Mental toughness & perseverance 1 - Daniel Gucciardi

Mental toughness & perseverance 1

Running head: Mental toughness & perseverance

When the going gets tough: Mental toughness and its relationship with behavioural

perseverance

1Daniel F. Gucciardi*, 2Peter Peeling, 1Kagan J. Ducker, and 2Brian Dawson

1School of Physiotherapy and Exercise Science, Curtin University

2School of Sport Science, Exercise & Health, The University of Western Australia

Author Notes

*Address correspondence to Daniel Gucciardi, School of Physiotherapy and Exercise

Science, Curtin University, GPO Box U1987, Perth, Australia, 6845. Email:

[email protected]

Word Count: 3060

Abstract Word Count: 208

Number of Tables: 3

Number of Figures: 0

Supplementary Material: yes

Manuscript accepted for publication in Journal of Science and Medicine in Sport on

December 6th, 2014

mailto:[email protected]


Abstract 1

Objectives: This study examined the association between self-reported mental toughness and 2

behavioural perseverance among a sample of male Australian footballers in a naturalistic 3

context. 4

Design: Cross-sectional study, with the multistage 20 m shuttle run test (MST) employed as a 5

proxy for behavioural perseverance. 6

Methods: 330 male Australian footballers aged between 15 and 18 years (M = 16.86; SD = 7

.71) with between 2 and 14 years playing experience (M = 9.32; SD = 2.51) participated. 8

Initially, footballers completed a mental toughness questionnaire, before having their height 9

and body mass measurements taken. Subsequently, a performance testing session was 10

completed, which included the 20 m sprint, Australian football-specific agility run, vertical 11

jump, and the MST. Bayesian estimation was employed to allow for the simultaneous 12

examination of existing findings with our new data in a way that provides an automatic meta-13

analysis of evidence in this area. 14

Results: The analysis indicated a 95% probability that the association between mental 15

toughness and behavioural perseverance lies between .14 and .34, even when controlling for 16

other factors known to influence MST performance, including age, height, body mass, and 17

years playing experience. 18

Conclusions: Taken together with previous research, these findings support the theoretical 19

proposition that persistence, effort or perseverance represents a behavioural signature of 20

mental toughness. 21

22

Keywords: Bayesian structural equation modelling; beep test; mentally tough; substantive-23

methodological synergy 24


Introduction 1

Coaches, athletes, and sport scientists agree that psychological characteristics – 2

alongside physical, technical, and tactical skills – are essential for optimal sport 3

performance1. Over the past decade, mental toughness has gained considerable attention as 4

the umbrella construct that appears to encapsulate several of the key psychological attributes 5

central to achieving sporting success. Initial research on this concept was founded in 6

professional practice knowledge, that is, information derived largely from the experiences 7

and perceptions of practitioners working in the field. The unsystematic nature of the 8

accumulation and communication of this professional practice knowledge was addressed in 9

the next wave of research, whereby scholars identified and described unobservable personal 10

attributes considered central to mental toughness (e.g., self-belief, emotion regulation) 11

through systematic investigations. As a collective, the work completed thus far suggests that 12

mental toughness represents as a psychological capacity to deliver high performance on a 13

regular basis despite varying degrees of situational demands2,3. 14

In an attempt to advance conceptualisations of mental toughness, researchers have 15

turned their attention to the behavioural features of this psychological concept3,4, that is, 16

observable behaviours or actions that are typically demonstrated in challenging or demanding 17

situations (i.e., person-situation interaction). Persistence, effort, or perseverance is often 18

reported as a behavioural signature of mentally tough individuals, akin to the psychological 19

concept of grit5; for example, pushing through challenging situations, refusing to give up or 20

quit, and sustaining high effort levels over time6,7,8. Recent research on mental toughness in 21

non-athlete samples has supported these initial contentions, particularly as it pertains to 22

sustained efforts towards longer-term goals. Among tertiary students, for example, self-23

reported mental toughness has been associated with higher levels of academic and social goal 24

progress over a university semester2. Furthermore, in a six-week selection course designed to 25


assess candidate’s suitability for entry into elite military training (i.e., Special Forces), results 1

indicated that for a one unit increase in self-reported mental toughness, the odds of passing 2

the selection test (versus failing) increased by a factor of 3.48, that is, just over three times as 3

