Effects of exercise-induced gradual versus instant avatar transformation on muscle strength and sense of embodiment in patients with chronic low back pain and healthy adults

1 Introduction

The effectiveness of exercise therapy is highly dependent on the patient’s motivation to proactively continue treatment (Maclean and Pound, 2000). In clinical practice, however, the monotony of treatment, difficulty in perceiving recovery progress, and pain and fear associated with exercise are major factors that could impact patients’ motivation to continue therapy (Colombo et al., 2007; Darlow et al., 2012). To address these challenges, exercise therapy utilizing virtual reality (VR) has recently attracted significant attention. VR not only allows for the reproduction of real-world tasks in a virtual space (Laver et al., 2017) but also holds the potential to improve motivation and therapeutic effects beyond traditional, real-world approaches. For example, integrating a VR rehabilitation system with “gamification” (i.e., introducing game elements like rewards or progress bars into the system) was found to increase patients’ engagement (Kern et al., 2019). Similarly, employing a VR game as a distraction during a single exercise session for patients with chronic low back pain resulted in a reduced perceived pain (Matheve et al., 2020). Furthermore, research is progressing on therapeutic interventions that leverage perceptual-motor experiences in VR that are impossible in the real world, such as mirroring the movement of a healthy arm onto the virtual representation of an impaired arm (Curiel et al., 2024).

One of the crucial elements in VR is the use of avatars, the virtual representation of users. When integrated with motion capture technologies, VR enables users to perceive the avatar as being located at the same spatial position and performing the same movements as their physical body. This experience can induce a sense of embodiment, a phenomenon wherein users perceive their avatars as a replacement for their own physical body (Kilteni et al., 2012). The sense of embodiment not only enhances the experience and immersion in the virtual space but also potentially triggers cognitive and behavioral effects that cannot be obtained with the physical body. A particularly noteworthy phenomenon is the “Proteus effect,” (Yee and Bailenson, 2007) where the appearance of one’s avatar shapes one’s self-image (i.e., the mental picture or perception an individual holds of themselves (Gallagher, 2000)), which in turn influence the user’s attitudes and behaviors. Previous studies have suggested that embodying an avatar appearing younger than the user can enhance motivation for continued exercise (Tammy Lin and Wu, 2021). Other studies have suggested that embodying a muscular avatar can reduce pain perception (Yim et al., 2024), decrease perceived exertion (i.e., how intense the user feels their physical effort was), or increase actual grip strength (Kocur et al., 2020). Their findings indicate that the Proteus effect can intervene in users’ perception of physical performance and even the physical performance itself. This raised the possibility that leveraging the Proteus effect could support exercise therapy, for example, by using avatars with a healthy and powerful appearance to influence how patients perceive their body and how much force they can exert.

Furthermore, embodying an avatar in VR, either via Proteus effect or not, has been shown to be effective in modulating a user’s body image (i.e., the subjective perception and attitude toward one’s own body) (Ferrer-García et al., 2018; Lin et al., 2021). This capability holds significant clinical promise for patients struggling with negative body images. For instance, patients with chronic low back pain often perceive their own back as “vulnerable” or “fragile”, and such a distorted body image contributes to their actual pain perception (Moseley, 2008). For this reason, virtual embodiment and the application of the Proteus effect may be especially relevant for these patients, by counteracting a negative body image and thereby supporting their engagement in exercise therapy and their physical performance.

Despite these promising findings, the literature currently lacks empirical evidence regarding the application of the Proteus effect within actual clinical or rehabilitation contexts. This gap motivated the present study to examine the utility of the Proteus effect in such settings. Furthermore, the observed effect sizes of the Proteus effect have on average been small to medium in previous studies (Ratan et al., 2020). Considering the expected application within an actual clinical or rehabilitation context, a stronger effect would be desirable. This prompted a search for more effective strategies for applying the Proteus effect to enhance its applicability in clinical rehabilitation.

In previous research on the Proteus effect, avatars were often assigned upon entering the virtual space or switched instantly during the experience (Praetorius and Görlich, 2020). Such approaches, however, lack the user’s own effort or a clear causal relationship in the acquisition of the avatar as their new body, thereby hindering self-identification with the avatar, that is, the process of imagining oneself as the character represented by the avatar and adopting that character’s perspectives and identity (Cohen, 2001). Given that the change in self-image via self-identification with the avatar is recognized as a major mechanism driving the Proteus effect (Praetorius and Görlich, 2020), these traditional approaches may be limited in their capacity to trigger a significant Proteus effect.

To address this, a screen-based system was developed that displayed a body-tracked avatar whose apparent muscle mass increased gradually in response to the user’s squatting effort. This gradual transformation approach was compared with the conventional method of instant avatar switching, using a squat-based exercise task in both patients with chronic low back pain and healthy individuals. The comparison aimed to evaluate whether an effort-linked, gradual avatar transformation strengthens the Proteus effect relative to the conventional instant-switching approach. Because the gradual method establishes a clear causal link between the user’s own effort and the avatar’s transformation, it was hypothesized that, compared with the instant method, it would produce a greater acute increase in maximal voluntary strength (H1), lead to a greater reduction in pain and fear of movement during squatting in the low back pain group (H2), and be rated as at least as usable (H3). Sense of embodiment was additionally measured to verify that participants experienced the body-tracked avatar as their own virtual body to some degree, as well as to serve as an exploratory outcome regarding whether embodiment differed between the two methods.

2 Related work

2.1 Proteus effect on motivation, perception and performance during physical activities

The Proteus effect is a psychological phenomenon that an individual’s attitudes and even behaviors change to conform to their image associated with their avatar’s appearance (Yee and Bailenson, 2007). This effect is primarily attributed to the sense of acquiring the avatar as a new representation of the self, or even a new “body”, resulting in a shift in the user’s self-image toward the characteristics suggested by the avatar’s appearance (Praetorius and Görlich, 2020). In the original study that first revealed the Proteus effect, participants using avatars with more attractive appearances behaved more intimately with others than those using less attractive avatars. In addition, participants using taller avatars negotiated more aggressively than those using shorter avatars (Yee and Bailenson, 2007).

