\section{Results}
\label{sec:results}

\subsection{Trial Measures}
\label{sec:results_trials}

All measures from trials were analysed using linear mixed models (LMM) or generalised linear mixed models (GLMM) with \factor{Visual Rendering}, \factor{Amplitude Difference} and their interaction as within-participant factors, and by-participant random intercepts.
%
Depending on the data, different random effect structures were tested.
%
Only the best converging models are reported, with the lowest Akaike Information Criterion (AIC) values.
%
Post-hoc pairwise comparisons were performed using the Tukey's Honest Significant Difference (HSD) test.
%
Each estimate is reported with its 95\% confidence interval (CI) as follows: \ci{\textrm{lower limit}}{\textrm{upper limit}}.


\subsubsection{Discrimination Accuracy}
\label{sec:discrimination_accuracy}

A GLMM was adjusted to the \response{Texture Choice} in the 2AFC vibrotactile texture roughness discrimination task, with by-participant random intercepts but no random slopes, and a probit link function (see \figref{results/trial_predictions}).
%
The points of subjective equality (PSEs, see \figref{results/trial_pses}) and just-noticeable differences (JNDs, see \figref{results/trial_jnds}) for each visual rendering and their respective differences were estimated from the model, along with their corresponding 95\% CI, using a non-parametric bootstrap procedure (1000 samples).
%
The PSE represents the estimated amplitude difference at which the comparison texture was perceived as rougher than the reference texture 50\% of the time. %, \ie it is the accuracy of participants in discriminating vibrotactile roughness.
%
The \level{Real} rendering had the highest PSE (\percent{7.9} \ci{1.2}{4.1}) and was statistically significantly different from the \level{Mixed} rendering (\percent{1.9} \ci{-2.4}{6.1}) and from the \level{Virtual} rendering (\percent{5.1} \ci{2.4}{7.6}).
%
The JND represents the estimated minimum amplitude difference between the comparison and reference textures that participants could perceive,
% \ie the sensitivity to vibrotactile roughness differences,
calculated at the 84th percentile of the predictions of the GLMM (\ie one standard deviation of the normal distribution)~\autocite{ernst2002humans}.
%
The \level{Real} rendering had the lowest JND (\percent{26} \ci{23}{29}), the \level{Mixed} rendering had the highest (\percent{33} \ci{30}{37}), and the \level{Virtual} rendering was in between (\percent{30} \ci{28}{32}).
%
All pairwise differences were statistically significant.

\begin{subfigs}{discrimination_accuracy}{%
        Generalized Linear Mixed Model (GLMM) results in the vibrotactile texture roughness discrimination task, with non-parametric bootstrap 95\% confidence intervals.
        %
        (a) Percentage of trials in which the comparison texture was perceived as rougher than the reference texture, as a function of the amplitude difference between the two textures and the visual rendering.
        %
        Curves represent predictions from the GLMM (probit link function) and points are estimated marginal means.
        %
        (b) Estimated points of subjective equality (PSE) of each visual rendering.
        %, defined as the amplitude difference at which both reference and comparison textures are perceived to be equivalent, \ie the accuracy in discriminating vibrotactile roughness.
        %
        (c) Estimated just-noticeable difference (JND) of each visual rendering.
        %, defined as the minimum perceptual amplitude difference, \ie the sensitivity to vibrotactile roughness differences.
    }
    \subfig[0.85]{results/trial_predictions}[]\\
    \subfig[0.45]{results/trial_pses}[]
    \subfig[0.45]{results/trial_jnds}[]
\end{subfigs}


\subsubsection{Response Time}
\label{sec:response_time}

A LMM analysis of variance (AOV) with by-participant random slopes for \factor{Visual Rendering}, and a log transformation (as \response{Response Time} measures were gamma distributed) indicated a statistically significant effects on \response{Response Time} of \factor{Visual Rendering} (\anova{2}{18}{6.2}, \p{0.009}, see \figref{results/trial_response_times}).
%
Participants took longer on average to respond with the \level{Virtual} rendering (\geomean{1.65}{s} \ci{1.59}{1.72}) than with the \level{Real} rendering (\geomean{1.38}{s} \ci{1.32}{1.43}), which is the only statistically significant difference (\ttest{19}{0.3}, \p{0.005}).
%
The \level{Mixed} rendering was in between (\geomean{1.56}{s} \ci{1.49}{1.63}).