likely to have passed the selection course2. 4

Only one study to date has examined persistence, effort, or perseverance as a 5

behavioural signature of mentally tough athletes. Compared with the aforementioned studies 6

in non-athlete samples2, behavioural perseverance was examined using a discrete task rather 7

than over an extended period of time among a sample of young elite cricketers9. In this study, 8

an intervention that systematically exposed cricketers to punishment-conditioned stimuli (i.e., 9

consequences for behaviour) using a multidisciplinary team and transformational delivery 10

approach (e.g., coaches, medical staff, and administrators expressed belief in the cricketers’ 11

ability to achieve the vision) improved the experimental groups’ behavioural perseverance as 12

measured by the 20-m multistage shuttle run test (MST) when compared with the control 13

group. Additionally, these improvements in behavioural perseverance over the 12 month 14

testing period were accompanied by increases in coach-rated mental toughness for the 15

experimental group, but not for the control group. 16

The current study was designed to extend this foundational work by examining the 17

extent to which these findings obtained with cricketers in a controlled setting would 18

generalise to Australian footballers in a naturalistic context. The inclusion of an objective 19

measure of behaviour also addressed one of the major criticisms of the mental toughness 20

literature, that is, the reliance on arbitrary metrics (e.g., correlating self-reported mental 21

toughness with self-reported stress). Arbitrary metrics provide little insight into the meaning 22

of self-reported psychological concepts for real-world behaviour and are therefore inadequate 23

for testing and informing psychological theories10. The Australian Football League (AFL) 24

National Draft Combine testing protocol is completed annually by potential draftees and 25


development players to assess a wide range of physical capacities. This testing battery 1

contains numerous assessments of physical qualities such as anthropometry (∑7 skinfolds, 2

height, body mass, arm length, hand span), speed (20 m sprint), speed-endurance (6 x 30 m 3

repeat sprint), agility (AFL-specific agility), power (vertical jump and running vertical jump) 4

and aerobic capacity (MST). Given the relationship between the MST and aerobic capacity as 5

measured by VO2max in elite and sub-elite athletes is imperfect (r = .61-78)11,12, psychological 6

factors may provide insight into the unexplained variance in this test (between 39% and 7

63%). Previously, the MST has been employed as a proxy of behavioural perseverance9, as it 8

requires individuals to sustain high levels of effort and push through physical and mental 9

demands in a single testing session. Consistent with experimental8 and prospective evidence2, 10

we hypothesised that mental toughness would be positively associated with behavioural 11

perseverance as measured by the MST. 12

This study also offers a methodological extension to previous research on mental 13

toughness, and more broadly the sport and exercise sciences. In contrast to traditionally 14

employed frequentist approaches (e.g., maximum likelihood, p values), we utilise Bayesian 15

estimation to explicitly integrate the prior findings of Bell and colleagues9 into our analytical 16

framework, thereby providing an automatic meta-analysis13. Other differences also exist 17

between Bayesian and frequentist approaches14. For example, the frequentist interpretation of 18

the 95% confidence interval is based on long run frequency, such that the 95% confidence 19

intervals of an infinite number of replications of the same experiment or study will capture 20

the fixed but unknown parameter estimate under the null hypothesis (i.e., true population 21

estimate). This interpretation differs within Bayesian statistics whereby the 95% confidence 22

interval (referred to as a credibility interval) indicates the probability that the unknown and 23

therefore random parameter estimate lies between the lower and upper values of the interval. 24

The parameter estimate is considered substantively important when the 95% credibility 25


interval does not contain zero, and therefore the null hypothesis is rejected. Additionally, 1

Bayesian analysis offers intuitive interpretations because of the direct focus on the probability 2

distribution of parameters, such that one can make statements about the probability of a 3

hypothesised theoretical model, given the data13. In contrast, frequentist approaches permit 4

statements about the probability of the data, given the hypothesised theoretical model. 5

Methods 6

Participants were 330 male Australian footballers aged between 15 and 18 years (M = 7

16.86; SD = .71). Footballers had played competitive football for between 2 and 14 years (M 8

= 9.32; SD = 2.51) at the time of participating, and were playing in the highest competition 9

within the state for their age group. Institutional ethics approval was obtained prior to the 10

commencement of this study. 11

Footballers were tested in groups of approximately 40 players during a 2-3 hour 12

session, which occurred over a 3-day period at the conclusion of the pre-season phase of the 13