The Proteus effect was investigated in various fields, including physical activities. Lin et al. investigated the Proteus effect on exercise outcomes among elderly individuals by comparing the effects of embodying a young versus an older avatar in VR. Their results indicated that female elderly individuals were more likely to be motivated to continue exercising, by measuring their self-efficacy for exercise, than male when embodying a young avatar, although this effect was temporary and limited to the day of the experiment (Tammy Lin and Wu, 2021). Yim et al. investigated how the Proteus effect influences acute pain perception by having participants embody avatars of different body types (muscular vs. normal) and genders while receiving a thermal pain stimulus. Their results showed that embodying a muscular avatar significantly reduced reported pain, especially for male participants, while all participants reported less pain when using an avatar whose gender matched their own (Yim et al., 2024). Kocur et al. investigated how an avatar’s muscular appearance affects physical performance by having participants embody avatars with varying levels of muscularity while completing an isometric force task and a grip strength measurement. The results indicated that embodying a muscular avatar significantly lowered participants’ perceived exertion during the physical task, and also increased the actual grip strength of male participants (Kocur et al., 2020).

These findings on the Proteus effect within the context of physical activity indicate that embodying avatars with appearances associated with health and power, such as a young or muscular physique, can motivate users to continue exercise tasks, cause users to perceive their physical condition as more robust (e.g., reporting less pain or lower perceived exertion), and enhance their actual physical performance (e.g., demonstrating greater grip strength). In the realm of exercise therapy, maintaining patient motivation is a critical factor for successful rehabilitation, and this motivation is closely linked to a patient’s perception of their physical improvement (Colombo et al., 2007). Although the application of the Proteus effect in clinical rehabilitation contexts remains largely unexplored, its demonstrated ability to alter not only the motivation toward exercise but also the perception of one’s physical condition makes it a promising area of investigation. Furthermore, the demonstrated ability of the Proteus effect to influence even the actual physical performance suggests its potential to affect the outcomes of exercise tasks. Accordingly, the present study evaluated how embodying a young, muscular avatar influences outcomes in an exercise task, in order to explore the utility of the Proteus effect within a rehabilitation framework.

2.2 Potential to enhance Proteus effect from the perspective of self-image

Previous review work suggested that the Proteus effect typically produces small to medium effect sizes (Ratan et al., 2020). A key mechanism driving this phenomenon is self-identification, a self-image changing process happened when an individual embodies an avatar, where the individual imagines oneself as the character represented by the avatar and adopting that character’s perspectives and identity (Cohen, 2001). Since this process is recognized as a major mechanism driving the Proteus effect (Praetorius and Görlich, 2020), the limited magnitude of the Proteus effect in many studies may be attributed to an inadequate depth of this self-identification. Given that stronger and more reliable effects are desirable for clinical and rehabilitation applications, it is worth exploring strategies for enhancing self-identification to improve the applicability of the Proteus effect in these contexts.

The present study therefore focused on the major factor of the Proteus effect, namely, the process of self-identification. In the fields of psychology and cognitive sciences, self-image has been discussed from the perspective of not only the “minimal self” but also the “narrative self” (Gallagher, 2000). The minimal self refers to the immediate, first-person experience of owning a body and controlling its actions in the present moment. In contrast, the narrative self is constructed over time, incorporating personal history and the causal relationships between one’s actions and their outcomes.

Conventional methods, such as immediately assigning an avatar upon entering the virtual environment or instantaneously switching its appearance (Praetorius and Görlich, 2020), primarily engage the minimal self. In these methods, the transformation is immediate and lacks a connection to the user’s own history or effort. This led to the hypothesis that if the process of embodying an avatar incorporates a narrative context that demonstrates causality for the embodiment, it could lead to a more significant change in self-image and subsequently a stronger self-identification with the avatar.

For this reason, the present study designed an avatar transformation method in which the avatar’s apparent muscle mass gradually increases in direct response to the user’s own squatting exercises, aiming to generate a narrative of achieving the muscular physique through personal effort. By providing this clear causal relationship for the acquisition of the transformed avatar, the method was intended to foster a stronger self-identification and thereby strengthen the resulting Proteus effect.

3 Materials and methods

3.1 Implementation of avatar gradual transformation system

First, standard-physique male and female humanoid avatars (Figures 1A,B) were created using “Character Creator 3,” a software for photorealistic 3D avatar creation ¹ . These avatars were then imported into “Blender,” a 3D modeling software ² , where the mesh was edited to create a muscular physique (Figures 1C,D). By registering this modified muscular mesh as a “shape key” for the original standard mesh, smooth interpolation between the two physiques was enabled.

These avatars, with the shape key of a muscular physique registered, were imported into the “Unity” ³ game engine. Within Unity, a program was implemented that detected the avatar’s squat motion by monitoring downward displacements in the height of the avatar’s waist and head. This program operated on the following algorithm.

• Squat Detection: Whenever a squat was detected, the program triggered a 30% incremental increase in the avatar’s shape key value, which was animated smoothly over 2 s.
• Decay: Immediately following the 2-s increase, the shape key value would then decay, decreasing by 10% over 1 s.
• Net Gain & Maximum: This sequence resulted in a 20% net increase in muscle mass in 3 s for each squat. Therefore, participants could reach the maximum (100%) muscular physique after 5 completed squats.
• Cap: To ensure the muscle mass did not exceed 100%, if a participant performed a 6th squat (or more) within the 30-s exercise period, the system triggered a 30% increase followed by a 30% decrease, effectively maintaining the 100% maximum.

When combined with motion capture technologies, this system enabled the avatar’s muscle mass to grow gradually and at a controlled, linear pace in real-time response to the participant’s actual squat exercises (as illustrated in Figure 2). This implementation was expected to create the sensation of muscular growth achieved through one’s own physical effort.

3.2 Setup of the avatar gradual transformation experience

The system was expected to be experienced by not only healthy individuals but also patients with chronic low back pain. A fully immersive VR system using head-mounted displays (HMDs) was considered likely to impose an excessive physical burden on patients. Therefore, to maximize comfort and accessibility, a non-immersive, screen-based setup was adopted, employing a camera for motion capture and a large external monitor to display the virtual environment rather than a head-mounted display.

The hardware configuration included a Microsoft Azure Kinect depth sensor camera ⁴ and a 65-inch LCD monitor, both connected to a notebook computer equipped with an NVIDIA GeForce RTX 4060 graphic card board ⁵ . The hardware configuration was established within a dedicated room (3 × 3 m 2), located inside a hospital’s rehabilitation facility. As shown in Figure 3, the user stood at the center of the dedicated room, facing the monitor positioned at the edge of the space. The user’s posture was captured by the camera and mapped onto the avatar, enabling the user to observe the avatar mirroring their movement on the monitor in real-time. This visual feedback facilitated the user’s sense of embodying the avatar within the virtual environment.

