Skip to content

Association analysis

Sample-level permutation-based association testing

Whereas most Luna commands operate on EDFs in a serial manner, this command instead operates at the sample level, not based on the original EDF/annotations but rather on derived metrics (i.e. the output of prior Luna commands). As such, association analyses in Luna do not require a sample-list as input.

Command Description
CPT Apply cluster-based permutation to one or more sets of metrics

CPT

Cluster-based permutation analysis

This command fits a set of linear models, using a permutation-based approach that allows for covariates, to generate pointwise empirical significance values, as well as family-wise corrected empirical p-values.

It additionally employs a simple clustering heuristic to find groups of adjacent predictors, and to evaluate the evidence for association over these clusters, based on the sum of test statistics. This command is set up to define clusters with respect to three types of adjacency:

  • by frequency (e.g. power for 4.5 Hz and 5.0 Hz might be tested jointly)

  • by spatial location (e.g. CZ and C1 might be considered nearby, and so tested jointly)

  • by pairs of channels (e.g. for connectivity measures, the pair C3-F1 might be considered adjacent to CZ-FZ)

These stratifiers are automatically determined by the presence of F, CH or CH1 and CH2 columns in the dependent variable files.

The idea behind cluster-based analyses non-parametric analysis (outlined here) is that power might be increased by searching for more modest effects that span "similar" predictors, with respect to topography or frequency.

This command can accept multiple types of DV in the same analysis: e.g. spindle density (DENS) as well as amplitude (AMP). With respect to multiple test correction (the PC and cluster-based empirical p-values below), this analysis will correct for both sets of values (accounting for any correlation between them). However, clustering only happens within a particular class of variable. That is, if each mesaure is present for 64 EEG channels, clusters of topographically adjacent channels may be formed for DENS, and separate clusters may be formed for AMP, but no cluster would contain both DENS and AMP measures (i.e. because there is no general way to specify the adjacency of different classes of variable).

The approach to permutation with nuissance variables is the Freedman-Lane method and follows an implementation described here.

Hint

For correctly-formated inputs, the CPT command can be used to analyse any types of numeric inputs, whether they come from prior Luna commands or not.

Parameters

Primary parameters to specify the data and any outlier actions for the dependent variables:

Option Example Description
iv-file demo.txt Name of a single, tab-delimited text file containing the primary independent variable and other covariates
iv DIS The primary IV (assumed to be a column in the iv-file)
covar AGE,SEX Covariates, coded numerically (binary 0/1 or real-valued, assumed to be columns in iv-file)
dv-file spec.txt,psd.txt One or more dependent variable files, in long-format (see below)
dv DENS,AMP The name of one or more DVs (assumed to be columns in the dv-file set
th 5 SD units for individual-level DV outlier removal (note: case-wise deletion)
winsor 0.02 Proportion (e.g. 2% here) for Winsorizing the DV (no outlier removal)

Parameters for the (cluster-based) permutation:

Option Example Description
nreps 1000 Number of permutations to perform
clocs clocs.txt File containing channel location information, described here
th-spatial 0.5 Threshold for defining adjacent channels (Euclidean distance, 0 to 2)
th-freq 1 Threshold for defining adjacent frequencies (Hz)
th-cluster 2 Absolute value of t-statistic for inclusion in a cluster

Secondary parameters

Option Example Description
abs Take the absolute value of all DVs
dB Take the log of all DVs
f-lwr 0.5 Ignore values for frequencies below 0.5 Hz
f-upr 25 Ignore values for frequencies above 25 Hz
complete-obs Instead of case-wise dropping individuals with missing data, flag an error
ex-ids id001,id002 List of individual IDs to exclude
inc-ids @{include.txt} List of individual IDs to include
1-sided Assume a 1-sided test (that B > 0 )
Outputs

Variable-level output (strata: VAR):

Variable Description
CH Channel name
CH1 First channel (for variables stratified by channel-pairs)
CH2 Second channel (for variables stratified by channel-pairs)
F Frequency (Hz) (for variables stratified by frequency)
B Beta from linear regression
T The correspondong t-statistic
PU Uncorrected empirical significance value
PC Family-wise corrected empirical significance value
CLST For variables assigned to a nominally-significant cluster (P<0.05), the cluster number K (else 0)

Cluster-level output (strata: K)

Variable Description
N Number of variables in this cluster
P Empirical significance value
SEED Seed variable (most significant)

Cluster-member output (strata: K x M)

Variable Description
VAR Variable name, i.e. member M of cluster K
Example

This command requires that both dependent variable (i.e. Luna sleep metrics) and the independent variable (i.e. demographics, disease state, etc) have a column labeled ID (case-sensitive) to link individuals across files.

