Put together estimation of enrollment #13

So there are a couple of models of enrollment that are in use. Of particular interest is the non-homogenious Poisson Process model.

In this paper, we discuss a principled framework that uses a more realistic and flexible model to monitor and predict patient accrual during clinical trials. We relax the assumption of constant accrual rate and use a non-homogeneous Poisson process (NHPP) to model patient accrual. - Zhang and Long 2010

They use a spline approach to model the rate parameter of the enrollment data.

I'm not quite sure how to use this or a similar approach in what I do because the enrollment data is only partially observed.

So there are a couple of models of enrollment that are in use. Of particular interest is the non-homogenious Poisson Process model. > In this paper, we discuss a principled framework that uses a more realistic and flexible model to monitor and predict patient accrual during clinical trials. We relax the assumption of constant accrual rate and use a non-homogeneous Poisson process (NHPP) to model patient accrual. - Zhang and Long 2010 They use a spline approach to model the rate parameter of the enrollment data. I'm not quite sure how to use this or a similar approach in what I do because the enrollment data is only partially observed.

The nhPP model would look something like this:


E = \sum^t E_\text{daily}
\\
E_\text{daily} \sim \text{Poisson}(\lambda(t)) \\

\lambda(t) \sim 
\begin{cases}
	0 & \text{If $s$ = not recruiting } \\
    f_\text{Recr}(t,e,\theta)  & \text{If $s$ = recruiting } \\
    f_\text{EBI}(t,e,\theta) & \text{If $s$ = enrolling by invitation }\\
\end{cases}

Where

f() is a currently unknown distribution
t is the elapsed duration
e is the planned enrollment
s is the current status
\theta is the parameters

The nhPP model would look something like this: $$ E = \sum^t E_\text{daily} \\ E_\text{daily} \sim \text{Poisson}(\lambda(t)) \\ \lambda(t) \sim \begin{cases} 0 & \text{If $s$ = not recruiting } \\ f_\text{Recr}(t,e,\theta) & \text{If $s$ = recruiting } \\ f_\text{EBI}(t,e,\theta) & \text{If $s$ = enrolling by invitation }\\ \end{cases} $$ Where - $f()$ is a currently unknown distribution - $t$ is the elapsed duration - $e$ is the planned enrollment - $s$ is the current status - $\theta$ is the parameters

https://mc-stan.org/users/documentation/case-studies/splines_in_stan.html

Resources for spline approach

https://mc-stan.org/users/documentation/case-studies/splines_in_stan.html Resources for spline approach

I think the better approach would be based on the paper from

Zhang, Xiaoxi, and Bo Huang. “A Simple and Robust Model for Enrollment Projection in Clinical Trials.” Contemporary Clinical Trials 123 (December 2022): 106999. https://doi.org/10.1016/j.cct.2022.106999.

In it, they assume that the per period enrollment is poisson, and that the rate is site specific, drawn from a gamma distribution.

I think I can use a similar model for enrollment, with the rate based on elapsed duration and random draws from a gamma distribution. The key step is that sums of poison RVs are distributed as RVs.

https://en.wikipedia.org/wiki/Poisson_distribution#Sums_of_Poisson-distributed_random_variables

Overall I think this is probably good enough for me.

I think the better approach would be based on the paper from > Zhang, Xiaoxi, and Bo Huang. “A Simple and Robust Model for Enrollment Projection in Clinical Trials.” Contemporary Clinical Trials 123 (December 2022): 106999. https://doi.org/10.1016/j.cct.2022.106999. In it, they assume that the per period enrollment is poisson, and that the rate is site specific, drawn from a gamma distribution. I think I can use a similar model for enrollment, with the rate based on elapsed duration and random draws from a gamma distribution. The key step is that sums of poison RVs are distributed as RVs. https://en.wikipedia.org/wiki/Poisson_distribution#Sums_of_Poisson-distributed_random_variables Overall I think this is probably good enough for me.

May require converting from mean/variance to shape/scale

https://en.wikipedia.org/wiki/Gamma_distribution

May require converting from mean/variance to shape/scale https://en.wikipedia.org/wiki/Gamma_distribution


N \sim \text{Poisson}(\lambda) \\
\lambda \sim \Gamma(\alpha,\beta) \\
\alpha = \left(\frac{\mu}{\sigma}\right)^2 \\
\beta = \frac{\mu}{\sigma^2} \\
\mu = \theta_0 + \theta_1  e + \theta_2 t + \theta_3 t e \\
\sigma = \eta_0 + \eta_1  e + \eta_2 t + \eta_3 t e \\

Probably use a gamma distribution for \theta and \eta priors. Gammas sum and they are positive as well. A truncated normal might be another option.

The effect sizes will probably be somewhere between zero and two.

$$ N \sim \text{Poisson}(\lambda) \\ \lambda \sim \Gamma(\alpha,\beta) \\ \alpha = \left(\frac{\mu}{\sigma}\right)^2 \\ \beta = \frac{\mu}{\sigma^2} \\ \mu = \theta_0 + \theta_1 e + \theta_2 t + \theta_3 t e \\ \sigma = \eta_0 + \eta_1 e + \eta_2 t + \eta_3 t e \\ $$ Probably use a gamma distribution for $\theta$ and $\eta$ priors. Gammas sum and they are positive as well. A truncated normal might be another option. The effect sizes will probably be somewhere between zero and two.

It might also be helpful to use enrollment information from the trials I wasn't able to match ICD10 codes to, when establishing1 priors. Need to think about that.

Based on an interview with Adam George, Clinical Trials will plan out the sites ahead of time, and so the number is already known when starting the trial (they just don't start at the same time). If I treat sites as exchangeable, and draw their properties from the same distribution, I can use observed sites to help model enrollment.

There are a variety of models that I can use, check the literature from urbas,sherlock, and metcalf 2022


N_c \sim \text{Poisson}(f(\lambda_c(e),t)) \\
\lambda_c \sim \Gamma(\alpha(e),\beta(e))

Based on an interview with Adam George, Clinical Trials will plan out the sites ahead of time, and so the number is already known when starting the trial (they just don't start at the same time). If I treat sites as exchangeable, and draw their properties from the same distribution, I can use observed sites to help model enrollment. There are a variety of models that I can use, check the literature from urbas,sherlock, and metcalf 2022 $$ N_c \sim \text{Poisson}(f(\lambda_c(e),t)) \\ \lambda_c \sim \Gamma(\alpha(e),\beta(e)) $$

After a lot of thinking, I believe I am going to try to estimate the when recruitment will close, and then estimate recruitment linearly from there.

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Put together estimation of enrollment #13