# Selected Publications

### A two-stage estimation procedure for non-linear structural equation models

Applications of structural equation models (SEMs) are often restricted to linear associations between variables. Maximum likelihood …

### Modeling the cumulative incidence function of multivariate competing risks data allowing for within-cluster dependence of risk and timing

We propose to model the cause-specific cumulative incidence function of multivariate competing risks data using a random effects model …

### Cox regression with missing covariate data using a modified partial likelihood method

Missing covariate values is a common problem in survival analysis. In this paper we propose a novel method for the Cox regression model …

### The liability threshold model for censored twin data

Family studies provide an important tool for understanding etiology of diseases, with the key aim of discovering evidence of family …

### Linear latent variable models: the lava-package

An R package for specifying and estimating linear latent variable models is presented. The philosophy of the implementation is to …

### Measuring early or late dependence for bivariate lifetimes of twins

We consider data from the Danish twin registry and aim to study in detail how lifetimes for twin-pairs are correlated. We consider …

### Convergence of Markov Chains in Information Divergence

Information theoretic methods are used to prove convergence in information divergence of reversible Markov chains. Also some ergodic …

# Recent Posts

### A simple ODE Class

A small illustration on using the armadillo C++ linear algebra library for solving an ordinary differential equation of the form $X’(t) = F(t,X(t),U(t)).$

The abstract super class Solver defines the methods solve (for approximating the solution in user-defined time-points) and solveint (for interpolating user-defined input functions on finer grid). As an illustration a simple Runge-Kutta solver is derived in the class RK4.

The first step is to define the ODE, here a simple one-dimensional ODE $$X’(t) = \theta\cdot\{U(t)-X(t)\}$$ with a single input $$U(t)$$:

rowvec dX(const rowvec &input, // time (first element) and additional input variables
const rowvec &x,     // state variables
const rowvec &theta) {   // parameters
rowvec res = { theta(0)*theta(1)*(input(1)-x(0)) };
return( res );
}


The ODE may then be solved using the following syntax

odesolver::RK4 MyODE(dX);
arma::mat res = MyODE.solve(input, init, theta);


with the step size defined implicitly by input (first column is the time variable and the following columns the optional different input variables) and boundary conditions defined by init.

### A probability game

Assume that two positive numbers are given, $$X$$ and $$Y$$, with unknown joint probability distribution $$P$$, and $$X\neq Y$$ a.s.

A player draws randomly one of the numbers and has to guess if the number is smaller or larger than the other unrevealed number, i.e., let $$U\sim Bernoulli(\tfrac{1}{2})$$ independent of $$X, Y$$, then the player sees $$Z_{1} = UX + (1-U)Y,$$ and $$Z_{2} = (1-U)X + UY$$ remains unseen.

A random guess (coin-flip) would due to the sampling $$U$$, indepedently of $$F$$, have probability $$\tfrac{1}{2}$$ of correct guessing. The question is if we can find a better strategy?

### Shift register (output)

The 74HC595: an 8-bit serial-in/serial or parallel-out shift register with a storage register and 3-state outputs.

If higher load is required there is also the TPIC6C595 (e.g., for driving LEDs), or it should be paired with for example ULN2803 or similar. For multiple inputs see the 74HC165.

The basic usage is to serially transfer a byte from a microcontroller to the IC. When latched the byte will then in parallel be available on output pins QA-QH (Q0-Q7).

### Regression models for the relative risk


Let $$Y$$ be the binary response, $$A$$ binary exposure, and $$V$$ a vector of covariates, then the target parameter is

\begin{align*} &\mathrm{RR}(v) = \frac{\pr(Y=1\mid A=1, V=v)}{\pr(Y=1\mid A=0, V=v)}. \end{align*}

Let $$p_a(V) = \pr(Y \mid A=a, V), a\in\{0,1\}$$, then the idea is to posit a linear model for $\theta(v) = \log \big(RR(v)\big)$ and a nuisance model for the odds-product $\phi(v) = \log\left(\frac{p_{0}(v)p_{1}(v)}{(1-p_{0}(v))(1-p_{1}(v))}\right)$ noting that these two parameters are variation independent which can be from the below L’Abbé plot. Similarly, a model can be constructed for the risk-difference on the following scale $\theta(v) = \mathrm{arctanh} \big(RD(v)\big).$

### Shift register (input)

The 74HC165 is an 8-bit parallel-load or serial-in shift register.

Wiring:

• Pin 8 (GND): GND
• Pin 16 (VCC): Voltage 5V
• Pin 11-14 & 3-6 (A-H): Input 1-8
• Pin 1 (SH/LD): shift/load pin
• Pin 2 (CLK): Clock pin
• Pin 9 (QH): Data output pin
• Pin 15 (CLK INH): Clock-inhibit pin (or GND)

Clocking is accomplished by a low-to-high transition of the clock (CLK) input while SH/LD is held high and CLK INH is held low. CLK INH can be wired to GND to save a pin on the microcontroller. Unused inputs pins should be grounded as well.

Multiple 74HC165 ICs can be daisy chained by wiring the serial-out pin 9 (QH) to pin 10 (SER) of the succeeding IC allowing us to tie multiple 74165 ICs together that can be controlled using only 3 pins.