-
The Examination of Residuals
-
Examination of ResidualsThe fitting of models to data is done
using an iterative approach. The first step is to fit a simple
model (usually a linear model). The next step is examine the
validity of the model. This is usually achieved by examination of
residuals. If the model proves to be incorrect, the residuals will
also point to improvements in the model.
-
The residuals are defined as the n differences :
-
Many of the statistical procedures used in linear and nonlinear
regression analysis are based certain assumptions about the random
departures from the proposed model. Namely; the random departures
are assumed i) to have zero mean,ii) to have a constant variance,
s2,iii) independent, andiv) follow a normal distribution.
-
Thus if the fitted model is correct, the residuals should
exhibit tendencies that tend to confirm the above assumptions, or
at least, should not exhibit a denial of the assumptions.
-
The Examination of Residuals
-
The residuals are defined as the n differences :
-
Many of the statistical procedures used in linear and nonlinear
regression analysis are based certain assumptions about the random
departures from the proposed model. Namely; the random departures
are assumed i) to have zero mean,ii) to have a constant variance,
s2,iii) independent, andiv) follow a normal distribution.
Thus if the fitted model is correct, the residuals should
exhibit tendencies that tend to confirm the above assumptions, or
at least, should not exhibit a denial of the assumptions.
-
The principal ways of plotting the residuals ei are:
1. Overall.3. Against the fitted values 2. In time sequence, if
the order is known.4. Against the independent variables xij for
each value of j
In addition to these basic plots, the residuals should also be
plotted5. In any way that is sensible for the particular problem
under consideration,
-
Overall Plot The residuals can be plotted in an overall plot in
several ways.
-
1.The scatter plot.2.The histogram.3.The box-whisker plot. 4.The
kernel density plot 5.a normal plot or a half normal plot on
standard probability paper.
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
0.7754533726256341
0.0
1.862208591774106
0.0
1.0515236681385431
0.0
0.29407146939774975
0.0
1.2108944247302134
0.0
0.3497621037240606
0.0
-0.3982631824328564
0.0
0.7180051397881471
0.0
0.7509925126214512
0.0
-3.1960098567651585
0.0
0.36359779187478125
0.0
-2.6168618205701932
0.0
1.523394530522637
0.0
3.273089532740414
0.0
0.9651830623624846
0.0
0.5838091965415515
0.0
-0.7133235158107709
0.0
-0.6271602615015581
0.0
3.2085154089145362
0.0
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
1.0820031093317084
0.0
0.8520942174072843
0.0
1.7942738850251772
0.0
-3.0199407774489373
0.0
1.2558848538901657
0.0
-0.2976003088406287
0.0
2.099936864397023
0.0
1.9526578398654237
0.0
-2.168603714380879
0.0
-0.1788066583685577
0.0
2.4729706638026983
0.0
-1.87520981853595
0.0
-1.266710114578018
0.0
0.495217591378605
0.0
-4.017347237095237
0.0
-2.8444719646358863
0.0
-2.2551239453605376
0.0
0.05352376319933683
0.0
0.26313273338018917
0.0
0.7754533726256341
1.862208591774106
1.0515236681385431
0.29407146939774975
1.2108944247302134
0.3497621037240606
-0.3982631824328564
0.7180051397881471
0.7509925126214512
-3.1960098567651585
0.36359779187478125
-2.6168618205701932
1.523394530522637
3.273089532740414
0.9651830623624846
0.5838091965415515
-0.7133235158107709
-0.6271602615015581
3.2085154089145362
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-0.7609469321323559
0.0
0.0
-1.626572156965267
-5.5
-5.75
1.0
1.0
-1.393418642692268
-5.0
-5.25
3.0
4.0
0.7211156116682105
-4.5
-4.75
2.0
6.0
-0.7201651897048578
-4.0
-4.25
3.0
9.0
1.7260526874451898
-3.5
-3.75
4.0
13.0
-0.8641063686809503
-3.0
-3.25
3.0
16.0
-2.1299138097674586
-2.5
-2.75
11.0
27.0
-4.468674887903035
-2.0
-2.25
10.0
37.0
-1.4340866982820444
-1.5
-1.75
18.0
55.0
0.4216190063743852
-1.0
-1.25
26.0
81.0
0.44700300350086764
-0.5
-0.75
19.0
100.0
-1.998710104089696
0.0
-0.25
22.0
122.0
2.5181407181662507
0.5
0.25
15.0
137.0
1.7004595065372996
1.0
0.75
24.0
161.0
0.4911225914838724
1.5
1.25
10.0
171.0
-1.4056968211662024
2.0
1.75
16.0
187.0
1.768139554769732
2.5
2.25
11.0
198.0
0.668051143293269
3.0
2.75
10.0
208.0
-0.03185050445608795
3.5
3.25
1.0
209.0
-0.9447057891520672
4.0
3.75
3.0
212.0
0.5262336344458163
4.5
4.25
3.0
215.0
1.9183653421350755
5.0
4.75
0.0
215.0
-2.7730311558116227
5.5
5.25
0.0
215.0
-0.8387360139749944
6.0
5.75
0.0
215.0
0.20523202692857012
-3.111563273705542
-1.071955466613872
-2.5757344701560214
-0.1353600964648649
1.29361751532997
1.492053343099542
1.2553755368571728
0.6399568519555032
-0.9320820026914589
1.403491296514403
-1.4491570254904218
1.7751017367118038
2.006909198826179
-1.3465455594996456
-1.2854252418037504
0.9757923180586658
-2.6994985091732815
2.0427114577614702
-1.2406144378473982
0.52531731853378
-2.581564331194386
-0.028712747734971344
-0.18336322682444006
-5.122274160385132
-0.27391706680646166
1.5407704268000089
-0.8782308213994838
4.151715984335169
-2.972419679281302
-3.3652031561359763
-1.703338057268411
2.6024781618616544
0.5973629413347226
1.544863152957987
-1.1312113201711327
-0.7246285349538084
0.5901574695599265
-1.2126588444516528
3.956065484089777
2.3232269086292945
-1.5989280655048788
-0.6628306437050924
0.5524771040654741
-0.4616549631464295
-1.0913595360761974
-1.296289156016428
-0.48644324124325067
-2.837700776581187
3.706009010784328
0.11545125744305551
1.3979934010421857
1.756684469000902
-0.45590240915771574
-1.2098371371394023
0.3803779691224918
-0.4790513230545912
-2.1246432879706845
2.306678652530536
0.5984702511341311
2.0880224838037975
-0.8000256457307842
-1.2409827832016163
-5.077963578514755
1.709113348624669
-0.5858396434632596
-1.3435419532470405
3.3579817682038993
1.5589512258884497
1.1152337719977368
-0.8661686479172204
-0.17148749975604005
-1.863350007624831
-1.1888005246873945
-4.851171979680657
-1.2707005225820467
-1.6546937331440859
2.465649231453426
-0.08426923159277067
0.9248947208106983
-3.8867074181325734
-0.3723334884853102
5.866240826435387E-4
2.166411832149606
0.3648938218248077
0.48830770538188517
2.4677274268469773
-0.577374521526508
-2.0672905520768836
1.3838712220604066
-2.7161877369508147
1.4576721696357708
-0.4098069439351093
-0.10425310392747633
0.3549985194695182
0.10529674909776077
2.4922428565332666
0.7604830898344517
-0.6875234248582274
2.715678419917822
-1.3058479453320615
-1.3478484106599353
-1.5066143532749265
-2.070873961201869
0.7558446668554097
2.687856976990588
-0.6649656825175043
0.7549078873125836
-1.6899775800993666
-1.946132215380203
-1.4684292182209902
2.330261850147508
-1.589610292285215
0.779939455242129
1.402686393703334
-4.90166712552309
-1.6785679690656252
2.028905328188557
-2.01275270228507
-1.2890495781903155
2.7654459699988365
-0.5773654265794903
-1.9590879674069583
-0.1479088496125769
-1.547050487715751
3.5262564779259264
1.6105059330584481
-0.02966771717183292
0.4604930836649146
2.72165380010847
2.573469828348607
-4.491230356507003
0.7309881766559556
-0.1618036549189128
-2.7942041924688965
0.45875822252128273
0.5252809387457091
-3.753921191673726
-0.4601224645739421
-1.5398700270452537
-0.483257736050291
-2.166179911000654
-2.5086819732678123
-3.6691017157863826
-2.425413185846992
-1.2863620213465765
-1.741368578223046
-0.09770246833795682
-0.5811693881696556
-4.103458195459098
0.4579396772896871
-3.569075488485396
-1.4326406017062254
1.514363248134032
0.9812833923206199
0.6715526978950948
4.028406692668796
-1.6374406186514534
4.294161044526845
-2.327778929611668
0.9046016202773899
-1.5397017705254257
0.7531048140663188
2.6890120352618396
-1.1060456017730758
0.07474454832845367
-2.5259487301809713
1.126868482970167
1.7686943465378135
-0.697345967637375
1.6053922990977298
-1.3977933122077957
2.8119120543124154
1.8203718354925513
-3.205332177458331
-0.029408511181827635
-2.521496753615793
0.7504172572225798
-5.170368240214884
-5.757174221798778
1.8998935047420673
-2.2092990548117086
0.8580173016525805
-0.48059519031085074
