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Unformatted text preview: 178 Chapter 15 / Simpie Linear Regression THE Begression procedure is designed To perform eiTher simple regression (ChapTer l5) or
muliiple regression (Chapier 16). We spliT The command inTo Two chapTers iargeiy for The sake
oi ciariTy. If The reader is anacquainTed wiTh mu/n'p/e regression, ihis chapTer, on simpie regres—
sion, will serve as an iniroduciion. Severai Things will be covered in The inTroduciory poriion of
This chapTer: (a) The concepT of predicTed values and The regression equaiion, (b) ihe reiaiion—
ship beTween bivariaie correlaiion and simpie regression, (c) The proporiion of variance in one
variable explained by anoTher, and (d) a TesT for curvilinear reiaiionships. Several words oi cauiion are appropriaTe here: FirsT, a number of Thick voiumes have been
wriTTen on regression analysis. We are in no way aTTempTing in a few pages To dupiicaie Those
efforis. The iniroducTions of These Two chapiers are designed primarily To give an overview and a concepTual feei for The regression procedure. Second, in This chapTer (and ChapTer T6), in addiTion To describing The sTandard linear relaiionships, we expiain how To conduci regression 2i
ThaT considers curvilinear Tendencies in The daia. We soggesT Thai Those less acquainied wi’rh regression should spend The Time To Thoroughly undersiand linear regression before aTTernpiing : The much less frequenily used TesTs for curviiinear Trends. PREDICTED VALUES AND THE REGRESSION EQUATIGN There are many Timeswhen, given informaiion aboai one characTerisiic of a parTicular phe nomenonqwe have some idea as To The naTure of anoTher characTerisTic. Consider The heighi ' ' and weighi of adulT humans. if we know Thai 0 person is 7 feeT (2T4 cm} Tall, we wouid suspeci ' 55' (wiTh a fair degree of ceriairiiy) Thai This person probably weighs more Than 200 pounds (91 if kg). ii a person is 4 feei 6 inches [l 37 cm) Tall, we would suspecT Thai such a person woald ' 3" weigh less Than iOO pounds (45 kg). There is a wide varieTy of phenomena in which, given . 3.;
inTormaTion abouT one variable, we have some clues aboui characierisiics of anoTher: l9 and academic success, oxygen upTake and abiiiTy To run a TasT mile, perceniage of TasTTwiTch muse cle fibers and speed in a TOOmeter race, Type of auTOmobiie one owns and moneTary nei : wor‘ih, average daily caioric inTake and body weighT, feelings OT sympaihy Toward a needy per son and likelihood oi helping ThaT person. ThroughouT a liTeTime humans make Thousands of i such inferences (e.g., he‘s faT, he musT eaT a loT). SomeTimes our inferences are correcT, oihe'i”: Times noi. 'Simple regression is designed To help us come to more accuraie inferences. [T can .; i335:
noT guaraniee Thai our inferences are correci, bui iT can deiermine The likelihood or probabﬂii‘y ' Thai our inferences are sound; and given a vaiue for one variable, iT can predici The mosT iikely value Tor The oiher variable based on available informaiion. ' ' To iilusTraie regression, we wili iniroduce our example aT This iime. While ii would be possible
io use The gradessav file To iilosTraTe simpie regression leg. The influence of previous GPA on
Tinai poinTs), we have chosen To creaTe a new doTa seT Thai is able To demonsiraTe boih linear
regression and curvilinear regression. The new file is tailed anxiety.sav and consisTs of a doTa
seT in which 73 sTudenTs are measured on iheir levei oi preexam anxieTy on a nonem To so”
vereli 0) scale, and Then measured on a lOO—poinT exam. The hypoThesis for a iinear relaTion«
ship mighT be ihai those wiTh very low anxieTy will do pooriy because They don'T care much and
ihai Those wiTh high anxieTy will do beTTer because They are moTivaied io spend more Time in
preparaiion. The dependeni (criTerion) variable is exam, and The independeni lpredicior) vari~
able is anxiety. ln oTher words we are aTTempTing To predici The exam score from The anxieiy Oi” es
0T {)6 en
se
"id
in
an "*9 Chapter 15 / Simple Linear Regression 179 score. Among oTher Things ThaT regression accomplishes, H is able To creaTe a regression equa—
