[R] Randomization Test

[Ogbos Okike]

Dear Kind List,

I am still battling with this. I have, however, made some progress
with the suggestions of Micheal and others. At least, I have a better
picture of what I want to do now as I will attempt a detailed
description here.

I am aware I should show you just a small part of my code and data.
But when I copied out a small portion and run to see what you get when
I send that,  I was not satisfied with the signal displayed. The epoch
analysis averages data and is quite sensitive to leveraging,
especially if a small sample is used.

So please permit/exercise patience  me to display the series of epoch
that give the averaged valued used. You can just run the code and see
the signal of interest. Here is the code and the data:

dta <- read.table( text ="n CR
 -5 8969
-4 8932
-3 8929
-2 8916
-1 8807
0 8449
1 8484
2 8148
3 8282
4 8305
5 8380
6 8530
7 8642
8 8780
9 8890
10 8962
-5 8929
-4 8916
-3 8807
-2 8449
-1 8484
0 8148
1 8282
2 8305
3 8380
4 8530
5 8642
6 8780
7 8890
8 8962
9 8949
10 8974
-5 8744
-4 8786
-3 8828
-2 8807
-1 8716
0 8520
1 8634
2 8640
3 8636
4 8658
5 8699
6 8682
7 8621
8 8626
9 8660
10 8737
-5 8592
-4 8612
-3 8628
-2 8589
-1 8318
0 8264
1 8294
2 8410
3 8442
4 8416
5 8389
6 8412
7 8453
8 8563
9 8581
10 8613
-5 8264
-4 8294
-3 8410
-2 8442
-1 8416
0 8389
1 8412
2 8453
3 8563
4 8581
5 8613
6 8647
7 8613
8 8508
9 7829
10 7499
-5 8613
-4 8647
-3 8613
-2 8508
-1 7829
0 7499
1 8213
2 7993
3 7821
4 8316
5 8460
6 8533
7 8584
8 8586
9 8567
10 8573
-5 8508
-4 7829
-3 7499
-2 8213
-1 7993
0 7821
1 8316
2 8460
3 8533
4 8584
5 8586
6 8567
7 8573
8 8617
9 8591
10 8661
-5 8851
-4 8893
-3 8858
-2 8803
-1 8790
0 8468
1 8545
2 8570
3 8568
4 8624
5 8669
6 8236
7 8190
8 8313
9 8389
10 8421
-5 8803
-4 8790
-3 8468
-2 8545
-1 8570
0 8568
1 8624
2 8669
3 8236
4 8190
5 8313
6 8389
7 8421
8 8468
9 8537
10 8580
-5 8570
-4 8568
-3 8624
-2 8669
-1 8236
0 8190
1 8313
2 8389
3 8421
4 8468
5 8537
6 8580
7 8605
8 8646
9 8690
10 8770
-5 8690
-4 8770
-3 8799
-2 8821
-1 8666
0 8539
1 8633
2 8617
3 8651
4 8693
5 8715
6 8738
7 8716
8 8677
9 8680
10 8700
-5 8756
-4 8632
-3 8662
-2 8596
-1 8552
0 8502
1 8633
2 8702
3 8745
4 8730
5 8708
6 8817
7 8724
8 8688
9 8693
10 8746
-5 8926
-4 8888
-3 8798
-2 8651
-1 8678
0 8578
1 8593
2 8598
3 8526
4 8181
5 8204
6 8373
7 8599
8 8773
9 8784
10 8746
-5 8678
-4 8578
-3 8593
-2 8598
-1 8526
0 8181
1 8204
2 8373
3 8599
4 8773
5 8784
6 8746
7 8747
8 8757
9 8749
10 8767
-5 8757
-4 8749
-3 8767
-2 8754
-1 8695
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1 8661
2 8653
3 8588
4 8562
5 8613
6 8595
7 8498
8 8404
9 8507
10 8599
-5 8695
-4 8631
-3 8661
-2 8653
-1 8588
0 8562
1 8613
2 8595
3 8498
4 8404
5 8507
6 8599
7 8592
8 8600
9 8637
10 8635
-5 8588
-4 8562
-3 8613
-2 8595
-1 8498
0 8404
1 8507
2 8599
3 8592
4 8600
5 8637
6 8635
7 8632
8 8674
9 8644
10 8687
-5 8595
-4 8498
-3 8404
-2 8507
-1 8599
0 8592
1 8600
2 8637
3 8635
4 8632
5 8674
6 8644
7 8687
8 8721
9 8747
10 8748
-5 8599
-4 8592
-3 8600
-2 8637
-1 8635
0 8632
1 8674
2 8644
3 8687
4 8721
5 8747
6 8748
7 8739
8 8763