The virtual environment was assembled within Unity as an indoor room filled with virtual objects of dumbbells and other gym equipment, resembling a gym setting intended to foster user motivation for exercise. Furthermore, the developed avatar gradual transformation system was integrated into this virtual environment. The system was activated when the user raised their hands so that the avatar’s hands reached the height of a dedicated virtual dumbbell positioned behind the avatar. Once activated, whenever the avatar squatted in synchrony with the user’s real-world movement, the system incrementally increased the avatar’s muscle mass.

3.3 User study

3.3.1 Study design

A user study was designed to compare the proposed gradual transformation approach with the typical instant avatar switching method. A mixed-subjects design was used, in which the participant group served as a between-subjects factor and the avatar transformation method as a within-subjects factor.

The participant groups were labeled and defined as follows.

• LBP: Patients with chronic low back pain that lasted 3 months or longer. The pain had a minimum intensity of 3 on an 11-point Numerical Rating Scale (NRS), with the scale ranging from 0 (“no pain”) to 10 (“worst imaginable pain”) (Euasobhon et al., 2022).
• Healthy: Participants having no prior history of LBP.

The avatar transformation conditions were labeled and defined as follows.

• Gradual: The proposed gradual transformation approach. As illustrated in Figure 2, participants performed squats with a standard-physique avatar during the exercise session, and the avatar’s apparent muscle mass increased in real-time and incrementally with each squat performed by the participant.
• Instant: The typical instant avatar switching method. As illustrated in Figure 4, participants performed squats with a standard-physique avatar during the exercise session, and the avatar’s apparent muscle mass remained unchanged regardless of the squats. The avatar then instantly switched to a muscular appearance after the exercise session ended.

Note that in both conditions, participants began the exercise session with a standard-physique avatar and concluded the session with a muscular-physique avatar, thereby maintaining control across the conditions. Finally, to mitigate order effects, the sequence in which participants experienced the two transformation conditions was counterbalanced across participants within each group.

3.3.2 Procedure

The user study procedure is outlined in Figure 5 and described in detail below.

• Before the exercise sessions:
• Upon arrival, participants received an explanation of the study procedure and signed a consent form.
• Participants underwent a series of physical performance measurements (detailed in Sec. 3.3.3.1) to establish a baseline.
• Participants completed questionnaires (items listed in Table1).
• During the first exercise session, conducted under one of the two avatar transformation methods (Gradual or Instant):
• Participants stood facing the monitor, observing a standard-physique avatar (matched to the participant’s gender) that mirrored their body movements in real time.
• Participants performed squat exercises at their own pace for 30 s; their performance during this phase determined the muscular changes subsequently applied to the avatar.
• Participants made poses and observed the resulting muscular avatar displayed on the monitor for 30 s.
• Participants again underwent a series of physical performance measurements.
• Participants completed questionnaires (items listed in Table 1).
• Participants rested for 10 min.
• The second exercise session followed the same procedure as the first, using the other avatar transformation method (e.g., if the first session used Gradual, the second used Instant).
• After both exercise sessions, a brief interview was conducted to gather participant feedback, which informed the later discussion of the quantitative results.

TABLE 1. Scales included in the pre-task and post-task questionnaires. The check mark (✓) indicates that a scale was administered, and the cross ( × ) indicates that it was not.

No prior familiarization trials were conducted before both exercise sessions. All measurements were administered using standardized verbal instructions, with no additional verbal encouragement. The same assessor (the second author) conducted every session, with one or two assistants helping set up the equipment and environment.

3.3.3 Measures

To evaluate the effects of the avatar transformation methods, objective performance data, subjective questionnaire ratings, and baseline participant characteristics were collected.

3.3.3.1 Physical performance measures

To objectively evaluate changes in physical performance, three measurements of maximal voluntary strength were adopted. These were isometric trunk extensor strength, isometric knee extensor strength, and grip strength. As detailed in Section 3.3.2, these physical performance measurements were conducted at baseline and following each of the two avatar transformation conditions. For every measurement, participants performed two maximum voluntary contractions (a standard measure of muscle strength, defined as the peak muscle strength an individual can voluntarily exert Meldrum et al. (2007)) lasting 5 s each, separated by a 60-s rest period. The higher value of the two attempts was recorded for subsequent analysis. The three measures (grip strength, isometric knee extensor strength, and isometric trunk extensor strength) were administered in a fixed order. The justification and specific procedures for each measurement are detailed below.

• Grip Strength: This was adopted based on a previous study by Kocur et al., which demonstrated a significant effect of muscular avatars on grip strength Kocur et al. (2020). Measured for the right hand using a digital grip dynamometer (T.K.K. 5401). Participants stood with their arms hanging naturally at their sides and performed two trials of 3-s maximum voluntary contractions, with a 60-s rest in between.
• Isometric Knee Extensor Strength: This was adopted because knee extensors are the primary muscles activated during the squat exercise task. While seated with their hip and knee joints at 90° flexion, the same dynamometer was fixed to the participant’s right leg, three finger-widths proximal to the lateral malleolus, and secured with a strap to a stable platform. They performed two trials of 5-s maximum voluntary contractions, with a 60-s rest period between trials.
• Isometric Trunk Extensor Strength: This measure was adopted because the system was expected to be experienced by patients with chronic low back pain, which often results in weakness of the trunk extensor muscles. Participants sat on the edge of a chair with their knees bent at 90° and arms crossed over their chest. To prevent posterior pelvic tilt, a towel was placed behind their sacrum. A belt-stabilized handheld dynamometer (Mobie Z, Sakai Medical Co., Ltd.) ⁶ was fixed at the level of the third thoracic vertebra (T3). Participants performed two trials of 5-s maximum isometric trunk extensions, separated by a 60-s rest.

3.3.3.2 Subjective measures

After completing each avatar transformation condition, participants rated their subjective experience using several questionnaires.

