INTERNATIONAL JOURNAL OF MULTIDISCIPLINARY RESEARCH AND ANALYSIS
ISSN(p in ): 2643-9840, ISSN(online): 2643-9875
Volume 08 Issue 10 Oc obe 2025
DOI: 10.47191/ijm a/ 8-i10-07, Impac Fac o : 8.266
Page No. 5547-5551
IJMRA, Volume 08 Issue 10 Oc obe 2025 www.ijm a.in Page 5547
Unde -U iliza ion o Analysis o Co a iance in Beha io al Resea ch
Denis Achung Uyanah, Ph.D.
Depa men O Educa ional Founda ions, Facul y o Educa ion Uni e si y o C oss Ri e S a e Calaba -Nige ia
ABSTRACT: One o he g ea es sou ce o alue o esea ch esul s is a iance con ol a ainable by applying a p inciple code-
named “MAXMINCOM”. This p inciple has h ee componen s: Maximiza ion o sys ema ic o desi able a iance, minimiza ion o
e o a iance and con ol o a iance a ising om he e ec o unwan ed a iables, gene ally called ex aneous a iables. When
he ex aneous o nuisance a iables can be emo ed, he esea ch design akes ca e o ha . When such comple e emo al o he
in luence o such ex aneous a iable is no possible o di icul , such a iable a e delibe a ely included so ha hei in luences
a e conside ed pa o he s udy. Thei in luences a e emo ed by pa ialing ou he a iance a ibu ed o such a iables, om
he o al a iance. Analysis o co a iance is one o he s a is ical analysis echniques ha is u ilized o accomplish his pu pose.
Many o he epo ed applica ion o analysis o co a iance ha e one ac o wi h wo le els and one co a ia e. This has
caused so many esea che s o hink ha his is a gene al ule. When he e a e mo e han wo le els o a ac o o ins ance; A1,
A2, A3, you ind he pai s A1 and A2, A1 and A3, A2, and A3, o h ee di e en hypo heses. I he aim om he beginning is o compa e
he ea men e ec o A1, A2, and A3, hen his can be handled on one single analysis wi h an app op ia e pos -hoc- es ha can
only be applied a e he obse ed mean alues o A1, A2, and A3, ha e been adjus ed o he e ec o he co a ia e(s). The e has
been sha p p o es om s uden s’ p ojec supe iso s and e en ex e nal examine s when s uden s ake he h ee le els o mo e
o he ac o e.g eaching me hods, oge he , ca y ou an adjus men o ea men means and applying pos -hoc es on adjus ed
means. S uden s who ca you such p ope p ocedu es ha e been denied g adua ion, down-g aded and some imes made o e-
analyze and in e p e he esul s o such pai s as di e en hypo heses. The ehemence wi h which his ejec ion o he legi ima e
p ocedu e is done, necessi a ed his pape . The pu pose was he e o e o p o ide heo e ical jus i ica ion o such de ailed analysis,
o sa e mainly he s uden s’ who a e he ic ims o he igno ance o he powe o ANCOVA and a cla i ica ion o p ojec
supe iso s, examine s and da a analysis s, ha a e con on ed o consul ed wi h da a om such s uden s and esea che s.
INTRODUCTION
The e ha e been e y ugly si ua ion ha s uden s ha e been subjec ed o in he cou se o execu ing hei esea ch p ojec
as a equi emen o ob aining a deg ee (Fi s , Second o Thi d) whe e analysis o co a iance was applied. This pape is a p oduc
o such ugly si ua ions. The s uden s had ca ied ou hei s udies wi h a single ac o (e.g. eaching me hod) wi h mo e han wo
me hods, meaning he ac o had mo e han wo le els. The s uden s mos imes ( he e ha e been se e al encoun e s obse ed
by his au ho ) logically ollowed he p ocedu es in analysis o co a iance, beginning wi h inding he eg ession coe icien o he
eg ession o he dependen a iable on he co a ia e o each g oup and es ing o he signi icance o hei di e ences. Whe e
he di e ences whe e no signi ican , a common eg ession coe icien was compu ed. Nex he s uden s ca ied ou he ANCOVA,
emo ing sum o squa es a ibu able o he co a ia e om he o al sum o squa es and es ing o signi icance o he e ec o
he ac o . The obse ed g oup means we e subsequen ly adjus ed o he e ec o he co a ia e be o e applying a pos -hoc es .
