Beyond Significant
Andrew Ellis
23/01/2019
To set the mood
IQ example
Summary: Two groups of people took an IQ test.
Group 1, \(N_1=47\), consumes a “smart drug”, and Group 2, \(N_2=42\), is a control group that consumes a placebo (Kruschke, 2013).
Key ideas
We will have a brief look at the theoretical background, then dive straight into a practical example.
Parameters are random variables. These are drawn from probability distributions, which reflect our uncertainty about the parameters.
The prior distribution is updated with the likelihood (data) to obtain a posterior distribution.
Bayesian workflow
Bayesian workflow
Bayesian model comparison
Generative model
Linear regression model as a probabilistic model
The graph on the right shows the data-generating process (dependencies among the random variables).
\(\alpha\) is our expectation for the placebo group, \(\beta\) is our expectation for the difference in means. \(\sigma\) is the variance of the outcome.
Inference using Stan
data {
int<lower=0> N;
vector[N] x;
vector[N] y;
}
parameters {
real alpha;
real beta;
real<lower=0> sigma;
}
model {
sigma ~ cauchy(0, 2.5);
alpha ~ normal(100, 15);
beta ~ normal(0, 10);
y ~ normal(alpha + beta * x, sigma);
}Estimated parameter
We obtain the marginal posterior distribution of the regression parameter \(\beta\)1
Criticize model (posterior predictive check)
We can draw samples from the posterior predictive distribution. The model predicts equal variances, and no outliers.
Predicted marginal means
## Hypothesis Tests for class b:
## Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio
## 1 (GroupSmartDrug)-... > 0 1.03 0.41 0.34 Inf 101.56
## Post.Prob Star
## 1 0.99 *
## ---
## '*': The expected value under the hypothesis lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.Posterior predictive distribution
Equal variance model 
Robust model 