In NicDubois/trainingrando: Training material for randomisation

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(trainingrando)

1. Generalized linear model

1.1 Normal distribution

1.1.1 Fixed effect only:

$$E[Y] = X\beta $$ Where:

• $\mathbf{X}$ is called the fixed-effect design matrix.

print(myRandoDataFrame[,c(2,3)])

• $\beta= [\beta_0, \beta_{age}, \beta_{TypeofTumor}]$

1.1.2 Linear mixed model

$$ \mathbf{Y} = \mathbf{X\beta} + \mathbf{Z\gamma} + \mathbf{\epsilon} $$

with

• $\mathbf{\gamma} \sim N(0,G)$

• $\mathbf{\epsilon} \sim N(0,R)$

• $ cov[\mathbf{\gamma},\mathbf{epsilon}] = \mathbf{0}$

Note: Matrix $\mathbf{G}$ is a covariance matrix for random errors and $\mathbf{R}$ is the covariance matrix for the random errors.

$\mathbf{Y}|\mathbf{\gamma} \sim N(\mathbf{X\beta}+\mathbf{Z\gamma},\mathbf{R})$ $\mathbf{Y} \sim N(\mathbf{X\beta}+\mathbf{Z\gamma},\mathbf{V})$ with $\mathbf{V} = \mathbf{ZGZ'} + \mathbf{R}$ $\mathbf{Z}$ is called the random-effect design matrix. $\mathbf{X}$ is called the fixed-effect design matrix.

\begin{eqnarray} X & & \mathrm{N}(0,1)\ Y & \sim & \chi^2_{n-p}\ R & \equiv & X/Y \sim t_{n-p} \end{eqnarray}

with repeated measurments

with random slope

1.2 Binomal distribution

Logistic regressions

$$ log(\frac{p}{1-p}) = X\beta $$

2. Survival models

2.1 Cox proportional hazards model

$$ \lambda(t|X_{i}) = \lambda_0(t) * exp(\beta_1 X_{i1} + ... + \beta_p X_{ip})$$ with

• $X_i = {X_{i1},...X_{ip}}$ be the realized values of covariates for subject i.

This expression gives the hazard fuction at time $ti$ for subject $i$ with covariate vector (explanatory variables) $Xi$.

The hazard function is $$ \lambda(t) = \frac{f(t)}{1-F(t)} $$ Where $f(t)$ is the time to (first) failure distribution (i.e. the failure density function).

Note: Although the failure rate, $\lambda (t)$, is often thought of as the probability that a failure occurs in a specified interval given no failure before time $t$ , it is not actually a probability because it can exceed 1. Erroneous expression of the failure rate in % could result in incorrect perception of the measure, especially if it would be measured from repairable systems and multiple systems with non-constant failure rates or different operation times. It can be defined with the aid of the reliability function, also called the survival function, $R(t)=1−F(t)$, the probability of no failure before time $t$.

NicDubois/trainingrando documentation built on March 13, 2020, 5:33 a.m.