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Lecture 13 (Week 7, Monday)

Sampling Distributions

  • doesn't really exist, something we have to invent
  • we learn a lot about data from their context

Be Science vs. Do Science

  • what is the cause of the mean difference between the Be Scientist group and the Do Science group?
    • both the experimental manipulation and randomness
    • based on size of PRE and F, there is an effect
    • also, if we had taken a different sample, there would have been a different result
    • sampling variation is always an issue
  • Rhodes 2019 Shuffled
ScienceData$Condition.shuffle <- shuffle(ScienceData$Condition)
gf_histogram(~Games, data=ScienceData, fill="darkorange", color "brown4", binwidth=1) %>%
  gf_facet_grid(Condition.shuffle~.) %>%
  gf_vline(xintercept = ~mean, data=favstats(Games~Condition.shuffle, data=ScienceData), color="black")
  • which of the following DGPs might account for the difference in means across the shuffled conditions?
    • randomness
    • we were the DGP
  • 4 shuffles: why does Be Science do better in the upper left?
    • randomness
  • what statistic could we use to summarize each shuffle?
    • statistic = something we calculate based on a sample
    • b1
  • a sampling distribution is a distribution of sample statistics, or estimates
  • any sample statistic can have a sampling distribution
  • in a sampling distributions, all samples have to be samples of the same size

Distribution Triad: What Distribution do Sample Statistics (Estimates) Come From?

  • sampling distribution = imaginary distribution made up of all samples of size n and all the Y-bars of those samples
  • not a particular number of samples; an infinitely large number of samples

Draw Pictures

  • what kind of questions require model of DGP vs. sampling distribution?
  • don't assume any of these have the same shape
  • mean of sample is not necessarily the same as mean of population
    • sampling variation
    • but can be used as a parameter estimate
    • gives a since of how off it can be
  • sample distribution provides context for interpreting single score
  • sample distribution provides context for interpreting sample statistic (parameter estimate)
    • although we collected a distribution of scores, we only have one mean
    • we need to know what distribution that mean comes from
  • sampling distribution is a model of the distribution of estimates
  • any parameter estimate can be thought of as coming from a sampling distribution of all possible samples of the same size that could have been drawn from the same DGP
  • could there be a sampling distribution of b1?
    • yes, it's statistic/estimate
    • we want to know about that sampling distribution for context

Modeling Sampling Distributions

  • with R
    • simulation (rnorm)
    • randomization (shuffle)
    • bootstrapping (resample)
  • mathematical models

Constructing a Sampling Distribution of b1

  • what does this code do?
outcome <- rnorm(n=10, mean=100, sd=15)
group <- rep(letters[1:2], each = 5)
study <- data.frame(group,outcome)
  • rnorm: mathematically construct a normal distribution with a mean of 100 and sd of 15 and randomly take 10 numbers from that distribution
  • rep: way to make a categorical variable (for simulations), makes 5 of A's and B's each
  • random sampling of numbers and random pairing of numbers with a grouping variable
    • simulating a distribution
    • asking 'if'
gf_histogram(~outcome, data=study, color="darkgreen") %>%
  gf_facet_grid(group~.) %>%
  gf_refine(scale_x_continuous(limits=c(50,150))) %>%
  gf_vline(xintercept = ~mean, data=favstats(outcome~group, data=study), color="blue")
lm(outcome~group, data=study)
supernova(lm(outcome~group, data=study))

  • trying to simulate b1

    • simulate a sample of data
    • simulate it into two groups
    • fit the model
    • calculate the mean of each group
    • difference between those group means is b1

  • what happens if I run this code again?

    • different random numbers every time will result in different distributions
    • blue vertical line (mean of distribution) is at a different value each time
    • different b1 each time

  • why does Group A do better in 3 of the 4 simulations

    • randomness

  • add a line of code to get b1

outcome <- rnorm(n=10, mean=100, sd=15)
group <- rep(letters[1:2], each = 5)
study <- data.frame(group,outcome)
b1(outcome~group, data=study)
  • gives the value of b1
  • every time you run it, you will get a different b1