\subsubsection{Finger Position and Speed}
\label{sec:finger_position_speed}

The frames analysed were those in which the participants actively touched the comparison textures with a finger speed greater than \SI{1}{\mm\per\second}.
%
A LMM AOV with by-participant random slopes for \factor{Visual Rendering} indicated only one statistically significant effect on the total distance traveled by the finger in a trial of \factor{Visual Rendering} (\anova{2}{18}{3.9}, \p{0.04}, see \figref{results/trial_distances}).
%
On average, participants explored a larger distance with the \level{Real} rendering (\geomean{20.0}{\cm} \ci{19.4}{20.7}) than with \level{Virtual} rendering (\geomean{16.5}{\cm} \ci{15.8}{17.1}), which is the only statistically significant difference (\ttest{19}{1.2}, \p{0.03}), with the \level{Mixed} rendering (\geomean{17.4}{\cm} \ci{16.8}{18.0}) in between.
%
Another LMM AOV with by-trial and by-participant random intercepts but no random slopes indicated only one statistically significant effect on \response{Finger Speed} of \factor{Visual Rendering} (\anova{2}{2142}{2.0}, \pinf{0.001}, see \figref{results/trial_speeds}).
%
On average, the textures were explored with the highest speed with the \level{Real} rendering (\geomean{5.12}{\cm\per\second} \ci{5.08}{5.17}), the lowest with the \level{Virtual} rendering (\geomean{4.40}{\cm\per\second} \ci{4.35}{4.45}), and the \level{Mixed} rendering (\geomean{4.67}{\cm\per\second} \ci{4.63}{4.71}) in between.
%
All pairwise differences were statistically significant: \level{Real} \vs \level{Virtual} (\ttest{19}{1.17}, \pinf{0.001}), \level{Real} \vs \level{Mixed} (\ttest{19}{1.10}, \pinf{0.001}), and \level{Mixed} \vs \level{Virtual} (\ttest{19}{1.07}, \p{0.02}).
%
%This means that within the same time window on the same surface, participants explored the comparison texture on average at a greater distance and at a higher speed when in the real environment without visual representation of the hand (\level{Real} condition) than when in VR (\level{Virtual} condition).

\begin{subfigs}{results_finger}{%
        Boxplots and geometric means of response time at the end of a trial, and finger position and finger speed measures when exploring the comparison texture, with pairwise Tukey's HSD tests: * is \pinf{0.05}, ** is \pinf{0.01} and *** is \pinf{0.001}.
        %
        (a) Response time of a trial.
        %
        (b) Distance traveled by the finger in a trial.
        %
        (b) Speed of the finger in a trial.
    }
    \subfig[0.32]{results/trial_response_times}[]
    \subfig[0.32]{results/trial_distances}[]
    \subfig[0.32]{results/trial_speeds}[]
\end{subfigs}


