- Descriptive Statistics
- Differences across Test Variables
- Test-retest Reliability
- Convergent Validity
- References
| Sex | N | Percentage |
|---|---|---|
| F | 18 | 60 |
| M | 12 | 40 |
| Mean | Median | Min | Max | SD | IQR |
|---|---|---|---|---|---|
| 30.51724 | 30 | 18 | 44 | 7.283154 | 13 |
| Mean | Median | Min | Max | SD | IQR |
|---|---|---|---|---|---|
| 15.45867 | 7.48 | 0 | 59.84 | 17.31337 | 18.7 |
| Mean | Median | Min | Max | SD | IQR |
|---|---|---|---|---|---|
| 9.858 | 4.77 | 0 | 47.7 | 13.13406 | 19.08 |
| Mean | Median | Min | Max | SD | IQR |
|---|---|---|---|---|---|
| 13.58267 | 7.58 | 0 | 60.64 | 15.72199 | 15.16 |
| Mean | Median | Min | Max | SD | IQR |
|---|---|---|---|---|---|
| 17.178 | 13.92 | 0 | 83.82 | 22.7664 | 27.84 |
To evaluate differences across sessions, clockwise/anti-clockwise direction and 1m/2m length, linear mixed effects models are fitted to the data from the standard triangle test and VR triangle test separately (Bates, Mächler, Bolker, & Walker, 2015; R Core Team, 2021). Three outcomes are evaluated: distance from the end, absolute angle of deviation and total distance traveled. These models evaluate the mean differences across these variables while accounting for the repeated measures with random intercepts fitted for individual participants. Normality and homogeneity of variance assumptions are evaluated with QQ-plots and fitted values vs residuals plots. If the model assumptions are violated, data is fitted with generalized linear mixed models with Gamma distribution and identity or log link. Means and mean differences are reported along with their standard errors. Statistical significance threshold is set at 0.05.
For standard triangle test, the mean distance from end is 49.2 ± SE 3.4 for session 1. There is statistically significant (z = -2.2, p = 0.03) decrease (-5.8 ± SE 2.7) in session 2. There are no statistical differences (p > 0.05) across clockwise/anti-clockwise direction or 1m/2m length. For VR triangle test, the mean distance from end is 46 ± SE 3.5 for session 1. There is statistically significant (z = -2.4, p = 0.016) decrease (-6.8 ± SE 2.8) in session 2. There are no statistical differences (p > 0.05) across clockwise/anti-clockwise direction or 1m/2m length. For standard triangle test, the mean angle of deviation is 9.1 ± SE 1.1 for session 1. There are no statistical differences (p > 0.05) across sessions, clockwise/anti-clockwise direction or 1m/2m length. For VR triangle test, the mean angle of deviation is 9.5 ± SE 1.1 for session 1. There is statistically significant (z = -2.6, p = 0.01) decrease (20% ± SE 1%) in session 2. There are no statistical differences (p > 0.05) across clockwise/anti-clockwise direction or 1m/2m length. For standard triangle test, the mean distance traveled is 198.7 ± SE 3.8 for session 1. There is statistically significant (z = 7.4, p < 0.0001) increase (15.9 ± SE 2.2) in session 2. There are no statistical differences (p > 0.05) across clockwise/anti-clockwise direction or 1m/2m length. For VR triangle test, the mean distance traveled is 220.5 ± SE 6.6 for session 1. There are no statistical differences (p > 0.05) across sessions, clockwise/anti-clockwise direction or 1m/2m length.
Generalized linear mixed model fit by maximum likelihood (Laplace
Approximation) [glmerMod]
Family: Gamma ( identity )
Formula: Dist.from.End ~ Time.Period + Is.Clockwise + Is.1m.First + (1 |
PartID)
Data: dataSource.s
Control: glmerControl(optimizer = c("bobyqa"))
AIC BIC logLik deviance df.resid
3297.3 3320.6 -1642.6 3285.3 354
Scaled residuals:
Min 1Q Median 3Q Max
-1.8225 -0.7078 -0.1428 0.6295 3.6654
Random effects:
Groups Name Variance Std.Dev.