-1.6400690583395772
-1.0274015949107707
-2.055480763374362
-0.7419339453917928
2.780416252790019
0.5223364496487193
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
0.8
-0.18796527001541108
0.7789594747009687
4.04941602027975
0.3169384399370756
-5.0332891987636685
-0.5626657184620854
1.8055789041682146
1.9452772903605364
-2.103088263538666
-3.717232175404206
3.81854988518171
-2.458327799104154
0.8514962246408686
1.310115749220131
2.8368776838760823
2.1837786334799603
-0.49400568968849257
1.080338734027464
0.17017100617522374
0.9300265446654521
-3.2165098673431203
-6.0
0.01190490841338242
1.7242604629427472E-12
1.2760587485511928E-16
2.706483492455394E-35
1.441410146577727E-14
0.24029297589712553
4.6426907276674395E-11
1.0610136880953273E-21
1.9017441196015713E-22
3.51034643115065E-6
0.00850661364579323
9.742855394188488E-34
2.766454225713994E-5
5.89398562666384E-17
3.687860857008131E-19
1.5934028779235595E-27
9.418048825266774E-24
2.581283782053885E-11
4.882900251596904E-18
6.0340568709299E-14
2.530485153170492E-17
0.0011723121753848354
-5.8
0.015936803218662438
1.0275979605199563E-11
1.0287721131036135E-15
6.06332058516196E-34
1.0058471003075666E-13
0.31504202327327946
2.461143221924536E-10
1.178960449550364E-20
2.2074444369079026E-21
1.1498880106515094E-5
0.016826564908533802
2.0307615884962307E-32
8.109928080452689E-5
4.860736272708787E-16
3.51002322773038E-18
2.4438168193452974E-26
1.177788553611052E-22
1.3980452264324114E-10
4.325423776423351E-17
4.0219341236273106E-13
2.138723694588788E-16
0.0027116808524875005
-5.6
0.02025890431151403
5.753077072239857E-11
7.791557718380369E-15
1.276063465715071E-32
6.593757280587995E-13
0.38801855573180954
1.2256334793577884E-9
1.230648652078067E-19
2.407044251845911E-20
3.538487844217202E-5
0.03126733140169225
3.9763830932283095E-31
2.2334030861947713E-4
3.7657513386699504E-15
3.138354863124289E-17
3.521018416204333E-25
1.3836629620397253E-21
7.113171569900592E-10
3.5994493663127253E-16
2.5183558765136975E-12
1.698095703744408E-15
0.005892376223558295
-5.4
0.024582588806354472
3.0257549022003674E-10
5.543524814761929E-14
2.5228452575279856E-31
4.060603079483127E-12
0.4489449560527856
5.733778854543468E-9
1.2067728184008788E-18
2.4656700682661934E-19
1.0229077407273656E-4
0.05458116023064165
7.314322276322613E-30
5.777950486227687E-4
2.7406767454457004E-14
2.636032148000878E-16
4.765676280874477E-24
1.5270381857583292E-20
3.3998671082486038E-9
2.8138443413070755E-15
1.48134373524785E-11
1.2665613769439218E-14
0.012028153372127107
-5.2
0.028677155431662032
1.4949404524643052E-9
3.705137201258185E-13
4.685603620097066E-30
2.3491173780852805E-11
0.4879668207637565
2.5198683782513572E-8
1.1116640300297322E-17
2.372697903851166E-18
2.7778693570582616E-4
0.08950582595866693
1.2639111933424719E-28
0.0014042263385883325
1.8737885697870967E-13
2.0799646345627474E-15
6.059508171590431E-23
1.5831646121256143E-19
1.5265716568656927E-8
2.066429838909499E-14
8.185612536636986E-11
8.874561683364196E-14
0.02306556200282752
-5.0
0.032457015275890455
6.938580592623962E-9
2.3263721390470607E-12
8.175175314376432E-29
1.2766607030835505E-10
0.4982463027978986
1.040330591718048E-7
9.620068896611382E-17
2.1448974105998393E-17
7.086695360127719E-4
0.1378848150205103
2.0517083364024035E-27
0.00320595194570326
1.2034829747643752E-12
1.5417636209625834E-14
7.23780381710084E-22
1.5419093604576116E-18
6.439157419013653E-8
1.425600550845698E-13
4.2491603169951625E-10
5.841497285985745E-13
0.04155140557350635
-4.8
0.03601693229092838
3.0253385064116596E-8
1.3721787089267806E-11
1.3399395246502331E-27
6.517827440610514E-10
0.47791919212873407
4.034795029432988E-7
7.820587980245917E-16
1.8214917063417304E-16
0.0016983699834516896
0.1995437694578481
3.128753013962142E-26
0.006875962460379448
7.2613256014837896E-12
1.073584498394883E-13
8.121436142855196E-21
1.410743997282017E-17
2.5515112354429025E-7
9.239141947469593E-13
2.0721044583339105E-9
3.6120856604084004E-12
0.07031759244555648
-4.6
0.03959024281651348
1.239178609329543E-7
7.603241054951158E-11
2.0631457794961377E-26
3.125984505290879E-9
0.43064702603929
1.4700367085187601E-6
5.9725146646621715E-15
1.4531299910623953E-15
0.0038236439267651363
0.2712792011937975
4.482120557658753E-25
0.013853725571199795
4.1157450675462387E-11
7.02281559728727E-13
8.560821769329295E-20
1.2125346889250747E-16
9.497789116569558E-7
5.624993428950072E-12
9.492416121662216E-9
2.0982078025455103E-11
0.11178894591934203
-4.4
0.04344129414778143
4.7681557809787217E-7
3.9577051663215234E-10
2.9842226935812124E-25
1.4084044173884012E-8
0.3645398651392283
5.031430489551248E-6
4.284810684687975E-14
1.0890262025156211E-14
0.0080868460168864
0.3464586580128102
6.03187567343786E-24
0.02622142313477617
2.1914805295754622E-10
4.315616976751826E-12
8.477243770038386E-19
9.790316410698371E-16
3.3212697875241347E-6
3.2171328424388446E-11
4.085060437751651E-8
1.144973800640278E-10
0.16695149958324754
-4.2
0.047737918800299116
1.7235486974321433E-6
1.935284021186126E-9
4.0549839201285796E-24
5.961074244687956E-8
0.2898846369558821
1.6177498548429093E-5
2.8877703082001897E-13
7.667058678856269E-14
0.01606710012495708
0.4156644996321826
7.625666781703677E-23
0.04662324872176327
1.09618375432027E-9
2.491329184618291E-11
7.885885792736755E-18
7.426015966058522E-15
1.0910442933781692E-5
1.7285126566465064E-10
1.6514932351936725E-7
5.869474168851378E-10
0.23422776930469802
-4.0
0.052461445259040035
5.852659741941154E-6
8.890016739916462E-9
5.1761116902853704E-23
2.3701633497549913E-7
0.21655188355164354
4.886386963912814E-5
1.8283117485146834E-12
5.070792506071836E-13
0.02998833966879192
0.4684799894389249
9.056488947684374E-22
0.07787631625465055
5.150930632675259E-9
1.3510640345862867E-10
6.891327495061686E-17
5.291412564985596E-14
3.366954164331939E-5
8.724342936069412E-10
6.272081700000066E-7
2.826567993677813E-9
0.308704553352856
-3.8
0.057392090849456276
1.8669796689051404E-5
3.836339571450534E-8
6.206899828342648E-22
8.852962415293067E-7
0.1519691122033458
1.386503399869834E-4
1.0874128184373649E-11
3.150500025411453E-12
0.052580406273710883
0.49601607425689204
1.0104119962001124E-20
0.1221982153008063
2.2737600729173176E-8
6.882993282107136E-10
5.657334922097504E-16
3.5419625483302886E-13
9.760872079602468E-5
4.136656771386838E-9
2.2377069433697854E-6
1.2787224004734609E-8
0.3822119692156139
-3.6
0.06216791871813373
5.594773329562978E-5
1.5552066306174886E-7
6.992016247982244E-21
3.1063869818828808E-6
0.10018559505544589
3.695818336439468E-4
6.075684886347856E-11
1.8388221645865706E-11
0.08660681119506738
0.49335217955545785
1.0589945400360927E-19
0.18012787307995745
9.428881239166282E-8
3.294089183902798E-9
4.362922376726725E-15
2.2272701679751147E-12
2.658254961045663E-4
1.84256510545202E-8
7.499828060229213E-6
5.434376258007752E-8
0.4445515469624385
-3.4
0.06638051448675492
1.575005222323803E-4
5.922645677657523E-7
7.399233216800549E-20
1.023950799574072E-5
0.06204572056714841
9.25458370215946E-4
3.18898612117111E-10
1.0082229229848092E-10
0.13400986026795633
0.46097242442782643
1.042666794855213E-18
0.2494328067883102
3.6730960924443363E-7
1.4809825974521624E-8
3.160819362563923E-14
1.3157047131853352E-11
6.800819365177919E-4
7.709971832747724E-8
2.3613262733348894E-5
2.1696002896442968E-7
0.4857318296942567
-3.2
0.06966173543661595
4.1652214403381423E-4
2.1188489580565508E-6
7.355760802152677E-19
3.170729668223727E-5
0.036097321545367614
0.0021770066439745837
1.572412950209299E-9
5.193139627774244E-10
0.19479511209466588
0.4046219480416458
9.64392801512157E-18
0.32447618065542916
1.3441910812484074E-6
6.254911257130995E-8
2.1511884764478417E-13
7.301306090154138E-11
0.0016344906532993293
3.0306747836993485E-7
6.984209958293116E-5
8.137038128107129E-7
0.49857166846885553
-3.0
0.07173684885227845
0.0010347865786791202
7.1209976341720745E-6
6.869499167381463E-18
9.223503948149884E-5
0.01972852962758952
0.004810821199648229
7.283450779207939E-9
2.5128121700024564E-9
0.2659965175081292
0.3336418508220302
8.37951672013294E-17
0.39652325617463885
4.621110183769036E-6
2.481698078508144E-7