• Pain and Fear of Movement (LBP group only): In both the pre-task and the post-task questionnaires, participants in the LBP group rated their pain intensity and fear of movement on an 11-point Numerical Rating Scale (NRS). For pain, the scale ranged from 0 (“no pain”) to 10 (“worst imaginable pain”) (Euasobhon et al., 2022), and for fear, it ranged from 0 (“no fear”) to 10 (“highest possible fear”) (Fujii et al., 2022).
• System Usability: In the post-task questionnaire, all participants rated the system’s usability on a 7-point Likert scale using the questions in Table 2, where a higher point meant greater agreement with the statement.
• Sense of Embodiment: In the post-task questionnaire, all participants completed the Virtual Embodiment Questionnaire (VEQ) (Roth and Latoschik, 2020). The VEQ assesses three subscales. These are Body Ownership (the feeling that the avatar is one’s own), Agency (the feeling of controlling the avatar), and Perceived Body Change (the feeling that one’s physical body has transformed). Each subscale includes four items rated on a 7-point Likert scale, where higher scores indicate a stronger sensation.

TABLE 2. Questionnaire items for system usability evaluation.

3.3.3.3 Participant characteristics

Prior to the task session, demographic data (age and BMI) were collected, and several scales were administered in the pre-task questionnaire to assess individual characteristics.

• Self-Esteem: The Japanese version of the Rosenberg Self-Esteem Scale (Gnambs et al., 2018; Mimura and Griffiths, 2007).
• Self-Concept Clarity: The Japanese version of the Self-Concept Clarity Scale (Campbell et al., 1996; Tokunaga and Horiuchi, 2012).
• Functional Disability (LBP group only): The Roland-Morris Disability Questionnaire (RDQ) (Roland and Morris, 1983), a 24-item scale where a higher number of “yes” responses to statements about daily limitations indicates greater disability.
• Body Perception Disturbance (LBP group only): The Japanese version of the Fremantle Back Awareness Questionnaire (FreBAQ) (Wand et al., 2014; Nishigami et al., 2018), which uses nine items on a 5-point Likert scale to assess perceptual distortions related to the back; a higher total score indicates more severe disturbance.

3.3.4 Hypotheses

The proposed Gradual method, by creating a stronger causal link between the user’s effort and the avatar’s transformation, was expected to enhance self-identification with the muscular avatar and thereby amplify the Proteus effect. This was anticipated to lead to a greater acute increase in physical performance and pain-related outcomes compared with the typical Instant method. The proposed Gradual method was also expected to be at least as usable as the typical Instant one, supporting its clinical viability. Accordingly, the following hypotheses were established for the user study.

• H1 (Muscle Strength): The Gradual condition will produce a significantly greater increase in muscle strength (trunk extensor, knee extensor, and grip strength) compared to the Instant condition.
• H2 (Pain and Fear of Movement): Among patients in the LBP group, the Gradual condition will lead to a significantly greater reduction in pain intensity and fear of movement during squatting compared to the Instant condition.
• H3 (Usability): The Gradual condition will receive subjective usability ratings equal to or higher than those of the Instant condition.

3.3.5 Participants

The required sample size was estimated a priori with G*Power (Faul et al., 2007) for the within-between interaction of a repeated measures ANOVA. The effect size was based on the influence of avatar body shape on grip strength reported by (Kocur et al., 2020) (partial η 2 = 0.109, corresponding to Cohen’s f = 0.35). With α = 0.05, power ( 1 − β ) = 0.95, two groups, two measurements, a correlation among repeated measures of 0.5, and a nonsphericity correction of ϵ = 1, the analysis indicated a required total sample of 30 (numerator df = 1, denominator df = 28, achieved power = 0.96).

To accommodate potential dropout, the user study included 32 Japanese adults, with 16 participants in the LBP group and 16 in the Healthy group.

Participants in the LBP group were recruited from outpatients at the pain and surgery clinic of an affiliated hospital. Participants in the Healthy group were recruited via advertisements within the same hospital and were matched to the LBP group for age and gender distribution. All participants in each group met the criteria defined in Section 3.3.1, with the LBP group criteria including chronic low back pain lasting 3 months or longer and a minimum pain intensity of 3 on an 11-point Numerical Rating Scale (NRS). Individuals with visual impairments, disqualifying organic diseases, or significant psychiatric or cognitive disorders (such as dementia) were excluded.

There were no significant baseline differences between the groups in terms of age (LBP: M = 56.9 , S D = 17.4 years; Healthy: M = 56.5 , S D = 16.8 years; p = . 943) or gender (9 males, 7 females per group; p = 1.000). Body Mass Index (BMI) distribution was also not significantly different ( p = . 516 ). For the LBP group, the median duration of pain was 120.0 months (Interquartile Range [IQR] = 66.8–238.5), the median pain intensity was 4.0 on the NRS over the past week (IQR = 3.0–6.0), the median functional disability (RDQ score) was 5.0 (IQR = 3.8–7.3), and the median body perception disturbance (FreBAQ score) was 6.5 (IQR = 4.8–10.3). These values indicate that the LBP sample had relatively mild-to-moderate pain and low functional disability.

The study protocol was approved by the Ethics Committee of Aichi Medical University (No. 2023-586). All participants were fully informed about the research and provided written informed consent prior to participation.

3.4 Statistical analysis

All analyses were performed in R (version 4.4.1). The significance level for all statistical tests was set at α = 0.05. Hypotheses H1 to H3 were treated as confirmatory, while all other analyses were exploratory and are reported as such, without correction for multiple comparisons. These exploratory analyses comprised the VEQ subscale analyses, the individual usability-item interactions, and the multiple regression models.

The physical performance and usability data were analyzed using a three-way mixed-model Analysis of Variance (ANOVA) with avatar transformation method (Gradual, Instant) as a within-subjects factor, and participant group (LBP, Healthy) and experience order (Instant → Gradual, Gradual → Instant) as between-subjects factors. The same model was applied to the sense of embodiment data. For measures collected exclusively from the LBP group (pain and fear of movement), a two-way repeated measures ANOVA with transformation method and experience order was used. The normality of each dependent variable was assessed with the Shapiro-Wilk test. Variables that violated this assumption (the five usability items, the VEQ Agency and Perceived Body Change subscales, and the number of squats performed) were analyzed with the Aligned Rank Transform (ART) procedure (Wobbrock et al., 2011) prior to ANOVA, using the ARTool package (version 0.11.2). Significant interactions were followed up with simple-effect comparisons of the estimated marginal means using the emmeans package ⁷ . Because all factors were two levels each, every simple-effect test comprised a single pairwise contrast, so the emmeans default (Tukey) adjustment did not alter the reported p-values.

Change scores for both conditions were calculated as the difference between the post-condition value and the single baseline value measured at the start of the session. For the condition performed second, this change score may therefore include any effect remaining from the first condition. Because experience order was modeled as a between-subjects factor, the cell sizes for higher-order (three-way) interactions were small (n ≈ 8 per cell). These interactions were therefore underpowered and are interpreted with caution.

