Summary of my notes on the lectures, practical lectures, formative tests and key points for each task. The notes on the practical lectures contain answers and the tables from the SPSS output where it was necessary and the formative test includes how got to the solutions.
Naturally, this is just a ...
PSY4107 –
Advanced
Statistics II
RMA Cognitive and Clinical Science Period 3
(Maastricht University)
This course focuses on repeated measures designs and starts with a review of oneway and
twoway within-subject designs, and split-plot designs with a covariate. This review is
followed by a treatment of mixed (multilevel) linear regression for nested and longitudinal
designs. We will start this treatment with so-called marginal models for repeated measures
as a flexible alternative to repeated measures ANOVA in case of missing data or within-
subject covariates, and end with random effects models for repeated measures and nested
designs. Part II concludes with the topic of optimal design and sample size.
This is a personal summary of this course. Therefore, this summary may contain errors and does
not replace the knowledge a student should acquire throughout this course.
,Topic 1 - Oneway within-subject ANOVA
Lecture 1
What is a WS design?
o K repeated measures of a (quantitative) outcome Y
o On the same N persons (or animals, families etc.)
o under K conditions or at K time points
Types of WS design
o WS exp, replications blocked, crossover
N = 40 students
K= 4 conditions (stand, rest, bonus, rest+bonus)
192 trials per conditions, presented in blocked order
condition order counterbalanced BS (Latin square)
outcome: mean RT
(per set of 6 trials, 32 sets per person per condition)
o WS exp, replications mixed, event-related design
N = 12 students
K = 4 angles of rotation (x same/different)
32 trials per angle (16 same, 16 diff), mixed
outcome: mean RT of all 32 trials
(per person per angle)
3
, o observational studies: growth curves (VGT – Progress test)
o repeated measures in BS exp (BS*WS = split-plot)
Within-subject versus between-subject:
o Advantages and drawbacks
Advantages:
much smaller N of persons needed
each person is his/her own control
Drawbacks:
not feasible in case of irreversible treatment effect
risk of „carry over„ effects (wash-out needed)
o Sample size
For comparing two conditions on a quantitative Y:
BS: unpaired t-test (or 1-way BS ANOVA)
WS: paired t-test (or 1-way WS ANOVA)
Due to smaller residual outcome variance, and observing
each subject in each condition, WS needs only (1-ρ)/2 × total
sample size of BS, where ρ = correlation between paired
samples
o Reduced SS(error)
4
, Univariate method
o The model
o Estimation
If only 1 observation: you cannot separate interaction
Interaction effect = (Yij –Yi – Yj + Ytotal)
With only 1 observation interaction effect and residual is same
o Example: raw data
o Example: SS(total)
Sum of squares
(-3)2 + (-1)2 = 10
Individual score (Yij) – Grand mean (Y)
6 – 10 = -4
5
,o Example: SS(condition)
Condition mean (Yj) – Grand mean (Y)
8 – 10 = -2
o Example: SS (person)
Person mean (Yi) – Grand mean (Y)
8 – 10 = -2
o Example: SS(residual)
Individual score (Yij) – Person/ Condition marginal mean
7–8=1
o Testing
Dividing by df gives the MS‟s for F-test, but:
Only 1 observation per cell (= person x condition
→ Interaction + error cannot be separated, MS(residual) is a
mix of interaction and error!
And person is random, not fixed → affects E(MS)
So what is the corrected F-test then?
6
, o Denominator of F
1-way WS design: treat fixed, person random, so:
if > 1 repli: test treat effect against interaction
if = 1 repli: test treat effect against residual (error+interaction pooled)
then: person and person*treat effects untestable. But who
wants to test these anyway?
(There would only be one time point at which person is tested
and to differentiate person effect and person*treatment effect
you would need at least 2 different time points)
→“You don’t have to understand the details, just believe it”
choice of denominator of F follows from the E(MS) table for that design
ANOVA of raw RTs ( > 1 replications per cell) gives the same F and p for the
condition effect as does ANOVA of average RT across trials !
in example: F= MS(cond) / MS(resid) = .67
o Sphericity
assumption: sphericity
= each pairwise difference has same variance
→ each pairwise comparison same SE (= SD / √n)
≈ compound symmetry: same variance in each condition, same
correlation in all pairs of conditions
Problem: Rarely valid if K > 2 conditions
larger type I error risk for F-test
too small / too large SE‟s for pairwise comparisons (higher risk of TI
errors for some and TII for others)
Solutions:
Epsilon-adjustment of df in univariate ANOVA:
o Multiply df(numerator) and df(denominator) with a factor
epsilon (ε) < 1
→ critical F-value higher
lower-bound ε = 1/(K-1) , is an overcorrection
(overcorrection more extreme with more conditions)
Better: GG (or HF) , ε lower (critical F higher) as
sphericity is more strongly violated.
“You do not have to know how it is computed”
multivariate ANOVA
From SUMMARY OF LECTURE
7
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