1.375351936757762E-12
3.806267227516807E-10
0.0036902878042878373
1.1191349905130234E-6
1.9405961647355065E-4
2.8668803419040156E-6
0.480745495055666
-2.8
0.07245289506725545
0.002415016343073242
2.248217265279699E-5
6.026694083386913E-17
2.5205148157403455E-4
0.010129104544329687
0.00998700512176007
3.169307382189872E-8
1.1422118166105283E-8
0.3412168279376385
0.2584450399436024
6.839755520150974E-16
0.45520927387440074
1.4924102835442008E-5
9.249821060974768E-7
8.260490171632482E-12
1.8640372659892636E-9
0.007826987303853426
3.882238465557136E-6
5.06535217666354E-4
9.488758146533295E-6
0.43547120740276885
-2.6
0.07181119071409517
0.005294756056849911
6.667949633938328E-5
4.966950294179795E-16
6.470519733888933E-4
0.0048854433721256104
0.01947636585549429
1.2955319553683753E-7
4.8774167534939205E-8
0.41118903663663287
0.1880669478628643
5.2446774036326464E-15
0.49091933079869915
4.5277948058732736E-5
3.2387267857621802E-6
4.660734770598343E-11
8.575639191095357E-9
0.015595008493629001
1.2651398542408127E-5
0.0012420546860069204
2.950297306414189E-5
0.370561471767902
-2.4
0.07002331484435864
0.01090506936677868
1.857817021873555E-4
3.845537719509694E-15
0.0015604349626131063
0.0022135712659360368
0.0356810125540007
4.97494738514706E-7
1.956544263360699E-7
0.4654887689409589
0.12856221496001616
3.777927128745412E-14
0.4973541666925923
1.290451942906652E-4
1.0652999086155987E-5
2.470356062278668E-10
3.706252511778699E-8
0.029189939629418335
3.8730353476055457E-5
0.002861069347203632
8.617450216053244E-5
0.29622224880204584
-2.2
0.06757638107283294
0.021099273608675414
4.862618068821634E-4
2.796925716387643E-14
0.0035351575503919912
9.421924688511959E-4
0.061407725458959936
1.7946734964433884E-6
7.373031411918068E-7
0.4950322377412137
0.08256021413879427
2.556495120069227E-13
0.47334520572523575
3.4550435793535204E-4
3.2917440657345645E-5
1.230045890744243E-9
1.5047352836179978E-7
0.05132599446934982
1.113835303987917E-4
0.006191168815182168
2.3645493273633092E-4
0.22244962680373206
-2.0
0.06526039434560513
0.03834980591371902
0.0011956222163409084
1.9110030645498765E-13
0.007523648021274306
3.76740528985988E-4
0.09928084101730976
6.081896166310598E-6
2.610111653766687E-6
0.49455472132139927
0.04980636766678447
1.6251481369530368E-12
0.4232011114798053
8.690040855952164E-4
9.555135007586438E-5
5.753600049438807E-9
5.739073662160109E-7
0.08478092365686038
3.009172556923535E-4
0.012585587769033822
6.095012048641302E-4
0.15692866609936476
-1.8
0.0641048585689198
0.06548101936768616
0.002761686569073507
1.226587204208978E-12
0.015041971474715926
1.4151473187483645E-4
0.15078720055367145
1.9361952409628898E-5
8.680179019953805E-6
0.4641430130357092
0.028226402134335256
9.70504459775056E-12
0.355444837949154
0.0020532730259414573
2.6055786622353265E-4
2.5282185838190585E-8
2.056269758284138E-6
0.13155746897651596
7.63712327297883E-4
0.024034271017883333
0.0014759014297744332
0.10399907579315344
-1.6
0.06520399386716703
0.1050326414837482
0.005992546246501506
7.395917720324138E-12
0.028251248590427623
4.993643762661913E-5
0.2151394748403076
5.790497656496391E-5
2.7117826230467376E-5
0.40920965971089746
0.015027362876080605
5.4445088923413743E-11
0.2804492081551
0.0045575157716777484
6.674644586385088E-4
1.0436290611182722E-7
6.921097583189855E-6
0.19177388669449913
0.001820828757506378
0.04311664900298833
0.0033573508891661104
0.0647460489702164
-1.4
0.06946570283278337
0.1582667698840475
0.01221532413943929
4.1893081539764616E-11
0.049845634357060326
1.6553508521736847E-5
0.2883582125862918
1.6268189660519026E-4
7.958617997350044E-5
0.3389194802477231
0.00751565122554816
2.869303462142317E-10
0.20787048037630707
0.009503120880205117
0.001606233562298566
4.0470102437785084E-7
2.18839871781847E-5
0.2626153007462353
0.004078166960578336
0.0726633877009813
0.007174517646150741
0.03786636857345775
-1.2
0.07736749497176458
0.22403289177603428
0.023391344865087753
2.2292005317711122E-10
0.08261771951284513
5.154887016751398E-6
0.3630790081720615
4.293576038161836E-4
2.1942041935631612E-4
0.2636961452772789
0.003531075933029271
1.4205314331881659E-9
0.14473981597659794
0.018614909182086907
0.003631163817057946
1.4742768233563092E-6
6.500316839621548E-5
0.3378369722503769
0.008580596793883623
0.11503841190434448
0.014402745203959634
0.02080418174355158
-0.9999999999999956
0.08881324661824018
0.29791366580929346
0.042078667928034136
1.1143267697451706E-9
0.1286399532416012
1.5080121482153499E-6
0.4294637884508458
0.0010645245585593041
5.68293929182356E-4
0.19273812323033526
0.0015584902537280549
6.6066577424483275E-9
0.09467597960943099
0.03425407030825213
0.007711512786461498
5.045223007633371E-6
1.8138409165390076E-4
0.40827324585576047
0.016960028687821354
0.17109082033486528
0.027161543277595984
0.010737525673482354
-0.7999999999999954
0.1031480417780182
0.3721565449835625
0.07110913062685714
5.232781304689233E-9
0.18816339438397175
4.1442612411135096E-7
0.47720892357349326
0.0024794126145430534
0.001382692437580858
0.13233906596388145
6.461862602283009E-4
2.886485394489394E-8
0.05817658089231821
0.05921340600048494
0.01538473141317562
1.621953099000806E-5
4.7546708569990267E-4
0.4635016257025822
0.03149142152632716
0.23903810348011312
0.04811939979749391
0.005206122541319606
-0.5999999999999952
0.11932118890966639
0.4367344911551622
0.11288736863706693
2.308390084129151E-8
0.25855384416715876
1.0699069361877814E-7
0.49813511370673236
0.005424984762689412
0.003160346878722927
0.0853621017049741
2.516911488219881E-4
1.1847139839435274E-7
0.03358251170065917
0.09615779492048106
0.02883346588843055
4.898383809682968E-5
0.0011708420644789485
0.4943200166429223
0.05493062528679206
0.3137358555260022
0.08008340973431634
0.002371270793492634
-0.399999999999995
0.13612476559215717
0.4814663253583251
0.16835341318789734
9.56626507145519E-8
0.33375167532058847
2.594785586417541E-8
0.48847501112265934
0.011150769380425564
0.006785791538623775
0.05172478908469987
9.209472630806528E-5
4.5678755166569607E-7
0.01821104010133282
0.1466917005672502
0.05076453059973289
1.3897091757432615E-4
0.0027085245120815264
0.4952468555759746
0.09001054993964538
0.38682788282210195
0.12520495172565632
0.001014622556365171
-0.19999999999999485
0.15241028027647768
0.49862142723332237
0.2358604356197115
3.7241933124625186E-7
0.4047179852625837
5.911715771355962E-9
0.4499809636417988
0.0215311754501769
0.013687460199828949
0.029443467346462308
3.1656155818373764E-5
1.6545183957571853E-6
0.009277114601892579
0.2102244310995784
0.08396155922354041
3.7038350223105477E-4
0.005886048261182656
0.46611368256028124
0.1385571310869037
0.44805154710597633
0.1838895496288403
4.0783497645933265E-4
5.329070518200751E-15
0.1672117999282318
0.4851014275694928
0.31041654573468597
1.3620047310272246E-6
0.4610395146271524
1.2652670812444626E-9
0.3894058990239429
0.03905595312145608
0.02593592966274708
0.015744750077359884
1.022205429071279E-5
5.629703666587754E-6
0.004439639301353384
0.28302019337098056
0.1304539430226225
9.273334759230367E-4
0.012016319353906487
0.41211514165360535
0.20036457733453283
0.48752263773058624
0.2537167782987414
1.5400010582322313E-4
0.2000000000000055
0.179763580569943
0.44335413442494426
0.3837878505106976
4.67930777491656E-6
0.4933786918634563
2.543943534550763E-10
0.31656835974308817
0.06655234100869029
0.046167607059550204
0.007909320695814414
3.100807361206565E-6
1.7995174617948566E-5
0.001995901239566704
0.3579383697217404
0.19041033647084463
0.00218110632108101
0.02304494627865656
0.34229604877664044
0.27218839308019716
0.49833132405901803
0.3288500573135632
5.462785710515344E-5
0.4000000000000057
0.18945758367110166
0.38064976649811966
0.4457529002331791
1.5102234212741146E-5
0.49599719889028177
4.8049549564664347E-11
0.24176257941616175
0.10653591181643056
0.07720216954191796
0.003732494648124753
8.836249791100615E-7
5.4035999631927066E-5
8.429212516724055E-4
0.4252611242134528
0.2610841007988465
0.004819193117394341
0.041518010372941366
0.2670802631522092
0.34735603887375627
0.478517893218417
0.4004084652658256
1.820387752996788E-5