Results are reported using the following conventions: Descriptive values in the text are reported as mean ± standard error, and error bars in the figures denote the standard error of the mean. For the confirmatory effects, 95% confidence intervals are also reported. Values of Cohen’s d, reported as effect sizes for the pairwise method contrasts, are interpreted following Cohen (Cohen, 1988) as small (0.2), medium (0.5), and large (0.8).

4 Results

No adverse events, such as exacerbation of pain, falls, or injury, occurred during squatting or strength testing. All data were therefore subjected to analysis. The statistical procedures, including the ANOVA models, the normality checks, and the effect-size conventions, are described in Section 3.4. Throughout this section, participants who experienced the Instant method before the Gradual method are labeled as Instant → Gradual, and those who experienced the Gradual method before the Instant method are labeled as Gradual → Instant. Where significance markers appear in the figures, they denote *p < . 05, p < . 01, *p < . 001, and ⁺ p < . 10.

4.1 Physical performance

Analysis revealed a significant main effect of the avatar transformation method on grip and trunk extensor strength. For both measures, the Gradual method produced a significantly greater acute increase than the Instant method in both absolute and percentage change. For the absolute change, grip strength increased by 0.823 ± 0.440 kgf under the Gradual method and by 0.166 ± 0.412 kgf under the Instant method (F ( 1,28 ) = 4.85, p = . 036, d = 0.555 (medium), 95% CI = [0.05, 1.27]), and trunk extensor strength increased by 1.86 ± 0.386 kgf versus 1.13 ± 0.352 kgf (F ( 1,28 ) = 9.85, p = . 004, d = 0.791 (medium), 95% CI = [0.25, 1.20]). For the percentage change, grip strength increased by 4.66 ± 1.77 % versus 2.08 ± 1.50 % (F ( 1,28 ) = 5.45, p = . 027, d = 0.588 (medium), 95% CI = [0.33, 4.83]), and trunk extensor strength increased by 12.48 ± 2.57 % versus 7.63 ± 2.19 % (F ( 1,28 ) = 13.27, p = . 001, d = 0.918 (large), 95% CI = [2.13, 7.57]). The absolute changes in grip and trunk extensor strength are shown in Figures 6, 7, and the percentage changes in Figures 8, 9. No main effect of the transformation method was found for knee extensor strength. Significant order effects and a method by order interaction were also observed, and they are described below and considered in the Discussion. The results thus provide partial support for H1.

The analysis also revealed a significant main effect of experience order on the percentage increase in grip and knee extensor strength. Participants in the Instant → Gradual group showed a greater percentage increase than those in the Gradual → Instant group for grip strength (7.06 ± 2.19 % versus − 0.31 ± 2.19 %; F ( 1,28 ) = 5.68, p = . 024, d = 0.601 (medium); Figure 10) and for knee extensor strength (18.81 ± 3.08 % versus 8.43 ± 3.08 %; F ( 1,28 ) = 5.66, p = . 024, d = 0.600 (medium); Figure 11).

A significant interaction between the avatar transformation method and experience order was observed for knee extensor strength, for both the absolute change (F ( 1,28 ) = 5.34, p = . 028) and the percentage increase (F ( 1,28 ) = 10.16, p = . 0035). Within the Instant → Gradual order, the simple effect of method was significant. The Gradual method produced a greater absolute increase in knee extensor strength than the Instant method (6.24 ± 1.02 kgf versus 3.91 ± 1.17 kgf; t ( 28 ) = 2.59, p = . 015, d = 0.923 (large)), as well as a greater percentage increase (22.92 ± 3.39 % versus 14.69 ± 3.20 %; t ( 28 ) = 3.55, p = . 0014, d = 1.263 (large); Figures 12, 13).

To examine whether the amount of exercise differed between conditions, the number of squats performed was also analyzed. The number of squats did not differ significantly between the Gradual (M = 8.75, SD = 2.79) and Instant (M = 8.75, SD = 2.82) conditions ( F ( 1,28 ) = 0.015 , p = . 905 ), and no other main effects or interactions were significant. This indicates that the number of squat repetitions performed was comparable across conditions and was not modulated by group or experience order. Squat depth, movement velocity, range of motion, and effort were not recorded, however, so differences among conditions in exercise intensity or movement quality cannot be excluded.

4.2 Subjective measures

The ANOVA on subjective measures from the LBP group revealed no significant differences between the conditions for either pain intensity or fear during squatting. Consequently, H2 was not supported.

Regarding system usability, the ART ANOVA revealed significant interactions for two items, which are described below.

For the item “It is easier to move my body with this system than moving my body in reality,” a significant interaction between experience order and avatar transformation method was observed (F ( 1,28 ) = 6.17 , p = . 019; Figure 14). The pattern indicated a crossover effect. In the Instant → Gradual order, the Gradual method tended to be rated higher ( 5.38 ± 0.375 ) than the Instant method ( 5.00 ± 0.418 ), whereas in the Gradual → Instant order, the Instant method tended to be rated higher ( 4.50 ± 0.365 ) than the Gradual method ( 4.31 ± 0.416 ). However, post hoc comparisons within each order did not reach significance (Instant → Gradual, t ( 28 ) = − 1.984 , p = . 114 , d = − 0.707 (medium); Gradual → Instant, t ( 28 ) = 1.529 , p = . 138 , d = 0.545 (medium)). Additionally, a significant three-way interaction among participant group, experience order, and avatar transformation method was observed for this item (F ( 1,28 ) = 4.80 , p = . 037; Figure 15). Post-hoc analysis revealed a marginally significant simple effect only for Healthy participants in the Instant → Gradual order (t ( 28 ) = 2.514 , p = . 072 , d = 1.185 (large)); all other subgroup comparisons were not significant. While the omnibus interactions reached significance, the absence of significant simple effects at the conventional level precludes strong conclusions about specific order-dependent patterns for this item.

