Chapter 8 Notes: Alternatives to Inverted-U Theory

Key terms (glossary)

  • Autotelic experience: a self-contained activity done for its own reward, not for some future benefit. In flow theory, this is a core aspect of why flow feels rewarding regardless of external outcomes.
  • Butterfly catastrophe model: a higher-order catastrophe model in which additional variables (e.g., self-confidence) can moderate the arousal-anxiety-performance relationship, producing butterfly-shaped patterns of catastrophe effects.
  • Catastrophe theory: a framework that allows for abrupt, discontinuous changes in performance when cognitive anxiety and physiological arousal interact, challenging the smooth inverted-U prediction.
  • Core flow: a concept related to the core、自 Flow state; often discussed in relation to peak or optimal experience.
  • Cusp catastrophe model: a three-dimensional catastrophe model used in sport psychology to show how arousal (physiological arousal) and cognitive anxiety interact to influence performance, with potential for sudden drops in performance.
  • Direction component of anxiety: part of anxiety theory focusing on whether anxiety facilitates or debases performance (directionality of anxiety effects).
  • Flow: a state of deep absorption, optimal experience, and automaticity in action; often described as being in the zone.
  • Flow Frequency component of anxiety: aspect of flow research that considers how often flow states occur or are expected to occur.
  • Hysteresis: a property of catastrophe models where the shift back to higher performance after a catastrophe requires a larger change in arousal than the shift into catastrophe, creating a lag effect.
  • Individual affect-related performance zone (IAPZ): a probabilistic extension of IZOF where performance likelihoods are linked to affect (emotions/affect) rather than just anxiety; expressed as curves predicting performance categories across arousal levels.
  • Individual zone of optimal functioning (IZOF): the individual-specific range of emotional states (traditionally pre-competition anxiety) where performance is optimal; varies across athletes.
  • Ipsative z-score: an intraindividual score that uses the person’s own mean and standard deviation (instead of a group mean/sd) to standardize observations.
  • Metamotivational: a concept from reversal theory referring to the underlying motivational stance (e.g., telic vs paratelic framing) that colors how people interpret and respond to arousal and anxiety.
  • Mindfulness: a present-focused awareness often discussed in the context of emotion regulation and performance anxiety.
  • Multidimensional anxiety theory (Martens et al.): theory proposing separate cognitive (thought-based) and somatic (physiological) components of state anxiety, with distinct relationships to performance.
  • Multiple baseline single-subject design: research design used to study changes in individuals over time with staggered baselines.
  • Orthogonal model of flow: a conceptual model that treats core flow dimensions as independent (orthogonal) factors influencing flow.
  • Paratelic-dominant individual: a person who tends to engage in activities for fun and arousal rather than for task completion; relates to reversal theory and motivational states.
  • Pleasant dysfunctional emotions: positive feelings that are dysfunctional in performance contexts (e.g., excessive cheerfulness that interferes with focus).
  • Pleasant functional emotions: positive emotions that support performance (e.g., confidence, determination).
  • Prestart state anxiety: anxiety experienced just prior to competition.
  • Probabilistic model: approach used to estimate the likelihood of different performance outcomes given levels of affect/arousal.
  • Protective frame: a concept from reversal theory referring to a stability mechanism that protects self-concept under pressure.
  • Psychological reversal: a reversal of motivational states (from telic to paratelic or vice versa) that changes how arousal is perceived and used.
  • Recall method: a method to assess optimal precompetitive anxiety by asking athletes to recall their best performances and rate their emotions at that time.
  • Reversal theory: a theory of motivation that emphasizes changes in how people perceive situations (telic vs paratelic) and how those perceptions influence arousal and behavior.
  • Standard deviation: a measure of dispersion around the mean; used in z-score calculations.
  • Telic-dominant individual: a person whose approach to activity is goal-directed and outcome-focused (teleology dominates).
  • Unpleasant dysfunctional emotions: negative emotions that impair performance (e.g., fatigue, fear, discouragement).
  • Unpleasant functional emotions: negative emotions that may be adaptive in some contexts (e.g., nervous energy that sharpens focus).
  • z-score: standardized score calculated as z=x−msdz = \frac{x - m}{sd}, where x is the observation, m is the mean of the group, and sd is the group standard deviation.
  • ipsative z-score: the same standardization as a z-score but using an individual’s own mean and standard deviation; formula z<em>ips=x−m</em>isd<em>iz<em>{ips} = \frac{x - m</em>i}{sd<em>i} where mi and sd_i are the individual’s mean and standard deviation of their own observations.