Western Australian Football League colts training program. Upon arrival, participants first 14

completed a brief questionnaire including demographic information (age, years playing) and 15

self-reported mental toughness in a large lecture theatre (seating capacity of 150). A semantic 16

differential approach was employed to directly measure mental toughness, using an 17

adaptation of the revised performance profile technique15. This approach contrasts with an 18

indirect approach whereby mental toughness is inferred from participants’ scores on 19

individual facets such as self-efficacy, optimism, and emotional regulation. There is evidence 20

to support the superiority of a direct approach to the measurement of mental toughness2. 21

Participants responded to 11 items on a 7-point scale anchored by bipolar construct 22

descriptions of the 11 key mental toughness components in Australian football5. The 7-point 23

scale included anchors of “100% of the time” for both poles (1 = contrast pole of mental 24

toughness and 7 = mentally tough pole) and “50% of the time” at the midpoint (see Table 2). 25


After participants completed the survey, their height and body mass was assessed. 1

Height was measured using a wall mounted stadiometer (Harpenden stadiometer, Holtain 2

Ltd. UK) and body mass recorded on a platform scale (UC300, A & D Mercury Pty Ltd. SA, 3

AUS). Following the completion of these anthropometric measures, a team-led warm-up was 4

completed before a performance testing session was commenced. The performance tests used 5

here were a subset of those implemented at the AFL National Draft Combine, which included 6

the 20 m sprint, AFL-specific agility run, vertical jump, and the MST16. The MST was 7

completed as per the standardised instructions published by Leger et al.17 in groups of 8

approximately 20 footballers. All tests were performed in an indoor gymnasium on a sprung 9

wooden floor. Only the MST data is reported in this study, given our focus on behavioural 10

perseverance rather than skill-related movement (e.g., vertical jump). 11

As correlation and multiple regression analyses do not take measurement error into 12

consideration (i.e., observed variables are assumed to be measured without error), the 13

relationship between mental toughness and behavioural perseverance (as measured by MST 14

performance) was tested within a Bayesian structural equation modelling (BSEM) 15

framework18 using Mplus 7.219. Age, height, body mass, and football experience were 16

included as covariates in the model, thereby permitting a test of the contribution of mental 17

toughness to behavioural perseverance while accounting for the influence of these variables. 18

The covariates were modelled as observed variables, whereas mental toughness was modelled 19

as a latent variable including those item indicators detailed in Table 2 and their error terms. 20

We drew from statistical recommendations regarding the quality of factor loadings to guide 21

the specification of priors for the mental toughness measurement model20, such that items 22

were specified to have a normal prior of .80 and a standard deviation + .34. The prior for the 23

structural path between mental toughness and behavioural perseverance was guided by Bell 24

and colleagues’ study with elite youth cricketers9, such that we converted their interaction 25


effect size into an unstandardised correlation coefficient (i.e., μ =.40, σ2 = .02, SD = + .28)21. 1

Weakly informative priors incorporate prior knowledge regarding the population parameter in 2

the model (e.g., specific value of the mean most likely) but with some degree of uncertainty 3

(i.e., moderate amount of variance around a specific value of the mean), and therefore do not 4

substantially influence the final parameter estimate in the posterior distribution once 5

combined with the data14. The Mplus syntax is provided in Table 3. 6

The posterior predictive p value (PPP) is computed to provide an indication of model 7

fit within BSEM; this index compares the deviation between the real and replicated data to 8

produce a 95% confidence interval for this discrepancy function22. A small positive PPP 9

value (e.g., 0.05) is indicative of poor fit, and a value around 0.5 and above is suggestive of 10

excellent fit18. Statistical criteria (i.e., potential scale reduction factor value is < 1.123) 11

alongside visual inspection of the trace plots (i.e., multiple chains converged to a similar 12

target distribution11) provided an indication of model convergence. See Appendix A of the 13

online supplementary material for further information on Bayesian estimation. The reliability 14

of the latent mental toughness factor was computed using McDonald’s24 omega coefficient 15

(ω). 16

Results 17

The probability of the hypothesised model, given the data, was acceptable (PPP = 18