The wo ks we e u ned down o ejec ed, some imes by hei supe iso , o in e nal examine o e en an ex e nal examine . The e
ha e been cases whe e he s uden s we e made o e-analyze he da a, aking he eaching me hods in pai s, emo ing he
eg ession componen o he analysis as well as he adjus men made be o e compa ison and epo he esul s pu ely on he sum
o squa es mean squa e and F- a io es and d awing conclusion om he e. The e we e cases whe e ex e nal examine s
ehemen ly ejec ed explana ions ha could disabuse hei mindse .
Such ins ances ha e been auma ic o bo h s uden s and hei in e nal supe iso s and examine s. The s uden s enco es
huge inancial loses, dampened mo ale and ea as well as w ong esol e, ei he no o ca y ou any s udy ha will esul in he
applica ion o ANCOVA o use he w ong app oach o pai ing eaching me hods, o example and epo ing he esul s, wi hou
p elimina y es s and any adjus men o ea men means. The epu a ion o p ojec supe iso s we e d ag down wi h a eeling
Unde -U iliza ion o Analysis o Co a iance in Beha io al Resea ch
IJMRA, Volume 08 Issue 10 Oc obe 2025 www.ijm a.in Page 5548
ha s uden s can no longe us hei expe ad ise in he cou se o hei esea ch wo k. Some examine s u he claimed ha
hey had done a su ey o applica ion o ANCOVA in academic jou nals and ha e no seen anywhe e eg ession analysis is
associa ed wi h ANCOVA nei he ha e hey seen any adjus men ca ied ou . Explana ions ha , esea che s whose wo ks a e
epo ed in jou nals may ha e educe hei s udies o wo le els o economic and simplici y pu poses and canno be aken as a
ule, ell on dea ea s. This exposi o y wo k on he capaci y o ANCOVA became e y impe a i e.
The pu pose o esea ch design gene ally and expe imen al design in pa icula is o ensu e ha he ou come and
conclusion d own ha e bo h in e nal and ex e nal alidi y. This is achie ed by maximizing he sys ema ic a iance, he a iance in
he dependen a iable(s) accoun ed o by he independen a iables o in e es o he esea che ; he minimiza ion o e o
a iance, i possible he educ ion in he e o o andom e o and con ol o a iances whose sou ce a e ex aneous o he s udy
o unwan ed. Some imes his con ol is easy h ough a ca e ul se ing o he expe imen . Howe e , he e a e a iables whose
in luence canno be elimina ed so easily. In ac , in some cases, i is close o impossibili y. Fo example, in a s udy o he e ec o
eaching me hod on s uden s’ achie emen in ma hema ics, he in luence o he di e ences in hei p e ious knowledge, hei
in elligence quo ien , hei ap i udes e c. a e e y di icul o emo e.
In such cases, i is almos a con en ion ha such a iables should be buil in o he design. App op ia e s a is ical
echniques a e hen applied o emo e he a iance a ibu able o such buil -in ac o s om he o al a iance, called pa ialing.
One such s a is ical echnique widely applied, especially in expe imen al s udies, is analysis o co a iance. The independen
a iable is adi ionally called a ac o and he e is always a dependen a iable o esponse a iable. The a iable buil -in whose
e ec o in luence is unwan ed is called a Co a ia e. When he e is only one ac o , i is one-way analysis o co a iance (ANCOVA).
So, he e could be wo-way, h ee-way e c. depending on he numbe o ac o s. When he dependen a iable is jus one, he
analysis is called Uni a ia e analysis o co a iance. When he e a e mo e han one dependen a iables, i is called Mul i a ia e
analysis o Co a iance (MANOCOVA). The numbe o co a ia es could be mo e han one.