PartID (Intercept) 54.5919 7.3886
Residual 0.2779 0.5272
Number of obs: 360, groups: PartID, 30
Fixed effects:
Estimate Std. Error t value Pr(>|z|)
(Intercept) 49.168 3.387 14.515 <2e-16 ***
Time.Period2 -5.773 2.678 -2.156 0.0311 *
Is.Clockwise1 -4.419 2.617 -1.689 0.0913 .
Is.1m.First2 4.132 2.727 1.515 0.1297
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) Tm.Pr2 Is.Cl1
Time.Perid2 -0.392
Is.Clockws1 -0.413 0.007
Is.1m.Frst2 -0.317 -0.122 -0.054
Generalized linear mixed model fit by maximum likelihood (Laplace
Approximation) [glmerMod]
Family: Gamma ( identity )
Formula: Dist.from.End ~ Time.Period + Is.Clockwise + Is.1m.First + (1 |
PartID)
Data: dataSource.vr
Control: glmerControl(optimizer = c("bobyqa"))
AIC BIC logLik deviance df.resid
3314.8 3338.1 -1651.4 3302.8 354
Scaled residuals:
Min 1Q Median 3Q Max
-1.6972 -0.6976 -0.1728 0.5004 4.3270
Random effects:
Groups Name Variance Std.Dev.
PartID (Intercept) 53.5506 7.3178
Residual 0.3216 0.5671
Number of obs: 360, groups: PartID, 30
Fixed effects:
Estimate Std. Error t value Pr(>|z|)
(Intercept) 46.005 3.502 13.135 <2e-16 ***
Time.Period2 -6.816 2.840 -2.400 0.0164 *
Is.Clockwise1 1.249 2.787 0.448 0.6541
Is.1m.First2 3.419 2.883 1.186 0.2357
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) Tm.Pr2 Is.Cl1
Time.Perid2 -0.486
Is.Clockws1 -0.357 -0.080
Is.1m.Frst2 -0.429 0.107 0.012
Generalized linear mixed model fit by maximum likelihood (Laplace
Approximation) [glmerMod]
Family: Gamma ( log )
Formula: Angle.of.Deviation ~ Time.Period + Is.Clockwise + Is.1m.First +
(1 | PartID)
Data: dataSource.s
Control: glmerControl(optimizer = c("bobyqa"))
AIC BIC logLik deviance df.resid
2340.7 2363.9 -1164.3 2328.7 348
Scaled residuals:
Min 1Q Median 3Q Max
-1.3109 -0.6891 -0.1594 0.4889 4.8146
Random effects:
Groups Name Variance Std.Dev.
PartID (Intercept) 0.1065 0.3263
Residual 0.5738 0.7575
Number of obs: 354, groups: PartID, 30
Fixed effects:
Estimate Std. Error t value Pr(>|z|)
(Intercept) 2.21069 0.11304 19.557 <2e-16 ***
Time.Period2 0.05160 0.08761 0.589 0.556
Is.Clockwise1 -0.07385 0.08620 -0.857 0.392
Is.1m.First2 0.13793 0.08747 1.577 0.115
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) Tm.Pr2 Is.Cl1
Time.Perid2 -0.348
Is.Clockws1 -0.363 -0.032
Is.1m.Frst2 -0.343 -0.088 -0.010
Generalized linear mixed model fit by maximum likelihood (Laplace
Approximation) [glmerMod]
Family: Gamma ( log )
Formula: Angle.of.Deviation ~ Time.Period + Is.Clockwise + Is.1m.First +
(1 | PartID)
Data: dataSource.vr
Control: glmerControl(optimizer = c("bobyqa"))
AIC BIC logLik deviance df.resid
2300.7 2324.1 -1144.4 2288.7 354
Scaled residuals:
Min 1Q Median 3Q Max
-1.2984 -0.7362 -0.1767 0.4944 4.5578
Random effects:
Groups Name Variance Std.Dev.