.228, Δobserved and replicated 2 95% CI [-29.21, 61.31]). Four chains were estimated and 19

in 23300 iterations reached an appropriate convergence criterion. Visual inspection of trace 20

plots (e.g., see Figures S1, S2, S3, and S4 of Supplementary Material) and an examination of 21

the PSR development over iterations (i.e., smooth decrease in PSR, last few thousand 22

iterations were close to 1) verified support for model convergence18. Standardised factor 23

loadings for the mental toughness items (λ > .635; M = .72, SD = .06) and its reliability were 24

excellent (ω = .92). Mental toughness was positively associated with football experience (β = 25


.19; 95% CI = .08, .30), and body mass was positively associated with height (β = .73; 95% 1

CI = .67, .77) and age (β = .12; 95% CI = .01, .23); all other correlations among the 2

covariates and mental toughness were not significant (see Table 1). Body mass (β = -.35; 3

95% CI = -.49, -.20) and mental toughness (β = .24; 95% CI = .14, .34) were substantively 4

important antecedents of MST performance (see Table 1). Mental toughness and the 5

covariates accounted for 17.2% of the variance in behavioural perseverance, whereby the 6

inclusion of mental toughness to the model accounted for an additional 5.4% of the explained 7

variance. Couched within a Bayesian framework, these findings indicated that there is a 95% 8

probability that the parameter value falls between the lower and upper limits of these 9

intervals. For example, the relationship between mental toughness and behavioural 10

perseverance lies between .14 and .34. 11

Discussion 12

Consistent with previous research2,9, we hypothesised that mental toughness would be 13

positively associated with behavioural perseverance, and this relationship would hold while 14

controlling for covariates including age, body mass, height, and years playing experience. In 15

testing this hypothesis, we employed an emerging methodology – Bayesian structural 16

equation modelling – to formally integrate prior findings into our analytical framework, 17

thereby providing an automatic meta-analysis13. Our hypothesis regarding the relationship 18

between mental toughness and behavioural perseverance was supported, indicating that this 19

psychological capacity accounted for an additional 5.4% of the variance in MST performance 20

beyond key covariates known to influence performance on this test (age, height, playing 21

experience, body mass). This result provides evidence that previous findings obtained in a 22

controlled setting with cricketers – whereby mental toughness was manipulated via a psycho-23

social intervention – generalise to a naturalistic, field-testing situation with Australian 24

footballers9. Taken together with previous research in non-athlete samples2 and descriptions 25


of mentally tough athletes6,7,8, there appears to be emerging evidence to support the notion 1

that persistence, effort, or perseverance represents a behavioural signature of mentally tough 2

individuals. From a practical standpoint, the MST is a key predictor of being selected for elite 3

football (AFL), with only small differences between those players who are ‘drafted’ and 4

those who compete in the National championships25. Nevertheless, additional work is 5

required to ascertain whether or not mental toughness provides incremental validity over and 6

above other pertinent individual difference variables (e.g., emotion regulation). 7

Mental toughness is a complex psychological concept that captures one’s personal 8

capacity to deliver high performance on a regular basis despite varying degrees of situational 9

demands2,3. From a transactional perspective of stress, mental toughness may play a role in 10

facilitating behavioural perseverance in performance situations that are goal-relevant and 11

therefore task-engaging through an appraisal system where situational demands are perceived 12

as either challenging or threatening for valued outcomes such as performance and well-13

being26. Previous research on mental toughness in sport has supported the importance of 14

distinguishing between negative and positive stressors5. The extent to which individuals 15

perceive features of the situation as challenging or threatening depends on their relative 16

evaluations of the demands of the situation alongside their available resources. When the 17

available resources are perceived as outweighing the situational demands, individuals 18

appraise the situation as challenging; this perception contrasts with threatening appraisals in 19

which individuals perceive the demands of the situation as exceeding their available 20

resources26,27. Challenge appraisals typically foster positive outcomes such as mastery, 21

growth, or gains, whereas hindrance stressors typically thwart the attainment of such positive 22

outcomes28. An empirical test of this theoretical speculation represents one way in which the 23

current study may be extended in future research. 24


A key strength of this study was the use of Bayesian estimation, which allowed us to 1

directly test our informative hypothesis that self-reported mental toughness would be 2

associated with behavioural perseverance, as well as integrate prior knowledge into the 3

analysis to facilitate estimation in the model with our new data. In this way, existing 4

knowledge is updated with new data to produce results that are an automatic meta-analysis13. 5

