Lecture notes Statistics 2 (P_BSTATIS_2)

Introduction to Multivariate Association

(0)​ Fundamentals in Stats1

●​ Sampling from a population → simple random, systematic, stratified, cluster, convenience etc
●​ Descriptive statistics: summarise sample or population data with numbers, tables and graphs
●​ Inferential statistics: make predictions about population parameters, based on sample data

(1)​ Associations and Causality

Spurious association: a connection between two variables that appears to be causal but is not
●​ Correlation ≠ causation




1.1​ Criteria for Establishing Causality

John Stuart Mill (1943)
We can only argue that B is caused by A if…
1.​ There is a relationship between A and B (association)
2.​ B must take place after A (appropriate time order)
3.​ The association between A and B is not explained by other factors (elimination of alternative
explanations)

​ 1.1.1​ Eliminating Alternative Explanations
​ ​ → controlling for other variables, eliminate its effects

Experimental Control: in research design
●​ Randomised Controlled Trial (RCT) is often considered the gold standard
○​ Time-order manipulated (criteria 2)
○​ Alternative explanations (partially) excluded through randomisation (criteria 3)
○​ Both observable and non-observable characteristics must be equal
Statistical Control: in data-analysis strategy
●​ Option 1: examine X-Y relationship within subgroups (based on other variables)
→ often unrealistic
●​ Option 2: include alternative explanations in statistical model

1.2​ Multivariate Associations
​ Involves evaluating multiple variables (more than two) to identify any possible association
among them

●​ Important to recognise relevant alternative explanations → know your theory
●​ Adjust your statistical analyses and interpretation accordingly → know your statistics
→ to avoid biassed results due to lurking variables

,Types of Multivariate Associations




​ 1.2.1​ Spurious Associations
​ ​ When both variables are also related to a third variable and the association between X
and Y disappears (mostly) when controlling for the third variable

Eg​ the association between height and maths skills is fully explained by school grade




→ Consequently, estimated association between variables can change dramatically depending on the
data analysis strategy

,​ 1.2.2​ Suppression
​ ​ Sometimes, we find (almost) no association between X and Y until we control for a
third variable




→ association between intervention and language skills were suppressed due to intervention group
scoring lower on pretest intervention language skills

​ 1.2.3​ Simpson’s Paradox – direct-indirect
​ ​ Relationship between X and Y is even reversed within levels of a third variable




→ on average, there is a negative association: experienced typists type faster and make fewer typos.
At the individual level, there is a positive experience: the faster you type, the more typos you make

, ​ 1.2.4​ Chain Relations – mediation
​ ​ X1 causes X2, X2 causes Y
→ important to identify the ‘working mechanisms’ of an intervention




​ 1.2.5​ Statistical Interaction (Moderation)
​ ​ The association between X1 and Y differs across levels of X2

Options
●​ No association between X and Y
○​ But it does exist at subpopulation based on X2: eg, positive and negative effects in
subpopulation cancels 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/non-existing within subpopulation based on
X2
Note: the average X-Y association does not necessarily reflect the association in all subpopulations