Stat

Longitudinal data

Same people, MULTIPLE times

Cross Sectional

Many People, One observation per time

Time series

One subject measured REPEATEDLY

Regular

everyone is measured at the SAME time points

Balanced

everyone has the SAME number of measurements

MCAR

missing completely at RANDOM → friend misses birthday party nothing to do with the data

MAR

Missing at Random → doing good on test scores, dont need to come back and continue

MNAR

missing not at random → doesnt answer depression test because they depressed

RM ANOVA

Does the average response change over time

Fixed effect

average/population effect (avg person bp starts at 120)

Random effect

How individuals differ from that average (sarahs bp runs 10 higher naturally)

Sphericity

equal variances of all pairwise differences between repeated measurements.

Mauchly's test

Is sphericity held?

IF sphericity is violated these are the corrections

Greenhouse-Geisser, Huynh-Feldt, Chi-Muller

Greenhouse-Geisser, Huynh-Feldt, Chi-Muller

Adjust degrees of freedom

RM ANOVA weakness

even one missing subject will be excluded from the data

LLMs

Random effect is added

Random effect

individual and cluster differences

Random intercept

Everyone can start somewhere different, but they change over time at the same rate.

Random intercept + random slope model:

People start at different points and change differently over time

Covariance Structures

how do we think repeated measurements are correlated with each other

Compound Symmetry (CS)

Every measurement has the same variance, and every pair has the same correlation.

AR(1)

Measurements closer together in time are more correlated.

(nearby = strongly related
far apart = less related)

Toeplitz

Correlation also depends on time lag, BUT it does not have to follow the neat exponential ρ, ρ², ρ³ pattern.

"I don't want to assume a pattern."

ICC = Intraclass Correlation Coefficient.

How much of the total variation comes from differences BETWEEN subjects/clusters?

ICC = 0

People within a cluster are not similar

Higher ICC

Stronger clustering and dependence

Maximum Likelihood (ML)

Lets you compare models with AIC/BIC

MODEL SELECTION

Restricted Maximum likelihood

Report Final Model

Likelihood ratio test

Models are NESTED

Nested basically means:

The simpler model can be created by removing parameters from the more complicated model.

H₀:

Reduced/simple model is enough.

Hₐ:

Full or complex model fits better

AIC/BIC

Smaller is better