1 Bootstrap Confidence Intervals for Three-way Component Methods Henk A.L. Kiers University of Groningen The Netherlands

2 i = 1. I j= J VARIABLES k=1 K SUBJECTs OCCASIONS three- way data X

3 Three-way Methods: Tucker3 X a = AG a (C  B) + E a A (I  P), B (J  Q), C (K  R) component matrices G a matricized version of G (P  Q  R) core array CP = Candecomp/Parafac X a = AG a (C  B) + E a G (R  R  R) superdiagonal Practice: three-way methods applied to sample from population goal: results should pertain to population i = 1. I j= J VARIABLES k=1 K SUBJECTs OCCASIONS three- way data X

4 Example (Kiers & Van Mechelen 2001): scores of 140 individuals, on 14 anxiousness response scales in 11 different situations Tucker3 with P=6, Q=4, R=3 (41.1%) Rotation: B, C, and Core to simple structure

5 Results for example data Kiers & Van Mechelen 2001: B 

6 C 

7 Core 

8 Is solutions stable? Is solution ‘reliable’? Would it also hold for population? Kiers & Van Mechelen report split-half stability results: Split-half results: rather global stability measures

9 How can we assess degree of stability/reliability of individual results?  confidence intervals (CI) for all parameters not readily available derivable under rather strong assumptions (e.g., normal distributions, full identification) alternative: BOOTSTRAP

10 BOOTSTRAP distribution free very flexible (CI’s for ‘everything’) can deal with nonunique solutions computationally intensive

11 Bootstrap procedure: Analyze sample data X (I  J  K) by desired method  sample outcomes  (e.g., A, B, C and G) Repeat for b=1:500 draw sample with replacement from I slabs of X  X b (I  J  K) analyze bootstrap sample X b in same way as sample  outcomes  b (e.g., A b, B b, C b and G b ) For each individual parameter  : 500 values available range of 95% middle values  “95% percentile interval” (  Confidence Interval)

12 Basic idea of bootstrap: distribution in sample = nonparametric maximum likelihood estimate of population distribution draw samples from estimated population distribution, just as actual sample drawn from population From which mode do we resample? Answer: mimic how we sampled from population sample subjects from population  resample A-mode

13 Three questions: How deal with transformational nonuniqueness? Are bootstrap intervals good approximations of confidence intervals? How deal with computational problems (if any)? Lots of possibilities, depends on interpretation Not too bad Simple effective procedure

14 1. How to deal with transformational nonuniqueness? identify solution completely identify solution up to permutation/reflection  for CP and Tucker3 identify solution up to orthogonal transformations identify solution up to arbitrary nonsingular transformations  only for Tucker3

15 Identify solution completely:  uniquely defined outcome parameters   bootstrap straightforward (CI’s directly available) CP and Tucker3 (principal axes or simple structure) - solution identified up to scaling/permutation Both cases: - further identification needed

16 does not affect fit Identify solution up to permutation/reflection  outcome parameters  b may differ much, but maybe only due to ordering or sign  bootstrap CI’s unrealistically broad !  how to make  b ’s comparable? Solution:  reorder and reflect columns in (e.g.) B b, C b such that B b, C b optimally resemble B, C

17 cannot fully mimic sample & analysis process more realistic solution Identified up to perm./refl. takes orientation, order, (too?!) seriously direct bootstrap CI’s Completely identified conspros e.g., two equally strong components  unstable order

18 Intermezzo What can go wrong when you take orientation too seriously? Two-way Example Data: 100 x 8 Data set PCA: 2 Components Eigenvalues: 4.04, 3.96, , (first two close to each other) PCA (unrotated) solutions for variables (a,b,c,d,e,f,g,h) bootstrap 95% confidence ellipses * *) thanks to program by Patrick Groenen (procedure by Meulman & Heiser, 1983)

19 Look at loadings for data and some bootstraps: … leading to standard errors:... What caused these enormous ellipses?

20 Conclusion: solutions very unstable, hence: loadings seem very uncertain Configurations of subsamples very similar So: We should’ve considered the whole configuration ! However ….

