Publications by Enwu Liu
Multiple imputation for missing data
Missing data in research Missing data in research can cause serious bias and decrease statistical power. Several statistical methods have been developed to deal with missing data, such as expectation-maximization (EM) algorithm and multiple imputation (MI). The key difference between the EM algorithm and the MI method is that EM algorithm dose ...
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Complementary log-log link
The Complementary Log-Log link has the following form: \[\eta_i = \log (-\log(1-\pi_i))\] Comparing with the logit link function with the form: \[\eta_i=logit(\pi_i)=\log\frac{\pi_i}{1-\pi_i}\]...
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Parameter approaches to estimate the Force of Infection
Notes to estimate the Force of Infection using linear parameter model. glm(cbind(Pos,Tot-Pos), we use cbind() here, because the data are not individual observation, it is a kind of weighted data. X<–matrix(rep(1,length(Age))), x was assigned to negative because FOI need to be positive and for the catalytic model we need to restrict coefficie...
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Newton Raphson and Fisher Scoring
For Fisher scoring we use the following iterative equation to calculate the \(\beta\)s \[\beta^{(t+1)}=\beta^{(t)}+\left [-El''(\beta^{(t)})\right ]^{-1}l'(\beta^{(t)}\] where \(l\) is the log-likelihood function for the entire sample \(y_1,...,y_N\) and \(-E(l''(\beta^{(t)})=-E(\frac{\partial l^2}{\partial \beta_j \partial \beta_k})=E\left[\f...
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SIR with vaccination
This is a note for SIR model with vaccination\(^1\). Suppose p proportion of population was vaccinated with 100% vaccine efficacy, we can write the dynamic system as the following: \[ \left\{\begin{array}{l} \frac{ds(t)}{dt}=\mu(1-p)-(\lambda(t)+\mu)s(t) & (1)\\ \frac{d\lambda(t)}{dt}=\nu\lambda(t)(R_0s(t)-1) &(2)\\ ...
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MSEIR
This is the note for the MSEIR model (Models with maternal antibodies and latent periods) from the book “Modeling infectious disease parameters based on serological and social contact data: a modern statistical perspectiv” in page 49, the authors showed the ODE system and the solution of the ODE by the figure 3.18. The ODE system can be so...
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Runge-Kutta Method
The Runge-Kutta method attempts to improve the Euler’s method. Each Runge–Kutta method is derived from an appropriate Taylor method (In fact, Euler’s method can also be seen derived from a simple Taylor method). These methods can be constructed for any order. By far, the most often used method is the classical fourth-order Runge-Kutta me...
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Calculate R0
\(R_0\), pronounced “R naught, is called the basic reproduction number in Epidemiology, it is defined as the expected number of secondary cases produced by a single case in a susceptible population.\(^1\) \(R_0\) is an important parameter for public health planning such as how quickly people can return to work or when to reopen schools, et a...
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Solving Differential Equations in R: Package deSolve
https://www.jstatsoft.org/article/view/v033i09...
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