Experiment 3.2

8.2.2.1 Building up the growth model

In Experiment 3.2, we had two groups. The Experienced group did Experiment 3.1 (walking) before Experiment 1 (heading judgements), whereas the Novel group only did Experiment 2 and did not do Experiment 1. We compared the thresholds between the two

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groups to investigate whether having taken part in the walking experiment would have any effect on the performance on the heading judgment experiments.

To build a growth model, we started with a baseline model (baseline) with only the intercept. Distance (7m, 6m, 4m and 3m from target) was first added into the model (distRI) as the level-1 predictor, followed by the random effect on the slope (which allows the slope varies between participants) in model distRS. Scene, group and relevant interaction terms were then sequentially added into the model as the level-2 predictor (scn.int, scn.slp, grp.int, grp.slp, scn.grp.int and scn.grp.slp). The R code is listed below with tested effects stated in the comments.

# Setting up a baseline model that includes only random intercepts

baseline <- lme(threshold ~ 1, random = ~1|SubjectNo/Scene, data = threshs,

method = "ML", na.action = na.exclude, control = list(opt="optim"))

# Adding in distance as the level-1 fixed effect

distRI <- update(baseline, .~. + distance) # Introducing random slopes

distRS <- update(distRI, random = ~ distance|SubjectNo/Scene) # Effects of Scene on the intercept

scn.int <- update(distRS, .~. + Scene) # Effects of Scene on the slope

scn.slp <- update(scn.int, .~. + Scene : distance) # Effects of Group on the intercept

grp.int <- update(scn.slp, .~. + Group) # Effects of Group on the slope

grp.slp <- update(grp.int, .~. + Group : distance) # Effects of Scene × Group interaction on the intercept

scn.grp.int <- update(grp.slp, .~. + Scene : Group) # Effects of Scene × Group interaction on the slope

scn.grp.slp <- update(scn.grp.int, .~. + Scene : Group : distance)

8.2.2.1.1 Model comparisons

Then we compared the fit of the models to test the effects of the predictors that were added sequentially into the models. The R code and output is listed below.

# Code

anova(baseline, distRI, distRS, scn.int, scn.slp, grp.int, grp.slp) # Output

## Model df AIC BIC logLik Test L.Ratio p-value ## baseline 1 4 1223.726 1239.887 -607.8629

## distRI 2 5 1193.992 1214.194 -591.9962 1 vs 2 31.73337 <.0001 ## distRS 3 9 1112.829 1149.192 -547.4147 2 vs 3 89.16297 <.0001 ## scn.int 4 12 1112.260 1160.743 -544.1301 3 vs 4 6.56914 0.0870 ## scn.slp 5 15 1088.743 1149.347 -529.3717 4 vs 5 29.51682 <.0001

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## grp.int 6 16 1090.727 1155.371 -529.3633 5 vs 6 0.01673 0.8971 ## grp.slp 7 17 1091.991 1160.675 -528.9954 6 vs 7 0.73596 0.3910 ## scn.grp.int 8 20 1097.465 1178.270 -528.7323 7 vs 8 0.52612 0.9131 ## scn.grp.slp 9 23 1100.466 1193.392 -527.2332 8 vs 9 2.99819 0.3919

From the output, we can see that group does not have an effect on the intercept [𝜒2(1) = 0.017, p = 0.90] or on the slope [𝜒2_{(1) = 0.74, p = 0.39]. Moreover, there was no}

significant interaction between group and scene [both ps > 0.1]. The results suggest that there was no substantial difference between the two groups in the pattern of results. For parsimony, we collapsed the data from the two groups by excluding the predictor group and associated interactions from the model. In other words, we chose the model scn.slpas the final model for parameter estimate evaluation.

Figure 8.11. Thresholds as a function of distance for both Experienced and Novel groups

8.2.2.1.2 Parameter estimates for the model scn.slp.

To assess whether the slope parameters estimated in the model scn.slp, a contrast matrix was created for simultaneous one-sample tests using multcomp package (Hothorn, Bretz, and Westfall 2008). The R code and output are shown below.