0.6000000000000059
0.19580775724047011
0.3070131429070126
0.4863553837602777
4.578860221989714E-5
0.46841916266741285
8.525655284684048E-12
0.1734471827366198
0.16020841094232674
0.12127694267331458
0.001654686780159113
2.3654718127728688E-7
1.5242875756967837E-4
3.3441946960951574E-4
0.4746349396126568
0.3363000449254048
0.010002957379579163
0.07026741240696209
0.195766465495914
0.4164249098717748
0.43165300832520287
0.4579996795530312
5.698627115745074E-6
0.800000000000006
0.19846431642777937
0.23261888101957662
0.49850540647136693
1.3041578681007013E-4
0.41557238223877613
1.4210940203432321E-12
0.11689663628319491
0.226324303872489
0.1789713730553132
6.891107035146896E-4
5.94872873191874E-8
4.0393103685428677E-4
1.2463863927387035E-4
0.4976457759022208
0.4069396620515412
0.01950469222294466
0.11171924285719996
0.13480046772300944
0.4689808852942392
0.3657867390731133
0.4921343654208355
1.6758423852068088E-6
1.0000000000000062
0.1972729746940306
0.16557301299803873
0.4800015200493028
3.4894705750322154E-4
0.3463500839146385
2.2252276616372887E-13
0.07401050604699569
0.3003541718382386
0.24811070041796607
2.6959931776420273E-4
1.4053585745320599E-8
0.0010055510730964305
4.3638562811277025E-5
0.49015962372849664
0.46258304407725287
0.035727808661168335
0.1668624444161429
0.08719690962440267
0.49616963073019255
0.29119085837273173
0.4967739406642562
4.6296983502767654E-7
1.2000000000000064
0.19233637113716476
0.11071099051927724
0.43418213263215966
8.770927274473954E-4
0.27116929462083966
3.2732764266104875E-14
0.04401911712688283
0.37444905488169705
0.32312012963769043
9.90843756190018E-5
3.1189378825178882E-9
0.002351568522268848
1.435306904933131E-5
0.45353555643407806
0.4939761650945082
0.06147948954557794
0.2341239345155384
0.052986762180342484
0.4931304384494252
0.21776301020560412
0.4710754960720886
1.2015139436187376E-7
1.4000000000000066
0.18403240265223966
0.06954221442139068
0.3689418126660617
0.0020710380327646494
0.19944461758740065
4.5232179074949254E-15
0.024594946197237316
0.4385391836616344
0.3953112090470066
3.4209612908703876E-5
6.502537984037003E-10
0.005166158630852909
4.434817004102023E-6
0.3942228037296118
0.4955401792903579
0.09938268388268495
0.30859552313091176
0.030247554376311915
0.46041560672713716
0.15298437649069335
0.4196418734519606
2.9292843403440244E-8
1.6000000000000068
0.17297395121716835
0.041035798629386375
0.29451024144702925
0.004593960256374807
0.13780365114381346
5.87176861281972E-16
0.012909428626606758
0.4824814979276925
0.4543293969645914
1.1095521976937835E-5
1.2735490173517583E-10
0.010661895743516007
1.2872509731456739E-6
0.32190575339032534
0.46699082485756355
0.15092034307028102
0.3821114254593285
0.016220705705859047
0.40382653980702693
0.10096402256634812
0.35117510035559424
6.708892600643215E-9
1.800000000000007
0.15993067946382158
0.022747509711256764
0.22085107686721875
0.009572888220560714
0.08944492892967112
7.160559380882354E-17
0.006365386218581205
0.4986657249187041
0.49052273675619384
3.38067880449436E-6
2.3431766304689008E-11
0.020670820728047944
3.5100018751412713E-7
0.24692913226855343
0.41342279338508253
0.2152987168432614
0.4444746747854804
0.008171575893470773
0.3327333050057361
0.06259545556095528
0.2760738095016226
1.443433379011997E-9
2.000000000000007
0.14574862012938752
0.01184571891468719
0.15558052603366068
0.018739385129911267
0.05453901838349248
8.203167187885369E-18
0.0029484861573541845
0.4841667580904762
0.497512551492854
9.676463849082882E-7
4.049962335997649E-12
0.03764762677022977
8.991001352941289E-8
0.17793956172568312
0.34382470663204673
0.2885304772382123
0.48569162287930695
0.0038672165094772205
0.25754568906202063
0.03645654925264589
0.2038840705928892
2.917421931450533E-10
2.2000000000000064
0.1312889825401171
0.00579489514397374
0.1029597655029712
0.034460714740179116
0.03124032229417955
8.828212334569712E-19
0.0012830099419862753
0.4416080820052118
0.47402968136563167
2.6018726801137967E-7
6.575874805645896E-13
0.06441308253207786
2.1635422093817178E-8
0.12045623861726204
0.268618760026855
0.3632440770644835
0.498575346612296
0.0017192845105353553
0.18727029330739212
0.01994641762330486
0.1414483801753779
5.5393440186108805E-11
2.4000000000000075
0.11737923273538692
0.0026630930167331514
0.06400832298690044
0.059531910420454554
0.016810486769059756
8.92525375222725E-20
5.244662488526635E-4
0.3783865166420753
0.4242908096252546
6.572218573955092E-8
1.0030271372423771E-13
0.1035302247357634
4.8907929413382245E-9
0.07660245242058852
0.19714791745472174
0.4295977359868676
0.480792386297294
7.180482181480013E-4
0.1279204832643036
0.010252053211969607
0.09218690952117888
9.880388620102772E-12
2.6000000000000085
0.10475186491813092
0.0011496977579344182
0.03738194961936744
0.09661219589146706
0.008497705336120912
8.47666400921603E-21
2.014010189410371E-4
0.30457263934039813
0.3567617689370373
1.5595329347458504E-8
1.4372371184845605E-14
0.15632082026498287
1.0386034859480503E-9
0.04576280587402719
0.13592665905826273
0.47728965479260127
0.43555294881634293
2.81718950618468E-4
0.08208577201463102
0.004950093366475776
0.056441315605938194
1.6555652833716103E-12
2.800000000000008
0.09395429263549265
4.662701656501879E-4
0.02050897906178112
0.14728912488640794
0.004035335432808755
7.562858261409711E-22
7.265447628791937E-5
0.23030465493252555
0.2818055793846451
3.476431672530249E-9
1.9346426666846217E-15
0.22172930139199057
2.0719385457862002E-10
0.02568261316918084
0.0880387131832578
0.4981483008447603
0.3706644454125526
1.0383293018807711E-4
0.04948257340420132
0.002245290424244219
0.03246247256097533
2.6060046704947205E-13
3.000000000000007
0.08524214260655952
1.7764302762306609E-4
0.010570188456584697
0.21094343087002257
0.0018001728562739123
6.338748896409432E-23
2.4621793503479297E-5
0.1635954196697265
0.209111280238361
7.279967492659258E-10
2.4464125330273183E-16
0.2954513283676545
3.882938919384852E-11
0.013540115568256204
0.05356724668076058
0.48841824673473194
0.2963312796464721
3.595098161874371E-5
0.0280216215257612
9.567274897100333E-4
0.017539720063824298
3.853547028873869E-14
3.200000000000008
0.07849813227289168
6.357923523286881E-5
0.005117737592604691
0.2838036153512859
7.544064137081711E-4
4.990886914792644E-24
7.838511279430577E-6
0.10916820241156801
0.1457679108031561
1.4321278059897083E-10
2.906131215432095E-17
0.36983277831713357
6.835980933618935E-12
0.006705978037387441
0.030618334112492273
0.44986447825646086
0.22255156764672746
1.1693458827349237E-5
0.014907020312711291
3.8296631525554535E-4
0.008902670804840085
5.353067653765027E-15
3.4000000000000092
0.07322831561046571
2.1376619544312676E-5
0.0023277154681044425
0.35869591999913986
2.969977299470288E-4
3.691547883748288E-25
2.344250867763212E-6
0.06843492144281078
0.09545594747468975
2.6466146299956132E-11
3.2430776002882165E-18
0.4348920294429945
1.1305706880562088E-12
0.0031200278971322605
0.016440703080568934
0.38924954982245064
0.15701473625177684
3.572990187872215E-6
0.007449806942519983
1.4400895354201417E-4
0.004244969425905436
6.985562710154709E-16
3.6000000000000085
0.06866669028425669
6.7517966053704E-6
9.94576895054387E-4
0.42588421621352673
1.0983923074024346E-4
2.565050250235711E-26
6.586143336001502E-7
0.0403010147810006
0.058721974611040596
4.594690215623333E-12
3.399821036010645E-19
0.48041227006095605
1.7565122989962959E-13
0.0013636767075055041
0.008293079126658385
0.31639610656556444
0.10406549758508916
1.0255982027792328E-6
0.0034974844191131608
5.0871554387793615E-5
0.0019014522048411654
8.563605533938352E-17
3.800000000000008
0.0639749911601833
2.0033474097421625E-6
3.992118592216623E-4
0.4750214793699185
3.8160884325434136E-5
1.6743253157179913E-27
1.738260517213414E-7
0.02229516986131877
0.0339355509186672
7.493390358611102E-13
3.348199812333378E-20
0.49854381057341074
2.56366493633818E-14
5.599135308176386E-4
0.003929776281501021
0.24159655494140153
0.06479324203555042
2.765535728655029E-7
0.0015424928278077686
1.688173644561062E-5
8.001158156540842E-4
9.862079688772778E-18
4.000000000000009
0.05847819677516397