9 8792
10 8558
-5 8600
-4 8637
-3 8635
-2 8632
-1 8674
0 8644
1 8687
2 8721
3 8747
4 8748
5 8739
6 8763
7 8792
8 8558
9 8442
10 8555
-5 8748
-4 8739
-3 8763
-2 8792
-1 8558
0 8442
1 8555
2 8622
3 8634
4 8698
5 8732
6 8713
7 8732
8 8681
9 8615
10 8624
-5 8698
-4 8732
-3 8713
-2 8732
-1 8681
0 8615
1 8624
2 8649
3 8656
4 8678
5 8723
6 8693
7 8548
8 7803
9 7801
10 7724
-5 8723
-4 8693
-3 8548
-2 7803
-1 7801
0 7724
1 7910
2 7829
3 7995
4 8156
5 8307
6 8377
7 8465
8 8506
9 8516
10 8536
-5 8548
-4 7803
-3 7801
-2 7724
-1 7910
0 7829
1 7995
2 8156
3 8307
4 8377
5 8465
6 8506
7 8516
8 8536
9 8574
10 8623
-5 8821
-4 8856
-3 8798
-2 8772
-1 8705
0 8682
1 8691
2 8720
3 8727
4 8789
5 8821
6 8811
7 8841
8 8849
9 8849
10 8860
-5 8835
-4 8829
-3 8826
-2 8799
-1 8775
0 8756
1 8793
2 8814
3 8847
4 8838
5 8833
6 8841
7 8847
8 8903
9 8933
10 8918
-5 8890
-4 8875
-3 8874
-2 8865
-1 8891
0 8839
1 8853
2 8888
3 8884
4 8890
5 8889
6 8839
7 8879
8 8908
9 8924
10 8882
-5 8853
-4 8888
-3 8884
-2 8890
-1 8889
0 8839
1 8879
2 8908
3 8924
4 8882
5 8910
6 8903
7 8859
8 8858
9 8863
10 8847
-5 8924
-4 8882
-3 8910
-2 8903
-1 8859
0 8858
1 8863
2 8847
3 8883
4 8869
5 8878
6 8897
7 8922
8 8895
9 8858
10 8858
-5 8910
-4 8903
-3 8859
-2 8858
-1 8863
0 8847
1 8883
2 8869
3 8878
4 8897
5 8922
6 8895
7 8858
8 8858
9 8736
10 8905
-5 8859
-4 8858
-3 8863
-2 8847
-1 8883
0 8869
1 8878
2 8897
3 8922
4 8895
5 8858
6 8858
7 8736
8 8905
9 8935
10 8974
-5 8897
-4 8922
-3 8895
-2 8858
-1 8858
0 8736
1 8905
2 8935
3 8974
4 8946
5 8952
6 9010
7 8980
8 8976
9 8970
10 8961
-5 9376
-4 9336
-3 9311
-2 9287
-1 9221
0 9087
1 9132
2 9175
3 9166
4 9240
5 9264
6 9271
7 9319
8 9324
9 9333
10 9351
-5 9287
-4 9221
-3 9087
-2 9132
-1 9175
0 9166
1 9240
2 9264
3 9271
4 9319
5 9324
6 9333
7 9351
8 9362
9 9385
10 9354
-5 9407
-4 9414
-3 9354
-2 9298
-1 9319
0 9147
1 9178
2 9196
3 9258
4 9303
5 9369
6 9382
7 9375
8 9389
9 9376
10 9264
-5 9386
-4 9396
-3 9424
-2 9391
-1 9284
0 9267
1 9278
2 9318
3 9334
4 9275
5 9306
6 9308
7 9358
8 9335
9 9373
10 9379
-5 9284
-4 9267
-3 9278
-2 9318
-1 9334
0 9275
1 9306
2 9308
3 9358
4 9335
5 9373
6 9379
7 9355
8 9340
9 9327
10 9320
-5 9327
-4 9320
-3 9315
-2 9336
-1 9371
0 9259
1 9330
2 9355
3 9334
4 9353
5 9370
6 9394
7 9400
8 9318
9 9037
10 8994
-5 9394
-4 9400
-3 9318
-2 9037
-1 8994
0 8943
1 8964
2 8997
3 9158
4 8964
5 8564
6 8736
7 8818
8 8938
9 9034
10 9132
-5 8943
-4 8964
-3 8997
-2 9158
-1 8964
0 8564
1 8736
2 8818
3 8938
4 9034
5 9132
6 9167
7 9200
8 9257
9 9266
10 9306
-5 9338
-4 9354
-3 9372
-2 9338
-1 9308
0 9282
1 9324
2 9318
3 9342
4 9370
5 9331
6 9327
7 9338
8 9381
9 9394
10 9332
-5 9372
-4 9338
-3 9308
-2 9282
-1 9324
0 9318
1 9342
2 9370
3 9331
4 9327
5 9338
6 9381
7 9394
8 9332
9 9331
10 9293
-5 9338
-4 9381
-3 9394
-2 9332
-1 9331
0 9293
1 9309
2 9325
3 9406
4 9409
5 9413
6 9426
7 9440
8 9449
9 9512
10 9494
-5 9361
-4 9354
-3 9299
-2 9282
-1 9250
0 9242
1 9254
2 9321
3 9390
4 9414
5 9435
6 9437
7 9426
8 9398
9 9383
10 9354
-5 9365
-4 9421
-3 9416
-2 9355
-1 9338
0 9324
1 9325
2 9322
3 9319
4 9381
5 9315
6 9314
7 9359
8 9403
9 9419
10 9474
