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Combined the two 6 km/h stats tables.
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moorepants committed Jun 4, 2024
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\usepackage{todonotes}

\def\kph{\kilo\meter\per\hour}
\def\mps{\meter\per\second}

\title{Bicycle Balance Assistance Reduces Probability of Falling at Low Speeds
When Subjected to Mechanical Perturbations}
Expand Down Expand Up @@ -377,60 +378,56 @@ \subsection{Statistics}
or off).

\begin{table}
\small
\centering
\caption{Statisitcal model variables.}
\begin{tabular}{lll}
\toprule
Variable & Units & Description \\
\midrule
\(L\) & \si{\newton\meter\second} & angular impulse of the applied
perturbation torque \\
\(L\) & \si{\newton\meter\second} & angular impulse of perturbation torque \\
\(c\) & integer & order number of perturbation \\
\(\sigma_\delta\) & standard deviation of steer angle during Phase X \\
\(k\) & TODO & gain of balance assist control \\
\(\delta_0\) & \si{\degree} & steer angle at start of perturbation \\
\(\phi_0\) & \si{\degree} & roll angle at start of perturbation \\
\(v\) & \si{\meter\per\second} & forward speed (6 or 10) \\
\(v\) & \si{\meter\per\second} & forward speed (1.7 or 2.8) \\
\(f\) & boolean & outcome of perturbation: did not fall, did fall \\
\bottomrule
\end{tabular}
\end{table}

We evaluate this hypothesis using a multivariate logistic regression model.
We evaluate this hypothesis using a multivariate logistic regression model in
the form

\begin{align}
f_{ij} | p_{ij} \sim \textrm{Bern}(p_{ij})
f_{ij} | p_{ij} \sim \textrm{Bern}(p_{ij}) \textrm{.}
\end{align}

\(f_{ij}\) is the binary outcome of perturbation \(j\) of participant \(i\)
which follows a Bernoulli distribution given the probability \(p_{ij}\) that a
fall occurs. The log-odds of the probabiliy is then a linear function of our
independent variables with \(\beta\) as the intercept and \(\alpha_k\) as the
linear coefficients.
Fall outcome \(f_{ij}\) is the binary outcome of perturbation \(j\) of
participant \(i\) which follows a Bernoulli distribution given the probability
\(p_{ij}\) that a fall occurs. The log-odds of the probabiliy is then a linear
function of our independent variables with \(\beta\) as the intercept and
\(\alpha_k\) as the linear coefficients.

\begin{align}
\log \left(\frac{p_{ij}}{1-p_{ij}} \right) = \beta + \sum_{k=0}^{K} \alpha_k x_{ij}^{k}
\end{align}

\todo[inline]{we already use k for gain}

We scalle all independent variables such they they have a mean of zero and a
We scale all independent variables such they they have a mean of zero and a
standard deviation of one. We use cluster-mean centering, as recommended by
\cite{Enders2007}, with the clusters being an individual subject because we are
interested in the association between the state of the balance-assist system
and the outcome of the perturbation.

Cluster-mean centering shows there to be no variation between participants,
i.e. that rider skill (the only participant differentiating factor) is not a
predictor. We expected there to be a variation between participants in how well
they are able to resist the perturbation. However, this was not true. This
allowed us to use a simpler single-level logistic regression model, instead of
a multilevel model. This left use with angular impulse, perturbation order,
balance assist state, and roll \& steer angles at the time of perturbation. We
also include interaction effects between the balance assist state and the other
four variables.

\todo[inline]{Marten writes ``The skill variable is not included, because it
was not collected correctly for all participants.'' This needs to be addressed.
What about skill being tied to participant clusters?}
Cluster-mean centering shows there to be no variation between participants. We
expected there to be a variation between participants in how well they are able
to resist the perturbation. However, this was not true. This allowed us to
stick with a simple single-level logistic regression model, instead of a
multilevel model. This left use with angular impulse, perturbation order,
balance assist state, and roll \& steer angles at the time of perturbation as
independen tvariables.. We also include interaction effects between the balance
assist state and the other four variables.

We divide the analysis into two separate model evalautions, one for the
6~\si{\kilo\meter\per\hour} \(k=10\) trials and one for the
Expand All @@ -440,61 +437,41 @@ \subsection{Statistics}
\section{Results}

The coefficient estimates for a single-level logistic regression at
6~\unit{\kilo\meter\per\hour} with gain $k=10$ are shown in Table
\ref{table:freq-coefs-6}. The angular impulse, perturbation order, and balance
assist state are all statistically significant predictors. Larger angular
impulse increases the probability to fall and enduring more perturbations or
having the balance assist on, decrease the probability to fall. The associated
multiplicative change in odds are shown in
6~\unit{\kilo\meter\per\hour} with gain $k=10$ are shown in
Table~\ref{table:freq-coefs-6}. The angular impulse, perturbation order, and
balance assist state are all statistically significant predictors. Larger
angular impulse increases the probability to fall and enduring more
perturbations or having the balance assist on, decrease the probability to
fall. The associated multiplicative change in odds are shown in
Table~\ref{table:freq-change-in-odds-6}.

