Variance Inflation Factor and Condition Number in multiple linear regression

The Variance Inflation Factor and the Condition Number are measures traditionally applied to detect the presence of collinearity in a multiple linear model. This paper presents the relation and the difference between both measures from theoretical and empirical perspectives by using Monte Carlo simu...

Full description

Saved in:
Bibliographic Details
Published inJournal of statistical computation and simulation Vol. 88; no. 12; pp. 2365 - 2384
Main Authors Salmerón, R., García, C. B., García, J.
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis Ltd 13.08.2018
Subjects
Online AccessGet full text
ISSN0094-9655
1563-5163
DOI10.1080/00949655.2018.1463376

Cover

Loading…
Abstract The Variance Inflation Factor and the Condition Number are measures traditionally applied to detect the presence of collinearity in a multiple linear model. This paper presents the relation and the difference between both measures from theoretical and empirical perspectives by using Monte Carlo simulations and taking special interest in the computational techniques.
AbstractList The Variance Inflation Factor and the Condition Number are measures traditionally applied to detect the presence of collinearity in a multiple linear model. This paper presents the relation and the difference between both measures from theoretical and empirical perspectives by using Monte Carlo simulations and taking special interest in the computational techniques.
Author García, J.
García, C. B.
Salmerón, R.
Author_xml – sequence: 1
  givenname: R.
  orcidid: 0000-0003-2589-4058
  surname: Salmerón
  fullname: Salmerón, R.
  organization: Department of Quantitative Methods for Economics and Business, University of Granada, Granada, Spain
– sequence: 2
  givenname: C. B.
  orcidid: 0000-0003-1622-3877
  surname: García
  fullname: García, C. B.
  organization: Department of Quantitative Methods for Economics and Business, University of Granada, Granada, Spain
– sequence: 3
  givenname: J.
  surname: García
  fullname: García, J.
  organization: Department of Economic and Business, University of Almería, Almería, Spain
BookMark eNqFkE1LAzEQhoNUsK3-BCHgeWuy-eriSYqthaIX9RrS7ERSttmaZA_-e3fbevHiaWB43neYZ4JGoQ2A0C0lM0rm5J6QildSiFlJ6HxGuWRMyQs0pkKyQlDJRmg8MMUAXaFJSjtCCKWiHKPNh4neBAt4HVxjsm8DXhqb24hNqPGiDbU_Ll-6_RYi9gHvuyb7QwO48QFMxBE-I6TUQ9fo0pkmwc15TtH78ult8VxsXlfrxeOmsKziubBCSeOAG1crXpGaSMe3VErFKWO0dramprRQC6BAoGSlLWErhBSVFcxZxqbo7tR7iO1XBynrXdvF0J_UJeFKMVVx1VMPJ8rGNqUITlufjx_maHyjKdGDPv2rTw_69FlfnxZ_0ofo9yZ-_5P7AdJxdMs
CitedBy_id crossref_primary_10_1002_pan3_10753
crossref_primary_10_1038_s41598_021_98256_6
crossref_primary_10_2139_ssrn_4433939
crossref_primary_10_1016_j_jecp_2024_105927
crossref_primary_10_1111_imig_12690
crossref_primary_10_1007_s10614_024_10575_8
crossref_primary_10_1016_j_jrmge_2025_01_012
crossref_primary_10_1142_S0219622022500948
crossref_primary_10_1108_K_03_2021_0197
crossref_primary_10_2147_JPR_S386440
crossref_primary_10_1007_s00191_019_00636_9
crossref_primary_10_1371_journal_pone_0319929
crossref_primary_10_3390_app10072573
crossref_primary_10_1016_j_technovation_2024_103000
crossref_primary_10_2139_ssrn_3782199
crossref_primary_10_1016_j_phro_2022_09_004
crossref_primary_10_4236_as_2023_146055