Bayesian estimation contrasts with the traditional frequentist approach in which one aims to 6

examine such informative hypotheses yet tests the null hypothesis. For example, the null 7

hypothesis testing approach would permit the following statement as it applies in the current 8

study; the probability of making an error with our finding that the relationship between 9

mental toughness and behavioural perseverance is not zero is below 5% (when p < .05). 10

Bayesian estimation has been advocated29 and demonstrated30 as an important 11

methodological consideration for knowledge development and accumulation in the sport and 12

exercise sciences. 13

Despite this methodological strength, it is important to acknowledge additional 14

avenues of future research that are grounded in the limitations of the current study. First, 15

although a casual direction is implied in a structural equation modelling framework of cross-16

sectional data, the non-experimental nature of this study limits our ability to infer causality 17

from the current findings. Longitudinal examinations and experimental manipulations of 18

mental toughness (or appraisals of the situational demands) would also assist interpretations 19

on the causal nature of these relations. For example, it would be interesting to ascertain 20

whether or not mental toughness moderates the relationship between behavioural 21

perseverance in a low and high pressure testing situation. Second, it should be considered that 22

as a result of the large number of athletes sampled in this investigation, it was not possible to 23

test all footballers at the same time of day. As a result, it is possible that the performance 24

outcomes may be biased by circadian influences. Third, our methodological design may have 25


introduced a ‘mere measurement effect’ whereby completing a mental toughness survey 1

immediately prior to the MST enhanced the accessibility of beliefs, attitudes, or motivations 2

about the target behaviour and therefore increased the likelihood of actioning that behaviour. 3

Fourth, we were unable to account for potential non-independence in the MST data (i.e., 4

footballers who performed the test as a group, e.g., social facilitation effects), as this 5

information was not recorded during data collection. Finally, given the field-based nature of 6

our measure of behavioural perseverance, we are unable to ascertain whether or not this 7

psychological capacity to deliver high performance despite varying degrees of situational 8

demands is related to one’s underlying fitness levels. A baseline measure of physical fitness 9

(i.e., maximal aerobic capacity) would need to be included in any statistical model alongside 10

mental toughness to account for this variable in the prediction of behavioural perseverance. 11

Conclusions 12

In this study, we provided cross-sectional support for the theoretical proposition that 13

perseverance is a behavioural signature of mentally tough athletes. By incorporating 14

empirical2,9 and subjective knowledge6,7,8 into our analysis, Bayesian estimation allowed for 15

the simultaneous examination of these ‘old’ results with our new data to provide an automatic 16

update on this aspect of mental toughness theory13. Delineating the mechanisms by which 17

mental toughness translates into behavioural perseverance represents an important 18

consideration for advancing theory and practice in this area (e.g., appraisal systems). 19

Practical Implications 20

• Persistence, effort or perseverance is a behavioural signature of mental toughness 21

• Physical and psychological attributes are both important considerations for MST 22

performance 23

• When athletes deliver high performances or even fail, reinforce aspects of their behaviour 24

that relate to persistence, effort, or perseverance 25


Acknowledgements

We wish to acknowledge the assistance of Jon Haines from the Western Australian

Football Commission. Appreciation is also extended to the following individuals for their

assistance with data collection: Kym Guelfi, Ray Davey, Chee Yong Low, Marcus Lee and

Bradley Kelly. No financial assistance was received for this study.


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Table 1. Descriptive statistics and standardised parameter estimates for structural equation model including mental toughness and covariates as

predictors of behavioural perseverance (Note: 95% credibility intervals are presented in parentheses; MST = multistage 20 m shuttle run test).

Correlations Structural

Path

M SD 1 2 3 4 5 MST

1 Age 16.86 .71 - -.07 (-.17, .04)

2 Playing experience 9.32 2.51 .18 (.07, .28) - .04 (-.06, .15)

3 Height 180.55 6.46 .02 (-.09, .13) -.02 (-.13, .09) - .08 (-.07, .22)

4 Body mass 74.11 8.37 .12 (.01, .23) -.05 (-.16, .06) .73 (.67, .78) - -.35 (-.49, -.20)

5 Mental toughness 5.67 .69 -.02 (-.14, .09) .19 (.08, .30) .02 (-.09, .13) .03 (-.09, .14) - .24 (.14, .34)

6 MST 12.61 1.02 - - - - - -


19

Table 2. Semantic differential approach to measuring mental toughness in Australian football.