For the item “This system is useful for moving my body and for rehabilitation,” a significant interaction between experience order and avatar transformation method was observed (F ( 1,28 ) = 8.32 , p = . 007; Figure 16). The pattern similarly indicated a crossover effect. In the Instant → Gradual order, the Gradual method tended to be rated higher ( 5.88 ± 0.221 ) than the Instant method ( 5.75 ± 0.233 ), whereas in the Gradual → Instant order, the Instant method tended to be rated higher ( 5.31 ± 0.384 ) than the Gradual method ( 5.12 ± 0.315 ). Post-hoc comparisons approached but did not reach conventional significance within each order (Instant → Gradual, t ( 28 ) = − 2.281 , p = . 061 , d = − 0.813 (large); Gradual → Instant, t ( 28 ) = 1.799 , p = . 083 , d = 0.641 (medium)). The overall crossover pattern is consistent with an order or practice effect, though the marginal nature of the simple effects warrants cautious interpretation.

For all remaining usability items, no significant effects of the avatar transformation method were observed. Overall, these results suggest that the Gradual method provides a user experience comparable to the Instant method, consistent with H3. The crossover patterns observed for perceived ease of body movement and usefulness point to a possible influence of experience order on these ratings, echoing the order effects observed in the physical performance data.

4.3 Sense of embodiment

Analysis of the Virtual Embodiment Questionnaire (VEQ) subscales revealed several significant effects. For the sense of agency, a significant main effect of the transformation method was observed, with the Gradual method rated higher than the Instant method overall (5.14 ± 0.175 versus 4.80 ± 0.196; F ( 1,28 ) = 8.85, p = . 006, d = 0.837 (large); Figure 17). No significant interaction with experience order or participant group was observed for this subscale, indicating that this advantage of the Gradual method showed no significant order dependence in the present sample. For body ownership, a significant interaction was found between the avatar transformation method and the experience order (F ( 1,28 ) = 16.09, p < . 001). Within the group that experienced the Instant → Gradual order, the simple main effect of method was significant, and the Gradual method was rated higher than the Instant method (4.55 ± 0.318 versus 3.93 ± 0.343; t ( 28 ) = 4.02, p < . 001, d = 1.433 (large); Figure 18). Finally, a significant interaction between participant group and avatar transformation method was observed for the perceived body change (F ( 1,28 ) = 4.29, p = . 048; Figure 19). The pattern suggested opposing trends between the groups. Healthy participants tended to report a stronger perceived body change under the Gradual method ( 4.05 ± 0.370 ) than under the Instant method ( 3.69 ± 0.289 ), whereas LBP patients reported similar levels across methods (Gradual 3.38 ± 0.362 and Instant 3.39 ± 0.379). However, post hoc comparisons did not reach significance for either group (Healthy t ( 28 ) = − 1.683, p = . 207, d = − 0.600 (medium); LBP t ( 28 ) = 1.247, p = . 223, d = 0.444 (small)), so these descriptive trends should be interpreted with caution.

4.4 Influence of participant characteristics

Multiple regression analyses were conducted to investigate the influence of baseline characteristics on outcome measures. The results indicated that self-esteem and age were significant predictors for several outcomes. Within the LBP group, self-esteem was positively correlated with fear experienced during the task (Std. β = 0.79 , p = . 036) and the post-task increase in fear (Std. β = 0.77 , p = . 037). Conversely, self-esteem was negatively correlated with the sense of agency (Std. β = − 0.52 , p = . 044). Age demonstrated a significant positive correlation with the perceived body change, indicating that older participants reported a stronger perception of transformation (Std. β = 0.72 , p = . 014).

5 Discussion

5.1 Acute strength gains from effort-linked avatar transformation

The Gradual method produced a significantly greater acute increase in grip and trunk extensor strength than the Instant method. Both conditions began with a standard-physique avatar and ended with a muscular one, and strength was measured while the avatar was in its muscular state. Exposure to the muscular avatar, which is the source of the Proteus effect, was therefore matched across the two conditions. The comparison thus isolates the added effect of linking the avatar’s transformation to the user’s own effort, rather than the effect of simply seeing a muscular avatar.

Several related processes may explain this added effect. The Instant switch primarily engages the minimal self, the immediate, first-person experience of owning and controlling a body in the present moment, which has served as the central focus of traditional studies on the Proteus effect (Gallagher, 2000; Praetorius and Görlich, 2020). In contrast, the Gradual method might additionally leverage the narrative self by incorporating a history of personal effort, so that the muscular physique is perceived as being “earned” through the user’s own exercise (Gallagher, 2000). This interpretation is supported by qualitative feedback, in which participants noted a distinct sense of actually exercising and of accomplishment, such as “I feel like doing an actual exercise,” “The gradual change provides a sense of improvement and a feeling of ‘doing it’,” or “It allows me to perceive the effort I am putting in”. Such a causal narrative may foster a deeper level of self-identification with the avatar (Cohen, 2001) and thereby amplify the Proteus effect.

Interview feedback also suggested a motivational route. Participants described a higher sense of achievement, such as “transforming gradually increases the sense of achievement,” a subjective feeling of greater strength, such as “I felt that it was easier to exert force,” and greater enjoyment, such as “it is fun to see muscles bulking up every time I move”. These experiences relate to performance accomplishments and emotional arousal, two of the sources of self-efficacy, defined as one’s belief in the capability to execute the actions required for specific performance attainments (Bandura, 1977). By making the link between effort and outcome visible, the system may have strengthened participants’ belief in their physical capability and helped them exert greater force during the maximal measurements (Bandura, 1977; Bandura, 1982). These interpretations, however, rest on qualitative feedback rather than a validated measure, because motivation was not assessed with a formal instrument. The one subjective item most related to continued engagement, “I want to continue using this system,” did not differ significantly between the two methods. The motivational account is therefore tentative and should be read as a direction for future study rather than an established effect.

It should be noted that these changes were measured immediately after a single short session. They therefore represent an acute change in maximal voluntary strength rather than a lasting improvement in physical function, and they may also be influenced by factors such as familiarization with the task or a temporary increase in arousal. Whether such acute effects accumulate into rehabilitation benefits over repeated sessions remains to be established.

5.2 Unchanged pain and fear, comparable usability

Among patients in the LBP group, neither pain intensity nor fear of movement differed between the Gradual and Instant methods, and H2 was not supported. One likely reason is that chronic pain and fear of movement develop over long periods and are unlikely to change measurably after a single short session. Because the study was not designed to capture longer-term change, these measures cannot indicate whether the method affects pain or fear over repeated use. Neither method increased pain or fear during the session, and no adverse events occurred (Section 5.7). This indicates that the Gradual method was tolerated over a single session regarding pain and fear, although its effect across repeated sessions remains to be tested.