Ti'orz To calculaTe a person‘s predfcfed score on The exam based on his or her anxieTy score. The regression equlaTion Tollows The model oT The general equaTion designed To predicT a sTudenT‘s
True or acTual score. The equaTion Tor The sTudenTs True score Tollows: exammue) = some constant + a coefficient >< anxiety + residual ThaT is, The True exam score is equal To a constant plus some weighTed number (coel'licienT)
Times The anxiety score plus The residual. The inclusion of The residual Term is To acknowledge
ThaT predfcfea’values in The social sciences are almosT never exachy correcT and ThaT To acquire
a True value requires The inclusion 0T 0 Term ThaT adiusTs Tor The discrepancy beTween The pre~
dicTeol score and The acTual score. This diTTerence is called The residual For iosTance, The equaTion based on our daTa seT (wiTh consTanT and coeTTicienT generoTed by The regression pro“
cedure) Tollows: ' examﬁme) : 64.247 + 2.8T8(anxiety) + residual To demonsTraTe The use 0T The equaTion, we will inserT The anxieTy value Tor subiecT #24, who
scored 6.5 on The anxieTy scale. examame) = 64.24? + 2.818(6.5) + residual exam(true) = 82.56 + residual The 82.56 is The sTudenT's pried/cred score based on his 6.5 onxieTy score. We know ThaT The
acTuol exam score Tor subiecT 24 was 94. We can now deTermine The value 0T The residual
(how Tar oTT our predicTed value was), buT we can do This only offer we know The True value of
The dependenT variable (exam in This case). The residual is simply The True value ménus The predicTed value (94 e 82.56), or ll.44. The equaTion wiTh all values inserTed now looks like
This: 94
this score 82.56 + 11.44
predicted score + residual We have included a brief descripTion oT The residual Term because you will see iT so Trequenle
in The sTudy oT sTaTisTics, buT we now Turn our oTTenTion To The issue oT predicTed values hosed on a calculaTed regressTon eqoaTTon. A more exTensive discussion of residuals Takes place in
ChapTer 28. The regression equoTion Tor The predicTed value OT exam is: exammredgcted) = 64.24? + 2.818(anxiety) To demonsTraTe compuTaTion, subiecTs 2, 43, and 72 scored )5, 7.0, and 9.0 anxieTy poinTs,
respecTively. CompuTaTion of The predicTed scores Tor each oT The Three Tollows. Following The
prechTed value is The acTual score achieved by The Three subiecTs (in parenTheses), To demon
sTroTe how well (or poorly) The equoTion was able To predicT The True scores: Subject 2: examwredicted) 2 64.247 + 2.818(?.5) = 68.47
Subject 43: exammredicted) = 64.24? + 2.818020) = 83.97
Subject 72: exammyedicteé) = 64.24? + 2.818(9.0) : 89.61 (actual score was 52)
(actual score was 87)
(actual score was 71) 180 Chapter 15/ Simple Linear Regression too high (68.47 vs. 52); tor subiect
ite close to the actual score (83.97 vs. 87); and tor subiect 72,
(89.61 vs. 7)). From this limited observation we
predict values is pretty good tor midrange
Or, we may conclude that there are tactors
y that intluence his or her exam score.
t is called mu/tfp/e regression and We notice that for subiect 2, the predicted value was much
43, the predicted value was go
the predicted value was also much too high
might conclude that the ability at our equation to
anxiety scores, but much poorer at the extremes. other than a measure at the subiect's preexarn anxiet
The issue at severe/factors intiuencing a variable at interes will be addressed in the next chapter. SIMPLE REGRESSION, AND AMOUNT OF VARIANCE EXPLAINED We are not left at the mercy at our intuition to determine whether or not a regression equation
is able to do a good (ob at predicting scores. The output generated by the ﬁegression com
mend calculates tour ditterent values that are ot particular interest to the researcher: l . SPSS generates a score that measures the strength ot relationship between the depend
ent variable (exam) and the independent variable (anxiety). This score is designated
with a capital R and is simply our old triend, the bivariate correlation (r). The capital R
is used (rather than a lower case I) because the ﬁegression command is usually used  to compute multiple correlations (that is, the strength ot relationship between several in
dependent variables and a single dependent variable). For a description at correlation, plpase reter to Chapter l0.