PartID (Intercept) 0.03844 0.1961
Residual 0.58027 0.7618
Number of obs: 360, groups: PartID, 30
Fixed effects:
Estimate Std. Error t value Pr(>|z|)
(Intercept) 2.24581 0.10030 22.391 <2e-16 ***
Time.Period2 -0.22638 0.08830 -2.564 0.0104 *
Is.Clockwise1 0.04019 0.08813 0.456 0.6483
Is.1m.First2 0.03902 0.08827 0.442 0.6584
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) Tm.Pr2 Is.Cl1
Time.Perid2 -0.451
Is.Clockws1 -0.444 -0.008
Is.1m.Frst2 -0.478 0.041 0.030
Generalized linear mixed model fit by maximum likelihood (Laplace
Approximation) [glmerMod]
Family: Gamma ( identity )
Formula: Dist.Travelled ~ Time.Period + Is.Clockwise + Is.1m.First + (1 |
PartID)
Data: dataSource.s
Control: glmerControl(optimizer = c("bobyqa"))
AIC BIC logLik deviance df.resid
3178.5 3201.7 -1583.3 3166.5 348
Scaled residuals:
Min 1Q Median 3Q Max
-7.7488 -0.5204 0.1037 0.6142 3.0567
Random effects:
Groups Name Variance Std.Dev.
PartID (Intercept) 67.792330 8.23361
Residual 0.008735 0.09346
Number of obs: 354, groups: PartID, 30
Fixed effects:
Estimate Std. Error t value Pr(>|z|)
(Intercept) 198.724 3.810 52.163 < 2e-16 ***
Time.Period2 15.937 2.157 7.388 1.49e-13 ***
Is.Clockwise1 1.237 2.140 0.578 0.563
Is.1m.First2 -2.928 2.147 -1.364 0.173
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) Tm.Pr2 Is.Cl1
Time.Perid2 -0.256
Is.Clockws1 -0.278 -0.017
Is.1m.Frst2 -0.277 -0.035 0.023
Generalized linear mixed model fit by maximum likelihood (Laplace
Approximation) [glmerMod]
Family: Gamma ( identity )
Formula: Dist.Travelled ~ Time.Period + Is.Clockwise + Is.1m.First + (1 |
PartID)
Data: dataSource.vr
Control: glmerControl(optimizer = c("bobyqa"))
AIC BIC logLik deviance df.resid
3782.5 3805.9 -1885.3 3770.5 354
Scaled residuals:
Min 1Q Median 3Q Max
-7.1658 -0.4409 0.0707 0.5154 2.5279
Random effects:
Groups Name Variance Std.Dev.
PartID (Intercept) 96.93413 9.8455
Residual 0.01908 0.1381
Number of obs: 360, groups: PartID, 30
Fixed effects:
Estimate Std. Error t value Pr(>|z|)
(Intercept) 220.5129 6.5302 33.768 <2e-16 ***
Time.Period2 0.1955 4.6548 0.042 0.966
Is.Clockwise1 -3.7617 4.6509 -0.809 0.419
Is.1m.First2 1.9222 4.6505 0.413 0.679
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) Tm.Pr2 Is.Cl1
Time.Perid2 -0.346
Is.Clockws1 -0.344 -0.032
Is.1m.Frst2 -0.356 0.015 -0.003
Between-session test-retest reliability is assessed separately for the standard triangle test and the VR triangle test. Three outcomes are evaluated: distance from the end, angle of deviation and total distance traveled. Reliability is evaluated for mean of 6 tests from each session with Pearson’s product moment correlation. This coefficient is interpreted as the consistency of the instrument across two time points. The magnitude of the coefficient (r) is interpreted as excellent (>0.900), good (0.750–0.899), moderate (0.500–0.749) and poor (<0.500) (Portney, Watkins, et al., 2009). If the reliability for an outcome is moderate or better, it is also assessed for the mean of first 4 tests from the two sessions. Normality of the outcome measures is evaluated with QQ-plots.
For standard triangle test, with average across six tests, distance from the end and angle of deviation show poor test-retest reliability (r < 0.5). Only the distance traveled metric shows moderate reliability (r=0.55 95% CI [0.23, 0.76]). With the first 4 tests, the reliability for distance traveled is also moderate (r=0.53 95% CI [0.2, 0.75]).
For VR triangle test, with average across six tests, distance from the end and angle of deviation show poor test-retest reliability (r < 0.5). Only the distance traveled metric shows moderate reliability (r=0.66 95% CI [0.4, 0.83]). With the first 4 tests, the reliability for distance traveled is also moderate (r=0.65 95% CI [0.38, 0.82]).