Martens’s Multidimensional Anxiety Theory (MDAT)

  • Core idea: Anxiety is multidimensional with two components that affect performance differently.
  • Components:
    • Cognitive state anxiety: thoughts, worry, fear of evaluation; has a negative linear relationship with performance: as cognitive anxiety increases, performance tends to decrease.
    • Somatic state anxiety: physiological arousal (e.g., heart rate, muscle tension); has an inverted-U relationship with performance: moderate arousal facilitates performance, too little or too much arousal impairs it.
  • Figure reference: The MDAT relationships are illustrated in a figure showing the two distinct patterns for cognitive vs somatic anxiety.
  • Empirical support: Some studies support the MDAT patterns (Burton 1988; Gould et al. 1987), but many investigations fail to confirm the theory consistently.
  • Relevance of z-scores: Intraindividualized scores (ipsative z-scores) are often used in MDAT contexts to compare an athlete to their own distribution rather than to a group.
  • Ipsative z-score concept (recap):
    • Standard z-score: z=x−msdz = \frac{x - m}{sd} with group mean m and group sd.
    • Ipsative z-score: z<em>ips=x−m</em>isd<em>iz<em>{ips} = \frac{x - m</em>i}{sd<em>i} with the athlete’s own mean mi and sd_i.
  • Practical note: Intraindividualized measures (ipsative) can be calculated across repeated anxiety observations to identify where a given score sits relative to an athlete’s own distribution.

Fazey and Hardy’s Catastrophe Theory (CATA)

  • Core idea: Cognitive anxiety and physiological arousal interact in a non-additive, catastrophic way to affect performance; simple smooth inverted-U fails to capture abrupt drops in performance.
  • Key assumptions:
    • Small increases in arousal typically lead to small changes in performance under low cognitive anxiety, but at higher cognitive anxiety, the same small arousal increase can trigger a large, abrupt performance drop (catastrophe).
    • After a catastrophe, small decreases in arousal do not immediately restore performance to the previous level (hysteresis).
  • The catastrophe model (simplified):
    • Cognitive anxiety (X) and physiological arousal (Y) interact to determine performance (Z).
    • At low cognitive anxiety, the arousal-performance curve is a smooth inverted-U.
    • At moderate cognitive anxiety, the curve becomes distorted and shows a potential drop with higher arousal, with a recovery path requiring a reduction in arousal.
    • At high cognitive anxiety, the relationship is highly distorted and exhibits a catastrophic drop as arousal increases further, with a recovery path needing substantial arousal reduction.
  • The 3D cusp model (Fazey & Hardy):
    • X-axis: cognitive anxiety; Y-axis: physiological arousal (or altered surrogate such as somatic anxiety or effort); Z-axis: performance.
    • The floor represents cognitive anxiety and arousal combinations; the surface represents performance levels; a catastrophe occurs when moving along X and Y crosses a bifurcation boundary.
  • Key term: hysteresis in catastrophe theory is the property that discontinuity (catastrophic drop) occurs at different arousal points depending on the direction of change in cognitive anxiety.
  • Empirical tests:
    • Hardy & Parfitt (1991) and Hardy, Parfitt, & Pates (1994): manipulating cognitive anxiety and arousal produced catastrophic decrements in basketball and bowling; small arousal changes under high cognitive anxiety did not recover performance.
    • Cohen, Pargman, & Tenenbaum (2003): arousal manipulated via treadmill walking and cognitive threat; results did not robustly support the catastrophe model, possibly due to manipulation difficulties.
    • Some studies replaced arousal with somatic anxiety or with effort as the controlling variable, still finding hysteresis in certain conditions (e.g., Krane et al. 1994; Durr 1996; Hardy et al. 2007).
  • Extensions:
    • Higher-order cusp models with personal factors like self-confidence can moderate the catastrophe; with high self-confidence, a catastrophe may not occur unless both somatic and cognitive anxiety are very high (butterfly catastrophe model).
  • Practical implications (Concept & Application boxes):
    • If cognitive anxiety is high, keeping physiological arousal in check is crucial to avoid catastrophic drops; when catastrophe occurs, rest to reduce arousal is recommended.
    • When cognitive anxiety is low, arousal can rise and still maintain performance within the usual inverted-U.
  • Summary takeaway: Catastrophe theory provides a plausible explanation for sudden drops in performance that inverted-U cannot account for, particularly in high-anxiety contexts.