The logic o analysis o Co a iance
When a a iable has been esidualized (such as inding he di e ence be ween he o iginal a iable y and he p edic ed y1) he
co ela ion be ween he p edic o a iable x and he p edic ed a iable y1 is always one while he co ela ion be ween he
p edic o a iable x and he esidual ye is ze o. So one may w i e his as:
Rxy1 = 1
xye = 0
whe e ye = y-y1
The esidualized a iable, acco ding o Ke linge (1973), is one om which wha e e a iance i sha ed wi h he p edic o
a iable has been emo ed. I one we e s udying he e ec o eaching me hods on s uden s achie emen in ma hema ics, and
wan s o adjus he ma hema ics achie emen sco e o di e ences in quan i a i e ap i ude, he independen a iable is eaching
me hod, he dependen a iable is ma hema ics achie emen and he co a ia e is quan i a i e ap i ude. One would i s use he
quan i a i e ap i ude o p edic hei ma hema ics achie emen , by eg essing ma hema ics sco e on ap i ude sco e. Suppose he
s uden s had been g ouped such ha yίȷ is he ac ual achie emen sco e o s uden s ί in g oup j, hen yίȷ is he p edic ed
ma hema ics sco e and yίȷ - ŷίȷ is he esidual. One can calcula e his esidual o all s uden s in he s udy, he esul ing se o sco es
ha e ze o co ela ion wi h he s uden s’ quan i a i e ap i ude.
A es o signi icance o di e ences o he g oups will indica e whe he he g oup means a e di e en when hei sco es
ha e been adjus ed (pu ged) o possible di e ences in ma hema ics ap i ude. This is he logic behind he celeb a ed analysis o
co a iance in design and analysis o expe imen .
One can he e o e w i e ha
yίȷ = 𝑦 + Tȷ + b (xίȷ - 𝑥 ) + eίȷ
whe e yίȷ he sco e ob ained by a ί h s uden unde j h ea men , 𝑦 is he g and mean ma hema ics achie emen , Tȷ is he e ec o
ea men j ( eaching me hod), b is he common eg ession coe icien when all ea men g oups a e pooled wi h y as dependen
a iable and x (Co a ia e) as independen a iable j xίȷ is he sco e ob ained by he ί h s uden in he j h g oup on he co a ia e j 𝑥
is he g and mean o he co a ia e and eίȷ is he e o in he sco e o he ί h s uden in he j h g oup. This Equa ion can be ew i en
as:
yίȷ - b (xίȷ - 𝑥 ) =𝑦 + Tȷ + eίȷ
This Equa ion shows clea ly ha a e he adjus men (LHS), wha e e is le is made up o he g and mean (𝑦), he
ea men e ec (Tȷ) and an e o e m (eίȷ). This also shows ha he end poin o ANCOVA is he adjus ed mean sco e (Ko ha i &
Go g, 2014). The implica ion is ha whe e he le el o a ac o o ea men g oups a e only wo, he adjus ed means a e me ely
epo ed. Some esea che s, ei he ou o igno ance o delibe a ely, do no e en epo he adjus men means. Howe e , i he
le els o he ac o a e mo e han wo, pos hoc compa ison is done using he adjus ed means and no he unadjus ed means.
Unde -U iliza ion o Analysis o Co a iance in Beha io al Resea ch
IJMRA, Volume 08 Issue 10 Oc obe 2025 www.ijm a.in Page 5549
This is whe e mos esea che s don’ ge o, ei he because hey do no know ha hey should ge o his poin o he analysis and
epo same o hey delibe a ely do no wan o. The o me eason looks mo e plausible. Hence, he need o his pape .
The use ulness o ge ing o his end canno be o e emphasized. I enables he esea che s o iden i y as well as ake
in o accoun sou ces o a iance a ibu able o concomi an ac o s he eby p o iding mo e o he much needed con ol and
inc easing he alidi y o he s a is ical conclusion. An adjus men o he e ec o he co a ia e(s) will hence lead na u ally o he
educ ion in he e o e m and consequen ly o an inc ease in he sensi i i y o all associa ed es s. Wha is accomplished is e y
compa able o wha designs like epea ed measu es o ea men -by-le els o andomized block designs do in he maximiza ion
o desi able sys ema ic a iances (Feld , 1958; Coch an, 1957; Elasho , 1969).
