Positional Variance Profiles: a new take on the speed accuracy tradeoff by Julien Gori
and Quentin Bellut from Sorbonne Université.
Fitts' model and experiment are the go-to studies for the speed accuracy tradeoff in
HCI.
In a Fitts' law experiment, the task is a time minimization task: accuracy is determined
by a target of given width, and the participant has to hit the target as fast as possible.
There are two versions of the law, the nominal, which uses W as accuracy measure, and the
effective law, which replaces W by the observed spread sigma.
Fitts' model is for endpoints only, and says nothing about trajectories, its experiment
requires crossing several values of D and W to evaluate a and b.
We recently proposed an alternative to Fitts' law called PVPs.
We'll start by reviewing PVPs, and explain how the present work is positioned.
The model applies to time minimization tasks that we have just seen, as well as dual minimization
tasks, where the participant aims toward a line at a given distance, and has to jointly
optimize movement time and precision.
If we get a participant to perform some movements in either task for a given condition, and
synchronize all trajectories, we get a plot like the one on the left, which represents
position of the cursor against time.
Now, if we compute the variance of these signals against time, we obtain a profile like the
one on the right with two phases: A first variance increasing phase with a small duration,
and a much longer second phase where variance decreases steadily.
In our previous work, we evaluated the features of these profiles on two datasets.
What we found is that: 1) The duration of the first phase was about 200-250ms, with
a small effect of distance; 2) The distance covered during the first phase was about two
thirds of the total distance to the target; and 3) the standard deviation at the end of
the first phase was proportional to the distance to the target.
For the second phase, we produced a theoretical model, which namely predicted that the standard
deviation of the trajectories decreases at best exponentially over time at rate C --- which
also means that the logarithm of the standard deviation decreases linearly in time.
This property is illustrated in the plots, which all have a log scale on the y-axis.
On the left panel, there are 8 variance profiles that correspond to a time-minimization task
for various values of D and W. On the middle panel, there are 5 variance
profiles obtained during a dual-minimization task for various strategies (from extreme
emphasis on speed, to extreme emphasis on accuracy).
In both panels, the first phase is very short, and has an almost constant duration.
Then, the second phase starts, and to reach a smaller width or accuracy constraint, you
have to remain longer in the second phase, which results in this plateauing that is obvious
especially in the middle plot.
The right panel shows how we estimate the slope of the second phase by fitting a spline
model.
The two phase-model also predicts Fitts' law, as movement time predicted by the model is
just the sum of the two phases' durations.
The model predicts the effective version of Fitts' law, and from there we can also express
the nominal version.
All terms in the intercept are constants, except for an alpha D term.
So our model predicts what Guiard calls the weak version of Fitts' law, where the slope
does not depend on D and W but the intercept does depend on D.
The value of alpha depends on the device and the participant, and can be very small as
we will see.
In comparison to experiments based on Fitts' model, PVPs have several advantages.
They describe whole trajectories and can be applied in tasks where W is not defined, whereas
Fitts' law is a model for endpoints that is applied to time-minimization protocols.
To evaluate Fitts' law parameters, you have to precisely determine the end of movements
and have several values of D and W. With PVPs, a single condition that emphasizes accuracy
is needed and movements don't even need to fully terminate for the model to be fitted.
Further we can better interpret the model's parameters
In the paper, the first thing we do is provide an extension to 2D because the 2D version
is needed in HCI.
We also replicate our original findings, but using more values of D (the dataset that we
had only had 2 values of D), and more devices (we had a mouse with constant CD gain and
a stylus).
We also address some problems with Fitts' law that we did not touch upon in our previous
work.
We also present a Python library that is easy to use so that people in HCI can engage with
PVPs.
To extend the PVP formulation, our strategy is to find a 1D proxy of the spread of the
data, and apply the regular PVP method to it.
The first step is to project all movements in a basis relative to the target and the
starting point.
alpha is the projection in the direction of the target, and beta the direction orthogonal
to it.
Then we compute the covariance matrix of the alpha beta vector at each timestamp.
The covariance matrix is the generalization of variance to N dimensions, and it can be
estimated by the sample covariance
Then, we reduce the covariance matrix to a single term.
There are infinitely may ways to do so, and we consider here 3 that are interesting:
PVP alpha, where we select the variance relative to the principal direction of movement.
PVP total which sums contributions from both directions, but no cross terms, and PVP generalized
which takes all terms into account.
No where do these metrics come from?
For PVP alpha, the idea is to consider only deviations in the principal direction of the
target.
This is also the ISO recommendation for Fitts' law for the endpoint case.
For PVP total, we use the so-called formula for total variation, which generalizes as
the square root of the trace of the covariance matrix.
In the endpoint case it is also equivalent to what Wobbrock and colleagues called the
distance from centroid formula.
For PVP generalized, we use Wilk's generalized variance, which generalizes to the 2nth root
of the determinant of the covariance matrix.
Total and Wilk's variance have interesting interpretations which I can't discuss here.
We performed an experiment to evaluate the 3 PVPs on a time minimization and dual minimization
paradigms, with three different devices (mouse, touchpad, controller) and 6 values of D. Because
we had a within subject design, to keep experiment duration low we did not fully cross D and
W in the time minimization condition.
The first result is that we found very little difference between the three PVPs.
You see that PVP alpha lies in between the two others.
This is something that we actually prove in the paper.
Differences are small because variance in the beta direction remains small.
Results are presented only for PVP alpha in the remainder.
We now come to the effects of D on PVP features.
On the left, we have tau, the duration of the first phase, against D. For the mouse
and the touchpad (blue and green), tau is almost constant, whereas for the controller in orange, you
see that tau scales with D, so that suggests a weak version of Fitts' law for the controller.
On the right, Dtau, the distance covered during the first phase is proportional to D.
For sigma 0, on the left, we see that the linear relation seems to hold as well, and
on the right C seems constant as the model predicts.
To quantify these results, we fitted multi level linear models to the PVP features, guided
by the previous boxplots.
Multi level models are similar to ANOVAs but for the case of regression of continuous variables
on a ratio scale.
In the first model for tau, D*W means we look at effects of D, W and their interaction,
while (1|participant) allows each participant to have its own intercept (similar to a rm
ANOVA).
Other simple effects, like "device" means that each device also has an own intercept
contribution.
Interactions effets, like device:D
allow different slopes per device for the model. Estimates of the effect sizes show that we replicated the results from our previous study, but more convincingly.
The third result is about Fitts' law predictions.
Namely, we can compute the values of a and b, and see how well they match to the one
that result from directly fitting Fitts' law.
Overall, estimates are pretty accurate except for the controller.
In the paper we discuss a few implications for Fitts' law.
First of all, we highlight the arbitrary nature of the ISO commendations that form the standard
for Fitts' law evaluations.
The 4.133 constant could be anything else.
Second, negative intercepts are possible: they are not just some constant positive term,
which we have here with tau 0, but also account for error rates, and scaling effects (alpha
and gamma). More implications in the paper
Finally, we have released a Python library for PVPs so that the community can use them
more easily.
The library is on PyPI the package index of Python, and can be easily installed.
It has a documentation and can be used very easily. This is an example code snippet, usually you would
create a PVP object, here a PVP alpha
and then you would loop over some dataset to add trajectories
once all trajectories for a given block have been added
we can simply compute the profiles
plot the profiles if you want them, and plot the fitted profiles
The conclusion is that PVPs are an alternative to Fitts' law evaluations
The protocol is simple, aiming at a small target is enough
there is no need to cross D and W
there is also no need to determine precisely the end of a movement
and we get more information than just a and b
We show that it can extend to any dimension and there is a library available