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Title: Modeling treatment events in disease progression

Paper content:
Ability to quantify and predict progression of a disease is fundamental for selecting an appropriate treatment.
Many clinical metrics cannot be acquired frequently either because of their cost (e.g. MRI, gait analysis) or because they are inconvenient or harmful to a patient (e.g. biopsy, x-ray).
In such scenarios, in order to estimate individual trajectories of disease progression, it is advantageous to leverage similarities between patients, i.e. the covariance of trajectories, and find a latent representation of progression.
Most of existing methods for estimating trajectories do not account for events in-between observations, what dramatically decreases their adequacy for clinical practice.
In this study, we develop a machine learning framework named Coordinatewise-Soft-Impute (CSI) for analyzing disease progression from sparse observations in the presence of confounding events.
CSI is guaranteed to converge to the global minimum of the corresponding optimization problem.
Experimental results also demonstrates the effectiveness of CSI using both simulated and real dataset.
The course of disease progression in individual patients is one of the biggest uncertainties in medical practice.
In an ideal world, accurate, continuous assessment of a patient's condition helps with prevention and treatment.
However, many medical tests are either harmful or inconvenient to perform frequently, and practitioners have to infer the development of disease from sparse, noisy observations.
In its simplest form, the problem of modeling disease progressions is to fit the curve of y(t), t ∈ [t min , t max ] for each patient, given sparse observations y := (ỹ(t 1 ), . . . ,ỹ(t n )).
Due to the highdimensional nature of longitudinal data, existing results usually restrict solutions to subspace of functions and utilize similarities between patients via enforcing low-rank structures.
One popular approach is the mixed effect models, including Gaussian process approaches (Verbeke, 1997; Zeger et al., 1988) and functional principal components (James et al., 2000) .
While generative models are commonly used and have nice theoretical properties, their result could be sensitive to the underlying distributional assumptions of observed data and hard to adapt to different applications.
Another line of research is to pose the problem of disease progression estimation as an optimization problem.
Kidzinski and Hastie.
Kidziński & Hastie (2018) proposed a framework which formulates the problem as a matrix completion problem and solve it using matrix factorization techniques.
This method is distribution-free and flexible to possible extensions.
Meanwhile, both types of solutions model the natural progression of disease using observations of the targeted variables only.
They fail to incorporate the existence and effect of human interference: medications, therapies, surgeries, etc.
Two patients with similar symptoms initially may have different futures if they choose different treatments.
Without that information, predictions can be way-off.
To the best of our knowledge, existing literature talks little about modeling treatment effect on disease progression.
In Kidziński & Hastie (2018) , authors use concurrent observations of auxillary variables (e.g. oxygen consumption to motor functions) to help estimate the target one, under the assumption that both variables reflect the intrinsic latent feature of the disease and are thus correlated.
Treatments of various types, however, rely on human decisions and to some extent, an exogenous variable to the development of disease.
Thus they need to modeled differently.
In this work, we propose a model for tracking disease progression that includes the effects of treatments.
We introduce the Coordinatewise-Soft-Impute (CSI) algorithm for fitting the model and investigate its theoretical and practical properties.
The contribution of our work is threefold: First, we propose a model and an algorithm CSI, to estimate the progression of disease which incorporates the effect of treatment events.
The framework is flexible, distribution-free, simple to implement and generalizable.
Second, we prove that CSI converges to the global solution regardless of the initialization.
Third, we compare the performance of CSI with various other existing methods on both simulated data and a dataset of Gillette Children's Hospital with patients diagnosed with Cerebral Palsy, and demonstrate the superior performances of CSI.
The rest of the paper is organized as follows.
In Section 2 we state the problem and review existing methods.
Next, in Section 3 we describe the model and the algorithm.
Theoretic properties of the algorithm are derived in Section 4.
Finally, in Section 5 and 6 we provides empirical results of CSI on the simulated and the real datesets respectively.
We discuss some future directions in Section 7.
In this paper, we propose a new framework in modeling the effect of treatment events in disease progression and prove a corresponding algorithm CSI.
To the best of our knowledge, it's the first comprehensive model that explicitly incorporates the effect of treatment events.
We would also like to mention that, although we focus on the case of disease progression in this paper, our framework is quite general and can be used to analyze data in any disciplines with sparse observations as well as external effects.
There are several potential extensions to our current framework.
Firstly, our framework could be extended to more complicated settings.
In our model, treatments have been characterized as the binary matrix I S with a single parameter µ.
In practice, each individual may take different types of surgeries for one or multiple times.
Secondly, the treatment effect may be correlated with the latent variables of disease type, and can be estimated together with the random effect w i .
Finally, our framework could be used to evaluate the true effect of a surgery.
A natural question is: does surgery really help?
CSI provides estimate of the surgery effect µ, it would be interesting to design certain statistical hypothesis testing/casual inference procedure to answer the proposed question.
Though we are convinced that our work will not be the last word in estimating the disease progression, we hope our idea is useful for further research and we hope the readers could help to take it further.

assistant: text: A novel matrix completion based algorithm to model disease progression with events