Descriptives vs. Inferentials

• Descriptives good
  • Simple & intuitive
    • Can efficiently describe the sample
  • Robust
    • Because they make no assumptions about the population
• Mean & SD
  • SD as average distance from the mean
  • SD as a standard unit of measurement
    • Standardized (z) scores
      • Why correlation is so popular
      • And covariance isn’t

Descriptives vs. Inferentials (cont.)

• Inferentials
  • Making assumptions about the population
  • Most importantly the distribution
    • Often assume approximates a normal distribution
      • But we know we’re wrong
    • Assumptions most robust against:
      • Kurtosis (& skew a bit)
      • Non-independence of measures (“multicollinearity”)
      • Changes in variance over time (“heteroscedasticity”)

Descriptives vs. Inferentials (end)

• Inferentials (cont.)
  • Assumptions not robust against:
    • Non-independence of participants
    • Multi-modality (more than one “hump”)
• Sample stats approximating population stats
  • Accuracy of sample stats improves when:
    • Larger sample sizes
    • More representative sampling
    • Multiple “draws” of samples

Central Limit Theorum

• “Multiple ‘draws’ of samples”
  • Sample stats never equal population stats
    • A sample stat always has some error to its measurement
  • But! (assuming consistent sampling techniques)
    • The error of measurement of sample stats greatly tends to be normally distributed
    • This leads to the Central Limit Theorem
      • Which undergirds—allows for—nearly every statistic you’ll use
      • So remember that, even if you rarely think about it

Variance & Covariance

• Variance = Information
• Seek to understand that information
  • The more we understand, the better
  • Often quantify “how much we understand” as a signal-to-noise ratio
    • \(\text{Variance understood} = \frac{\text{Variance accounted for}}{\text{Variance }not \text{ accounted for}}\)

Variance & Covariance (cont.)

• Variance = Information
• Seek to understand that information
  • The more we understand, the better (cont.)

Partialing Out Variance

• Ways to increase the size of the signal to the size of the noise:
• Increase size of signal
  • Bigger effects
  • Greater range of measurements of effects

Ways of Communicating

• Merging visuals with text
  • What is best described where
  • Visuals as “conversation pieces”
    • Text as highlighting what to focus on in visuals
    • Especially vis-à-vis theory

Basic Strategy

1. Assume the relationships between variables are linear
2. Find a line that best accounts for all variables in the model
   • Either via “ordinary least squares”
   • Or “maximum likelihood”
3. If the line—the linear regression—accounts for enough of the total enough, declare significance of model
   • Measure, e.g., with R²
4. And/or look at some/all of the included predictors to see which of them are significant contributor to that linear regression
   • Amount of variance accounted for by that variables vs. total variance

Partialing Out Variance

• At minimum, we separate out the variance associated with our predictor(s) from “error”
  • And, perhaps, the “intercept,” the starting, pre-intervention values for each participant

\[Y = b_{0} + b_{1}X_{1} + e\]

Partialing Out Variance

• Error can also be separated
  • E.g., if we know the sources of those errors (same hospital, neighborhood, etc.)

\[Y = b_{0} + b_{1}X_{1} + b_{2}X_{2} + e_{1} + e_{2}\]

Common Sources of Variance

• Similar predictors may share too much variance
  • If left un-addressed, can lead to “multicollinearity”
  • Which leads to unstable models terms
    • E.g., terms will flip from being significant to not & back depending on what other terms are added to the model
• Usually addressed by removing one of the multicollinear terms

Combining Sources of Variance (cont.)

• We do this all the time, in fact
  • Adding up responses to items on the same survey
  • Taking the average of a results from a blood draw

Model Fit

• We can group them into “families” of variables within the model…

Ostensible & Non-Ostensible

• Some Things We Can See…
  • Neighborhoods & paychecks
  • Blood pressure & adipose tissue
  • Smiles and cortisol levels

Ostensible & Non-Ostensible (cont.)

• Things we can observe empirically are sometimes called ostensible
• While the underlying “construct” they are manifestations of are non-ostensible

ZIP & Salary Example (cont.)

• This is usually done not using R²,
  • But instead the information criterion
  • Determined fro maximum likelihood estimations (MLEs)
    • So need to use them
    • But they are more robust than ordinary least squares
    • And ordinary least squares are MLEs when all assumptions are met

Model Testing (cont.)

• AIC & BIC are measures of misfit
  • So larger numbers are worse

Model Testing (cont.)

• We test this difference against a χ² with degrees of freedom equal to the difference in dfs between the models
• E.g.:

\(b_{1}X_{1}\)

\(b_{1}X_{1} + ( b_{ZIP}X_{ZIP} + b_{Salary}X_{Salary} )\)

• The second model as 2 more dfs than the first
  • So text χ² = 10 with df = 2
  • (Which would be significant; critical χ² \(\approx\) 6 for one-tailed tests)