In he main able o ANCOVA esul s, he sum o squa es due o he co a ia e is emo ed by sub ac ion, om he o al
a iance, jus as he sum o squa es due o o he ac o s in he ANCOVA model a e sub ac ed o ge he e o e m. The mean
squa es a e compu ed and he F- alues a he end, oge he wi h hei associa ed P- alues. When he ea men e ec is
signi ican , i is necessa y o de e mine speci ically which o hem di e signi ican ly om each o he . This is done by applying a
sui able mul iple compa isons es , which may be pai wise o a combina ion o ea men means, usually p ede e mine a he
design s age o he s udy. The mul iple compa ison, acco ding o Edwa ds (1972) and Ke linge and Pedhazu (1973) is done
be ween adjus ed means no he unadjus ed means. When he le el o he ac o (s) is (a e) jus wo (2) he unadjus ed and he
adjus ed a e included in he able o he ANCOVA esul s. I he ac o (s) ha e mo e han wo le els, he mul iple compa ison
esul s may be p esen ed in a sepa a e (de ached] able. Such a able should show only he adjus ed means, he mean di e ences,
he con idence limi s (i desi ed) and he associa ed P- alues. The P- alues a e used o de e mine i a pai ed compa ison is
signi ican o no .
I should equally be ema ked ha , when all ea men g oups ha e equal mean on he co a ia e, no adjus men o
means akes place, since he sum o squa es due o co a ia e will be ze o. This explains why he p elimina y eg ession analysis
o he in luence o he co a ia e on he dependen a iable, is e y impo an . In o he wo ds, i he β - alue is ze o o e y
close o ze o, co a iance analysis may no be necessa y. The no mal ANOVA will be enough. I should equally be no ed ha an
ANOVA o he co a ia e wi h he ea men as ac o will equally se e he pu pose o he eg ession analysis in pa , since u he
analysis may be equi ed o de e mine he alue o β when he means o he co a ia e by ea men a e signi ican ly di e en .
Tha means, i he ANOVA esul s a e signi ican , β will s ill be calcula ed. Tha is why eg ession analysis is p e e ed a ha
p elimina y s age o he analysis.
Basic assump ions o ANCOVA
Fo he applica ion o ANCOVA and all i s’ ex ensions o si ua ions whe e he e a e many ac o s o many esponse
a iables o many co a ia es, wi h he conclusion d awn he e om, o ha e bo h in e nal and ex e nal alidi y, he ollowing
assump ions should be sa is ied:
1. The e is some kind o ela ionship be ween he dependen a iable(s) and he co a ia es(s). When he eg ession
coe icien i ze o (0) o close o Ze o (0), he ANCOVA ceases o be use ul. The e o e eg ession analysis should be ca ied
ou as a p elimina y es . Whe e he eg ession o he dependen a iable (y) on he co a ia e (x) is signi ican , he p ocess
should con inue. Whe e he eg ession analysis yield no signi ican esul s, he ANCOVA p ocess may no be e y use ul
o a was ed e o s and esou ces (Ko ha i & Ga g 2014).
2. The ela ionship be ween he dependen a iable(s) and he co a ia es) is he same in all he ea men g oups. This
assump ion should always be es ed o checked. This assump ion is e e ed o in li e a u e o ANCOVA as he
Homogenei y o eg ession coe icien s. Acco ding o Ke linge and Pedhaze (1973), es ing his assump ion p oceeds
in exac ly he same manne as he es ing o he di e ence be ween eg ession coe icien and cons an s. The ma e s
o be esol ed a e: Does using sepa a e eg ession coe icien o each expe imen al g oup add signi ican ly o he
eg ession sum o squa es, when compa ed o he eg ession sum o squa es ob ained using a common eg ession
coe icien ? I should be poin ed ou ha when he eg ession lines a e pa allel, he bs a e iden ical, implying ha he
sum o squa es ob ained om using each b o i s own g oup is he same as he eg ession sum o squa es ob ained om
using a common b o all g oups. The e o e, he di e ence in he wo sums o squa es is a alid measu e o he di e ences
in he b’s, ha is due o depa u e om pa allelism o he eg ession lines o he sepa a e g oups. I also means ha
when his di e ence is no signi ican , i can be concluded ha , he e a e no di e ences be ween he b’s and a common
b will se e he pu pose. I he e a e h ee expe imen al g oup) and le S1 S2 and S3 be hei espec i e sums o squa es,
hen, hei sum S will be S1 + S2 + S3. The da a is hen pooled and sum o squa es compu ed. Le his be Sp. The Sp-S
ep esen s he disc epancy, which can be es ed o signi icance using he - a io ex . The equi emen is ha his es
should no be signi ican . When i is no signi ican , he use o common’ b’ is jus i ied.
