Historias con Impacto

Something to always keep in mind: the stories we tell come from our perspectives, shaped by our experiences and driven by our personal choices. Being open and transparent about this is part of telling an impactful story.

Our old friend: Simple Linear Regression

Basic setup:

$$y={\beta }_0+{\beta }_{1}x+ϵ$$

$$ϵ\sim N(0,{\sigma }^2)$$

Altogether, we have

$$f\left(y|x\right)\sim N({\beta }_0+{\beta }_{1}x,{\sigma }^2)$$

Intrepretation of the parameters ${\beta }_0,{\beta }_1,{\sigma }^2$:

• ${\beta }_0$ = the expected value of y when x=0

• ${\beta }_1$ = the expected change in y when you increase the value of x by 1

• ${\sigma }^2$ = a measure of the variability around the mean ( ${\beta }_0+{\beta }_{1}x$ )

SLR is an abstract concept

Like all statistical models. One of the best ways to learn about a model is to simulate from it.

#R Code
set.seed(17)
beta0 <- 1
beta1 <- 2
std.dev <- 5
x <- rt(n = 150, df = 3)
y <- beta0 + beta1*x + rnorm(n=150, mean = 0, sd=std.dev)

Simulating data from a SLR when the sample size $N=50,200,500$

Simulating data from a SLR when the sample size of $N=200$ and parameter values of ${\beta }_0=1,{\beta }_1=-2,\sigma =5$

Generalized Linear Model: Poisson Log-Linear Model

Here, we imply that:

$$y|x_i\sim Poisson\left({\lambda }_i\right)$$

$$log\left({\lambda }_i\right)={\beta }_0+{\beta }_1{x}_i$$

A special property of the Poisson distribution is that $E\left(y|x_i\right)=Var\left(y|x_i\right)={\lambda }_i$.

Poisson distribution with $\lambda =5$:

The story of the Poisson Log-Linear Model

For every value of $x$, we expect to see observations $y$ that are generated according to the Poisson distribution with ${\lambda }_x=e^{{\beta }_0+{\beta }_{1}x}$.

Simulating from a Poisson Log-Linear Model

pbeta0 <- 0.1
pbeta1 <- 0.2
px <- runif(n=100, min=-5, max=5)
xlambda <- exp(pbeta0 + pbeta1*px)
py <- rpois(n=100, lambda=xlambda)

Hierarchical Model

One example

$$y_{ij}\sim Poisson\left({\lambda }_{ij}\right)$$

$$log\left({\lambda }_{ij}\right)={\beta }_{0,i}+{\beta }_{1,i}{x}_{ij}$$

$${\beta }_{0,i}\stackrel{iid}{\sim }N({\mu }_0,{\sigma }_0^2)$$

$${\beta }_{1,i}\stackrel{iid}{\sim }N({\mu }_1,{\sigma }_1^2)$$

Story

Similar to the Poisson Log-Linear Model from before, but now we assume that some parameters vary across groups. However(!), the parameters ${\beta }_{0,i}$ and ${\beta }_{1,i}$ are linked across groups!

Simulating from a Hierarchical Model

## 10 individuals
pb0h <- rnorm(n = 10, mean = 0.1, sd = 0.1)
pb1h <- rnorm(n=10, mean=0.2, sd=0.1)
pxh <- matrix(data=NA, nrow=10, ncol=20) 
pyh <- matrix(data=NA, nrow=10, ncol=20)
for(j in 1:10){
  pxh[j,] <- runif(n=20, min=-5, max=5)
  pyh[j,] <- rpois(n=20, lambda=exp(pb0h[j] + pb1h[j]*pxh[j,]))
}

Simulation -- The Underused Tool

Advantages of simulation:

• No data needed.

• We can learn about what are models actually do, e.g. how different parameter values affect the outcome.

• We build intuition about what different models imply.

If after simulation it's hard to understand what these parameters mean in the overall context...

It'll be hard to interpret the parameters if we don't have a good idea of what they actually mean for our process.

Assessing/Interpreting a Model

Mostly done in the context of 'posterior predictive checks' when conducting Bayesian inference, this idea falls into the class of simulation-based model assessment.

Simulate data from the fitted model: $y_{rep}$

• Draw values of the parameters ${\theta }^i$ from their distributions

• Plug in ${\theta }^i$ into the model to simulate values of $y_{rep}^i$

• Repeat above steps multiple times (say, 100 times, $i=100$)

• Compare $y_{rep}$ to $y$

The key idea is: if our model is the data generating mechanism, could it generate data like ours?

Of course, there are still residuals and other traditional measures of model evaluation.

Interpeting our Model Results

We need to make sure we interpret the results in the context of:

• How we quantified the story we want to tell.

• What information was collected.

• What information was NOT collected

• What kinds of data can our model produce? And what does that imply about the real world?

• What structure did we not capture? What data not collected could affect the outcomes?

Historias con Impacto

Something to always keep in mind: the stories we tell come from our perspectives, shaped by our experiences and driven by our personal choices. Being open and transparent about this is part of telling an impactful story.