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
    • Central tendency
    • Variability
    • Shape (e.g., skew, kurtosis, normality)

Example of Describing a Distribution

• A set of systolic blood pressure (SBP) measurements:

Describing a Distribution:
Central Tendency

• What is the typical score in the distribution?

\(\mathsf{{Mean} = \overline{X} = \frac{\Sigma X_{i}}{N}}\)

\(\mathsf{\overline{X} = \frac{118 + 126 + 132 + 110 + 144 + 120}{6} = \frac{750}{6} = 125}\)

Describing a Distribution:
Variability

• To what degree do the scores differ from each other?
• Described as either:
  1. Variance
     • The degree to which scores deviate (differ) from the mean
     • Designed to accentuate larger distances

\[\mathsf{{Variance} = s^{2} = \frac{\Sigma{(X_{i} - \overline{X})^2}}{n-1}}\]

Describing a Distribution:
Variability (cont.)

• Or—more often in manuscripts—as:
  2. Standard deviation
     • How far, on average, individual scores deviate from the sample mean
     • Computed directly from variance

\[\mathsf{{SD} = s = \sqrt{Variance}}\]

Describing a Distribution:
Variability (cont.)

\(\mathsf{s^2 = \frac{(X_{1} - \overline{X})^2 + (X_{2} - \overline{X})^2 + (X_{3} - \overline{X})^2 + (X_{4} - \overline{X})^2 + (X_{5} - \overline{X})^2 + (X_{6} - \overline{X})^2}{N - 1}}\)

\(\ \ \ \ \mathsf{= \frac{(118 - 125)^2 + (126 - 125)^2 + (132 - 125)^2 + (110 - 125)^2 + (144 - 125)^2 + (120 - 125)^2}{6 - 1}}\)

\(\ \ \ \ \mathsf{= \frac{(-7)^2 + (1)^2 + (7)^2 + (-15)^2 + (19)^2 + (-5)^2}{5}}\)

\(\ \ \ \ \mathsf{= \frac{49 + 1 + 49 + 225 + 361 + 25}{5}}\)

\(\ \ \ \ \mathsf{= \frac{710}{5}}\)

\(\ \ \ \ \mathsf{= 142}\)

Describing a Distribution:
Standard Deviation

\[\mathsf{Standard\ Deviation = \sqrt{Variance} = \sqrt{\frac{\Sigma(X_{i} - \overline{X})^{2}}{N - 1}}}\]

\[\mathsf{s = \sqrt{s^{2}} = \sqrt{142} = 11.92}\]

• On average, individuals’ scores were 11.92 units (mmHg) 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:
Covariance

• But covariance does not provide clear information about the magnitude of association
  • Since its units of measurement are hard to interpret
    • Measuring the covariance between BMI and blood pressure in what units? This?—

\[\mathsf{\left( \frac{\text{kg} / \text{m}^{2}}{\text{mmHg}} \right)^{2}}\]

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
  • Stronger association as
    r gets closer to ±1

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
  • Called a point-biserial correlation
  • Abbreviated r_pb

Spearman’s rho (ρ)

• 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 (τ)

• Used when data severely violate the assumptions of Pearson’s r, e.g., normality
• Its formula adjusts for non-normality:

\[\mathsf{\tau = \frac{(N_{\mathsf{Concordant\ Pairs}}) - (N_{Discordant\ Pairs})}{{Total\ Number\ of\ Pairs}}}\]

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

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 BMI & blood pressure
  • While removing the effect of age
• Or, e.g., the correlation between time to burnout
  & nurse-to-patient ratio
  • While removing the effect of number of hours worked

Partial Correlation (cont.)

Formula (for Pearson’s r) is:

\[ \mathsf{r_{Y,X_1 \cdot X_2} = \frac{r_{Y,X_1} - r_{Y,X_2} r_{X_1,X_2}}{\sqrt{(1 - r_{Y,X_2}^2) \times (1 - r_{X_1,X_2}^2)}}} \]

E.g., the partial corr. between BMI & BP controlling for Age:

\[ \mathsf{r_{BMI\ \&\ BP\ \cdot\ Age} = \frac{r_{BMI\ \&\ BP} - (r_{BMI\ \&\ Age} \times r_{BP\ \&\ Age})}{\sqrt{(1 - r_{BMI\ \&\ Age}^2) \times (1 - r_{BP\ \&\ Age}^2)}}} \]

Partial Correlation (cont.)

Partial Correlation (end)

Semipartial Correlation

• Also called a “part” correlation
• Removes the effect of a third variable from only one of the two correlated variables
  • Partial correlation removes the influence of X₂ from both Y and X₁
  • Semipartial correlation removes the influence of X₂ only from X₁
    • E.g., the correlation between BMI & BP—removing only the effect of Age on BP

Semipartial Correlation (cont.)

• This is what, in fact, is done in linear regression
  • We remove the effect of Predictor A on Predictor B
    • (And vice versa)
    • While still allowing Predictors A & B to each be associated with the outcome

Semipartial Correlation (cont.)

Formula semipartial correlation:

\[ \mathsf{sr_{Y,X_1 \cdot X_2} = \frac{r_{Y,X_1} - r_{Y,X_2} r_{X_1,X_2}}{\sqrt{1 - r_{X_1,X_2}^2}}} \]

Semipartial Correlation (end)

E.g., semipartial corr. between BMI & BP, removing the effect of Age from BP:

\[ \mathsf{sr_{BMI\ \&\ BP\ \cdot\ Age} = \frac{r_{BMI\ \&\ BP} - (r_{BMI\ \&\ Age} \times r_{BP\ \&\ Age})}{\sqrt{1 - r_{BP\ \&\ Age}^2}}} \]

The Problem of Causality

• Problem is proving that one thing caused an other
• Traditionally, viewed as 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 have graduated from college
      if I hadn’t burned out.”

The Problem of Causality (cont.)

• We 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 get close to studying counterfactuals in science (cont.)
• E.g., mediated effects
  • Comparing events without versus with
    considering some possibly-mediating 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)

Causality without Counterfactuals

• Hill (1965) proposed nine guidelines to help evaluate whether an observed association may be causal
  • N.b., these are not strict rules
    • “What I do not believe … is that we can usefully lay down some hard-and-fast rules of evidence [for] cause and effect.” (p. 299)
• Offers a practical framework for causal reasoning
  • Esp. where counterfactuals are difficult to study
    • E.g., epidemiology, public & environmental health, policy analysis, etc.