\todo{Combine tables 3 and 4 into one}

\begin{table}
\centering
\caption{Logistic regression coefficient estimates at
6~\unit{\kilo\meter\per\hour} and gain $k=10$.}
6~\unit{\kilo\meter\per\hour} and gain $k=10$. Multiplicative change in odds of predictor variables at 6
\unit{\kilo\meter\per\hour} and gain $k=10$ based on frequentist
single-level logistic regression.}
\label{table:freq-coefs-6}
\begin{tabular}{lrrr}
\begin{tabular}{lrrrrrr}
\toprule
Variable & Estimate & Standard error & p-value \\
Variable & Estimate & Standard error & p-value & MCIO & 2.5\% & 97.5\% \\
\midrule
Intercept & -0.29 & 0.17 & 0.09 \\
Angular impulse & 1.69 & 0.27 & 0.00 \\
Perturbation order & -0.77 & 0.22 & 0.00 \\
Balance assist state & -0.64 & 0.27 & 0.02 \\
Roll angle & -0.25 & 0.21 & 0.24 \\
Steer angle & -0.14 & 0.21 & 0.51 \\
Balance-assist on $\times$ roll angle & 0.52 & 0.34 & 0.12 \\
Balance-assist on $\times$ steer angle & -0.41 & 0.34 & 0.22 \\
Balance-assist on $\times$ angular impulse & 0.41 & 0.41 & 0.32 \\
Balance-assist on $\times$ perturbation order & -0.53 & 0.34 & 0.12 \\
Intercept & -0.29 & 0.17 & 0.09 & 0.75 & 0.53 & 1.05 \\
Angular impulse & 1.69 & 0.27 & 0.00 & 5.40 & 3.18 & 9.16 \\
Perturbation order & -0.77 & 0.22 & 0.00 & 0.46 & 0.30 & 0.72 \\
Balance assist state & -0.64 & 0.27 & 0.02 & 0.53 & 0.31 & 0.89 \\
Roll angle & -0.25 & 0.21 & 0.24 & 0.78 & 0.51 & 1.18 \\
Steer angle & -0.14 & 0.21 & 0.51 & 0.87 & 0.58 & 1.32 \\
Balance-assist on $\times$ roll angle & 0.52 & 0.34 & 0.12 & 1.68 & 0.86 & 3.29 \\
Balance-assist on $\times$ steer angle & -0.41 & 0.34 & 0.22 & 0.66 & 0.34 & 1.28 \\
Balance-assist on $\times$ angular impulse & 0.41 & 0.41 & 0.32 & 1.51 & 0.67 & 3.38 \\
Balance-assist on $\times$ perturbation order & -0.53 & 0.34 & 0.12 & 0.59 & 0.30 & 1.15 \\
\bottomrule
\end{tabular}
\end{table}

\todo[inline]{Combine the columns of the multiplicative change in odds into the
coefficient table}

\begin{table}
\centering
\caption{Multiplicative change in odds of predictor variables at 6
\unit{\kilo\meter\per\hour} and gain $k=10$ based on frequentist
single-level logistic regression.}
\label{table:freq-change-in-odds-6}
\begin{tabular}{lllll}
\textbf{Variable} & \textbf{Multiplicative change in odds} & \textbf{2.5\%} & \textbf{97.5\%} & \textbf{p-value} \\ \hline
Intercept & 0.75 & 0.53 & 1.05 & 0.09 \\
Angular impulse & 5.40 & 3.18 & 9.16 & 0.00 \\
Perturbation order & 0.46 & 0.30 & 0.72 & 0.00 \\
Balance-assist on & 0.53 & 0.31 & 0.89 & 0.02 \\
Roll angle & 0.78 & 0.51 & 1.18 & 0.24 \\
Steer angle & 0.87 & 0.58 & 1.32 & 0.51 \\
Balance-assist on $\times$ roll angle & 1.68 & 0.86 & 3.29 & 0.12 \\
Balance-assist on $\times$ steer angle & 0.66 & 0.34 & 1.28 & 0.22 \\
Balance-assist on $\times$ angular impulse & 1.51 & 0.67 & 3.38 & 0.32 \\
Balance-assist on $\times$ perturbation order & 0.59 & 0.30 & 1.15 & 0.12 \\
\end{tabular}
\end{table}

The coefficient estimates for a single-level logistic regression at
10~\si{\kilo\meter\per\hour} with gain $k=8$ are shown in Table
\ref{table:freq-coefs}. The angular impulse and perturbation order are
Expand All @@ -503,6 +480,8 @@ \section{Results}
to fall. The associated multiplicative change in odds are shown in
Table~\ref{table:freq-change-in-odds}.