The command expects metrics in the long format, as would come from the -r option of destrat for example:

ID         CH     F      PSD
subj-01    AF3    0.5    393.263017810112
subj-01    AF3    0.75   483.725395908896
subj-01    AF3    1      359.270993287664
subj-01    AF3    1.25   248.829826621191
subj-01    AF3    1.5    161.100179146164
subj-01    AF3    1.75   103.629116449848
subj-01    AF3    2      67.5148408348248
subj-01    AF3    2.25   49.2884864501349
subj-01    AF3    2.5    38.0206381125483
...

This file has 1,029,990 rows: one for each individual, channel and frequency combination.

The file descr.txt has 130 rows, containing the descripive statistics (independent variables):

ID         C1   C2   X1
subj-01    1    1    28
subj-03    1    0    43
subj-04    1    1    33
subj-05    1    0    24
subj-06    1    1    35
subj-07    1    1    31
subj-09    1    1    31
subj-11    1    1    46
subj-12    1    0    28
...

The CPT command is invoked as follows, piping the arguments into Luna via standard input (here from the shell echo command).

echo 'iv-file=descr.txt iv=X1  covar=C1,C2
      dv-file=n2-spec.txt  dv=PSD  dB=PSD 
      nreps=1000
      clocs=~/luna/clocs
      th-spatial=0.5 th-freq=0.5 th-cluster=2
      f-upr=20 ' | luna --cpt -o out.db

This specifics a set of regressions of the IV X1 in the file descr.txt, with covariates C1 and C2. The dependent variable(s) are the values of PSD from the file n2-spec.txt. These DVs will be log-transformed (due to the dB option). We further specify 1000 permutations, with clustering based on channel locations in the file clocs and the given set of cluster-defining thresholds. We restrict the analysis to power values for frequencies under 20 Hz (i.e. if the file n2-spec.txt might contain higher values).

Only people present in both the IV and DV files will be included in analysis (based on the ID field). The order of individuals need not be the same across files.

The output written to the console is as follows:

===================================================================
+++ luna | v0.25.5, 31-Mar-2021 | starting 20-May-2021 14:15:07 +++
===================================================================
  read 130 people from sample_descr.txt (of total 130 data rows)
  reading metrics from n2-spec.txt
  converting input files to a single matrix
  found 130 rows (individuals) and 4503 columns (features)
  identified 0 of 130 individuals with at least some missing data
  finished making regular data matrix on 130 individuals
  final datasets contains 4503 DVs on 130 individuals, 1 primary IV(s), and 2 covariate(s)
  read 64 channel locations from ~/luna/clocs
  defining adjacent variables...
  on average, each variable has 25.348 adjacencies
  0 variable(s) have no adjacencies
  running permutations, assuming a two-sided test...
  found 199 clusters, maximum statistic is 107.218
  .......... .......... .......... .......... ..........  50 perms
  .......... .......... .......... .......... ..........  100 perms

  ...

  .......... .......... .......... .......... ..........  950 perms
  .......... .......... .......... .......... ..........  1000 perms
  2 clusters significant at corrected empirical P<0.05
  all done.
-------------------------------------------------------------------
+++ luna | finishing 20-May-2021 14:15:25                       +++
===================================================================

Based on the constraints, there are 4,503 power values per individual (i.e. frequency values from 0.5 Hz to 20 Hz in 0.25 Hz bins gives 79 bins; for 57 EEG channels (as this dataset contains) this yields 79 * 57 = 4,503 values per individual).

Taking advantage of the Eigen library for matrix operations, the implementation is relatively efficient: including loading all the data, and forming the adjacency map for the 4,503 measures, the above command fits just under 4.5 million linear models (each one for N=130 individuals) including the original data and the 1000 null permutations, but the entire analysis takes approximately 17 seconds running on a single macOS laptop.

The output notes that on average, each variable has about 25 'adjacent' neighbours (in both frequency and topographical space); no variables have 0 adjacent points. Overall, 199 clusters are extracted from the original data, given a minimim t-statistic threshold of 2.

The point-wise (i.e. variable-by-variable) output can be extracted as follows:

destrat out.db +CPT -r VAR 
ID  VAR           B             CH    CLST   F       PC        PU         T
.   AF3~0.5~PSD   -0.07959794   AF3   0      0.5     0.68131   0.02297   -2.339
.   AF3~0.75~PSD  -0.09944737   AF3   0      0.75    0.09191   0.00299   -3.426
.   AF3~1~PSD     -0.08683515   AF3   0      1       0.28371   0.00299   -2.923
.   AF3~1.25~PSD  -0.07991619   AF3   0      1.25    0.47152   0.00899   -2.614
.   AF3~1.5~PSD   -0.07243292   AF3   0      1.5     0.58441   0.01198   -2.460
.   AF3~1.75~PSD  -0.06208055   AF3   0      1.75    0.75424   0.02497   -2.225
.   AF3~10~PSD     0.05459445   AF3   0      10      0.99600   0.12387    1.548
.   AF3~10.25~PSD  0.05860681   AF3   0      10.25   0.98901   0.098909   1.642

Hint

The ID field in the output of the CPT command is always ., i.e. as this reflects sample-level output rather than any one observation.