Unde -U iliza ion o Analysis o Co a iance in Beha io al Resea ch
IJMRA, Volume 08 Issue 10 Oc obe 2025 www.ijm a.in Page 5550
Tha he ‘b’s a e no signi ican ly di e en , only shows ha he eg ession lines a e pa allel. The lines need o be
coinciden ie he eg ession cons an s should a mos no be signi ican ly di e en . I hey, a e di e en hen an in e ac ion exis
among he independen a iables and u he analysis may be equi ed, making he pic u e clea . I hey a e no signi ican ly
di e en , he di e ence among he in e cep s is in es iga ed. Tes ing he di e ence among he in e cep s is he same as es ing
he di e ence be ween ea men e ec s o he ac o . This is done by es ing he di e ence be ween he Rs o be ween wo
p opo ions o a iance. I his es is signi ican hen he di e ence be ween he ea men means is signi ican . When he e a e
wo g oups, he analysis ends he e. Howe e , i he e a e mo e han wo g oups, an app op ia e pos hoc ex is ca ied ou on
he adjus ed means.
Analysis o Co ance wi h mul iple co a ia es
The p ocedu es o ANCOVA wi h single co a ia e can be easily ex ended o wo o mo e co a ia es. The logic emains
he same. P elimina y analysis will equi e he de e mina ion o he collec i e in luence o he co a ia es on he dependen
a iable. I his yield esul ha a e no signi ican , o he collec i e in luence is app oxima ely ze o ANCOVA may be discon inued.
I he collec i e in luence is signi ican , he squa ed mul iple co ela ion coe icien (R12) is no ed. Ano he squa ed mul iple
co ela ion o he dependen a iable wi h he co a ia es and he ec o s ep esen ing he ea men s (R22) is hen compu ed.
The di e ence be ween hese wo R-squa ed alues (R12 – R22) indica es he p opo ion o he a iance in he dependen a iable
accoun ed o by he ea men s when he in luence o he co a ia es ha e been emo ed o adjus ed o . This di e ence is es ed
o signi icance using he F- a io es .
Wi h he ad en o compu e p og ammes all you need o do is o en e he dependen a iable, he ac o ( ixed o
andom) and he co a ia es in hei dialogue boxes. Speci y wha you need in he ou pu : he model ype o sums o squa es, pos
hoc es s ( ixed o andom) and he co a ia es in hei dialogues boxes. Speci y wha you need in he ou pu : he model, ype o
sums o squa es, pos hoc es s (i any) e c as is done o one co a ia e. The esul s come wi h unadjus ed and adjus ed means,
he R- squa ed, adjus ed R-squa ed, he βs and he mean alue o he co a ia es used in adjus ing he obse ed mean o he
dependen a iable (Using SPSS)
ANCOVA wi h mul iple ac o s (ca ego ical a iable)
The logic o analysis o co a iance can be ex ended o si ua ions whe e he e a e mo e han one independen a iable.
Fo example, a s udy may be conduc ed o de e mine he e ec o eaching me hod and a ea o specializa ion on s uden s’
achie emen in ma hema ics. Teache s who s udied social sciences, physical sciences, enginee ing and ma hema ics a e o en
d a ed o each ma hema ics. So he a ea o specializa ion is he second ac o wi h ou (4) le els o specializa ion. In his case,
he e a e ow ca ego ical a iables- eaching me hods and a ea o specializa ion. The e may be co a ia es-p emeasu ed. This is a
ac o ial analysis o co a iance case.