5.584056214422462E-7
1.505307056015119E-4
0.4977273957602402
1.2454777483752016E-5
1.026692536538264E-28
4.30978046409758E-8
0.011586764605485259
0.018423229068824813
1.1480401419110406E-13
3.0975852927219447E-21
0.4860144273964295
3.5150210231902906E-15
2.1596685195860991E-4
0.001749348914875187
0.17330334231339492
0.037897385718567045
7.005479893725573E-8
6.390680609937497E-4
5.262787310340818E-6
3.16283758935068E-4
1.0669325274984973E-18
4.20000000000001
0.05185280103401381
1.46217680195001E-7
5.332162243725768E-5
0.48992143201078014
3.818652246906948E-6
5.9142211633160316E-30
1.0038111735165476E-8
0.005656791575081706
0.009395787692538146
1.6523132483116732E-14
2.6921036346467914E-22
0.44509384544821884
4.527424170607214E-16
7.825459133864363E-5
7.315460129570368E-4
0.1167830288661034
0.02082310014313406
1.6670671863862918E-8
2.4872971247331293E-4
1.5412429984882314E-6
1.1745121750104389E-4
1.0843313113109335E-19
4.400000000000009
0.04419905964807744
3.5967196858610415E-8
1.774345577128557E-5
0.45302057403643003
1.0998686987413156E-6
3.200451954716131E-31
2.196369973103851E-9
0.002594387037125455
0.004501499773562611
2.2340057743097394E-15
2.1979452496202148E-23
0.3829222583237568
5.478113652942663E-17
2.6637235643535195E-5
2.873845723984673E-4
0.07392802873275621
0.010748258884594736
3.726704092407896E-9
9.09420662533131E-5
4.240167465995686E-7
4.097271561529333E-5
1.0352461800031325E-20
4.6000000000000085
0.03597520483855461
8.311317812399076E-9
5.546635872304397E-6
0.39351927419277394
2.9759674022729573E-7
1.6269779149673978E-32
4.514561409258582E-10
0.0011177790070356164
0.0020259926490267172
2.8374798913177554E-16
1.685770930465837E-24
0.3094754683600448
6.226836840224841E-18
8.5177535936053E-6
1.0605761285303583E-4
0.043963788720467564
0.005211796428114282
7.826242710500395E-10
3.123622639257601E-5
1.0958509717401085E-7
1.3427296236437375E-5
9.284999175013796E-22
4.80000000000001
0.027827798900440917
1.8042212007204402E-9
1.6288372727639395E-6
0.32112246279993784
7.564358300429006E-8
7.76977778938975E-34
8.717304710581145E-11
4.5241159487156987E-4
8.565941276330375E-4
3.385617481314999E-17
1.2146099820724095E-25
0.23496243867366032
6.649064242372305E-19
2.5586895005059664E-6
3.676857850881963E-5
0.024560527662235314
0.0023740691484333354
1.5439680503849558E-10
1.0078801713695923E-5
2.660581577395903E-8
4.133700302220387E-6
7.823060649603428E-23
5.000000000000011
0.020389774150558763
3.679309052374137E-10
4.493475067467359E-7
0.24616820458777378
1.8062281597691984E-8
3.4857170762841476E-35
1.581268005488683E-11
1.7201570506382208E-4
3.402271092356734E-4
3.794893386478941E-18
8.221133985757273E-27
0.1675819533024171
6.669759385291574E-20
7.220490426806088E-7
1.1974804376884124E-5
0.012889523683690084
0.0010159115071689314
2.8614087503145396E-11
3.0550325045395195E-6
6.068177236441374E-9
1.1954901372719373E-6
6.191959681321672E-24
5.20000000000001
0.014119772099463357
7.048542960248933E-11
1.1645109289132117E-7
0.1772759515988484
4.051629463229366E-9
1.4690354106798895E-36
2.6945443461290747E-12
6.144111206600486E-5
1.2694609102160298E-4
3.995929939347409E-19
5.2273695226247E-28
0.11228264901766921
6.285160890765295E-21
1.9141342793087148E-7
3.663672203997514E-6
0.006354664690118223
4.0838989703341255E-4
4.981705929815338E-12
8.699200977084374E-7
1.3001591874663887E-9
3.247952007606755E-7
4.604008717412733E-25
5.400000000000009
0.00922695641731718
1.2684957837950986E-11
2.8350543957403164E-8
0.11992902067890024
8.537749873914172E-10
5.816061494873438E-38
4.313420342097268E-13
2.061610561419857E-5
4.4496550505363045E-5
3.952689916154009E-20
3.12241965088978E-29
0.07067318224231024
5.563898861834891E-22
4.766885278377991E-8
1.0529830650685807E-6
0.0029430999078619764
1.542235496643034E-4
8.147659462819076E-13
2.327016637625041E-7
2.616925724689814E-10
8.289528024457342E-8
3.21588677446E-26
5.60000000000001
0.005683928495690103
2.1445456728014057E-12
6.483892685308233E-9
0.07621761315696446
1.690105164911729E-10
2.1631284535283278E-39
6.486565735783584E-14
6.498465056189925E-6
1.4651765243951225E-5
3.673027850171354E-21
1.7520880858563747E-30
0.04178815902201864
4.626990942678517E-23
1.1152021404266195E-8
2.8430386150908946E-7
0.0012804834379337932
5.471205089277322E-5
1.2518268013194846E-13
5.847580473927065E-8
4.9481500216038573E-11
1.987497539617068E-8
2.1101919252800203E-27
5.800000000000011
0.00329824917935448
3.4059492042059624E-13
1.3930504192799334E-9
0.04550331070752603
3.142972820761861E-11
7.557744834128628E-41
9.16356490000328E-15
1.924294508289541E-6
4.5322113445224635E-6
3.206360072177516E-22
9.235856919283436E-32
0.023211778911532786
3.614720038464666E-24
2.4509197040833833E-9
7.211086833817074E-8
5.233587878553004E-4
1.8233577592203503E-5
1.8068088816583632E-14
1.3804145268972965E-8
8.789231738724283E-12
4.476514444466644E-9
1.3007677442511032E-28
6.000000000000011
0.0018019541511603541
5.08156667493393E-14
2.81160590832002E-10
0.025520386221924252
5.4906545029246546E-12
2.480611399146781E-42
1.2161039282203665E-15
5.35289701195418E-7
1.3170035643152454E-6
2.6294018069485594E-23
4.573567056260555E-33
0.012112121168551135
2.6528169196622273E-25
5.060124090089446E-10
1.718206220112928E-8
2.009470667585609E-4
5.7084391921263935E-6
2.449834706232827E-15
3.061254450958346E-9
1.4666131245032604E-12
9.471744523519124E-10
7.532413550521545E-30
-
2.The Chi-square goodness of fit test The standard statistical
test for testing Normality are:1.The Kolmogorov-Smirnov test.
-
The empirical distribution function is defined below for n
random observationsThe Kolmogorov-Smirnov testThe
Kolmogorov-Smirnov uses the empirical cumulative distribution
function as a tool for testing the goodness of fit of a
distribution. Fn(x) = the proportion of observations in the sample
that are less than or equal to x.
-
Let F0(x) denote the hypothesized cumulative distribution
function of the population (Normal population if we were testing
normality)If F0(x) truly represented distribution of observations
in the population than Fn(x) will be close to F0(x) for all values
of x.
-
The Kolmogorov-Smirinov test statistic is := the maximum
distance between Fn(x) and F0(x). If F0(x) does not provide a good
fit to the distributions of the observation - Dn will be large.
Critical values for are given in many texts
-
Let fi denote the observed frequency in each of the class
intervals of the histogram.
The Chi-square goodness of fit test The Chi-square test uses the
histogram as a tool for testing the goodness of fit of a
distribution. Let Ei denote the expected number of observation in
each class interval assuming the hypothesized distribution.
-
m = the number of class intervals used for constructing the
histogram).
The hypothesized distribution is rejected if the statistic:
is large. (greater than the critical value from the chi-square
distribution with m - 1 degrees of freedom.
-
Note. The in the above tests it is assumed that the residuals
are independent with a common variance of s2. This is not
completely accurate for this reason:Although the theoretical random
errors ei are all assumed to be independent with the same variance
s2, the residuals are not independent and they also do not have the
same variance.
-
They will however be approximately independent with common
variance if the sample size is large relative to the number of
parameters in the model.It is important to keep this in mind when
judging residuals when the number of observations is close to the
number of parameters in the model.
-
Time Sequence Plot The residuals should exhibit a pattern of
independence. If the data was collected in time there could be a
strong possibility that the random departures from the model are
autocorrelated.
-
Namely the random departures for observations that were taken at
neighbouring points in time are autocorrelated.This autocorrelation
can sometimes be seen in a time sequence plot.The following three
graphs show a sequence of residuals that are respectively i)
positively autocorrelated , ii) independent and iii) negatively
autocorrelated.