In terms of usability, the Gradual method was broadly comparable to the Instant method, which is a positive outcome for clinical adoption given the greater complexity of the Gradual method, and it is consistent with H3. Qualitative feedback revealed divergent preferences. Some participants found it “easier to perceive the difference when the transformation occurs all at once at the end,” whereas others found it “easier to understand the change when it occurs one by one.” These opposing views likely reflect individual differences in cognitive processing style, and they suggest that letting patients choose the feedback style could further improve the user experience, as discussed in Section 5.5.

5.3 Sense of embodiment

The sense of agency showed a significant main effect of the transformation method, with higher ratings for the Gradual method, and this advantage did not vary significantly with experience order in the present sample. This stronger sense of agency has a plausible basis in the real-time, effort-linked feedback of the Gradual method. The sense of agency is strengthened when the discrepancy, or prediction error, between the predicted sensory outcome of an action and the actual feedback is reduced (Synofzik et al., 2013). In the Gradual condition, the incremental muscle growth may have acted as a direct and predictable consequence of each squat. Such real-time, step-by-step feedback may have allowed the brain to associate the visual transformation with the motor command more easily than an abrupt, end-of-session switch, thereby reducing prediction errors. This interpretation is consistent with feedback such as “I felt like the results of the exercise were showing for what I was doing.” Because the sense of agency is a core component of embodiment (Kilteni et al., 2012), this result is consistent with the view that the Gradual method deepened the users’ identification with the muscular avatar.

For body ownership, a significant interaction with experience order was found, and the advantage of the Gradual method appeared only in the Instant → Gradual order. Body ownership is therefore not treated here as clear evidence for the Gradual method, and this order effect is considered in Section 5.4.

For perceived body change, a significant interaction between participant group and method was found. Healthy participants tended to report a stronger perceived body change under the Gradual method, whereas LBP patients reported similar levels across the two methods. The simple effects did not reach significance, so this pattern is interpreted with caution. It may reflect a larger gap between the patients’ own body perception and the avatar, a possibility that is examined further in Section 5.5.

5.4 Influence of experience order

Several effects of experience order were found. Experience order had a significant main effect on the percentage increase in grip and knee extensor strength, and a significant interaction between the transformation method and experience order was found for knee extensor strength and for body ownership. Because the strength changes were calculated from a single baseline and both conditions were performed in one short session, these order effects may partly reflect warm-up, getting used to the task, tiredness, or effects remaining from the first condition, in addition to any difference between the two sequences.

The counterbalanced design helps to separate these possibilities. Any effect that acts equally on both conditions leaves the main effect of the transformation method unchanged, so a method main effect that appears with no order main effect and no interaction is easier to interpret than one that is qualified by order. That said, given the current sample size, the absence of significant interaction cannot fully rule out order-dependent carryover effects. With that caution in mind, trunk extensor strength and sense of agency met this criterion and are therefore treated here as the more interpretable basis for comparing the two methods. Knee extensor strength and body ownership, in contrast, showed a method effect only in interaction with experience order, and are therefore not treated as evidence that the Gradual method is superior. In particular, the knee extensors are the main muscles used in the squat, so the knee measurement taken right after the exercise is the one most affected by tiredness and by the muscle having just been used. This is the likely reason the interaction appears for the knee but not for the trunk.

The order effects followed a consistent pattern. Strength gains and subjective ratings tended to be larger when the Gradual method was experienced after the Instant method. One possible reason is that experiencing the Instant method first, which gave no feedback during the exercise, made the Gradual method feel more impactful by comparison. However, effects remaining from the first condition cannot be ruled out, and a single session cannot separate these explanations. A practical implication is that the order of experiences is itself a design choice, and that starting with the Instant method may increase the perceived impact of the Gradual method that follows.

Beyond physical performance, crossover interactions were also found for the perceived ease of body movement and perceived usefulness, with the method experienced second tending to receive higher ratings. Because the simple effects for these items were only marginal, these patterns are interpreted with caution, although their consistency with the order effects in the strength data gives them tentative support.

5.5 Participant characteristics and personalization

Previous literature suggested that individuals with lower self-esteem are more susceptible to self-image manipulation and are therefore more strongly influenced by the Proteus effect (Banakou et al., 2018). In the LBP group, this study did not directly test this relationship, but a consistent pattern was observed. Lower self-esteem was associated with a higher sense of agency, which the Gradual method also increased, and with less fear of movement, which was the expected effect for the muscular avatar to produce through the Proteus effect. However, fear itself did not differ significantly between the two methods, so this association should be read with caution. Moreover, both associations were also limited to the subjective measures, as self-esteem was not associated with the objective strength gains that distinguished the two methods (e.g., trunk extensor strength, grip strength). This pattern is therefore treated as suggestive, indicating that the subjective experience of avatar-based feedback in LBP patients may depend in part on personal characteristics such as self-esteem.

In addition, age was positively correlated with the perceived body change, as older participants reported a stronger perception of transformation. This could be due to a larger difference between their own bodies and the avatar with a younger appearance.

These findings suggest that adjusting avatar characteristics based on user traits such as age or self-esteem could help control for these factors or improve the sense of embodiment and, possibly, the Proteus effect. Future work could develop adaptive systems that deliver personalized avatar transformation feedback based on individual patient characteristics, providing a more tailored treatment experience and potentially improving the effectiveness of avatar-based rehabilitation.

5.6 Strengths and practical implications for exercise therapy

The proposed effort-linked Gradual avatar transformation method has several strengths. First, it introduces a design principle in which the avatar’s transformation is tied in real time to the user’s own effort rather than switched instantly, extending how the presentation of an avatar can be designed to influence the user. Second, it brings the Proteus effect and avatar-based feedback, which so far have had limited clinical application, into an exercise-therapy context for patients with chronic low back pain. Third, it runs on a non-immersive, monitor-based setup that avoids the physical burden of head-mounted displays and can be deployed in settings such as a standard hospital rehabilitation room, which supports practical implementation. Fourth, it neither increased pain or fear of movement nor produced any adverse events, which suggests that it was well tolerated, although effects across repeated sessions still require validation.

These features point to practical applications in exercise therapy. Making the relationship between exercise effort and visible bodily change explicit may help patients perceive otherwise hard-to-detect progress. This could plausibly support engagement in exercise, although the present study did not measure engagement directly and this link remains to be tested. Beyond chronic low back pain, the same principle of linking real-time avatar transformation to the user’s own effort could be explored for other rehabilitation tasks, particularly those involving exercise therapy. For instance, the avatar’s body segments corresponding to the limb used during the exercise could be transformed, such as the arms during reaching tasks in stroke rehabilitation (Thielman et al., 2004) or the legs during chair-stand exercises in sarcopenia rehabilitation (Yoshimura et al., 2022).

