Write w9 lecture (continuous predictors, interactions, and The Grand Vision)

[elizabethpankratz]

Learning objectives

• How do we incorporate continuous predictors into our model?
  • as normal
  • but centre them
• How do we interpret model estimates for continuous predictors?
  • change in outcome for unit change in predictor (all others equal to zero)
• How do we think about interactions involving continuous predictors?
  • change in predictor A's effect for unit change in predictor B (and vice versa)
• (?) What are best practices for reporting results from Bayesian models?

Content

• interpreting continuous predictors (coef = change on outcome scale for one unit change on predictor scale, a la slopes, a la lines, a la school math)

  • also a la treatment coding, where we moved from 0 to 1: also a change of one

• recap of hypothesis testing versus "new statistics" effect size estimation, and why we advocate for the latter

Recap of grand vision / learning outcomes of course overall

• What are statistical models useful for? (And what are they not useful for?)
• What are statistical populations, samples and distributions?

Things not addressed in this course:

• group-level effects, a.k.a. random effects
• how to select and set your own priors (have been dealing with brms defaults)
• contrast coding beyond treatment coding (e.g., sum coding, but also funkier stuff for more complex hypotheses)
• hypothesis testing/evidence for effects from Bayes Factors

[elizabethpankratz]

Arc 1: How do we use/interpret a continuous predictor?

Model: continuous ~ continuous. log_v1_dur ~ speech_rate.

You need

• So far, only categorical predictors, with diff numbers of levels (e.g., Relation_type three levels, Branching two levels)
• But sometimes interesting predictors might be continuous.
• How do we incorporate those into the model?

Go search

• The good news is: no contrast coding (the whole point was to translate things into numbers, but now we are already using numbers)
• Before fitting model: In principle, how do we interpret the betas for a continuous predictor?
  • change in y for unit change in x
  • plot some basic lines w misc diff slopes
  • show linear expression y = b + mx, say this is what m represents, say that in our models, m is beta
  • reference treatment coding and show how that's also a unit change, from 0 to 1

Find take

• After fitting model (one continuous predictor): practice interpretation: what do the numbers tell us?
  • for change of one unit of x (i.e., predictor), what's the assoc'd change in y (i.e., outcome var)
  • but: intercept is the mean estimate when continuous input is at 0 ... which ... nonsense often. esp for speech rate.
• Solution: center! yet another transformation.
  • pretty much always a good idea, for this reason, and for other mathsy reasons that we won't have a chance to get into (bc lessens correls btwn vars, esp impt when multiplicative terms like interactions are involved)
  • How? subtract mean.
  • Can show plots on orig scale x axis and on centered x axis to convince.
  • now, a value of 0 represents the mean of the variable.
• Fit model again, now with centered variable. See how intercept interpretation/esitm? changes. Take-away: Now that we have done centering, it is more interpretable. p much always a good idea.

Return change

• The case of continuous predictors drives home the way that the beta coef is the slope of the line.

Arc 2: How do we fit + interpret interactions with continuous predictors?

continuous categorical interaction (from dur_ita_pol: `log_v1_dur ~ c2_duration vowel`)

You need

• Last week: interactions. We saw it between two categorical variables.
• Can also have interactions that involve continuous variables.
• This section of the lecture: worked example of a continuous * categorical interaction.
  • connects to concepts from last week.
  • Focus of tutorial: continuous * continuous interaction (and binary outcome, so how to deal with non-linear backtransformation when reporting continuous variables)

Go search

• Variables: We're going to take speech rate and see how it interacts with the vowel that is in each word (subsetting to "a" and "o")
• work out what we think the betas will mean now: how does treatment coding, and also the multiplication of the interaction term, play together with a continuous variable?

Find take

• fit the model
• interpret the results (students should be able to interpret the non-interaction terms on their own now, so can ask the room)
• walk through the two interpretations of the interaction's beta. speech_rate_c:vowelo a.k.a. $\beta_3$: a positive coefficient, meaning:
  • 1) there is a positive adjustment to the effect of speech_rate_c when we move from vowel == 'a' (baseline) to vowel == 'o' (non-baseline).
  • because the effect of speech_rate_c is negative, a positive adjustment means that the effect is weaker (brings it closer to zero).
  • we can see this in the plot: the slope of vowel == 'a' is more strongly negative than the slope of vowel == 'o'.
  • 2) there is a positive adjustment to the effect of moving from vowel == 'a' to vowel == 'o' when we increase one unit of speech_rate_c.
  • because the effect of vowel is negative, a positive adjustment means that the effect is weaker (brings it closer to zero).
  • we can see this in the plot: the difference between vowel == 'a' and vowel == 'o' gets smaller as we move from L to R on x axis of speech rate.

Return change

• [reprise this week's learning outcomes here, then break before big conclusion]

Arc 3: The Grand Vision

You need

• inference process: slide 13 of week 1
  • get sample from population
  • make inference based on sample about population
  • tool for doing that: statistical analysis.
• in this class, from that data, we've done quant analysis (slide 10 of week 1)

Go search

• in literature, will often see "effect is significant" or "effect is not significant"
• but what is more interesting is how big hte effect is, how much of a difference htings make
• the Bayesian way of modelling is a really good way of making this happen, because it's about
  • considering a bunch of different possibilities
  • e.g., for binomial outcomes: the possible probabilities of success
  • and using the information gained from the data to allocate our belief across a range of those possibilities
  • the posterior distributions, the 95% CrIs

Find take

• my goal (and stefano's goal): to give you a solid foundation for the rest of your statistics journey.
  • this is hard stuff. unintuitive stuff. but by being here, you're doing the hard work, and it will pay off when it comes time to build on this knowledge that you've gained here.
• we've looked at the fundamentals.
  • you can look anywhere and find more complexity.
  • but best practice: keep it simple. 2x2 analyses are your friend. (shravan's advice.)
• so: you are equipped for the kinds of analyses that you'll probably want to do in your dissertations.

Return change

• in the papers you read, you probably won't see a lot of Bayesian analyses. so you might be asking yourself, why are we doing this. what good is this if this isn't what researchers are using.
• that's because it's up and coming. the frequentist method is much more entrenched by now, because the bayesian method was computationally really difficult to do until extremely recently.
• you're on the cutting edge of the future of applied statistics by being here.