-
i) Positively auto-correlated residuals
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
-3.8131292967591435
1.0
-1.3848894013790414
2.0469269657041878
2.0
2.294309979333775
0.45207571020000614
3.0
-1.5721877844043775
-1.979055923584383
4.0
-2.3974421310413163
-0.6162917998153716
5.0
-2.04982643481344
-1.4951638149796054
6.0
-4.199441718810704
-2.8537942853290588
7.0
-2.523142484278651
0.045272372517501935
8.0
-0.5006497758586193
-0.541394911124371
9.0
2.4359878807445057
2.9232433007564396
10.0
4.780823473993223
2.1499045033124276
11.0
1.9658687051560264
0.030954652174841613
12.0
-0.7412004379148129
-0.7690596248721704
13.0
-0.9959931048797444
-0.303839442494791
14.0
1.192831859953003
1.466287358198315
15.0
0.5005495040677488
-0.8191091183107346
16.0
1.1439142326707952
1.8811124391504563
17.0
2.1946011656837072
0.5015999704482965
18.0
-1.6781827980594244
-2.1296227714628913
19.0
-2.7697392397385556
-0.8530787454219535
20.0
1.289251486014109
2.0570223568938673
21.0
3.867242412525229
2.0159222913207486
22.0
4.378122866910417
2.563792804721743
23.0
3.380217094672844
1.0728035704232752
24.0
2.4010059860302135
1.4354827726492658
25.0
4.694918061431963
3.4029835660476238
26.0
5.7579636631999165
2.695278453757055
27.0
1.0829771781573072
-1.3427734302240424
28.0
-3.2550001378695015
-2.0465040506678633
29.0
-0.4978683136869222
1.3439853319141548
30.0
-0.7606151939398842
-1.9702019926626235
31.0
-5.366886398405768
-3.593704605009407
32.0
-3.0741675800527446
0.16016656445572153
33.0
-1.96417249753722
-2.1083224055473693
34.0
-4.0489544517186005
-2.151464286725968
35.0
-3.586846105463337
-1.6505282474099658
36.0
-0.4647858986572828
1.0206895240116864
37.0
2.2165286281961016
1.2979080565855838
38.0
1.5931423149595503
0.4250250640325248
39.0
-2.0358284018584527
-2.418350959487725
40.0
0.4915050340059679
2.6680208975449204
41.0
2.4771070457063615
0.07588823791593313
42.0
-0.7320832082768902
-0.80038262240123
43.0
-0.0814425220596604
0.6389018381014466
44.0
-0.14392117009265348
-0.7189328243839554
45.0
-3.556685896910494
-2.9096463549649343
46.0
-1.209506081067957
1.4091756384004839
47.0
4.2788674363691825
3.010609361808747
48.0
4.506086952460464
1.7965385268325917
49.0
-2.073080850095721
-3.6899655242450535
50.0
-3.6417250157683156
-0.32075604394776747
-
ii) Independent residuals
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
-3.306922735646367
1.0
-0.13322505765245296
-0.13322505765245296
2.0
-0.6969753485464025
-0.6969753485464025
3.0
2.732704160735011
2.732704160735011
4.0
-0.2537376531108748
-0.2537376531108748
5.0
3.691366146085784
3.691366146085784
6.0
-0.8638517101644538
-0.8638517101644538
7.0
0.5020137905376032
0.5020137905376032
8.0
2.9104103305144235
2.9104103305144235
9.0
-1.8366154108662158
-1.8366154108662158
10.0
-0.42434294300619513
-0.42434294300619513
11.0
0.5949550541117787
0.5949550541117787
12.0
-1.2282521311135497
-1.2282521311135497
13.0
0.6066238711355254
0.6066238711355254
14.0
-2.346628207305912
-2.346628207305912
15.0
-4.090634320164099
-4.090634320164099
16.0
0.9111226972891018
0.9111226972891018
17.0
1.2959344530827366
1.2959344530827366
18.0
-0.23471557142329402
-0.23471557142329402
19.0
3.4192817111033946
3.4192817111033946
20.0
-0.6747632141923532
-0.6747632141923532
21.0
-0.0026329871616326272
-0.0026329871616326272
22.0
0.2726255843299441
0.2726255843299441
23.0
-2.1584082787740044
-2.1584082787740044
24.0
0.46552713683922775
0.46552713683922775
25.0
0.06414211384253576
0.06414211384253576
26.0
-0.8128972694976255
-0.8128972694976255
27.0
-4.103640094399452
-4.103640094399452
28.0
0.2521460373827722
0.2521460373827722
29.0
-0.7082667252689134
-0.7082667252689134
30.0
0.9214363672072068
0.9214363672072068
31.0
2.991055225720629
2.991055225720629
32.0
-0.5736728780902922
-0.5736728780902922
33.0
-2.4717928681639023
-2.4717928681639023
34.0
-0.9373434295412153
-0.9373434295412153
35.0
0.09074028639588505
0.09074028639588505
36.0
-0.44186890590935946
-0.44186890590935946
37.0
-2.8558224585140124
-2.8558224585140124
38.0
-2.880233296309598
-2.880233296309598
39.0
3.8568396121263504
3.8568396121263504
40.0
-4.397115844767541
-4.397115844767541
41.0
-1.2978080121683888
-1.2978080121683888
42.0
5.5438431445509195
5.5438431445509195
43.0
2.0406514522619545
2.0406514522619545
44.0
2.2628591977991164
2.2628591977991164
45.0
2.782435331027955
2.782435331027955
46.0
-2.473916538292542
-2.473916538292542
47.0
-0.6019445208949037
-0.6019445208949037
48.0
-3.0258433980634436
-3.0258433980634436
49.0
1.2795499060302973
1.2795499060302973
50.0
-1.4287070371210575
-1.4287070371210575
-
iii) Negatively auto-correlated residuals
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
-0.9
0.2088040673697833
1.0
0.7658543381694471
0.9537779988022521
2.0
-3.311280124762561
-2.4528799258405343
3.0
3.2642474252497777
1.056655491993297
4.0
-1.9678914213727694
-1.0169014785788022
5.0
-0.7538433237641584
-1.6690546544850804
6.0
5.319280262483517
3.8171310734469444
7.0
-2.6139327928831335
0.8214851732191164
8.0
1.5800658275111346
2.3194024834083393
9.0
-2.6572333808871917
-0.5697711458196864
10.0
0.24065502657322213
-0.2721390046644956
11.0
-1.2370737749733962
-1.4819988791714422
12.0
1.9656613403640222
0.6318623491097242
13.0
-1.96097153093433
-1.3922954167355783
14.0
0.5481938387674745
-0.704872036294546
15.0
3.4693616726144683
2.834976839949377
16.0
-2.530444817239186
0.021034338715253398
17.0
2.7536000288819196
2.7725309337256476
18.0
-1.398709173372481
1.0965686669806018
19.0
2.37374388234457
3.3606556826271117
20.0
-6.204674718901515
-3.1800846045371145
21.0
1.9847025214403402
-0.8773736226430628
22.0
0.8103636446321616
0.020727384253405035
23.0
-0.9610002962290309
-0.9423456504009664
24.0
2.128806954715401
1.2806958693545312
25.0
-2.8761050998582505
-1.7234788174391724
26.0
1.884726771095302
0.3335958354000468
27.0
-1.91078311218007
-1.610546860320028
28.0
3.7049617276352365
2.2554695533472113
29.0
-3.2489751902176067
-1.2190525922051165
30.0
1.4232671219360782
0.3261197889514733
31.0
1.1417430414439877
1.4352508515003137
32.0
-0.7751327757432591
0.5165929906070232
33.0
-0.07451035344274715
0.39042333810357377
34.0
0.8216261448978912
1.1730071491911076
35.0
-1.4390175238077063
-0.38331108953570947
36.0
-0.3026029844477307
-0.6475829650298692
37.0
3.3289893508481327
2.7461646823212504
38.0
-1.0144785846932791
1.4570696293958463
39.0
-1.2494056136347353
0.061957052821526304
40.0
-3.080167743974016
-3.0244063964346424
41.0
5.890818101761397
3.168852344970219
42.0
-0.769233338360209
2.082733772112988
43.0
-2.759539711405523
-0.885079316503834
44.0
2.5120966711256187
1.7155252862721682
45.0
-4.182544216746464
-2.638571459101513
46.0
4.35619404015597
1.9814797269646078
47.0
1.4903762348694727
3.2737079891376197
48.0
-7.989958248799667
-5.043621058575809
49.0
1.8652626749826595
-2.6739962777355686
50.0
0.24590099201304838
-2.1606956579489633
-
There are several statistics and statistical tests that can also
pick out autocorrelation amongst the residuals. The most common
are: ii)The autocorrelation functioni)The Durbin Watson statistic
iii)The runs test
-
The Durbin Watson statistic :If the residuals are serially
correlated the differences, ei - ei+1, will be stochastically
small. Hence a small value of the Durbin-Watson statistic will
indicate positive autocorrelation. Large values of the
Durbin-Watson statistic on the other hand will indicate negative
autocorrelation. Critical values for this statistic, can be found
in many statistical textbooks.The Durbin-Watson statistic which is
used frequently to detect serial correlation is defined by the
following formula:
-
The autocorrelation function:This statistic measures the
correlation between residuals the occur a distance k apart in time.
One would expect that residuals that are close in time are more
correlated than residuals that are separated by a greater distance
in time. If the residuals are independent than rk should be close
to zero for all values of k A plot of rk versus k can be very
revealing with respect to the independence of the residuals. Some
typical patterns of the autocorrelation function are given
below:The autocorrelation function at lag k is defined by :
-
This statistic measures the correlation between residuals the
occur a distance k apart in time. One would expect that residuals
that are close in time are more correlated than residuals that are
separated by a greater distance in time. If the residuals are
independent than rk should be close to zero for all values of k A
plot of rk versus k can be very revealing with respect to the
independence of the residuals.