crossref_primary_10_3390_math8040605
crossref_primary_10_1002_fes3_505
crossref_primary_10_1007_s12046_023_02403_w
crossref_primary_10_3390_app13116639
crossref_primary_10_1007_s10489_023_05005_5
crossref_primary_10_1007_s41651_023_00142_4
crossref_primary_10_1002_dug2_12067
crossref_primary_10_1016_j_scs_2024_105700
crossref_primary_10_5397_cise_2023_01053
crossref_primary_10_1016_j_jtcvs_2023_05_044
crossref_primary_10_1021_acs_jcim_1c01567
crossref_primary_10_1080_1351847X_2022_2038648
crossref_primary_10_1021_jasms_3c00168
crossref_primary_10_1186_s12889_023_16502_0
crossref_primary_10_3390_su16219316
crossref_primary_10_1016_j_envint_2024_108520
crossref_primary_10_1007_s00127_022_02403_x
crossref_primary_10_1016_j_jenvman_2023_118112
crossref_primary_10_1080_02687038_2023_2183076
crossref_primary_10_1016_j_ssresearch_2022_102779
crossref_primary_10_1080_00128775_2019_1618194
crossref_primary_10_1002_jum_15399
crossref_primary_10_1007_s10826_023_02605_0
crossref_primary_10_1016_j_physa_2019_122556
crossref_primary_10_2139_ssrn_4129886
crossref_primary_10_1002_csr_2342
crossref_primary_10_1016_j_amper_2023_100112
crossref_primary_10_3390_su15086549
crossref_primary_10_1080_15567249_2020_1868619
crossref_primary_10_3390_su17010019
crossref_primary_10_1109_ACCESS_2020_3043317
crossref_primary_10_1186_s12889_025_21686_8
crossref_primary_10_3390_w15122269
crossref_primary_10_1007_s10901_024_10124_7
crossref_primary_10_1016_j_envsoft_2023_105853
crossref_primary_10_1016_j_autcon_2021_103629
crossref_primary_10_1142_S234573762150007X
crossref_primary_10_21511_ppm_18_2__2020_34
crossref_primary_10_55905_cuadv16n10_194
crossref_primary_10_1177_0308518X19865550
crossref_primary_10_1016_j_jtrangeo_2024_103828
crossref_primary_10_1080_14631377_2023_2236874
crossref_primary_10_3390_s20020444
crossref_primary_10_1016_j_chb_2025_108557
crossref_primary_10_1016_j_jweia_2021_104792
crossref_primary_10_1080_14719037_2022_2037014
crossref_primary_10_3390_atmos14030438
crossref_primary_10_3389_fams_2023_1282975
crossref_primary_10_4081_gh_2022_1072
crossref_primary_10_1108_JAEE_10_2020_0257
crossref_primary_10_1080_20964471_2022_2136610
crossref_primary_10_1016_j_jhazmat_2024_134284
crossref_primary_10_1088_2752_5309_ac8ac2
crossref_primary_10_1029_2022GH000764
crossref_primary_10_1177_09721509231213235
crossref_primary_10_1002_agj2_21222
crossref_primary_10_1016_j_jclepro_2024_142686
crossref_primary_10_3390_fire6050197
crossref_primary_10_1080_20964471_2023_2291215
crossref_primary_10_1016_j_respol_2019_103856
crossref_primary_10_1080_24694452_2022_2161988
crossref_primary_10_1038_s41593_021_00914_5
crossref_primary_10_1016_j_cities_2023_104572
crossref_primary_10_3389_fcvm_2021_744354
crossref_primary_10_1002_csr_2083
crossref_primary_10_1080_03610918_2022_2110594
crossref_primary_10_1080_10618600_2020_1779729
crossref_primary_10_1016_j_apnr_2021_151413
crossref_primary_10_1007_s00180_020_01047_2
crossref_primary_10_33333_rp_vol51n2_05
crossref_primary_10_1002_cem_3292
crossref_primary_10_3390_agronomy14122932
crossref_primary_10_1214_21_BJPS516
crossref_primary_10_1177_1358863X231226216
crossref_primary_10_3389_fpubh_2024_1384561
crossref_primary_10_3390_s23146326
crossref_primary_10_1016_j_jik_2023_100422
crossref_primary_10_1088_2053_1591_ac144b
