Three Building Blocks of Analysis

1. Variability: The degree of differences within a set of scores
2. Covariability: The degree to which variability in one set of scores corresponds with variability in another set
3. Interpretation: Making sense of the arbitrary scores of tests

The Nature of Variability

• We assume that people differ (or might differ) with respect to their behaviors, genetics, attitudes/beliefs, etc.
  • Inter-individual differences: differences between people (e.g., in their levels of an attribute)
  • Intra-individual differences: differences emerging in one person over time or in difference circumstances (e.g., change)

Importance of Individual Differences

• For health & social science research:
  • We seek to understand differences among people (causes, consequences, etc.)
• For applied health & social science science:
  • Important decisions/interventions are based upon differences among people

Variability and Distributions of Scores

• A set of test scores (from different people) is a “distribution” of scores.
• The differences within that distribution are called “variability”
• How can we quantitatively describe a distribution of scores, including its variability?
  • At least three kinds of information

Describing a Distribution:
An Example Distribution

Describing a Distribution:
Central Tendency

• What is the typical score in the distribution, which score is most representative of the entire distribution?

\(\text{Mean} = \overline{X} = \frac{\Sigma X}{N}\)

\(\overline{X} = \frac{100 + 120 + 100 + 90 + 130 + 110}{6} = \frac{660}{6} = 110\)

\(\overline{X} = \frac{660}{6} = 110\)

Describing a Distribution:
Variability

• To what degree do the scores differ from each other?
  • Variance
  • Standard deviation
  • In terms of “the degree to which scores deviate (differ) from the mean of the distribution”

\(\text{Variance} = s^{2} = \frac{\sum{(X - \overline{X})}}{N}\)

Describing a Distribution:
Variability (cont.)

\(s^{2} = \frac{\sum{(X - \overline{X})}}{N}\)

\(\ \ \ \ = \frac{(110 - 110)^{2} + (120 - 110)^{2} + (100 - 110)^{2} + (90 - 110)^{2} + (130 - 110)^{2} + (110 - 110)^{2}}{6}\)

\(\ \ \ \ = \frac{(0)^{2} + (10)^{2} + (-10)^{2} + (-20)^{2} + (20)^{2} + (0)^{2}}{6}\)

\(\ \ \ \ = \frac{0 + 100 + 100 + 400 + 400 + 0}{6}\)

\(\ \ \ \ = \frac{1000}{6}\)

\(\ \ \ \ = 166.67\)

Describing a Distribution:
Standard Deviation (cont.)

\(\text{Standard Deviation} = s = \sqrt{s^{2}} = \sqrt{\frac{\sum(X - X^{2})}{N}}\)

\(s = \sqrt{s^{2}} = \sqrt{166.67} = 12.91\)

• Standard deviation is (simply) a measure of how far scores are—on average—from the sample mean
  • It is the average distance from the mean

Association Between Distributions

• Covariability: The degree to which two distributions of scores (e.g., X and Y) vary in a corresponding manner
• Two types of information about covariability:
  1. Direction: positive/direct or negative/inverse
  2. Magnitude: strength of association

Association Between Distributions:
Covariance

Association Between Distributions:
Correlation

• Correlation (e.g., r_xy, or just r): a standardized index of covariability
• Direction of association is the same as C_xy
  • r > 0 Positive association
  • r < 0 Negative association
  • r = 0 No association
• Magnitude of association
  • -1 ≤ r ≤ 1
  • As r gets closer to |1|, stronger association

Partial Correlation

• A partial correlation allows us to see the correlation between two variables
  • With a shared correlation (covariate) removed
  • E.g., the correlation between
    • General weight (e.g., BMI) &
    • Maximum rate of oxygen consumption (VO₂ max)
    • While removing the effect of age
  • Or the correlation between
    • Time to burnout among ICU nurses &
    • Nurse-to-patient ratio
    • While removing the effect of number of hours worked

Semipartial Correlation

• Also called a “part” correlation
• Removes the effect of a third variable from one of the two correlated variables

• Why do such a thing?

Pearson Product-Moment Correlation

• The one you know about
• The correlation between two continuous (interval or ratio) variables
• Abbreviated as r
• The formula is also equal to the formula for a correlation between a continuous variable & a dichotomous variable
  • Call a point-biserial correlation
  • Abbreviated r_pb

Spearman’s \(\rho\) (“rho”)

• Spearman’s rank correlation (ρ)
  • Summarizes correspondences between ranks
  • Used for ordinal-ordinal comparisons
  • And non-normal data
• Often, however, people use Pearson’s r assuming ordinal data is in fact interval
• Admittedly, the results are often similar to r
  • But ρ is more robust than r

Kendall’s \(\tau\) (“tau”)

• Used When data severely violates the assumptions of the Pearson r—e.g., normality
• Its formula adjusts for non-normality:

\(\tau = \frac{\text{(Number of Concordant Pairs)} - (\text{Number of Discordant Pairs})}{\text{Total Number of Pairs}}\)

• But is inefficient: It does not use all of the available information
  • And so may be less accurate

The Problem of Causality

• Proving that one thing caused an other
• Traditionally, viewed a nearly impossible to establish
• Requiring evidence for “counterfactuals”
  • Proof of what would have happened in the same situation
    • But with different influences
  • I.e., saying X caused Y,
    • Means Y would not have happened without X
  • “I would not have bought that coffee
      if it wasn’t so cheap.”

The Problem of Causality (cont.)

• We can get close to studying counterfactuals in science
• E.g.:
  • Closely-matched participants in experimental designs (e.g., with controls)
  • Longitudinal designs
    • Especially following a “cohort” through time
    • And alternating which undergoes what events

The Problem of Causality (cont.)

• We can get close to studying counterfactuals in science (cont.)
• E.g.:
  • Mediated effects
    • Comparing events with and without
      considering some possibly-moderating effect

The Problem of Causality (cont.)

The Problem of Causality  (cont.)

• Tests of mediated effects are currently considered among the best measures of causality
  • Since they are closest to being able to test a counterfactual
• But there is also a general trend
  towards accepting causal
  explanations in some areas
  of research

The Problem of Causality  (end)

• E.g., Bulfone et al. (2022)
  • Found that there is a direct effect of academic self-efficacy on academic success
  • And that burn-out mediated part of that relationship
  • (The effects were similar among males & females)