# Code contrast.matrix <- rbind( "Slope: Line - 0" = c(0, 1, 0, 0, 0, 0, 0, 0), "Slope: Outline - 0" = c(0, 1, 0, 0, 0, 1, 0, 0), "Slope: Cloud - 0" = c(0, 1, 0, 0, 0, 0, 1, 0), "Slope: Room - 0" = c(0, 1, 0, 0, 0, 0, 0, 1) ) comps <- glht(scn.slp, contrast.matrix) summary(comps, test = adjusted("bonferroni"))

171 # Output

## Simultaneous Tests for General Linear Hypotheses ##

## Fit: lme.formula(fixed = threshold ~ distance + Scene + distance:Scene, ## data = threshs, random = ~distance | SubjectNo/Scene, method = "ML", ## na.action = na.exclude, control = list(opt = "optim"))

##

## Linear Hypotheses:

## Estimate Std. Error z value Pr(>|z|) ## Slope: Line - 0 == 0 -0.41454 0.06169 -6.720 7.25e-11 *** ## Slope: Outline - 0 == 0 -0.21442 0.06097 -3.517 0.00174 ** ## Slope: Cloud - 0 == 0 -0.03880 0.06135 -0.633 1.00000 ## Slope: Room - 0 == 0 0.01678 0.06153 0.273 1.00000 ## ---

## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## (Adjusted p values reported -- bonferroni method)

The column Estimateshows the estimated slope for each condition. Here we can see that the slope was the steepest in the Line [estimated slope = -0.41] and Outline [estimated slope = -0.21] conditions. In contrast, the slopes are relatively flat in the Cloud [estimated slope = -0.039] and Room conditions [estimated slope = 0.017]. The associated z- and p- values indicate that only in the Line and Outline conditions the slopes were significantly different from 0 [both Bonferroni ps < 0.001].

For pairwise comparisons on the slopes between the conditions, another contrast matrix was used. The R code and output are shown below.

# Code contrast.matrix <- rbind( "Outline - Line " = c(0, 0, 0, 0, 0, 1, 0, 0), "Cloud - Line " = c(0, 0, 0, 0, 0, 0, 1, 0), "Room - Line " = c(0, 0, 0, 0, 0, 0, 0, 1), "Cloud - Outline" = c(0, 0, 0, 0, 0, -1, 1, 0), "Room - Outline" = c(0, 0, 0, 0, 0, -1, 0, 1), "Room - Cloud " = c(0, 0, 0, 0, 0, 0, -1, 1) ) comps <- glht(scn.slp, contrast.matrix) summary(comps, test = adjusted("bonferroni")) # Output

## Simultaneous Tests for General Linear Hypotheses ##

## Fit: lme.formula(fixed = threshold ~ distance + Scene + distance:Scene, ## data = threshs, random = ~distance | SubjectNo/Scene, method = "ML", ## na.action = na.exclude, control = list(opt = "optim"))

##

## Linear Hypotheses:

## Estimate Std. Error z value Pr(>|z|) ## Outline - Line == 0 0.20012 0.08129 2.462 0.0829 . ## Cloud - Line == 0 0.37574 0.08157 4.606 2.46e-05 *** ## Room - Line == 0 0.43133 0.08172 5.278 7.83e-07 *** ## Cloud - Outline == 0 0.17562 0.08104 2.167 0.1813 ## Room - Outline == 0 0.23121 0.08113 2.850 0.0262 * ## Room - Cloud == 0 0.05559 0.08150 0.682 1.0000

172 ## ---

## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## (Adjusted p values reported -- bonferroni method)

The column Estimateshows the estimated difference in slope between any two conditions. The associated z- and p-values test the hypothesis of whether the difference is different from 0. Here we can see that the slope in the Line condition is significantly steeper than in the Cloud and Room conditions [both Bonferroni ps < 0.0001]. The slope in the Outline condition is significantly steeper than in the Room condition [Bonferroni p < 0.05]. There is also a marginal difference between the Line and Outline conditions [Bonferroni p = 0.083].