Instructions: The items below capture various beliefs, attitudes, and behaviours that people hold or display. Using the scale provided, please indicate the extent to which these

attributes provide a reflection of you as footballer. For example, if you always believe in your physical and mental abilities, mark the box on the right-hand side (X). If you sometimes

believe in your ability, and in other circumstances you doubt your abilities, then mark yourself within the middle sections of the continuum based on approximate percentages; for

example, you believe in yourself approx. 50% of the time (O) or only 25-30% of the time (*). If you always have doubts in yourself, then mark the box on the left-hand side (#).

100% 50/50 100%

100% doubt in my physical my physical and mental ability

under pressure.

# * O X 100% belief in my physical and mental ability under pressure.

Succumb to physical fatigue and niggly injuries, and dislike

50/50 contests.

100% ability to play through physical fatigue, play whilst carrying a

niggly injury, and enjoy 50/50 situations.

100% lack of motivation and an attitude of only doing the

basics and enough to get by.

A philosophy characterized by always working hard and pushing yourself

through (physically and mentally) demanding situations in competition

and training.

Lack “footy smarts” or an understanding of the game and the

way it is played.

100% awareness of the pressures, adversities, and challenges in football

and an understanding of the game.

Poor integrity and personal philosophy on life and football,

and generally conform without any care in the world.

I place great importance and significance on personal values relevant to

one becoming a better person and footballer.

Lack an awareness of my emotions, how they facilitate my

performance, and how to regulate them to perform well.

100% awareness of my emotions, and ability to manage my emotions to

enhance performance across all situations.

Extrinsically and/or unmotivated thereby requiring an

external source to get me going 100% of the time.

Internal motivation and desire for competitive challenges and team

success, and having the desire to put the necessary things into practice to

achieve my goals 100% of the time.

Suboptimal performance under pressure as a result of

anxiety, nervousness, and threat.

I am 100% able to execute skills and procedures under pressure and

stress.

A weak attitude characterized by laziness, easily intimidated

and giving in too easily, and succumbing to pressure

A 100% unshakeable, tough attitude directed towards becoming a

champion of the game.

Easily distracted and unfocused on the job at hand 100% of

the time.

100% focus and concentrate on the job at hand and what you want to

achieve despite internal or external pressures, obstacles, or adversities.

An inability to adapt to pressure, being easily broken, and

not putting up a fight in the face of adversity.

100% ability to overcome adversities with an exceptional work ethic and

persevering determination to showcase my mental and physical ability.


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Table 3. Overview of Mplus specifications for Bayesian analysis (Note: text in green and preceded

by an exclamation mark is not read by Mplus when executing the analysis).

MODEL:

! measurement model of mental toughness

MT BY mt1* mt2 mt3 mt4 mt5 mt6 mt7 mt8 mt9 mt10 mt11 (f1l1-f1l11);

MT@1;

! structural model

shuttle ON MT (b1);

shuttle ON height weight age yrsplay;

height weight age yrsplay MT WITH height weight age yrsplay MT (cf1-cf10);

! name residual variances

mt1 mt2 mt3 mt4 mt5 mt6 mt7 mt8 mt9 mt10 mt11 (rv1-rv11);

! name the correlated residuals

mt1-mt11 WITH mt1-mt11 (cr1-cr55);

ANALYSIS:

! tells Mplus to use Bayesian estimation

ESTIMATOR = BAYES;

! specify the value of the Gelman-Rubin convergence criterion to be .01

BCONVERGENCE = .01;

! specify the use of 4 processors (to speed up computations when available)

PROCESSORS = 4;

! specify 4 independent chains of the MCMC procedure

CHAINS = 4;

MODEL PRIORS:

! informative prior for structural parameter

! where the mean is set at 0.40 and the variance is .02

! see DOI: 10.1037/a0033129

b1~N(.40,.02);

! informative priors for factor loadings of mental toughness model

! where the mean is set at 0.80 and the variance is .03

f1l1-f1l11~N(.80,.03);

! residual variances

rv1-rv11~IW(1,17);

! correlated residuals for mental toughness items

cr1-cr55~IW(0,17);

OUTPUT:

STDYX CINTERVAL TECH1 TECH8;

PLOT:

TYPE = PLOT2 PLOT3;