-5 9355
-4 9338
-3 9324
-2 9325
-1 9322
0 9319
1 9381
2 9315
3 9314
4 9359
5 9403
6 9419
7 9474
8 9525
9 9501
10 9447
-5 9325
-4 9322
-3 9319
-2 9381
-1 9315
0 9314
1 9359
2 9403
3 9419
4 9474
5 9525
6 9501
7 9447
8 9424
9 9396
10 9388
-5 9447
-4 9424
-3 9396
-2 9388
-1 9396
0 9346
1 9358
2 9353
3 9350
4 9378
5 9372
6 9354
7 9349
8 9392
9 9440
10 9467
-5 9388
-4 9396
-3 9346
-2 9358
-1 9353
0 9350
1 9378
2 9372
3 9354
4 9349
5 9392
6 9440
7 9467
8 9519
9 9550
10 9565
-5 9353
-4 9350
-3 9378
-2 9372
-1 9354
0 9349
1 9392
2 9440
3 9467
4 9519
5 9550
6 9565
7 9565
8 9497
9 9500
10 9472
-5 9522
-4 9529
-3 9492
-2 9432
-1 9382
0 9355
1 9361
2 9350
3 9382
4 9451
5 9491
6 9506
7 9529
8 9543
9 9556
10 9553
-5 9492
-4 9432
-3 9382
-2 9355
-1 9361
0 9350
1 9382
2 9451
3 9491
4 9506
5 9529
6 9543
7 9556
8 9553
9 9502
10 9470
-5 9551
-4 9505
-3 9389
-2 9406
-1 9377
0 9284
1 9365
2 9424
3 9412
4 9403
5 9384
6 9394
7 9404
8 9413
9 9407
10 9405
-5 9579
-4 9576
-3 9543
-2 9451
-1 9421
0 9361
1 9394
2 9400
3 9387
4 9366
5 9346
6 9360
7 9385
8 9435
9 9443
10 9430
-5 9361
-4 9394
-3 9400
-2 9387
-1 9366
0 9346
1 9360
2 9385
3 9435
4 9443
5 9430
6 9454
7 9531
8 9547
9 9581
10 9540
-5 9510
-4 9546
-3 9564
-2 9508
-1 9422
0 9369
1 9395
2 9438
3 9423
4 9392
5 9368
6 9366
7 9348
8 9340
9 9375
10 9391
-5 9423
-4 9392
-3 9368
-2 9366
-1 9348
0 9340
1 9375
2 9391
3 9466
4 9545
5 9574
6 9564
7 9527
8 9513
9 9494
10 9542
-5 9511
-4 9491
-3 9457
-2 9453
-1 9402
0 9382
1 9407
2 9437
3 9403
4 9404
5 9425
6 9486
7 9457
8 9451
9 9423
10 9401
-5 9425
-4 9486
-3 9457
-2 9451
-1 9423
0 9401
1 9429
2 9422
3 9431
4 9462
5 9475
6 9474
7 9487
8 9493
9 9495
10 9499
-5 9404
-4 9385
-3 9363
-2 9399
-1 9411
0 9355
1 9357
2 9363
3 9382
4 9387
5 9408
6 9429
7 9456
8 9487
9 9526
10 9487
-5 9493
-4 9439
-3 9400
-2 9378
-1 9371
0 9369
1 9374
2 9305
3 9298
4 9298
5 9325
6 9381
7 9477
8 9508
9 9496
10 9517
-5 9371
-4 9369
-3 9374
-2 9305
-1 9298
0 9298
1 9325
2 9381
3 9477
4 9508
5 9496
6 9517
7 9561
8 9570
9 9546
10 9544
-5 9510
-4 9506
-3 9530
-2 9441
-1 9427
0 9393
1 9420
2 9444
3 9468
4 9484
5 9525
6 9542
7 9557
8 9548
9 9550
10 9593
-5 9589
-4 9598
-3 9527
-2 9417
-1 9390
0 9374
1 9386
2 9407
3 9453
4 9447
5 9419
6 9386
7 9373
8 9364
9 9376
10 9389
-5 9453
-4 9447
-3 9419
-2 9386
-1 9373
0 9364
1 9376
2 9389
3 9376
4 9375
5 9370
6 9391
7 9458
8 9446
9 9456
10 9463
-5 9364
-4 9376
-3 9389
-2 9376
-1 9375
0 9370
1 9391
2 9458
3 9446
4 9456
5 9463
6 9500
7 9486
8 9474
9 9495
10 9531
-5 9491
-4 9441
-3 9388
-2 9380
-1 9369
0 9354
1 9367
2 9369
3 9341
4 9305
5 9308
6 9324
7 9385
8 9451
9 9496
10 9527
-5 9369
-4 9354
-3 9367
-2 9369
-1 9341
0 9305
1 9308
2 9324
3 9385
4 9451
5 9496
6 9527
7 9544
8 9543
9 9535
10 9536
-5 9586
-4 9583
-3 9572
-2 9533
-1 9454
0 9392
1 9420
2 9451
3 9475
4 9514
5 9561
6 9542
7 9502
8 9461
9 9468
10 9463
-5 9587
-4 9562
-3 9530
-2 9445
-1 9404
0 9395
1 9417
2 9449
3 9467
4 9470
5 9524
6 9512
7 9448
8 9398
9 9431
10 9467
-5 9467
-4 9470
-3 9524
-2 9512
-1 9448
0 9398
1 9431
2 9467
3 9490
4 9517
5 9526
6 9574
7 9573
8 9562
9 9563
10 9566
",header=TRUE)