\todo{Combine tables 4 and 5 into one}

\begin{table}
\centering
\caption{Logistic regression coefficient estimates at
Expand All @@ -528,20 +507,25 @@ \section{Results}

\begin{table}
\centering
\caption{Multiplicative change in odds of predictor variables at 10 \unit{\kilo\meter\per\hour} based on
frequentist single-level logistic regression.} \label{table:freq-change-in-odds}
\caption{Multiplicative change in odds of predictor variables at 10
\unit{\kilo\meter\per\hour} based on frequentist single-level logistic
regression.}
\label{table:freq-change-in-odds}
\begin{tabular}{lllll}
\textbf{Variable} & \textbf{Multiplicative change in odds} & \textbf{2.5\%} & \textbf{97.5\%} & \textbf{p-value} \\ \hline
Intercept & 0.78 & 0.57 & 1.07 & 0.12 \\
Angular impulse & 10.92 & 6.23 & 19.13 & 0.00 \\
Perturbation order & 0.31 & 0.21 & 0.48 & 0.00 \\
Balance-assist on & 0.64 & 0.40 & 1.03 & 0.07 \\
Roll angle & 1.31 & 0.85 & 2.04 & 0.22 \\
Steer angle & 0.69 & 0.43 & 1.10 & 0.12 \\
Balance-assist on $\times$ roll angle & 0.54 & 0.28 & 1.07 & 0.08 \\
Balance-assist on $\times$ steer angle & 1.76 & 0.88 & 3.50 & 0.11 \\
Balance-assist on $\times$ angular impulse & 1.59 & 0.68 & 3.74 & 0.29 \\
Balance-assist on $\times$ perturbation order & 0.69 & 0.37 & 1.30 & 0.25 \\
\toprule
Variable & Multiplicative change in odds & 2.5\% & 97.5\% & p-value \\ \hline
\midrule
Intercept & 0.78 & 0.57 & 1.07 & 0.12 \\
Angular impulse & 10.92 & 6.23 & 19.13 & 0.00 \\
Perturbation order & 0.31 & 0.21 & 0.48 & 0.00 \\
Balance-assist on & 0.64 & 0.40 & 1.03 & 0.07 \\
Roll angle & 1.31 & 0.85 & 2.04 & 0.22 \\
Steer angle & 0.69 & 0.43 & 1.10 & 0.12 \\
Balance-assist on $\times$ roll angle & 0.54 & 0.28 & 1.07 & 0.08 \\
Balance-assist on $\times$ steer angle & 1.76 & 0.88 & 3.50 & 0.11 \\
Balance-assist on $\times$ angular impulse & 1.59 & 0.68 & 3.74 & 0.29 \\
Balance-assist on $\times$ perturbation order & 0.69 & 0.37 & 1.30 & 0.25 \\
\bottomrule
\end{tabular}
\end{table}

Expand All @@ -557,9 +541,9 @@ \section{Discussion}
\subsection{Balance-assist}

Turning the balance-assist system on significantly \((p<0.05)\) reduces the
odds that a fall occurs while cycling at a speed of 6~\si{\kph} but at
10~\si{\kph} that is not the case \((p=0.07)\). If all other factors remain the
same at 6~\si{\kph}, the balance-assist system halves the odds that a
odds that a fall occurs while cycling at a speed of 1.7~\si{\mps} but at
2.8~\si{\mps} that is not the case \((p=0.07)\). If all other factors remain
the same at 1.7~\si{\mps}, the balance-assist system halves the odds that a
perturbation results in a fall. How much this changes the probability that a
fall occurs depends on the values of the other predictor variables: if the odds
that a perturbation results in a fall are a 1000:1, turning on the
Expand All @@ -573,11 +557,10 @@ \subsection{Balance-assist}

\begin{figure}
\centering
\subcaptionbox{6~\si{\kph}}{
\includegraphics[width=70mm]{example-image-a}
%\includegraphics[width=\textwidth]{figures/predicted_fall_probability-6.png}
\subcaptionbox{1.7~\si{\mps}}{
\includegraphics[width=70mm]{example-image-a}
}
\subcaptionbox{10~\si{\kph}}{
\subcaptionbox{2.8~\si{\mps}}{
\includegraphics[width=70mm]{example-image-b}
}
\caption{Comparison of predicted fall probability for balance-assist on or
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