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Supplementary Material

Appendix A – Additional Information on Bayesian Structural Equation Modelling

Broadly speaking, there are three key components to a Bayesian analysis (for reviews, see

van de Schoot & Depaoli, 2014; van de Schoot et al., 2014; Zyphur & Oswald, in press). First,

analysts must define (un)certainty about the parameters of the model in the prior distribution. This

distribution reflects background knowledge prior to collecting new data, which can be derived from

sources such as meta-analyses, related empirical research, theoretical expectations, or experts’

subjective views. As detailed in the methods section of the main document, our prior distribution

was informed by previous research on similar data (Bell, Hardy, & Beattie, 2014). The variance of a

parameter estimate captures the degree of (un)certainty in this background knowledge, whereby a

small prior variance reflects a high degree of certainty that the estimate reflects the population mean

(i.e., informative prior), and a large prior variance reflects a high degree of uncertainty in prior

beliefs of the estimate (i.e., non-informative prior). Priors can occur anywhere along this continuum

of (un)certainty from low to high precision. Second, new data is collected so that it can be

compared against these prior beliefs; in other words, what is the likelihood or probability of the

observed data given existing beliefs as captured in the parameters of the prior distribution? Third,

new data is mixed or combined with existing beliefs via Bayes theorem thereby producing a

posterior distribution. For interested readers, Zyphur and Oswald (in press) have provided a

technical but accessible description of Bayes’ rule. Essentially, the posterior distribution represents

an update in the current state of affairs that encompasses a compromise between prior beliefs and

new data. The median of the posterior distribution reflects the most likely estimate of a parameter,

whereby the shape of the peak provides an indication of the (un)certainty of this estimate (e.g.,

shallow peak indicates a large 95% range of possible estimates).

Markov chain Monte Carlo (MCMC) methods are typically performed to obtain the

posterior distribution (van de Schoot et al., 2014; Zyphur & Oswald, in press). These simulations

involve an iterative process whereby “the conditional distribution of one set of parameters given


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other sets can be used to make random draws of parameter values, ultimately resulting in an

approximation of the joint distribution of all the parameters” (Muthén & Asparouhov, 2012, p.

334). In other words, all parameters of the model are repeatedly estimated to generate a distribution

that is built up from the interaction of prior beliefs with new data. As can be seen in Table 3 of the

main document, we specified the MCMC convergence criterion using the Gelman-Rubin potential

scale reduction value of .01, rather than the default Mplus formula based on the value of .05.

Interested readers are referred elsewhere for a detailed review and application of Bayesian

estimation in the sport and exercise sciences (Gucciardi & Zyphur, in press).


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References

Bell, J. J., Hardy, L., & Beattie, S. (2014). Enhancing mental toughness and performance under

pressure in elite young cricketers: A 2-year longitudinal intervention. Sport, Exercise and

Performance Psychology, 2, 281-297. doi: 10.1037/a0033129

Gucciardi, D.F., & Zyphur, M.J. (in press). Exploratory structural equation modelling and Bayesian

estimation. In N. Ntoumanis & N.D. Myers (Eds.), An introduction to intermediate and

advanced statistical analyses for sport and exercise scientists. Wiley.

Muthén, B. O., & Asparouhov, T. (2012). Bayesian structural equation modeling: A more flexible

representation of substantive theory. Psychological Methods, 17, 313-335. doi:

10.1037/a0026802

van de Schoot, R., & Depaoli, S. (2014). Bayesian analyses: Where to start and what to report. The

European Health Psychologist, 16, 75-84.

van de Schoot, R., Kaplan, D., Denissen, J., Asendorpf, J. B., Neyer, F. J., & van Aken, M. A. G.

(2014). A gentle introduction to Bayesian analysis: Applications to developmental research.

Child Development, 85, 842-860. doi: 10.1111/cdev.12169

Zyphur, M. J., & Oswald, F. L. (in press). Bayesian estimation and inference: A user’s guide.

Journal of Management. doi: 10.1177/0149206313501200


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Appendix B – Figures of Trace Plots

Figure S1. Four chains specified for the Gibbs sampler of the regression of shuttle run performance

on mental toughness.

Figure S2. Four chains specified for the Gibbs sampler of the regression of shuttle run performance

on weight.


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Figure S3. Four chains specified for the Gibbs sampler of the correlation between mental toughness

and years playing experience.

Figure S4. Four chains specified for the Gibbs sampler of the mean score for shuttle performance.

Mental toughness & perseverance 1 - Daniel Gucciardi

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