 data<-matrix(c(dta$CR),ncol=71)
A<-matrix(rep(-5:10,71))
B<-matrix(data)

 oodf<-data.frame(A,B)
 a<--5:10
oodf<-data.frame(A,B)
library(plotrix)
std.error<-function(x) return(sd(x)/(sum(!is.na(x))))
oomean<-as.vector(by(oodf$B,oodf$A,mean))
oose<-as.vector(by(oodf$B,oodf$A,std.error))
plot(-5:10,oomean,type="l",ylim=c(8890,9100),
 )
A<-oomean-1.96*oose
 B<-oomean+1.96*oose
lines(a,A,col="red")
 lines(a,B,col="red")

 My Question:
I wish to conduct a randomization test of significance (90 and 99
percentile) of the reductions/decreases as displayed by the signal.

I am attempting using:
x<-sample(8890:9500,1136,replace=T )

to generate the random numbers, where 8890, 9500 and 1136 are the
minimum  and maximum of the signal and 1136 the length of sample data.
Q1: Please how do I generate many samples as x above, say up to 5000
or 10,000? I manually generated and stored as x1,x2, x3 up to x100.

Q2: Please how do I use this randomly generated numbers to test the
statistical significance level of the signal generated by
plot(-5:10,oomean,type="l",ylim=c(8890,9100),  )?

I wish to test for 90% and 99% percentile.

I am sorry that this is too long.