8.2.2.2 Head orientation

We recorded the head orientation (yaw, pitch and roll) for investigating whether the participants strategically rotated their head in order to facilitate the use of some cues. Here, therefore, the difference in the yaw between the beginning and end of a trial is of the primary interest. The yaw data in the right-shifting trials were flipped around so as to combine with the data from the left-shifting trials. Positive yaw differences indicate that the head was turning to the same direction of the simulated movement, and negative yaw differences the opposite. The data were illustrated in Figure 8.12.

Figure 8.12 The change of head yaw respectively for the left-shifting (left-handed panel) and right-shifting (right-handed panel) trials. Error bars indicate 95% confidence intervals.

173 8.2.2.2.1 Build a growth model

We examined the yaw data as a function of distance from the target (7m, 6m, 4m and 3m), the direction of the simulated movement (right- and leftwards) and cue richness (Line, Outline, Cloud and Room). A taxonomy of models was fitted on the yaw data with predictors added in sequentially. On the basis of model baseline, model dst.int included the level-1 predictor distance. Model dst.slp included random effects to allow the slopes of the yaw data against distance to vary among participants. Model scn.int and scn.slp respectively included the level-2 predictor Scene and interaction term Scene × distance so that the effect of cue richness on the magnitude of yaw and slope can be tested. Model drt.int and drt.slp respectively included the level-2 predictor Direction and interaction term Distance × distance to test the effect of simulation direction on the magnitude of yaw and slope. Model scn.drt.int and scn.drt.slp respectively included the interaction term Scene × Direction and Scene × Direction × distance to test the effect of Scene × Direction interaction on the magnitude of yaw and slope. The R code and output are listed below.

# Code

baseline <- lme(abs_meandYaw ~ 1, random = ~ 1|SubjectNo/Scene/Direction, data = yaw_diff_means, method = "ML", na.action = na.exclude, control = list(opt="optim "))

dst.int <- update(baseline, .~. + distance)

dst.slp <- update(dst.int, random = ~ distance|SubjectNo/Scene/Direction) scn.int <- update(dst.slp, .~. + Scene)

scn.slp <- update(scn.int, .~. + Scene : distance) drt.int <- update(scn.slp, .~. + Direction)

drt.slp <- update(drt.int, .~. + Direction : distance) scn.drt.int <- update(drt.slp, .~. + Scene : Direction)

scn.drt.slp <- update(scn.drt.int, .~. + Scene : Direction : distance)

anova(baseline, dst.int, dst.slp, scn.int, scn.slp, drt.int, drt.slp, scn. drt.int, scn.drt.slp)

# Output

## Model df AIC BIC logLik Test L.Ratio p-value ## baseline 1 5 -278.9667 -255.2054 144.4834 ## dst.int 2 6 -277.6862 -249.1725 144.8431 1 vs 2 0.719435 0.3963 ## dst.slp 3 12 -265.6867 -208.6595 144.8434 2 vs 3 0.000564 1.0000 ## scn.int 4 15 -259.8828 -188.5988 144.9414 3 vs 4 0.196089 0.9782 ## scn.slp 5 18 -257.0048 -171.4639 146.5024 4 vs 5 3.121961 0.3732 ## drt.int 6 19 -255.1623 -164.8691 146.5811 5 vs 6 0.157504 0.6915 ## drt.slp 7 20 -253.4774 -158.4320 146.7387 6 vs 7 0.315097 0.5746 ## scn.drt.int 8 23 -253.5568 -144.2546 149.7784 7 vs 8 6.079459 0.1078 ## scn.drt.slp 9 26 -248.7469 -125.1879 150.3735 8 vs 9 1.190102 0.7554

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As shown in the output, none of the predictors significantly improves the model fit. The results suggest that there was no effect of distance to target, simulation direction or cue richness on the rotation of the head during simulation. It seems that the participants did not rotate their head significantly to help judge the heading of the simulated movement.

8.3 Chapter 4 - Supplementary Materials