crossref_primary_10_3389_fpsyg_2022_921797
crossref_primary_10_1111_rode_13191
crossref_primary_10_1016_j_jclepro_2020_124188
crossref_primary_10_1371_journal_pone_0286408
crossref_primary_10_1093_annweh_wxaa143
crossref_primary_10_1016_j_techsoc_2021_101742
crossref_primary_10_1111_gove_12707
crossref_primary_10_1016_j_jtrangeo_2023_103744
crossref_primary_10_1108_BL_08_2023_0241
crossref_primary_10_1007_s40313_022_00951_w
crossref_primary_10_1016_j_socnet_2025_02_001
crossref_primary_10_1016_j_agrformet_2024_110377
crossref_primary_10_3389_fpsyg_2023_1150421
crossref_primary_10_3390_su11184935
crossref_primary_10_1080_07420528_2021_1886111
crossref_primary_10_3390_rs15143660
crossref_primary_10_1007_s40201_021_00684_z
crossref_primary_10_1016_j_softx_2024_101926
crossref_primary_10_1080_02508281_2023_2185733
crossref_primary_10_2139_ssrn_4054805
crossref_primary_10_1016_j_jad_2021_10_028
crossref_primary_10_56982_dream_v2i10_159
crossref_primary_10_21511_imfi_18_1__2021_28
crossref_primary_10_1016_j_identj_2024_04_004
crossref_primary_10_1007_s11356_022_22649_x
crossref_primary_10_1007_s11469_022_00951_6
crossref_primary_10_3390_su141811310
crossref_primary_10_1111_insr_12575
crossref_primary_10_3390_math8060931
crossref_primary_10_1080_15389588_2019_1707194
crossref_primary_10_3233_JIFS_189487
crossref_primary_10_1007_s12665_024_11871_0
crossref_primary_10_1007_s10661_020_08644_0
crossref_primary_10_56982_dream_v1i06_59
crossref_primary_10_1007_s00520_024_08772_0
crossref_primary_10_1016_j_jclepro_2023_138154
crossref_primary_10_56982_dream_v1i06_55
crossref_primary_10_1515_ijsl_2023_0084
crossref_primary_10_1007_s00521_023_08313_6
crossref_primary_10_1186_s12889_020_09989_4
crossref_primary_10_1016_j_ssmph_2023_101515
crossref_primary_10_1007_s10551_022_05228_5
crossref_primary_10_1002_cmtd_202100108
crossref_primary_10_2139_ssrn_4160654
crossref_primary_10_1007_s12061_024_09633_3
crossref_primary_10_3233_JIFS_189495
crossref_primary_10_3390_su13084318
crossref_primary_10_1016_j_jhazmat_2023_131408
crossref_primary_10_3389_fpubh_2022_945846
crossref_primary_10_1016_j_jenvman_2024_121294
crossref_primary_10_1108_BL_07_2023_0230
crossref_primary_10_1155_2021_9969322
crossref_primary_10_1155_2023_7818070
crossref_primary_10_1073_pnas_2307214121
crossref_primary_10_1002_sres_3140
crossref_primary_10_1016_j_xnsj_2023_100261
crossref_primary_10_1177_01600176221125692
crossref_primary_10_3390_healthcare11091207
crossref_primary_10_1021_acs_est_0c03285
crossref_primary_10_1016_j_uclim_2022_101203
crossref_primary_10_1080_02664763_2019_1701638
crossref_primary_10_3390_app8112121
crossref_primary_10_1111_dom_15546
crossref_primary_10_2196_42782
crossref_primary_10_3390_su13158133
crossref_primary_10_1177_23326492241268598
crossref_primary_10_1080_00949655_2018_1543423
crossref_primary_10_1080_02286203_2021_1905375
crossref_primary_10_3390_s20113264
crossref_primary_10_54097_hset_v8i_1125
crossref_primary_10_3390_hydrology9020036
crossref_primary_10_1016_j_jweia_2024_105936
crossref_primary_10_1186_s40621_020_00246_1
crossref_primary_10_1016_j_scs_2023_104882
crossref_primary_10_1038_s41539_024_00251_9
crossref_primary_10_1109_TEM_2024_3425656
crossref_primary_10_3390_su15021552
crossref_primary_10_1007_s10579_021_09547_3