The variable names (under VAR) are automatically constructed from the specified dv variables (i.e. PSD here) and any frequency and channel stratifiers, with the ~ character separating them. To ease parsing, the output also includes the channel (CH) and frequency (F) associated with each variable.

Note that Luna's generic output mechanism sorts by the VAR stratifier, which is alphanumeric rather than numeric. This means the order of rows may not be in strictly increasing numeric order (i.e. 1 , 10 , 11 , 2 , 3 , ... ). To facilitate plots etc, you can simply sort the table by F, e.g. if in the R package:

d <- d[ order(d$F) , ]

We'll use the lunaR ltopo.xy() routine to produce a plot of these data, first defining an additional variable as the signed -log10 of the empirical p-value:

d$S <- sign( d$B ) * -log10( d$PU )

The Luna graph command:

ltopo.xy( c=d$CH, x=d$F, y=d$S, z=d$S, pch=20, col=rbpal, cex=0.4, xline=15, yline=c(-2,0,2), y.symm=T) 
then produces this plot where the Y-axis and also color-scaling show this scaled verson of the empirical significance:

img

Directly examining the PC column, we see there are 46 variables significant at 0.05 after correction:

table( d$PC < 0.05 ) 
and we can see which frequencies/channels they correspond to:
table( d$F[ d$PC < 0.05 ] , d$CH[ d$PC < 0.05 ] ) 
        C1 C2 CP1 CP2 CP3 CPZ CZ F1 F2 F4 FC1 FC2 FC3 FC4 FCZ Fp1 Fp2 FPZ FZ P2 POz PZ
  0.5    1  0   0   0   0   0  1  0  0  0   1   0   0   0   1   0   0   0  0  0   0  0
  0.75   1  1   0   1   1   1  1  1  1  1   1   1   1   1   1   1   1   1  1  1   0  1
  1      1  1   0   0   0   1  1  0  0  0   1   1   0   0   1   1   1   0  0  1   0  1
  1.25   0  0   0   0   0   0  1  0  0  0   0   0   0   0   0   0   0   0  0  0   0  0
  3      0  0   0   0   0   0  0  0  0  0   0   0   0   0   0   1   0   0  0  0   0  0
  3.25   0  0   0   0   0   0  0  0  0  0   0   0   0   0   0   1   0   0  0  0   0  0
  3.5    0  0   0   0   0   0  0  0  0  0   0   0   0   0   0   1   0   0  0  0   0  0
  15     0  0   0   0   0   0  0  0  0  0   0   0   0   0   0   0   0   0  0  0   1  0
  15.25  0  0   0   0   0   0  0  0  0  0   0   0   0   0   0   0   0   0  0  0   1  0
  15.5   0  0   1   0   0   0  0  0  0  0   0   0   0   0   0   0   0   0  0  0   1  0
  15.75  0  0   1   0   0   0  0  0  0  0   0   0   0   0   0   0   0   0  0  0   0  0
  16     0  0   1   0   0   0  0  0  0  0   0   0   0   0   0   0   0   0  0  0   0  0
  16.25  0  0   1   0   0   0  0  0  0  0   0   0   0   0   0   0   0   0  0  0   0  0

The console log notes there are two clusters significant at the empirical 0.05 level (after correction for all multiple testing, i.e. based on comparing the maximum of the cluster sum-statistics in the observed versus the permuted data). The p-values and seed variables can be obtained as follows:

destrat out.db +CPT -r K
ID   K   N    P         SEED
.    1   30   0.04495   CP4~16~PSD
.    2   33   0.01498   POz~15.25~PSD

(which note, does not include a cluster around 1 or 3 Hz). The members of each cluster can be found as follows:

destrat out.db +CPT -r K M
ID  K   M    VAR
.   1   1    C4~15.5~PSD
.   1   2    C4~15.75~PSD
.   1   3    C4~16.25~PSD
.   1   4    C4~16.5~PSD
.   1   5    C4~16~PSD
.   1   6    CP2~15.5~PSD
.   1   7    CP2~15.75~PSD
.   1   8    CP2~16.25~PSD
.   1   9    CP2~16.5~PSD
.   1   10   CP2~16~PSD
...
(or alternativelty, from the CLST variable in the point-wise output (-r VAR).

Warning

Further guidance on best practice on cluster thresholds, and implementing alternative methods to more intelligently select adaptive thresholds will be implemented in future releases. As is, please consider the clustering of the CPT as a secondary feature that has not been optimized in Luna yet: the primary application is to efficiently generate point-wise, corrected empirical p-values.