The analysis begins wi h he de e mina ion o he na u e o he ela ionship (in luence) be ween he dependen a iable
and co a ia e(s). This leads o ei he s op he ANCOVA p ocess o “p oceed” o de e mine o compu e he mul iple R-squa ed.
Nex , he wo ac o s a e ep esen ed by wo coded ec o s-one o each ac o . Mul iple R-squa ed is he compu ed using he
co a ia es and he wo coded ec o s (p oduc ec o s be ween he ca ego ical a iables a e gene a ed o p oduce o ep esen
he in e ac ion). The di e ence be ween he mul iple R-squa ed ha includes all he co a ia es and all he coded ec o s on one
hand and he mul iple R-squa ed wi h co a ia e only, is compu ed. This di e ence ep esen s he e ec s o he wo ac o s, when
he in luences o he co a ia es ha e been adjus ed o . The esul s can be es ed o signi icance using he no mal F- a io es .
This p ocedu e is o manual compu a ions.
Wi h he ad en o s a is ical packages like he SPSS, i is much easie . En e he da a in he da a sp ead shee and de ine he
a iables in he a iable sp ead shee . En e he dependen a iable in i s dialogue box. En e he ac o s in he ac o s dialogue
box in his example, all he ac o s a e ixed. So, bo h eaching me hod and eache s’ quali ica ion will en e he same box. Take
he co a ia e o co a ia es o co a ia es dialogue box. Speci y wha you need in he ou pu , desc ip i e s a is ics, pos -hoc es s,
le el o signi icance, plo s, he in e ac ions ha a e o in e es o you, else he sys em will p oduce all in e ac ion, indica e e ec
size, i you wan and click OK. You a e done. You will lea n how o in e p e and he necessi y o each esul .
CONCLUSION
This pape , wi hin he limi o p esen a ion due o delibe a e exclusion o compu a ional de ails has shown ha i is g a e
e o o limi he applica ion o ANCOVA o one ac o wi h jus wo le els; i is an e o o compa e unadjus ed means in pos hoc
es s; ANOCOVA can be ca ied ou wi h mul iple ac o s, mul iple ac o s le els and mul iple co a ia es; he endpoin is he
adjus men o ea men means; he ANCOVA p ocess may be e mina ed i all ea men g oups ha e equal mean on he co a ia e
o when he in luence o he co a ia e on he dependen a iable is ze o o e y close o ze o.
Unde -U iliza ion o Analysis o Co a iance in Beha io al Resea ch
IJMRA, Volume 08 Issue 10 Oc obe 2025 www.ijm a.in Page 5551
REFERENCES
1) Ke linge , F.N. & Pedhazu , E.J. (1073). Mul iple eg ession in beha io al esea ch. New Yo k: Hol , Rineha and Wins on.
2) Elashol , J.D. (1969). Analysis o co a iance: a delica e ins umen . Ame ican Educa ional Resea ch Jou nal, 6,383-401
3) Coch an, W.G. (1957). Analysis o co a iance: i s na u e and uses. Biome ics, 13, 261-281
4) Feld , L.S. (1958). A compa ison o he p ecision o h ee expe imen al designs employing a concomi an a iable.
Psychome ika, 23, 335-353.
5) Edwa ds, A.L. (1972). Expe imen al design in psychological esea ch (4 h ed). New Yo k: Hol , Rineha and Wins on, Inc.
6) Ko ha i, C.R. & Ga g, G. (2014). Resea ch Me hodology: me hods and echniques (3 d ed). New Delhi: New Age
In e na ional L d.
The e is an Open Access a icle, dis ibu ed unde he e m o he C ea i e Commons
A ibu ion – Non Comme cial 4.0 In e na ional (CC BY-NC 4.0)
(h ps://c ea i ecommons.o g/licenses/by-nc/4.0/), which pe mi s emixing, adap ing and
building upon he wo k o non-comme cial use, p o ided he o iginal wo k is p ope ly ci ed.