2. Along with the computation at it, SPSS prints out a probability value (,0) associated with Rip indicate the signiﬁcance ot that association. Once again, a p < .05 is generally interpreted as indicating a statistically signiticant correlation. it ,o > .05, the strength at ' '5
association between the two variables is usually not considered statistically signiﬁcant; or the relationship between the two constructs is considered weak or nonexistent. 3. quuare (or R2) is simply the square at R, but it has special signiﬁcance. The R2 value is
the proportion at variance in one variable accounted tor (or explained) by the other variable. tor the relationship between anxiety and exam, SPSS calculated values at R
t .48794 (,0 < .OOOl) and R2 = .23808. The Rsauare value indicates that 23.8% ot
the variance in the exam score is accounted tor by pretest anxiety. The standard dis
claimers must be inserted here: With a correlation, be cautious about interring causaw
tion. in this case the direction ot causation is sale to assume because an exam score cannot intluence pre—exarn anxiety. 4. SPSS calculates the constant
equation. As already noted, the constant and coetticient computed tor the regression equation identitying the relationship between anxiety and exam were 64.247 and
2.8l8, respectively. TESTING FOR A CURVILINEAR RELATIONSHIP Most knowledgeable people would consider it toolishness to thinl< that higher pretest anxielil
"ll will produce higher scores on an exam. A widely held position is that very low level anxiety Wt
result in poor scores (due to lack ot motivation) and that as anxiety scores increase, motivation and the coel‘licient (called Bwvalues) tor the regressioa ; . Chapter 15 / Simple Linear Regression 131 ,ci ; To do well increases and higher scores resul’r. However, There comes a poinT when addiTional
2, anxieTy is deTrirnenTal To perlormance, and or The Top end of The anxieTy scale There would once
ve again be a deCrease ol perlorrnance. Regression analysis (whelher il be simple or mulriple)
ye measures a Ifnear relaiionship beTween ibe independeni variable(s) and The dependenr vari—
>r5 able. In The liciional daTa seT presenied earlier There is a lairly sTrong linear relaiionship be—
e. Tween anxiety and The exam score ( m“: .484, ,a < .000?) BUT perhaps The regression equa
sd Tion would generale more accuraTe predicied values (yielding a belier "iii" ol The Clara) ii a quadraTic equaTion were employed Thai included an anxietysquared (anxiety?) Term. Usually, belore one TesTs Tor a curvilinear Trend, There needs To be evidence (TheoreTical or
compuTaTional) ’rhaT such a relaTionship exisTs. Frankly, in The social Sciences, curvilinear Trends
happen in only a limiTed number of circumsiances, buT They can be criTical To underslanding The 3n daTa when They do occur. To demonsTraTe, we produce The scaTTerploT beTween exam and
W anxieTy. The graph (below) shows The exam scores on The verfical axis (The scale ranges Trorn 40 To
C), HO), and anxiety on The horizonial axis wiTh a range ol 0 To l0. lniiial visual inspeclion re
ad ' veals whaT appears To be a curvilinear Trend. For mid—range anxieTy values TheTesT scores are
R highesi, and al The exTremes They Tend To be lower. One needs To be cauiioned when aﬁempT
3d ing To read a scaTTergram. A relaiionship may appear To exisl, buT when TesTed is rial sTaTisii
n cally significant More lrequenlly, visual inspecTion alone does noT reveal a curvilinear Trend
lg, buT a sTaTisTical resT does. When exploring The relaTionship beiween Two variables, Begression is able To reveal whe’rher There is a significanl linear Trend, or significani curvilinear Trend, signifi
ifh canT linear and curvilinear Trends, or neiTher.
ll); i Saat’cerpia’r or Exam Score by Anxiety l 11 «—— mm
a. Samp/e Scaﬁer p/of demonsfrar—  Bl ] : r'ng a con/rilmear frenc/ ' mu 9 j E 3 I so ° 3 “ ° ° e 4, ‘ l5 s i g ‘ , ‘l e t l
18!, 5 ea . 0 ‘ g ,, , 4 , e l
R lg 70 " “ 4' ‘ e 50 50 l v e at s.
ol ‘ r g, «r
.
i o“
. : ¢ 1,
' ul—
: G 0 ° )
ire m 5"” "1' {WM 5 m" is
ANXIETY
n .