\subsection{Questionnaires}
\label{sec:questions}

%\figref{results/question_heatmaps} shows the median and interquartile range (IQR) ratings to the questions in \tabref{questions} and to the NASA-TLX questionnaire.
%
Friedman tests were employed to compare the ratings to the questions (see \tabref{questions}), with post-hoc Wilcoxon signed-rank tests and Holm-Bonferroni adjustment, except for the questions regarding the virtual hand that were directly compared with Wilcoxon signed-rank tests.
%
\figref{question_plots} shows these ratings for questions where statistically significant differences were found (results are shown as mean $\pm$ standard deviation):
%
\begin{itemize}
    \item \response{Hand Ownership}: participants slightly feel the virtual hand as their own with the \level{Mixed} rendering (\num{2.3 +- 1.0}) but quite with the \level{Virtual} rendering (\num{3.5 +- 0.9}, \pinf{0.001}).
    \item \response{Hand Latency}: the virtual hand was found to have a moderate latency with the \level{Mixed} rendering (\num{2.8 +- 1.2}) but a low one with the \level{Virtual} rendering (\num{1.9 +- 0.7}, \pinf{0.001}).
    \item \response{Hand Reference}: participants focused slightly more on their own hand with the \level{Mixed} rendering (\num{3.2 +- 2.0}) but slightly more on the virtual hand with the \level{Virtual} rendering (\num{5.3 +- 2.1}, \pinf{0.001}).
    \item \response{Hand Distraction}: the virtual hand was slightly distracting with the \level{Mixed} rendering (\num{2.1 +- 1.1}) but not at all with the \level{Virtual} rendering (\num{1.2 +- 0.4}, \p{0.004}).
\end{itemize}
%
Overall, participants' sense of control over the virtual hand was very high (\response{Hand Agency}, \num{4.4 +- 0.6}), felt the virtual hand was quite similar to their own hand (\response{Hand Similarity}, \num{3.5 +- 0.9}), and that the virtual environment was very realistic (\response{Virtual Realism}, \num{4.2 +- 0.7}) and very similar to the real one (\response{Virtual Similarity}, \num{4.5 +- 0.7}).
%
The textures were also overall found to be very much caused by the finger movements (\response{Texture Agency}, \num{4.5 +- 1.0}) with a very low perceived latency (\response{Texture Latency}, \num{1.6 +- 0.8}), and to be quite realistic (\response{Texture Realism}, \num{3.6 +- 0.9}) and quite plausible (\response{Texture Plausibility}, \num{3.6 +- 1.0}).
%
Participants were mixed between feeling the vibrations on the surface or on the top of their finger (\response{Vibration Location}, \num{3.9 +- 1.7}); the distribution of scores was split between the two poles of the scale with \level{Real} and \level{Mixed} renderings (42.5\% more on surface or on finger top, 15\% neutral), but there was a trend towards the top of the finger in VR renderings (65\% \vs 25\% more on surface and 10\% neutral), but this difference was not statistically significant neither.
%
The vibrations were felt a slightly weak overall (\response{Vibration Strength}, \num{4.2 +- 1.1}), and the vibrotactile device was perceived as neither distracting (\response{Device Distraction}, \num{1.2 +- 0.4}) nor uncomfortable (\response{Device Discomfort}, \num{1.3 +- 0.6}).
%
%Finally, the overall workload (mean NASA-TLX score) was low (\num{21 +- 14}), with no statistically significant differences found between the visual renderings for any of the subscales or the overall score.

%\figwide{results/question_heatmaps}{%
%
%    Heatmaps of the questionnaire responses, with the median rating and the interquartile range in parentheses on each cell.
%
%    (Left) point Likert scale questions (1=Not at all, 2=Slightly, 3=Moderately, 4=Very, 5=Extremely).
%
%    (Middle) point Likert scale questions (1=Extremely A, 2=Moderately A, 3=Slightly A, 4=Neither A nor B, 5=Slightly B, 6=Moderately B, 7=Extremely B) with A and B being the two poles of the scale.
%
%    (Right)  Load Index (NASA-TLX) questionnaire (lower values are better).
%}

\begin{subfigs}{question_plots}{%
        Boxplots of responses to questions with significant differences and pairwise Wilcoxon signed-rank tests with Holm-Bonferroni adjustment: * is \pinf{0.05}, ** is \pinf{0.01} and *** is \pinf{0.001}.
    }
    \subfig[0.24]{results/questions_hand_ownership}[]
    \subfig[0.24]{results/questions_hand_latency}[]
    \subfig[0.24]{results/questions_hand_reference}[]
    \subfig[0.24]{results/questions_hand_distraction}[]
\end{subfigs}