Many thanks for your kind contributions

Best
Ogbos







On Sun, Feb 10, 2019 at 3:55 PM Ogbos Okike <giftedlife2014 using gmail.com> wrote:
>
> Dear Michael,
> This is great! Thank you.
>
> I have not really got any response other than yours.
>
> I have long before now included what I have in a paper submitted to a journal.
>
> I am awaiting the feedback of the reviewer. I will compare the
> comments with your input here and determine the corrections to make
> and probably return to the list for additional help.
>
> Best wishes
> Ogbos
>
> On Fri, Feb 8, 2019 at 4:31 PM Meyners, Michael <meyners.m using pg.com> wrote:
> >
> > Ogbos,
> >
> > You do not seem to have received a reply over the list yet, which might be due to the fact that this seems rather a stats than an R question. Neither got your attachment (Figure) through - see posting guide.
> >
> > I'm not familiar with epoch analysis, so not sure what exactly you are doing / trying to achieve, but some general thoughts:
> >
> > * You do NOT want to restrict your re-randomizations in a way that "none of the dates corresponds with the ones in the real event" - actually, as a general principle, the true data must be an admissible re-randomization as well. You seem to have excluded that (and a lot of other randomizations at the same time which might have occurred, i.e. dates 1 and 2 reversed but all others the same), thereby rendering the test invalid. Any restrictions you have on your re-randomizations must've applied to the original randomization as well.
> > * If you have rather observational data (which I suspect, but not sure), Edgington & Onghena (2007) would rather refer to this as a permutation test - the difference being that you have to make strong assumptions (similar to parametric tests) on the nature of the data, which are designed-in to be true for randomization tests. It might be a merely linguistic discrimination, but it is important to note which assumptions have to be (implicitly) made.
> > * I'm not sure what you mean by "mean differences" of the events - is that two groups you are comparing? If so, that seems reasonable, but just make sure the test statistic you use is reasonable and sensitive against the alternatives you are mostly interested in. The randomization/permutation test will never proof that, e.g., means are significantly different, but only that there is SOME difference. By selecting the appropriate test statistic, you can influence what will pop up more easily and what not, but you can never be sure (unless you make strong assumptions about everything else, like in many parametric tests).
> > * For any test statistic, you would then determine the proportion of its values among the 5000 samples where it is as large or larger than the one observed (or as small or smaller, or either, depending on the nature of the test statistic and whether you aim for a one- or a two-sided test). That is your p value. If small enough, conclude significance. At least conceptually important: The observed test statistic is always part of the re-randomization (i.e. your 5000) - so you truly only do 4999 plus the one you observed. Otherwise the test may be more or less liberal. Your p value is hence no smaller than 1/n, where n is the total number of samples you looked at (including the observed one), a p value of 0 is not possible in randomization tests (nor in other tests, of course).
> >
> > I hope this is helpful, but you will need to go through these and refer to your own setup to check whether you adhered to the principles or not, which is impossible for me to judge based on the information provided (and I won't be able to look at excessive code to check either).
> >
> > Michael
> >
> > > -----Original Message-----
> > > From: R-help <r-help-bounces using r-project.org> On Behalf Of Ogbos Okike
> > > Sent: Montag, 28. Januar 2019 19:42
> > > To: r-help <r-help using r-project.org>
> > > Subject: [R] Randomization Test
> > >
> > > Dear Contributors,
> > >
> > > I conducting epoch analysis. I tried to test the significance of my result using
> > > randomization test.
> > >
> > > Since I have 71 events, I randomly selected another 71 events, making sure
> > > that none of the dates in the random events corresponds with the ones in
> > > the real event.
> > >
> > > Following the code I found here
> > > (https://www.uvm.edu/~dhowell/StatPages/R/RandomizationTestsWithR/R
> > > andom2Sample/TwoIndependentSamplesR.html),
> > > I combined these two data set and used them to generate another 5000
> > > events. I then plotted the graph of the mean differences for the 5000
> > > randomly generated events. On the graph, I indicated the region of the
> > > mean difference between the real 71 epoch and the randomly selected 71
> > > epoch.
> > >
> > > Since the two tail test shows that the mean difference falls at the extreme of
> > > the randomly selected events, I concluded that my result is statistically
> > > significant.
> > >
> > >
> > >
> > > I am attaching the graph to assistance you in you suggestions.
> > >
> > > I can attach both my code and the real and randomly generated events if you
> > > ask for it.
> > >
> > > My request is that you help me to understand if I am on the right track or no.
> > > This is the first time I am doing this and except the experts decide, I am not
> > > quite sure whether I am right or not.
> > >
> > > Many thanks for your kind concern.
> > >
> > > Best
> > > Ogbos
> > > ______________________________________________
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> > > https://stat.ethz.ch/mailman/listinfo/r-help
> > > PLEASE do read the posting guide http://www.R-project.org/posting-
> > > guide.html
> > > and provide commented, minimal, self-contained, reproducible code.