Publications by Enwu Liu
Conduct natural spline regression 'by hand'
The use of Natural (restricted) spline regression model has been very popular to model non-linear effects of continuous covariates. Statistical software such as R, SAS, STATA and SPSS, et al all can be used to perform the natural spline regression. However, the output results by these software sometimes are quite ‘confusing’,therefore, if ...
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Notes for limiting distribution
For i.i.d random variables, we can always say \(X_n\overset{d}{\rightarrow}X\) since \[\begin{align}%\label{eq:union-bound} F_{X_n}(x)=F_X(x), \qquad \textrm{ for all }x. \end{align}\] \[\therefore\] \[\begin{align}%\label{eq:union-bound} \lim_{n \rightarrow \infty} F_{X_n}(x)=F_X(x), \qquad \textrm{ for all }x. \end{align}\] For i.i.d random ...
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Converge in distribution
For i.i.d random variables, we can always say \(X_n\overset{d}{\rightarrow}X\) since \[\begin{align}%\label{eq:union-bound} F_{X_n}(x)=F_X(x), \qquad \textrm{ for all }x. \end{align}\] \[\therefore\] \[\begin{align}%\label{eq:union-bound} \lim_{n \rightarrow \infty} F_{X_n}(x)=F_X(x), \qquad \textrm{ for all }x. \end{align}\] ...
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Power analysis
The hypothesis tests, sample size calculations(power analyses) and diagnosis tests are all have the same mathematical base but with many different terms in Statistics and Epidemiology. Here I will review their calculations and their relationships. First, we summarize these terms into a table and a figure.(To Be Continued…) ...
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Random effect meta-analysis
The calculations for random effect model meta-analysis are quite straight forward, the followings are steps we need to conduct a random effect model meta analysis using inverse variance weight method. Random effect meta analysis calculations: Suppose we got the coefficient(\(\beta_i\)) for a predictor and its variance (\(v_i\)), \(i=1,2,...n\)...
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Variance of mixture normal distributions
There are several statistical methods to deal competing risk in survival analysis. Mixture model also can be used when there was competing risk in survival analysis. The following paper introduced how to use mixture model to deal with competing risk in survival analysis. Larson, M. G., & Dinse, G. E. (1985). A mixture model for the regressio...
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Notes for Student t distribution
Let \(W\) denote a random variable with standard normal distribution that is \(N(0, 1)\); let \(V\) denote a random variable that is \(\chi^2(r)\); and let \(W\) and \(V\) be independent. Then the joint pdf of \(W\) and \(V\) ,say \(h(w, v)\), is the product of the pdf of \(W\) and that of \(V\) or \[ h(w,v)=\begin{cases} \frac{1}{\sqr...
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Notes for student's theorem
In proving Student’s Theorem, a linear transformation by matrix can be used,i.e \[W=\begin{bmatrix} \mathbf{\bar{X}}\\ \mathbf{Y} \end{bmatrix}=\begin{bmatrix} \mathbf{v'}\\ \mathbf{I-1v'} \end{bmatrix} \] where, \(\mathbf{v'}=(\frac{1}{n},\frac{1}{n},...,\frac{1}{n})=\frac{1}{n}\mathbf{1'}\) The covariance matrix of the multivariate no...
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General format of mixture distribution
The mixture distribution in general is defined as following: Suppose that we have \(k\) distributions with respective pdfs \(f_1(x), f_2(x), . . . , f_k(x)\),with supports \(\mathcal{S_1, S_2, . . . , S_k}\), means \(\mu_1, \mu_2, . . . , \mu_k\), and variances \(\sigma_1^2, \sigma_2^2,...,\sigma_k^2\), with positive mixing probabilities \(...
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loggamma distribution
There are several formats of loggamma distributions, such as in this PhD thesis https://macsphere.mcmaster.ca/bitstream/11375/6816/1/fulltext.pdf or here for online discussions https://stats.stackexchange.com/questions/370880/what-is-the-expected-value-of-the-logarithm-of-gamma-distribution. Here, we derive mean and variance of another format o...
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