Statistics 2 Lecture 1

Criteria for establishing causality: Jon Stuart Mill

• We can only argue that B is caused by A if:

  • There is a relationship between A and B (association)

  • B bust take place after A (appropriate time order)

  • The association between A and B is not explained by other factors (elimination of alternative explanations)

Eliminate alternative explanations

• Can (in part) be achieved by controlling for other variables → eliminate their effect

• Two ways:

  • Experimental control

  • Statistical control

Eliminate alternative explanations: Experimental control

• In the research design

• RCT often considered the gold standard

  • Time order manipulated (Crit. 2)

  • Alternative explanations (partially) excluded through randomization (Crit. 3)

• Both observable and non-observable characteristics must be equal

• → Feasible and realistic?

Eliminate alternative explanations: Statistical control

• In the data-analysis strategy

• Option 1: Examine x-y relationship within subgroups (based on other variables) → often unrealistic

• Option 2: include alternative explanations in your statistical model

Multivariate associations

• Important to recognize relevant alternative explanations → Know your theory

• And adjust your statistical analyses and interpretation accordingly → Know your statistics

• So that you can avoid biased results due to lurking variables: Variables that are not included in a study, but do explain or influence the association under study

Multivariate associations: Spurious associations

• When both variables are also related to a third variable and the association between x and y disappears (mostly) when controlling for this third variable

• → Estimated association between variables can thus change dramatically depending on the data analysis strategy

• Association between x and y disappears

Multivariate associations: Suppression

Sometimes we find (almost) no association between x and y until we control for a third variable

Multivariate association: Simpson’s paradox

• Sometimes the relationship between x and y is even reserved within levels of a third variable

• Example: Typing speed and typo’s

  • → On average: negative association - Experienced typists type faster and make few typos

  • → At the individual level: Positive association - the faster you type the more typo’s you make

Multivariate associations: Chain relations

• Another reason for a disappearing association between x1 and y when controlling for x2 is a chain relation (mediation):

• x1 has an indirect effect on y, via mediator x2

• In other words:

  • x1 causes x2

  • x2 causes y

• Often a research topic in intervention studies:

  → Important to identify the ‘working mechanisms’ of an intervention

• Example: The association between years of education and lifespan (x1) disappears or weakens when we account for income

Multivariate associations: Statistical Interaction

• In addition, we often see interaction effects between predictors: The association between x1 and y differs across levels of x2

• Various options:

  • No association between x and y

    • But it does exist at subpopulations based on x2.x1

    • For example: Positive and negative effects in subpopulations cancel each other out.

  • Positive association between x and y

    • But different strengths or even negative/non-existing within subpopulation based on x2

  • Negative association between x and y

    • But different strengths or even positive or non-existing within subpopulation based on x2

• Again: The average x – y association does not necessarily reflect the association in all subpopulations!

Multivariate associations: Multiple causes

• However, differences in a criterion variable usually have multiple causes:

  • Several variables together explain differences in y

• These predictors are mostly correlated:

  • This is called confounding.

  • The estimated association between x and y changes when an extra x is added to the regression model.

  • For example, for spurious associations, suppression and Simpson’s paradox.

    And sometimes not correlated:

  • The estimated association between x and y does not change when the extra x is added to the regression model.

  • Nevertheless, there may be statistical interaction