-
Some typical patterns of the autocorrelation function are given
below:Auto correlation pattern for independent residuals
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
1.0
-0.038773132079313655
2.0
-0.0314914959556063
3.0
0.02484996999055511
4.0
0.10438702724172799
5.0
-0.012173708989394072
6.0
-0.005164234839185156
7.0
0.0848813943625828
8.0
-0.07699421507413717
9.0
-0.10735449310118383
10.0
-0.025644996726441605
11.0
-0.05668110038300256
12.0
-0.11225511146813005
13.0
0.09538202149474273
14.0
0.04966484986834985
15.0
0.06132230706384689
16.0
0.049209424040327576
17.0
0.08858525005712181
18.0
0.04857256099330698
19.0
0.0839532652678372
20.0
0.05784994542955246
21.0
-0.05285418636526629
22.0
-0.028132543280207756
23.0
0.013124195079626588
24.0
-0.05444837298728089
25.0
0.06536064191436708
26.0
-0.040304009000919905
27.0
-0.022840648034389233
28.0
-0.015172595736657968
29.0
0.04276902028209406
30.0
0.08737994044577135
-
Various Autocorrelation patterns for serially correlated
residuals
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
1.0
-0.6
2.0
0.36
3.0
-0.216
4.0
0.1296
5.0
-0.07776
6.0
0.046655999999999996
7.0
-0.027993599999999997
8.0
0.016796159999999997
9.0
-0.010077695999999999
10.0
0.006046617599999999
11.0
-0.0036279705599999994
12.0
0.0021767823359999995
13.0
-0.0013060694015999995
14.0
7.836416409599997E-4
15.0
-4.701849845759998E-4
16.0
2.8211099074559984E-4
17.0
-1.692665944473599E-4
18.0
1.0155995666841595E-4
19.0
-6.0935974001049565E-5
20.0
3.656158440062974E-5
21.0
-2.1936950640377843E-5
22.0
1.3162170384226705E-5
23.0
-7.897302230536022E-6
24.0
4.738381338321613E-6
25.0
-2.843028802992968E-6
26.0
1.7058172817957808E-6
27.0
-1.0234903690774685E-6
28.0
6.140942214464811E-7
29.0
-3.6845653286788867E-7
30.0
2.210739197207332E-7
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
1.0
0.6
2.0
0.36
3.0
0.216
4.0
0.1296
5.0
0.07776
6.0
0.046655999999999996
7.0
0.027993599999999997
8.0
0.016796159999999997
9.0
0.010077695999999999
10.0
0.006046617599999999
11.0
0.0036279705599999994
12.0
0.0021767823359999995
13.0
0.0013060694015999995
14.0
7.836416409599997E-4
15.0
4.701849845759998E-4
16.0
2.8211099074559984E-4
17.0
1.692665944473599E-4
18.0
1.0155995666841595E-4
19.0
6.0935974001049565E-5
20.0
3.656158440062974E-5
21.0
2.1936950640377843E-5
22.0
1.3162170384226705E-5
23.0
7.897302230536022E-6
24.0
4.738381338321613E-6
25.0
2.843028802992968E-6
26.0
1.7058172817957808E-6
27.0
1.0234903690774685E-6
28.0
6.140942214464811E-7
29.0
3.6845653286788867E-7
30.0
2.210739197207332E-7
-
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
1.0
0.6876644452187053
2.0
0.2232647784358995
3.0
-0.1897750616705145
4.0
-0.4223119274199373
5.0
-0.4437053124999999
6.0
-0.3051203675609047
7.0
-0.09906376828614406
8.0
0.08420420304322236
9.0
0.18738204572834055
10.0
0.19687440434072256
11.0
0.13538352803872608
12.0
0.04395512026483116
13.0
-0.03736185222510641
14.0
-0.0831424091567826
15.0
-0.08735421910125163
16.0
-0.06007039061577546
17.0
-0.019503120373081995
18.0
0.01657765231711965
19.0
0.036890728636913075
20.0
0.038759531084514326
21.0
0.026653551440169714
22.0
0.008653638119863463
23.0
-0.007355592401883918
24.0
-0.016368612278194212
25.0
-0.01719780985220789
26.0
-0.011826322370995327
27.0
-0.0038396652062359315
28.0
0.003263715425300528
29.0
0.007262840226087496
30.0
0.007630759594789478
-
Comment:It can be shown that the Durbin Watson statistic, D, is
equivalent to the autocorrelation at lag 1, r1.
-
The runs test:This test uses the fact that the residuals will
oscillate about zero at a normal rate if the random departures are
independent. If the residuals oscillate slowly about zero, this is
an indication that there is a positive autocorrelation amongst the
residuals. If the residuals oscillate at a frequent rate about
zero, this is an indication that there is a negative
autocorrelation amongst the residuals.
-
In the runs test, one observes the time sequence of the sign of
the residuals: + + + - - + + - - - + + +and counts the number of
runs (i.e. the number of periods that the residuals keep the same
sign). This should be low if the residuals are positively
correlated and high if negatively correlated.
-
Plot Against fitted values and the Predictor Variables XijIf we
"step back" from this diagram and the residuals behave in a manner
consistent with the assumptions of the model we obtain the
impression of a horizontal "band " of residuals which can be
represented by the diagram below.
Sheet: Model
Sheet: Variance
Sheet: Ideal
Sheet: Independent
Sheet: Positive Autocorr
Sheet: negative auto corr
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
-0.5
0.06333333333333316
0.20000000000000004
0.16333333333333316
0.2633333333333332
-0.48
0.04373333333333315
0.2
0.14373333333333316
0.24373333333333316
-0.46
0.02493333333333317
0.2
0.12493333333333317
0.22493333333333318
-0.44
0.006933333333333153
0.2
0.10693333333333316
0.20693333333333316
-0.42
-0.010266666666666868
0.2
0.08973333333333314
0.18973333333333314
-0.4
-0.02666666666666681
0.2
0.0733333333333332
0.1733333333333332
-0.38
-0.04226666666666684
0.2
0.057733333333333164
0.15773333333333317
-0.36
-0.05706666666666685
0.2
0.04293333333333316
0.14293333333333316
-0.34
-0.07106666666666682
0.2
0.028933333333333186
0.12893333333333318
-0.32
-0.08426666666666684
0.2
0.01573333333333317
0.11573333333333317
-0.3
-0.09666666666666685
0.2
0.0033333333333331605
0.10333333333333317
-0.28
-0.10826666666666683
0.2
-0.008266666666666825
0.09173333333333318
-0.26
-0.11906666666666683
0.2
-0.01906666666666683
0.08093333333333318
-0.24
-0.12906666666666683
0.2
-0.029066666666666838
0.07093333333333317
-0.22
-0.13826666666666684
0.2
-0.03826666666666684
0.06173333333333317
-0.2
-0.14666666666666683
0.2
-0.04666666666666683
0.05333333333333318
-0.18
-0.15426666666666683
0.2
-0.05426666666666684
0.04573333333333317
-0.16
-0.16106666666666686
0.2
-0.06106666666666684
0.03893333333333317
-0.14
-0.16706666666666684
0.2
-0.06706666666666683
0.032933333333333176
-0.12
-0.17226666666666685
0.2
-0.07226666666666684
0.027733333333333166
-0.09999999999999964
-0.17666666666666692
0.2
-0.07666666666666691
0.023333333333333095
-0.07999999999999963
-0.1802666666666669
0.2
-0.0802666666666669
0.019733333333333103
-0.05999999999999961
-0.18306666666666688
0.2
-0.08306666666666689
0.01693333333333312
-0.03999999999999959
-0.18506666666666688
0.2
-0.08506666666666687
0.014933333333333132
-0.019999999999999574
-0.18626666666666686
0.2
-0.08626666666666685
0.013733333333333153
4.440892098500626E-16
-0.18666666666666684
0.2
-0.08666666666666684
0.01333333333333317
0.020000000000000462
-0.1862666666666668
0.2
-0.08626666666666681
0.013733333333333195
0.04000000000000048
-0.18506666666666682
0.2
-0.0850666666666668
0.014933333333333201
0.0600000000000005
-0.18306666666666677
0.2
-0.08306666666666677
0.01693333333333323
0.08000000000000052
-0.18026666666666674
0.2
-0.08026666666666675
0.019733333333333256
0.10000000000000053
-0.17666666666666675
0.2
-0.07666666666666673
0.023333333333333275
0.12000000000000055
-0.1722666666666667
0.2
-0.0722666666666667
0.027733333333333304
0.14000000000000057
-0.1670666666666667
0.2
-0.06706666666666668
0.03293333333333333
0.1600000000000006
-0.16106666666666664
0.2
-0.061066666666666644
0.03893333333333336
0.1800000000000006
-0.1542666666666666
0.2
-0.054266666666666616
0.04573333333333339
0.20000000000000062
-0.14666666666666658
0.2
-0.046666666666666586
0.05333333333333342
0.22000000000000064
-0.13826666666666657
0.2
-0.03826666666666655
0.06173333333333345
0.24000000000000066
-0.12906666666666652
0.2
-0.02906666666666652
0.07093333333333349
0.2600000000000007
-0.11906666666666649
0.2
-0.019066666666666482
0.08093333333333352
0.2800000000000007
-0.10826666666666646
0.2
-0.00826666666666645
0.09173333333333356
0.3000000000000007
-0.09666666666666641
0.2
0.0033333333333335907
0.1033333333333336
0.32000000000000073
-0.08426666666666638
0.2
0.015733333333333627
0.11573333333333363
0.34000000000000075
-0.07106666666666633
0.2
0.028933333333333672
0.12893333333333368
0.36000000000000076
-0.057066666666666294
0.2
0.04293333333333371
0.14293333333333372
0.3800000000000008
-0.04226666666666626
0.2
0.05773333333333375
0.15773333333333375
0.4000000000000008
-0.0266666666666662
0.2
0.0733333333333338
0.1733333333333338
0.4200000000000008
-0.010266666666666147
0.2
0.08973333333333386
0.18973333333333386
0.44000000000000083
0.006933333333333902
0.2
0.10693333333333391
0.2069333333333339
0.46000000000000085
0.024933333333333946
0.2
0.12493333333333395
0.22493333333333396
0.48000000000000087
0.043733333333333985
0.2
0.143733333333334
0.243733333333334
0.5000000000000009