[1] "r=0.33 95% CI [-0.04, 0.62] t[28]=1.83, p=0.077, Poor"
[1] "r=0.14 95% CI [-0.23, 0.47] t[28]=0.74, p=0.467, Poor"
[1] "r=0.36 95% CI [0, 0.64] t[27]=2.03, p=0.052, Poor"
[1] "r=0.09 95% CI [-0.28, 0.43] t[28]=0.47, p=0.645, Poor"
[1] "r=0.55 95% CI [0.23, 0.76] t[27]=3.4, p=0.002, Moderate"
[1] "r=0.53 95% CI [0.2, 0.75] t[27]=3.21, p=0.003, Moderate"
[1] "r=0.66 95% CI [0.4, 0.83] t[28]=4.69, p=<0.001, Moderate"
[1] "r=0.65 95% CI [0.38, 0.82] t[28]=4.52, p=<0.001, Moderate"
Convergent validity is only assessed for the reliable outcomes across the two instruments. First, the convergence between two instruments is evaluated using Pearson’s product moment correlation. The magnitude of the coefficient (r) is interpreted as excellent (>0.900), good (0.750–0.899), moderate (0.500–0.749) and poor (<0.500) (Portney et al., 2009). If the outcomes have moderate or better convergence, their absolute agreement is evaluated with a Bland-Altman plot. In addition to a qualitative assessment of the plot, the bias and limits of agreement are reported. The bias is interpreted as the systematic error between the two instruments. The limits of agreement are interpreted as the range of values which explain 95% of the differences in scores from the two instruments. The limits of agreement include both the systematic difference and the random differences across the two instruments. The percentage limits of agreement are also reported which are obtained by expressing the limits of agreement as a percentage of the mean of scores across the instruments. The absolute maximum of the percentage limits of agreement are interpreted as excellent (0.0–4.9%), good (5.0–9.9%), moderate (10.0–49.9%) and poor (>50.0%) absolute agreement.
For VR triangle test, with average across six tests from session 1, distance traveled shows show poor convergent validity against distance traveled on the standard vertical test.
For VR triangle test, with average across six tests from session 2, distance traveled shows show moderate convergent validity against distance traveled on the standard vertical test (r=0.64 95% CI [0.37, 0.81]).
[1] "r=0.45 95% CI [0.09, 0.7] t[27]=2.59, p=0.015, Poor"
[1] "r=0.64 95% CI [0.37, 0.81] t[28]=4.45, p=<0.001, Moderate"
`geom_smooth()` using formula 'y ~ x'
Number of comparisons: 30
Maximum value for average measures: 259.1667
Minimum value for average measures: 183.3417
Maximum value for difference in measures: 31.05
Minimum value for difference in measures: -38.18333
Bias: -6.277222
Standard deviation of bias: 17.93638
Standard error of bias: 3.274719
Standard error for limits of agreement: 5.659641
Bias: -6.277222
Bias- upper 95% CI: 0.4203307
Bias- lower 95% CI: -12.97478
Upper limit of agreement: 28.87808
Upper LOA- upper 95% CI: 40.45334
Upper LOA- lower 95% CI: 17.30281
Lower limit of agreement: -41.43252
Lower LOA- upper 95% CI: -29.85725
Lower LOA- lower 95% CI: -53.00779
Derived measures:
Mean of differences/means: -2.544087
Point estimate of bias as proportion of lowest average: -3.423784
Point estimate of bias as proportion of highest average -2.422079
Spread of data between lower and upper LoAs: 70.31059
Bias as proportion of LoA spread: -8.927847
Bias:
-6.277222 ( -12.97478 to 0.4203307 )
ULoA:
28.87808 ( 17.30281 to 40.45334 )
LLoA:
-41.43252 ( -53.00779 to -29.85725 )
Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. https://doi.org/10.18637/jss.v067.i01
Portney, L. G., Watkins, M. P., et al. (2009). Foundations of clinical research: Applications to practice (Vol. 892). Pearson/Prentice Hall Upper Saddle River, NJ.
R Core Team. (2021). R: A language and environment for statistical computing. Retrieved from https://www.R-project.org/
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