Hanin’s IZOF Theory (Individual Zone of Optimal Functioning)

  • Origin: Yuri Hanin (1986, 1989) proposed IZOF to explain that optimal performance occurs at athlete-specific emotional states, not at a universal “moderate anxiety” level.
  • Original IZOF (anxiety-focused):
    • Different athletes have different optimal prestart state anxiety levels.
    • Example: 46 elite female rowers had mean optimal prestart state anxiety around 43.80, with a wide range (26–67).
    • Measurement: State anxiety measured with Spielberger’s State Anxiety Inventory (SAI).
    • Application: If optimal prestart anxiety can be determined, arousal control techniques can help move toward that ideal level.
    • Direct vs recall methods:
    • Direct: measure optimal prestart anxiety directly across competitions.
    • Recall: determine optimal anxiety retrospectively by asking athletes to recall their best performances and rate their feelings then.
  • Recall method findings:
    • Correlations between actual optimal precompetitive anxiety and recalled optimal anxiety are high (roughly r ≈ .95) when recall is within about two days of the target competition; correlations decline with longer recall intervals.
    • If actual data are unavailable, recalled scores can be a viable proxy, though accuracy may vary.
  • IZOF definition and zone:
    • An IZOF is the athlete’s optimal prestart state anxiety level, plus/minus a zone of confidence around it, defined as ±4 points on the SAI (roughly ±0.5 SD of observed precontest scores).
    • This yields a zone of optimal functioning rather than a single optimum level.
  • Practical steps for IZOF establishment:
    • Direct measurement of optimal anxiety (best but time-consuming).
    • If direct measurement is impractical, use recall method to estimate the optimal prestart anxiety.
    • After determining the IZOF, intervene to move an athlete’s prestart anxiety into the zone.
  • Evidence supporting IZOF:
    • Prapavessis & Grove (1991); Raglin & Turner (1993); Turner & Raglin (1996) found IZOF predictions superior to other models (e.g., Morgan’s mental health model) or to inverted-U models in predicting performance.
  • IZOF evolution: emotion-inclusive IZOF
    • Emotions beyond anxiety: Hanin expanded IZOF to include multiple emotions (positive and negative; functional and dysfunctional) to reflect more complex affective states.
    • Example: For certain sports, peak performance aligns with high levels of pleasant functional emotions (e.g., confidence, determination) and low levels of unpleasant dysfunctional emotions (e.g., fatigue, discouragement).
  • IZOF for emotions (8.4–8.5):
    • Emotions can be categorized into four quadrants by affective intensity and category:
    • Unpleasant dysfunctional emotions (e.g., fatigue, fear, discouragement).
    • Unpleasant functional emotions (e.g., nervous, concerned, anxious, tense, angry).
    • Pleasant functional emotions (e.g., excited, determined, confident).
    • Pleasant dysfunctional emotions (e.g., cheerful, calm, relaxed).
    • Best performance tends to occur when pleasant functional emotions are high and unpleasant dysfunctional emotions are low.
    • PANAS (Positive and Negative Affect Schedule) and Affect Grid (Russell et al.) are used to measure emotion and arousal; heart rate is sometimes included as a physiological arousal indicator.
  • IZOF model involving emotions (figures/illustrations):
    • The model can be illustrated as a curve showing that performance is optimal within a zone of emotional states, not just around a single anxiety level.
    • The concept supports applying arousal-and-emotion regulation techniques to move into the individual’s emotional IZOF.
  • Real-world implications:
    • An athlete’s IZOF for precompetitive emotions can be established directly or via retrospective recall; once set, techniques can help keep emotions within the IZOF.
    • Studies have shown IZOF-based predictions can outperform inverted-U in predicting performance across various sports.
  • Emerging idea: IZOF across multiple emotions
    • Robazza et al. (2006); Hagtvet & Hanin (2007) proposed methods to generalize IZOF to multiple emotions using aggregated emotion scales and intensity-versus-category analyses.
    • Example study with ice hockey players (Hagtvet & Hanin, 2007) found a general inversion of the relationship between emotional intensity and category of emotion across best performance exemplars.
  • Practical takeaway:
    • Determine an athlete’s emotional IZOF (for precompetitive emotions) and use emotion regulation strategies to stay within that zone for peak performance.

