This document summarizes all test material for the midterm exam of ARMS. Some parts of the Grasple lessons are written in Dutch, but overall in English.
With this summary i got an 8 for the exam.
,Lecture 1 Multiple Linear Regression (MLR)
Frequentist framework = tests how well the data fits the null hypothesis (NHST)
- P-values
- Confidence intervals (=if we were to repeat this experiment many times and calculate
a CI each time, 95% of the intervals will include the true parameter value, and 5%
won’t)
- Effect sizes
- Power analysis
Bayesian framework = probability of the hypothesis given the data, taking prior information
into account
- Bayes factor (BFs)
- Priors (expectation beforehand)
- Posteriors (=prior and data)
- Credible intervals (=there is 95% probability that the true values is in the interval)
Empirical research = uses collected data to learn from, information is captured in a likelihood
function. →frequentist
X-axis: values for population mean
→for example height: 140 and 230 cm height for an adult are less likely than 165 cm for an
adult.
Y-axis: probability of the observed data for each value of population mean (µ)
Bayesian approach = prior knowledge is updated with information in the data and together
provides the posterior distribution for µ
- Advantage = accumulating knowledge (today’s posterior is tomorrow’s prior)
- Disadvantage = results depend on choice of prior
The posterior distribution of the parameters of interest provides all desired estimates:
- Posterior mean or mode
- Posterior SD (comparable to frequentist standard error)
- Posterior 95% credible interval (providing the bounds of the part of the posterior in
which 95% of the posterior mass is)
Results depend on things not observed and on the sampling plan (how you test).
Bayesian probability = probability that hypothesis Hj is supported by the data.
→Pr(Hj|data)
Frequentist probability = probability of observing same or more extreme data given that the
null hypothesis is true (p-values).
→Pr(data|H0)
PMP = Posterior Model Probability; the (Bayesian) probability of the hypothesis after
observing the data
→are also relative probabilities
, →PMPs are updates of prior probabilities for hypotheses with the BF
Bayesian probability of a hypothesis being true depends on two criteria:
- The prior = how sensible it is, based on prior knowledge
- The data = how well it fits the new evidence
Bayesian testing is comparative: hypotheses are tested against one another
Bayes Factor (BF) = 10 → support for H1 is 10 times stronger than for H0
Bayes Factor (BF) = 1 → support for H1 is as strong as support for H0
Both frameworks use probability theory, but:
- Frequentist: probability is the relative frequency of events
→more formal
- Bayesian: probability is the degree of belief
→more intuitive
→this leads to debate (=same word is used for different things)
→and leads to differences in the correct interpretation of statistical results (like confidence
and credible interval)
Multiple linear regression (MLR)
‘normal’ linear regression:
^Y = B0 + B1 x X
^Y = intercept + slobe x X-value
→so we use X to predict Y
Residual = distance from the line = e
Multiple linear regression = with more predictors (Y = observed, Y^ = predicted)
Y = B0 + B1 x X + B2 x X + e
Y = intercept + slobe 1 x X-value + slobe 2 x X-value + residual
→Life satisfaction decreases by age, but increases by years of education
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