21 Identify solution up to orthogonal transformations Tucker3 solution with A, B, C columnwise orthonormal:  any rotation gives same fit (counterrotation of core)  outcome parameters  b may differ much, but maybe only due to coincidental ‘orientation’  bootstrap CI’s unrealistically broad Make  b ’s comparable:  rotate B b, C b, G b such that they optimally resemble B, C, G How? minimize f 1 (T)=||B b T–B|| 2 and f 2 (U)=||C b U–C|| 2 counterrotate core: G b (U  T) minimize f 3 (S)=||SG b –G|| 2 use B b* = B b T, C b* = C b U, G b* = SG b to determine 95%CI’s comparable across bootstraps

22 Notes: first choose orientation of sample solution (e.g., principal axes or other) order of rotations (first B and C, then G): somewhat arbitrary, but may have effect

23 Identify solution up to nonsingular transformations....analogously.....  transform B b, C b, G b so as to optimally resemble B, C, G

24 Expectation: the more transformational freedom used in bootstraps  the smaller the CI’s Example: anxiety data set (140 subjects, 14 scales, 11 situations) apply 4 bootstrap procedures compare bootstrap se’s of all outcomes

25 Bootstrap Method mean se (B) mean se (C) mean se (G) Principal Axes Simple Structure Orthog Matching Oblique Matching Some summary results:

26 Now what CI’s did we find for Anxiety data Plot of confidence ellipses for first two and last two B components

27 Confidence intervals for Situation Loadings

28 Confidence intervals for Higest Core Values

29

30 2. Are bootstrap intervals good approximations of Confidence Intervals? 95%CI should contain popul.values in 95% of samples  “coverage” should be 95% Answered by SIMULATION STUDY Set up: construct population with Tucker3/CP structure + noise apply Tucker3/CP to population  population parameters draw samples from population apply Tucker3/CP to sample and construct bootstrap CI’s check coverage: how many CI’s contain popul. parameter

31 Design of simulation study: noise: low, medium, high sample size (I):20, 50, size conditions: (J=4,8, K=6,20, core: 222, 333, 432) Other Choices number of bootstraps: always 500 number of populations: 10 number of samples 10 Each cell: 10  10  500 = Tucker3 or CP analyses (full design: 3  3  6=54 conditions)

32 Here are the results Should be close to 95%

33 Some details: ranges of values per cell in design (and associated se’s) Some cells really low coverage Most problematic cases in conditions with small I (I=20)

34 3. How deal with computational problems (if any) Is there a problem? Computation times per 500 boostraps: (Note: largest data size: 100  8  20) CP: min 4 s, max 452 s Tucker3 (SimpStr): min 3 s, max 30 s Tucker3 (OrthogMatch): min 1 s, max 23 s Problem most severe with CP

35 Idea: Start bootstraps from sample solution Problem: May detract from coverage Tested by simulation: CP with 5 different starts per bootstrap vs Fast bootstrap procedure How deal with computational problems for CP?

36 Results: Fast method about 6 times faster (as expected) Coverage Optimal method:B: 95.5%C: 95.1% Fast method: B: 95.3%C: 94.7% Time gain enormous Coverage hardly different

37 Conclusion & Discussion Bootstrap CI’s seem reasonable Matching makes intervals smaller Computation times for Tucker3 acceptable, for CP can be decreased by starting each bootstrap in sample solution

38 Conclusion & Discussion What do bootstrap CI’s mean in case of matching? 95% confidence for each value ? - chance capitalization - ignores dependence of parameters (they vary together) Show dependence by bootstrap movie...!?! Develop joint intervals (hyperspheres)...? Sampling from two modes (e.g., A and C) ? some first tests show that this works some first tests show that this does not work