The Probabilistic Model for Determining the IAPZ (Individual Affect-Related Performance Zone)

  • Shift from IZOF to IAPZ:
    • Kamata, Tenenbaum, and Hanin (2002) introduced a probabilistic model, dropping IZOF terminology in favor of IAPZ (Individual Affect-Related Performance Zone).
    • IAPZ emphasizes probabilistic associations between affect (emotions and arousal) and performance categories, not just a single “zone.”
  • Core idea: For a given athlete and observed affect (e.g., felt arousal), one can estimate the probability of different performance categories (poor, moderate, optimal) occurring.
  • Example: Archery, felt arousal as a predictor
    • An archery example used a 1–9 felt arousal scale and showed that certain arousal levels are associated with specific performance categories with particular probabilities.
  • Step-by-step procedure used to derive IAPZ (as demonstrated by Johnson, Edmonds, Kamata, & Tenenbaum, 2009):
    1) Collect data: For each series of arrows, record performance and complete Affect Grid (pleasure and felt arousal scores).
    2) Categorize performance: Rank performances and label bottom third as poor (P), middle third as moderate (M), top third as optimal (OP).
    3) Compute mean felt arousal for the optimal performance group (example mean ≈ 6.06 on a 1–9 scale).
    4) Reclassify observations into IAPZ categories using felt arousal relative to the optimal mean (e.g., optimal stays OP; moderate below/above based on whether arousal is below/above 6.06; poor below/above based on arousal relative to 6.06).
    5) Perform logistic ordinal regression with observed arousal as the independent variable and performance category as the dependent variable to obtain slopes and coefficients.
    6) Create IAPZ curves: Use slopes and regression weights to plot probability curves for each performance category across arousal levels. This yields probability versions of the IZOF concept, indicating the likelihood that a given arousal level will be associated with each performance category.
  • Practical significance:
    • IAPZ provides probabilistic, data-driven curves that help practitioners predict which arousal levels are most likely to yield which performance outcomes for a given athlete.
    • The approach can guide arousal regulation to maximize the probability of optimal performance.
  • Archery example (illustrative):
    • At a felt arousal level of 7.0, probabilities for P/B, Mo/B, OP, Mo/A, and P/A were approx. 0.04, 0.20, 0.45, 0.32, 0.10 respectively, suggesting that arousal around 6–7 is most conducive to OP for this archer,
    • The model also indicated that very low arousal (near 1) yields high probabilities of poor performance, while very high arousal (near 9) increases risk of suboptimal performance.
  • Practical takeaway:
    • The probabilistic IAPZ framework offers a sophisticated, athlete-specific way to link affective states to performance outcomes and to tailor interventions to shift arousal into the most favorable probabilistic zones.

Flow: The Psychology of Optimal Experience (Mihaly Csikszentmihalyi) and Applications to Sport