:n _ A simple procedure oh‘ered by SPSS wilhin The coniexi of The Begression command is a quick
qd : TesT To check lor linear or curvilinear Trends. You idenTiTy The dependen’r variable (exam), The
' independenl variable (anxiety), Then lrom The resalTing dialog box, selecT Linear and Quad
ratio. An 0K click will produce a Twoline oquuT (shown on The Tollowing page) Thai indicaTes
ii linear and/or curvilinear Trends exis’r. The Bvalues are also included so Thar ET is possible To
wriTe prediciednvalue equaTions for eiTher linear or curvilinear relaTionships. This process also
E creaies a chad showing The scaTTergram (dale poinTs are all connecied), ihe linear regression
3W line (The sTraighT one) and The curvilinear regression line (The curved one). NoTice The similariiy
Jill ; beTween The Two charts in Screens 15.l and 15.2. Also noie Thai The consTanT and coellicienis 3a in The equaTions (lollowing page) uTilize values lrorn The Two lines of oquuT. 132 Chapter 15 / Simpie Linear Regression 110 .m
Sample Scarfer p/of demonsfraﬁng Too 4
a linear and a cum/linear Trend 90.1
80 '
l
TD §
ea  “ Observed
so: Linear
«so, “i r __ i Mi '1 Quadratic:
8 1D _ $ 0 2 4 f3
Dependent Method I _ quuare d.f. F I Sig of F I. b0
EXAM smear .233 71 22.19 .000 ' 34.247
EXAM quadratic .841 70 . The linear and curvilinear regression equaTions now Tallow: Linear equaTion (The sTraighT line): exammd) m 84.25 + 2.82{anxiety) QuadraTic equaTion (The curved line): exammred) : 30.38 + 18.93(anxiety) + —T.52(arzxiety)2 Observe ThaT in The ouTiauT (above), The P value Tor The linear equaTioa indicaTes ThaT anxiety
explains 23.8% of The exam parlormance, while The R2 value Tor The quadraﬁc equa’rion {where
boTh The linear and The curvilinear Trend influences The ouTcome) indicafes ThaT 64.1% of The
variahce in exam is explained by anxiety and The square ol anxiety. Under Sig of F, The .000
Tor boTh The linear and The curvilinear equaTion indi'caTe ThaT boTh Trends are sTaTisTically signiliI canT. We would like To see iT The quadra’ric equaTion is more successTol aT predicTing acTual scores
Than was The linear equaTion. To do so we subsTiTuTe The anxiety values for The same subiecTs' (numbers 2, 43, and 72) used To illusTraTe The linear equaTion: SubjecT 2: exam (med) = 30.38 + 18.93(1.5)+ —‘T.52(1.5)2 = 55.31 (acTual score, 52)
SubiecT 43: exam (med) 3 30.38 + 18930.8) + ~1.52(7.0)2 : 88.30 (acTuai score, 87).
SubiecT 72; exam (med) = 30.38 + 13.93(9.0) + —1.52(9.0)2 = 77.49 (acTual score, 78‘ A quick check of The resulTs from The linear eqaaTéoa demonsTraTes The subsTaaTially superior.
predicTive abiliTy oT The qaadraTic eqaaTion. NoTe The chariL below: Subtnber _ Acuaisoore _ ' I Predicted Tiear score A number of books are available ThaT cover boTh simple, curvilinear, and mulTiple regression?
 Goni'a Several ThaT The authors Teel are especialiy good include: Chaﬁeriee and Price (1 999),
and Smi‘rh (i993); Schulman “998); Sea and SrivasTava (T997); and Weisberg (i985l. Please see The reTerence secTion Tor more deTailed inTormaTion on These resources. Chapter 15 / Simple Linear Regression 183 STEP BY STEP Simple Linear and Curviiinear Regression l Creare and name a dafa fi/e or edn‘ {irrnecessan/j an a/ready exisring rife {see Chaprer 3) To enrer SPSS, a click on Start in five Task bar {boﬂom ofscreen) acrivai‘es rhe sran‘ menu: bioThis" .. .. . I Scree'n__ Affer clicking rhe SPSS program icon, Screen i appears on The moniror. Screens 7 and 2 (dis
p/ayed on The inside froni‘ caved a//ow you To access The dare We used in conducfing The analy
. sis of inferesi‘, The fof/owing sequence accesses The anxietysav file for funfher analyses: DiniThis' I l: isﬁiiiﬁssﬁiiiiFeniréadiis' _.> l_n__'$I;:gr_Een_ "issue ~>Qpen assuage [crisis]
: type anxiety. sav > [orQQanxietysav ] "J Wherher z‘irsf enrering SPSS or rerurning from earlier operai‘ions The srandard menu of com— mands across The Top is required {shown below). As /ong as if is visih/e you may perform any
analyses. If is nor necessary for rhe daia Window ro he visib/e. sees _:...sis : This menu pi commands disappears or modiﬁes when using pivoT Tables or ecliiing graphs. To
uncover The siandarol menu 0T commands simply click on The or The icon. Afrer comp/erion of 5fep 3, a screen Wiih the desired menu bar appears. When you c/ic/< a
command (from ihe menu bar], a series of opi‘ions WW appear (usua/M hie/ow fhe se/ecfed
command. l/l/n‘h each new ser of opfions, c/ick The desired ifem. The sequence io access linear
regression begins of any screen wifh fhe menu of commands visible: "MThis . " ' ' 'ilh_s:c:re.éiz'_ .. . .. .. ..