crossref_primary_10_1080_14992027_2023_2291633
crossref_primary_10_1016_j_ecoenv_2020_111059
crossref_primary_10_1108_IJOEM_03_2022_0490
crossref_primary_10_7717_peerj_7552
crossref_primary_10_1007_s11056_024_10055_9
crossref_primary_10_1108_TR_02_2022_0100
crossref_primary_10_1061_JCEMD4_COENG_12971
crossref_primary_10_1007_s10708_024_11037_2
crossref_primary_10_3389_fpsyg_2022_921237
crossref_primary_10_3390_ijerph192214842
crossref_primary_10_18502_ijrm_v20i5_11052
crossref_primary_10_3390_bs14060498
crossref_primary_10_3390_ijerph19137943
crossref_primary_10_1080_00031305_2019_1635527
crossref_primary_10_1177_26338076221105899
crossref_primary_10_1007_s10661_024_13069_0
crossref_primary_10_1007_s10639_020_10412_z
Cites_doi 10.1080/01621459.1981.10477619
10.2307/1267205
10.1016/S0304-4076(01)00144-0
10.1002/0471458503
10.1016/0304-4076(82)90020-3
10.2307/1926450
10.1214/ss/1177013439
10.2307/1923926
10.1111/insr.12099
10.1111/j.2517-6161.1969.tb00813.x
10.2307/2286476
10.1080/02664763.2015.1120712
10.1214/ss/1177013442
10.1007/BF00426854
10.1007/s11135-006-9018-6
10.1080/00224065.1973.11980577
10.1002/wics.84
10.1080/00031305.1984.10483170
10.1090/S0025-5718-1992-1106966-2
10.1016/S0313-5926(10)50034-1
10.1080/01621459.1975.10479882
10.1080/01621459.1990.10474941
ContentType Journal Article
Copyright 2018 Informa UK Limited, trading as Taylor & Francis Group
Copyright_xml – notice: 2018 Informa UK Limited, trading as Taylor & Francis Group
DBID AAYXX
CITATION
7SC
8FD
JQ2
L7M
L~C
L~D
DOI 10.1080/00949655.2018.1463376
DatabaseName CrossRef
Computer and Information Systems Abstracts
Technology Research Database
ProQuest Computer Science Collection
Advanced Technologies Database with Aerospace
Computer and Information Systems Abstracts – Academic
Computer and Information Systems Abstracts Professional
DatabaseTitle CrossRef
Computer and Information Systems Abstracts
Technology Research Database
Computer and Information Systems Abstracts – Academic
Advanced Technologies Database with Aerospace
ProQuest Computer Science Collection
Computer and Information Systems Abstracts Professional
DatabaseTitleList Computer and Information Systems Abstracts
DeliveryMethod fulltext_linktorsrc
Discipline Statistics
Mathematics
Computer Science
EISSN 1563-5163
EndPage 2384
ExternalDocumentID 10_1080_00949655_2018_1463376
GroupedDBID .7F
.QJ
0BK
0R~
29L
30N
4.4
5GY
5VS
8VB
AAENE
AAGDL
AAHIA
AAJMT
AALDU
AAMIU
AAPUL
AAQRR
AAYXX
ABCCY
ABFIM
ABHAV
ABJNI
ABLIJ
ABPAQ
ABPEM
ABTAI
ABXUL
ABXYU
ACGEJ
ACGFS
ACGOD
ACTIO
ADCVX
ADGTB
ADXPE
AEISY
AENEX
AEOZL
AEPSL
AEYOC
AFKVX
AFRVT
AGDLA
AGMYJ
AHDZW
AIJEM
AIYEW
AJWEG
AKBVH
AKOOK
ALMA_UNASSIGNED_HOLDINGS
ALQZU
AQRUH
AVBZW
AWYRJ
BLEHA
CCCUG
CE4
CITATION
CS3
DGEBU
DKSSO
DU5
EBS
EJD
E~A
E~B
F5P
GTTXZ
H13
HF~
HZ~
H~P
IPNFZ
J.P
KYCEM
M4Z
MS~
NA5
NY~
O9-
P2P
PQQKQ
QWB
RIG
RNANH
ROSJB
RTWRZ
S-T
SNACF
TASJS
TBQAZ
TDBHL
TEJ
TFL
TFT
TFW
TN5
TTHFI
TUROJ
TWF
UPT
UT5
UU3
YQT
ZGOLN
ZL0
~S~
7SC
8FD
ADYSH
JQ2
L7M
L~C
L~D
ID FETCH-LOGICAL-c394t-c576afe4afd7490d06f4b166741331dfcd1a2ced5e1e0e232c2eb55659c53fc33
ISSN 0094-9655
IngestDate Mon Jul 14 08:14:02 EDT 2025
Thu Apr 24 23:04:40 EDT 2025
Sun Aug 03 02:37:22 EDT 2025
IsDoiOpenAccess false
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Issue 12
Language English
LinkModel OpenURL
