) and ‘Knowledge of Convention’ (‘Know’ vs. ‘Don’t Know’) was performed on the

same dependent variable, and means are shown in Table D2. This analysis showed a main effect

of attention [F(1, 42) = 25.23, p = 9.90e-06, ges = 0.116], but no main effect of ‘Knowledge’

[F(1, 42) = 0.25, p = 0.619, ges = 0.005] nor an interaction [F(1, 42) = 0.99, p = 0.324, ges =

0.005].

Table D2. Probability of encoding shown relative to Attention condition and Knowledge of ‘Tie Goes to Runner’ Convention (standard deviations are shown in parentheses).

Attended Convention Knowledge

Rho

Unattended Don’t Know .943 (.066)

Attended Don’t Know .973 (.040)

Unattended Know .937 (.114)

Attended Know .939 (.138)


ENDOGENOUS VISUAL PRIOR ENTRY 38 Fidelity of Encoding A paired t-test on the log transform of kappa (fidelity of encoding) parameter showed no effect

of attention on performance [t(43) = 0.92, p = 0.364]. Analysis of variance (ANOVA) on

attention (‘attended’ vs. ‘unattended’) and ‘Knowledge of Convention’ (‘Know’ vs. ‘Don’t

Know’) was performed on the same dependent variable, and means are shown in Table D3. This

analysis showed no main effect of attention [F(1, 42) = 0.87, p = 0.357, ges = 0.005], no main

effect of ‘Knowledge’ [F(1, 42) = 2.99, p = 0.091, ges = 0.051; see Footnote 8] nor an

interaction [F(1, 42) = 2.36, p = 0.132, ges = 0.014].

Table D3. Fidelity of encoding shown relative to Attention condition and Knowledge of ‘Tie Goes to Runner’ Convention (standard deviations are shown in parentheses).


Kappa

Unattended Don’t Know 14.09 (5.87)

Attended Don’t Know 15.12 (4.91)

Unattended Know 12.18 (2.88)

Attended Know 11.03 (1.67)

Point of Subjective Simultaneity A paired t-test on PSS showed no effect of locus of attention on performance [t(43) = 1.83, p =

0.075]. Analysis of variance (ANOVA) on locus of attention (‘Glove’ vs. ‘Base’) and

‘Knowledge of Convention’ (‘Know’ vs. ‘Don’t Know’) was performed on the same dependent

variable, and means are shown in Table D4. This analysis showed no main effect of locus of

attention [F(1, 42) = 3.27, p = 0.08, ges = 0.014; see Footnote 8], no main effect of ‘Knowledge’

[F(1, 42) = 0.29, p = 0.59, ges = 0.006] nor an interaction [F(1, 42) = 0.10, p = 0.75, ges =

0.000].


ENDOGENOUS VISUAL PRIOR ENTRY 39 Table D4. Point of Subjective Simultaneity shown relative to Locus of Attention condition and Knowledge of ‘Tie Goes to Runner’ Convention (standard deviations are shown in parentheses).


PSS (ms)

Glove Don’t Know -17.8 (60.2)

Base Don’t Know -5.4 (39.7)

Glove Know -23.5 (35.3)

Base Know -15.6 (42.8)

Just-Noticeable Difference A paired t-test on the log transform of JND showed a main effect of locus of attention on

performance [t(43) = 2.33, p = 0.025], where observers were more sensitive to the temporal order

of events when attending the base (118 ms) than when attending the glove (128 ms). Analysis of

variance (ANOVA) on locus of attention (‘Glove’ vs. ‘Base’) and ‘Knowledge of Convention’

(‘Know’ vs. ‘Don’t Know’) was performed on the same dependent variable, and means are

shown in Table D5. This analysis showed a main effect of locus of attention [F(1, 42) = 5.41, p =

0.025, ges = 0.013], but no main effect of ‘Knowledge’ [F(1, 42) = 1.98, p = 0.166, ges = 0.041]

nor an interaction [F(1, 42) = 0.86, p = 0.360, ges = 0.002].

Table D5. Just-Noticeable Difference shown relative to Locus of Attention condition and Knowledge of ‘Tie Goes to Runner’ Convention (standard deviations are shown in parentheses).


JND (ms)

Glove Don’t Know 133.1 (54.1)

Base Don’t Know 122.5 ( 65.7)

Glove Know 112.7 (70.1)

Base Know 105.0 (56.4)



Appendix D - Figure 1. Plot of all possible correlation coefficients (r). Each of four outcome variables (JND, PSS, probability, and fidelity) have a intercept and a locus of attention effect parameter (means). Thus, there are eight parameters and therefore 28 unique, non-trivial, correlations. These correlations were performed on the non-transformed values.

doi: 10.1037/cep0000118 safe or out: ralph s. redden

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