  • Definition and origin:
    • Flow is a state of complete absorption in an activity, with effortless and autonomous functioning; originally defined as an autotelic experience—the activity is rewarding in itself, not for external outcomes.
    • Csikszentmihalyi (1990) emphasized that flow is an end in itself, but later literature connected flow to Hanin’s IZOF concepts through broader affective contexts.
  • Autotelic experience:
    • A self-contained activity that provides intrinsic reward; the doing itself is the reward.
  • Susan Jackson’s role:
    • Major advocate for applying flow concepts to sport and exercise; defined flow as a state of optimal experience involving total absorption and awareness, often achieving peak performance.
  • Nine defining characteristics of flow (Csikszentmihalyi, 1990):
    1) Challenge/skill balance
    2) Merging of action and awareness (automaticity, spontaneity)
    3) Clear goals
    4) Clear, unambiguous feedback
    5) Total concentration on the activity
    6) Sense of control (paradox of control)
    7) Loss of self-awareness (being one with the activity)
    8) Altered sense of time
    9) Autotelic experience (flow as its own reward)
  • Measurement of flow:
    • Flow State Scale-2 (FSS-2): a 36-item instrument measuring situation-specific flow
    • Dispositional Flow Scale-2 (DFS-2): a 36-item instrument measuring flow as a personality trait
    • Short forms exist for both scales for time-constrained contexts
  • Core flow and core concepts:
    • Flow can occur even without peak performance; it is a state of optimal experience that may occur with both high and moderate performance outcomes.
    • Core flow is sometimes used to refer to a focused, deeply immersive state that is central to peak functioning in sport.
  • Flow in practice (Conditions for Flow; table 8.1):
    • Positive mental attitude
    • Positive affect
    • Attentional focus
    • Perception of being well prepared physically
    • Oneness with teammates and/or coach
    • These attributes should be cultivated to increase the likelihood of entering flow states and potentially experiencing peak performance.
  • Core connections to other theories:
    • Flow intersects with IZOF in recognizing individual, emotion-rich optimal states rather than universal “moderate arousal” prescriptions.
    • The best performance in some contexts aligns with flow, though flow is not strictly required for peak performance.
  • Practical implications for coaching and performance:
    • Training programs can emphasize the development of skills and conditions that balance challenge and ability, provide clear feedback, and facilitate focused attention to promote flow experiences.
    • Use of flow scales (FSS-2/DFS-2) to monitor and cultivate flow states in training and competition.

Connections, Implications, and Practical Takeaways

  • Across theories, a common goal is to explain why some athletes perform best at certain emotional/arousal states and why performance can deteriorate abruptly in others.
  • The inverted-U model is a simplification; catastrophe theory, IZOF, IAPZ, and flow provide more nuanced explanations that accommodate individual differences, abrupt performance changes, and the role of multiple emotions.
  • Measurement and application:
    • Ipsative z-scores help tailor interpretations to individuals.
    • Direct and recall methods help establish individual optimal states when real-time data are not feasible.
    • Probabilistic models (IAPZ) provide actionable, quantitative guidance for tuning arousal and affect to maximize the probability of optimal performance.
    • Flow scales (FSS-2, DFS-2) and the nine flow characteristics offer practical tools for recognizing and fostering conditions that support immersive performance.
  • Ethical and practical implications:
    • Emphasis on individual differences supports personalized coaching, but also requires careful data collection and interpretation to avoid overfitting to a single competition or athlete.
    • The interaction of cognitive anxiety, somatic arousal, and effort highlights the need for ethical handling of arousal manipulation and emotional states during testing and training.

Equations and Key Calculations (LaTeX)

  • Z-score (group-based):
    z=x−msdz = \frac{x - m}{sd}
    where x is the observation, m is the group mean, and sd is the group standard deviation.
  • Ipsative z-score (intraindividualized):
    z<em>ips=x−m</em>isd<em>iz<em>{ips} = \frac{x - m</em>i}{sd<em>i} where mi and sd_i are the individual's mean and standard deviation across repeated observations.
  • IZOF zone around optimal prestart anxiety (example):
    • IZOF = optimal prestart anxiety ± 4 points on the SAI.
    • This corresponds to approximately ±0.5 standard deviation of observed precontest anxiety scores.
  • IAPZ probability example (archery):
    • At felt arousal level 7.0, approximate category probabilities were:
      P( ext{P/B}) = 0.04,\n P( ext{Mo/B}) = 0.20,
      P( ext{OP}) = 0.45,
      P( ext{Mo/A}) = 0.32,
      P( ext{P/A}) = 0.10.
    • This illustrates how arousal levels map to performance category probabilities rather than a single optimum.

Quick study prompts

  • What are the two components of MDAT, and how do they relate to performance?
  • How does catastrophe theory differ from inverted-U in predicting performance changes under arousal?
  • How is the IZOF zone determined, and what is the purpose of the recall method?
  • What is the difference between IZOF and IAPZ?
  • List and describe the nine defining characteristics of flow.
  • How can flow measurement tools (FSS-2, DFS-2) be used to improve training outcomes?
  • How can the concepts of directionality and metamotivational states influence coaching strategies?