MergedId FETCHMERGED-LOGICAL-c394t-c576afe4afd7490d06f4b166741331dfcd1a2ced5e1e0e232c2eb55659c53fc33
Notes ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ORCID 0000-0003-2589-4058
0000-0003-1622-3877
OpenAccessLink https://hdl.handle.net/10481/86563
PQID 2047737947
PQPubID 53118
PageCount 20
ParticipantIDs proquest_journals_2047737947
crossref_citationtrail_10_1080_00949655_2018_1463376
crossref_primary_10_1080_00949655_2018_1463376
PublicationCentury 2000
PublicationDate 2018-08-13
PublicationDateYYYYMMDD 2018-08-13
PublicationDate_xml – month: 08
  year: 2018
  text: 2018-08-13
  day: 13
PublicationDecade 2010
PublicationPlace Abingdon
PublicationPlace_xml – name: Abingdon
PublicationTitle Journal of statistical computation and simulation
PublicationYear 2018
Publisher Taylor & Francis Ltd
Publisher_xml – name: Taylor & Francis Ltd
References Belsley DA (CIT0005) 2005
CIT00016
CIT00015
CIT00014
CIT00013
CIT00012
CIT00011
CIT00010
CIT00019
CIT00018
CIT00017
Salmerón R (CIT00023) 2016
Belsley DA. (CIT00024) 1991; 4
CIT00027
Berk KN. (CIT00021) 1977; 72
CIT00026
CIT00025
CIT00022
CIT00020
Besley DA. (CIT0001) 1991
CIT0003
CIT0002
CIT0004
CIT0007
CIT0006
Silvey SD. (CIT00029) 1969
CIT0009
CIT0008
Kendall MG. (CIT00028) 1957
References_xml – ident: CIT00020
– start-page: 1
  year: 2016
  ident: CIT00023
  publication-title: Comput Stat
– ident: CIT00026
  doi: 10.1080/01621459.1981.10477619
– volume-title: Conditioning diagnostics: collinearity and weak data in regression
  year: 1991
  ident: CIT0001
– ident: CIT00015
  doi: 10.2307/1267205
– ident: CIT00019
  doi: 10.1016/S0304-4076(01)00144-0
– ident: CIT00014
  doi: 10.1002/0471458503
– ident: CIT0004
– ident: CIT0006
  doi: 10.1016/0304-4076(82)90020-3
– ident: CIT00012
  doi: 10.2307/1926450
– ident: CIT00010
  doi: 10.1214/ss/1177013439
– ident: CIT00011
  doi: 10.2307/1923926
– ident: CIT00022
  doi: 10.1111/insr.12099
– start-page: 539
  year: 1969
  ident: CIT00029
  publication-title: J Royal Stat Soc Series B
  doi: 10.1111/j.2517-6161.1969.tb00813.x
– volume: 72
  start-page: 863
  issue: 360
  year: 1977
  ident: CIT00021
  publication-title: J Am Stat Assoc
  doi: 10.2307/2286476
– volume-title: Regression diagnostics: identifying influential data and sources of collinearity. Vol. 571
  year: 2005
  ident: CIT0005
– ident: CIT00027
  doi: 10.1080/02664763.2015.1120712
– volume-title: A course in multivariate analysis: Charles Griffen & Co
  year: 1957
  ident: CIT00028
– ident: CIT0007
  doi: 10.1214/ss/1177013442
– volume: 4
  start-page: 33
  issue: 1
  year: 1991
  ident: CIT00024
  publication-title: Comput Sci Econ Manag
  doi: 10.1007/BF00426854
– ident: CIT00013
  doi: 10.1007/s11135-006-9018-6
– ident: CIT00016
  doi: 10.1080/00224065.1973.11980577
– ident: CIT0003
– ident: CIT00017
  doi: 10.1002/wics.84
– ident: CIT0002
  doi: 10.1080/00031305.1984.10483170
– ident: CIT0009
  doi: 10.1090/S0025-5718-1992-1106966-2
– ident: CIT00018
  doi: 10.1016/S0313-5926(10)50034-1
– ident: CIT00025
  doi: 10.1080/01621459.1975.10479882
– ident: CIT0008
  doi: 10.1080/01621459.1990.10474941
SSID ssj0001152
Score 2.5717442
Snippet The Variance Inflation Factor and the Condition Number are measures traditionally applied to detect the presence of collinearity in a multiple linear model....