0.06333333333333405
0.20000000000000004
0.16333333333333405
0.2633333333333341
0.08666666666666684
-0.1
0.09999999999999999
11.538461538461515
0.08333333333333333
0.09333333333333341
-0.5
-0.1
0.2
0.1
-0.48
-0.10800000000000001
0.21600000000000003
0.10800000000000001
-0.46
-0.116
0.232
0.116
-0.44
-0.124
0.248
0.124
-0.42
-0.132
0.264
0.132
-0.4
-0.14
0.28
0.14
-0.38
-0.14800000000000002
0.29600000000000004
0.14800000000000002
-0.36
-0.156
0.312
0.156
-0.34
-0.164
0.328
0.164
-0.32
-0.17200000000000001
0.34400000000000003
0.17200000000000001
-0.3
-0.18
0.36
0.18
-0.28
-0.188
0.376
0.188
-0.26
-0.196
0.392
0.196
-0.24
-0.20400000000000001
0.40800000000000003
0.20400000000000001
-0.22
-0.21200000000000002
0.42400000000000004
0.21200000000000002
-0.2
-0.22
0.44
0.22
-0.18
-0.228
0.456
0.228
-0.16
-0.23600000000000002
0.47200000000000003
0.23600000000000002
-0.14
-0.24400000000000002
0.48800000000000004
0.24400000000000002
-0.12
-0.252
0.504
0.252
-0.09999999999999964
-0.26
0.52
0.26
-0.07999999999999963
-0.268
0.536
0.268
-0.05999999999999961
-0.276
0.552
0.276
-0.03999999999999959
-0.28400000000000003
0.5680000000000001
0.28400000000000003
-0.019999999999999574
-0.29200000000000004
0.5840000000000001
0.29200000000000004
4.440892098500626E-16
-0.3
0.6
0.3
0.020000000000000462
-0.30800000000000005
0.6160000000000001
0.30800000000000005
0.04000000000000048
-0.316
0.632
0.316
0.0600000000000005
-0.324
0.648
0.324
0.08000000000000052
-0.332
0.664
0.332
0.10000000000000053
-0.34
0.68
0.34
0.12000000000000055
-0.348
0.696
0.348
0.14000000000000057
-0.356
0.712
0.356
0.1600000000000006
-0.364
0.728
0.364
0.1800000000000006
-0.372
0.744
0.372
0.20000000000000062
-0.38
0.76
0.38
0.22000000000000064
-0.388
0.776
0.388
0.24000000000000066
-0.396
0.792
0.396
0.2600000000000007
-0.404
0.808
0.404
0.2800000000000007
-0.41200000000000003
0.8240000000000001
0.41200000000000003
0.3000000000000007
-0.42
0.84
0.42
0.32000000000000073
-0.42800000000000005
0.8560000000000001
0.42800000000000005
0.34000000000000075
-0.43600000000000005
0.8720000000000001
0.43600000000000005
0.36000000000000076
-0.44400000000000006
0.8880000000000001
0.44400000000000006
0.3800000000000008
-0.45199999999999996
0.9039999999999999
0.45199999999999996
0.4000000000000008
-0.46
0.92
0.46
0.4200000000000008
-0.46799999999999997
0.9359999999999999
0.46799999999999997
0.44000000000000083
-0.476
0.952
0.476
0.46000000000000085
-0.484
0.968
0.484
0.48000000000000087
-0.492
0.984
0.492
0.5000000000000009
-0.5
1.0
0.5
-0.09999999999999964
-0.07999999999999963
-0.05999999999999961
-0.03999999999999959
-0.019999999999999574
0.020000000000000462
0.04000000000000048
0.0600000000000005
0.08000000000000052
0.10000000000000053
0.12000000000000055
0.14000000000000057
0.1600000000000006
0.1800000000000006
0.20000000000000062
0.22000000000000064
0.24000000000000066
0.2600000000000007
0.2800000000000007
0.3000000000000007
0.32000000000000073
0.34000000000000075
0.36000000000000076
0.3800000000000008
0.4000000000000008
0.4200000000000008
0.44000000000000083
0.46000000000000085
0.48000000000000087
0.5000000000000009
-
Individual observations lying considerably outside of this band
indicate that the observation may be and outlier. An outlier is an
observation that is not following the normal pattern of the other
observations. Such an observation can have a considerable effect on
the estimation of the parameters of a model. Sometimes the outlier
has occurred because of a typographical error. If this is the case
and it is detected than a correction can be made. If the outlier
occurs for other (and more natural) reasons it may be appropriate
to construct a model that incorporates the occurrence of
outliers.
-
If our "step back" view of the residuals resembled any of those
shown below we should conclude that assumptions about the model are
incorrect. Each pattern may indicate that a different assumption
may have to be made to explain the abnormal residual pattern.
b)a)
Sheet: Model
Sheet: Variance
Sheet: Ideal
Sheet: Independent
Sheet: Positive Autocorr
Sheet: negative auto corr
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
-0.5
0.06333333333333316
0.20000000000000004
0.16333333333333316
0.2633333333333332
-0.48
0.04373333333333315
0.2
0.14373333333333316
0.24373333333333316
-0.46
0.02493333333333317
0.2
0.12493333333333317
0.22493333333333318
-0.44
0.006933333333333153
0.2
0.10693333333333316
0.20693333333333316
-0.42
-0.010266666666666868
0.2
0.08973333333333314
0.18973333333333314
-0.4
-0.02666666666666681
0.2
0.0733333333333332
0.1733333333333332
-0.38
-0.04226666666666684
0.2
0.057733333333333164
0.15773333333333317
-0.36
-0.05706666666666685
0.2
0.04293333333333316
0.14293333333333316
-0.34
-0.07106666666666682
0.2
0.028933333333333186
0.12893333333333318
-0.32
-0.08426666666666684
0.2
0.01573333333333317
0.11573333333333317
-0.3
-0.09666666666666685
0.2
0.0033333333333331605
0.10333333333333317
-0.28
-0.10826666666666683
0.2
-0.008266666666666825
0.09173333333333318
-0.26
-0.11906666666666683
0.2
-0.01906666666666683
0.08093333333333318
-0.24
-0.12906666666666683
0.2
-0.029066666666666838
0.07093333333333317
-0.22
-0.13826666666666684
0.2
-0.03826666666666684
0.06173333333333317
-0.2
-0.14666666666666683
0.2
-0.04666666666666683
0.05333333333333318
-0.18
-0.15426666666666683
0.2
-0.05426666666666684
0.04573333333333317
-0.16
-0.16106666666666686
0.2
-0.06106666666666684
0.03893333333333317
-0.14
-0.16706666666666684
0.2
-0.06706666666666683
0.032933333333333176
-0.12
-0.17226666666666685
0.2
-0.07226666666666684
0.027733333333333166
-0.09999999999999964
-0.17666666666666692
0.2
-0.07666666666666691
0.023333333333333095
-0.07999999999999963
-0.1802666666666669
0.2
-0.0802666666666669
0.019733333333333103
-0.05999999999999961
-0.18306666666666688
0.2
-0.08306666666666689
0.01693333333333312
-0.03999999999999959
-0.18506666666666688
0.2
-0.08506666666666687
0.014933333333333132
-0.019999999999999574
-0.18626666666666686
0.2
-0.08626666666666685
0.013733333333333153
4.440892098500626E-16
-0.18666666666666684
0.2
-0.08666666666666684
0.01333333333333317
0.020000000000000462
-0.1862666666666668
0.2
-0.08626666666666681
0.013733333333333195
0.04000000000000048
-0.18506666666666682
0.2
-0.0850666666666668
0.014933333333333201
0.0600000000000005
-0.18306666666666677
0.2
-0.08306666666666677
0.01693333333333323
0.08000000000000052
-0.18026666666666674
0.2
-0.08026666666666675
0.019733333333333256
0.10000000000000053
-0.17666666666666675
0.2
-0.07666666666666673
0.023333333333333275
0.12000000000000055
-0.1722666666666667
0.2
-0.0722666666666667
0.027733333333333304
0.14000000000000057
-0.1670666666666667
0.2
-0.06706666666666668
0.03293333333333333
0.1600000000000006
-0.16106666666666664
0.2
-0.061066666666666644
0.03893333333333336
0.1800000000000006
-0.1542666666666666
0.2
-0.054266666666666616
0.04573333333333339
0.20000000000000062
-0.14666666666666658
0.2
-0.046666666666666586
0.05333333333333342
0.22000000000000064
-0.13826666666666657
0.2
-0.03826666666666655
0.06173333333333345
0.24000000000000066
-0.12906666666666652
0.2
-0.02906666666666652
0.07093333333333349
0.2600000000000007
-0.11906666666666649
0.2
-0.019066666666666482
0.08093333333333352
0.2800000000000007
-0.10826666666666646
0.2
-0.00826666666666645
0.09173333333333356
0.3000000000000007
-0.09666666666666641
0.2
0.0033333333333335907
0.1033333333333336
0.32000000000000073
-0.08426666666666638
0.2
0.015733333333333627
0.11573333333333363
0.34000000000000075
-0.07106666666666633
0.2
0.028933333333333672
0.12893333333333368
0.36000000000000076
-0.057066666666666294
0.2
0.04293333333333371
0.14293333333333372
0.3800000000000008
-0.04226666666666626
0.2
0.05773333333333375
0.15773333333333375
0.4000000000000008
-0.0266666666666662
0.2
0.0733333333333338
0.1733333333333338
0.4200000000000008
-0.010266666666666147
0.2
0.08973333333333386
0.18973333333333386
0.44000000000000083
0.006933333333333902
0.2
0.10693333333333391
0.2069333333333339
0.46000000000000085
0.024933333333333946
0.2
0.12493333333333395
0.22493333333333396
0.48000000000000087
0.043733333333333985
0.2
0.143733333333334
0.243733333333334
0.5000000000000009
0.06333333333333405
0.20000000000000004
0.16333333333333405
0.2633333333333341
0.08666666666666684
-0.1
0.09999999999999999
11.538461538461515
0.08333333333333333
0.09333333333333341
-0.5
-0.1
0.2
0.1
-0.48
-0.10800000000000001
0.21600000000000003
0.10800000000000001
-0.46
-0.116
0.232
0.116
-0.44
-0.124
0.248
0.124
-0.42
-0.132
0.264
0.132
-0.4
-0.14
0.28
0.14
-0.38
-0.14800000000000002
0.29600000000000004
0.14800000000000002
-0.36
-0.156
0.312
0.156
-0.34
-0.164
0.328
0.164
-0.32
-0.17200000000000001