5.7 Limitations and future work

This study has several limitations. First, a limitation concerns the multiplicity of the confirmatory hypotheses. H1 combined trunk extensor, knee extensor, and grip strength; H2 combined pain intensity and fear of movement; and H3 combined five usability items, and each outcome within a hypothesis was tested at α = 0.05 without correction for multiple comparisons. This raises the family-wise Type I error rate for each hypothesis above the nominal 0.05 level of any single test. Statements that a hypothesis was or was not supported should therefore be read as a summary across several separately tested outcomes rather than as one error-controlled test of that hypothesis.

Regarding the mechanism this study evaluated, the study did not include a control without a muscular avatar and so did not test for the Proteus effect itself, which was not its aim. Since both conditions began with a standard-physique avatar and ended with a muscular one, with strength measured while the avatar was muscular, exposure to the muscular avatar was matched across conditions. The comparison in this study thus isolated the additional effect of linking the avatar’s transformation to the user’s own effort, beyond the shared exposure to the muscular avatar. What the comparison did not cover is that the effort-linked manipulation might have bundled several components, such as introducing real-time visual feedback and a reward-like signal that the present design cannot separate. Doing so would require additional controls, such as a gradual transformation not linked to the user’s own effort, a no-transformation condition, or a standard-avatar control, which remain for future work.

Several factors limit the strength comparison. First, because the measures were repeated within one short session, carryover such as warm-up, familiarization, or tiredness cannot be fully excluded despite the counterbalancing and the 10-min rest. Second, strength was also tested without familiarization trials and in a fixed order. A single assessor conducted all strength measurements and gave the standardized verbal instructions to every participant, with one or two assistants helping with equipment and environment setup, which likely reduced variability in test administration across participants and conditions. However, this same assessor was not blinded to the avatar transformation condition, so assessor expectancy cannot be excluded, although the within-participant design makes practice effects an unlikely explanation for the condition differences. In addition, although squat count did not differ between conditions, squat depth, movement velocity, range of motion, and effort were not recorded, so subtle differences in exercise behavior cannot be entirely ruled out. Future studies should use blinded or automated strength measurement and a design with a longer washout and fresh baselines.

Regarding generalizability, the system was non-immersive and screen-based, presenting the avatar in the third person, and body ownership and agency may differ under immersive, first-person embodiment, so the results may not transfer to head-mounted VR. Moreover, the sample also had relatively mild-to-moderate pain and low disability (median NRS = 4, RDQ = 5), so the findings should not be generalized to patients with severe pain, high disability, neurological involvement, post-surgical conditions, or balance impairment.

Finally, some technical and temporal factors point to future work. Occasional tracking jitter occurred, and participants felt the link was somewhat unnatural because the squat engages the legs while the whole avatar grew, which may have weakened the effort-transformation coupling and the resulting effect. The intervention was also limited to a single session, so long-term effects are unknown, although multi-session VR embodiment interventions for chronic low back pain have reduced pain and disability (Harvie et al., 2020; Trujillo et al., 2020). Future work could use more targeted and higher-fidelity transformation and tracking, examine immersive systems, and test whether longitudinal use yields benefits beyond the acute changes observed here.

5.8 Summary of the discussion

First, the Gradual method is argued to leverage the narrative self through a history of personal effort, so the muscular physique is perceived as “earned,” deepening self-identification with the avatar and amplifying the Proteus effect. The stronger sense of agency likewise reflects real-time, effort-linked feedback reducing prediction error and, as a core component of embodiment, suggests deepened identification with the muscular avatar. Second, given the counterbalanced design, trunk extensor strength and sense of agency showed no significant order main effect or interaction and offer the more interpretable basis for comparison, though this does not guarantee the absence of some carryover. Measures like knee extensor strength, whose method effect interacted with experience order, indicate that the order of experiences may be a design choice, especially for measures sensitive to carryover effects (e.g., tiredness). Third, chronic pain and fear of movement were unlikely to change after a single short session, but neither method increased them, indicating the Gradual method was tolerated over a single session; usability was broadly comparable to the Instant method, a positive outcome for clinical adoption given the Gradual method’s greater complexity. Fourth, the subjective experience of avatar-based feedback may also depend on personal characteristics such as self-esteem and age, pointing toward personalized avatar transformation feedback. Finally, the method ties transformation in real time to the user’s own effort, brings the Proteus effect into exercise therapy for chronic low back pain, and operates on a non-immersive, monitor-based setup suited to a standard hospital rehabilitation room. However, since the study did not test the Proteus effect itself and the effort-linked manipulation might have bundled several components, together with the acute, single-session nature of these effects, controlled comparisons and longitudinal examination remain for future study.

6 Conclusion

This study evaluated a non-immersive avatar system in which an avatar’s apparent muscle mass increased gradually in response to a user’s own squatting effort, and compared it with the conventional approach of switching the avatar to a muscular physique instantaneously. Compared with the Instant method, the effort-linked Gradual method exhibited significantly larger acute increases in grip strength (4.7% versus 2.1%) and trunk extensor strength (12.5% versus 7.6%) and a higher sense of agency ( d = 0.84 ). Of these, the advantages in trunk extensor strength and sense of agency showed no significant order dependence in the present sample. Significant order effects on other measures indicate, however, that the overall comparison should be interpreted in light of experience sequence rather than as a context-free advantage of the Gradual method. Pain and fear of movement did not change significantly over the single session. The method was, however, well tolerated and as usable as the Instant approach over this single session. Overall, the preliminary findings indicate that how an avatar’s appearance changes over time merits consideration as a design choice, and that making the link between effort and visible bodily change explicit was associated with acute increases in maximal voluntary strength and in the sense of agency. With future confirmation in multi-session, longitudinal studies, this effort-linked avatar approach could contribute to exercise therapy in rehabilitation.

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Footnotes

Edited by: Joseph Lobo, Bulacan State University, Philippines

Reviewed by: Víctor Hernández-Beltrán, University of Extremadura, Spain

Gang Qin, Hanyang University, Republic of Korea

Footnotes

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https://www.nvidia.com/

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https://cran.r-project.org/web/packages/emmeans/index.html