SourceID proquest
crossref
SourceType Aggregation Database
Enrichment Source
Index Database
StartPage 2365
SubjectTerms Collinearity
Computer simulation
Title Variance Inflation Factor and Condition Number in multiple linear regression
URI https://www.proquest.com/docview/2047737947
Volume 88
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
link http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3Nb9MwFLfKdhkHPgqIjYF8gNPkqo6dr-PoVspUyoEUKi5R7NjTJMimtuPAX8_zR7yUlc9LFNlNUvn9_PxsP_9-CL3MWGKI8iXRcZURLqQmIksZBHLQ9aSESmmzLWbJZM7PFvGi1_vWyVq6XouB_L71XMn_WBXKwK7mlOw_WDa8FArgHuwLV7AwXP_Kxh9homtT_t822uW0HY2tfo7dEhhdmu1oUzizsh9maSPkD5rosloeLdW5S4RtfhGlmgNHlsvZ8ogYBQj3HbvefvHVi3-FdRqnxjJir16zZiMd8Q10KLspf2Kj1Y7a8081Z911CGp4rYk7RmqRU9ySBOnkJVnXm3OSJ46Td6C8t00Yian3cN4dZ1kXdlHXuTInK3HL67dpkrkhv49Nvl5mRgDG0i0s27P35Xg-nZbF6aK4g3YjmF6AQ989npx8_hTGcOq0msKfbs9-GVb2bZ_ZjGo2B3UbqRQP0D1vPHzs8PIQ9VTTR_db-Q7svXkf3X0XKHtXfbT3obX06hGatsjCAVnYIQuD4XFAFnbIwhcNbpGFHbLwDbIeo_n4tBhNiNfdIJLlfE0kzEErrXil65Tnw3qYaC5okkDwyRittaxpFUlVx4qqoYKQXEZKxDAzyGXMtGTsCdppLhv1FGFOBdPDqM5rmXORKYjnBatgjKGGOTKT-4i37VZKT0pvtFG-lDRw17rmLk1zl76599EgPHblWFn-9MBha5TSd-AV_ICnKYMBKT34ffUztHeD-EO0s15eq-cQi67FCw-cH6O2g0U
linkProvider Library Specific Holdings
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Variance+Inflation+Factor+and+Condition+Number+in+multiple+linear+regression&rft.jtitle=Journal+of+statistical+computation+and+simulation&rft.au=Salmer%C3%B3n%2C+R&rft.au=Garc%C3%ADa%2C+C+B&rft.au=Garc%C3%ADa%2C+J&rft.date=2018-08-13&rft.pub=Taylor+%26+Francis+Ltd&rft.issn=0094-9655&rft.eissn=1563-5163&rft.volume=88&rft.issue=12&rft.spage=2365&rft_id=info:doi/10.1080%2F00949655.2018.1463376&rft.externalDBID=NO_FULL_TEXT
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0094-9655&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0094-9655&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0094-9655&client=summon