@ﬁnalyze ﬁgegression —>%ml:inear AT This poinT, a new dialog box opens (Screen 5.3, following page) ThaT allows you To conduci
regression analysis. Because This box is much more lrequeniiy used To conduci mu/rip/e regres
sion analysis Than simple linear regression, There are many opTions available Thai we will noT
discuss oniil nexi chapier. A warning To porenis wiih young children, however: Under no cir
cumsiances allow a child less Than l3 years of age To click on The §taﬁstics or Blots pushbui
ions. Windows open wiih opTions so Terrifying Thai some have never recovereaE Tram The Trauma.
The lisT To The ieTT will iniiially conTain only Two variables, anxiety and exam (The anxietyz vari—
able will be creaTecl iaier). The procedure is To seieci exam anal pasie iT inTo The erendem‘
box, seleci anxiety and pasie iT inio The independenﬂs) box, Then click on The UK burion. The
program will Then run yielding Rand Ra values, Fvalues and Tesis OT significance, The Bvalues 184 Chapter 15 / Simple Linear Regression ThaT provide‘coasTanTs and coefficienis Tor The regression equaTion, and BeTa (E3) values To show
The sTrengTh ol associaTion beTween The Two variables. Some of The Terms may be unfamiliar To
you and wiil be explained in The oquuT secTion. The sTep—by—s’rep sequence Toilows Screen l5.3. 
The iniiia/ linear Regression
Win do w To conducf simple /inear regression Wiin a dependenf variab/e of exam and an independenf
variable of anxiety perform The fol/owing sequence of sieps. Begin wii/T Screen i5.3,' resu/is _
from fT’Tis analysis are in The Oufpuf secfion. DbThiS _ I . exam “5 upper In. Screen Curvilinear Trends We wili show Two sequences ThaT deal wi’rh curvilinear Trends. The TirsT deals wiTh The simpie Twoline oquuT reproduced in The inTroducTion, and creaiion of The graph ThaT displays linear 
and curvilinear Treads (also displayed in The iniroducTion). The second considers The more Tor—
mal procedure of creoTirig a quadraTic variable ThaT you may use in a number of diTTerenT
analyses. While much 0T wha‘r Takes place in The second procedure could be produced by The"
TirsT, we Teel H is imporTariT ThaT you undersiand whaT is realiy Taking place when you TesT Tor a
curvilinear Trend. Furihermore, alThaugh we presené curviiinear Treads in The conTexT 0T simpie‘
regression, The same principies (and access procedures) apply To The nexT chap’rer on mulTiple regression. To access fire Curve Estimation chan‘ requires a diﬁfereni siep—4 sequence. r...— ln Screen
:'
A new diaiog box now opens (Screen 15.4, Toilowing page) ThaT allow a number of opTions (of curve esTimaTion. The early procedure is idenTical To ThaT shown in The previous sequence Oi;
sTeps (sequence sTep 5): Selec’r exam and pasTe iT inTo The erendent box, selec‘r anxiety and _Do' This. gegression —> gurve Estimation Analyze w) Oi’
H'
rﬂi
We ile
ile or
of id Chapter 15 / Simple Linear Regression 135 posTe iT inTo The independent box. ATTer The dependenT and 'independenT variables are se
lecTecl, noTice‘ThaT Three 0T The opTions in This diaiog box are already selec’red by deTaelT: in
clude constant in equation provides The consTanT, necessary To creaTe a regression equaTion.
The Pigt modeis opTion creaTes The graph ThaT was iilusTraTed in The inTroducTion. The ginear model refers To TesTing To:r and Then including as a Tine on The graph any linear Trend in your
daTa. ' I I i [3]me Estimation I The Curve EsTimoTion Window WiThiri The Modeis box a varieTy oi Toys exisT Tor The maThemaTical or sTaTisTical wizards. Ail 0T
The curvilinear models included here have Their unique applico’rions, buT none are used Tre
quenle in The social sciences, and only The Quadratic oprion will be addressed here. For The presenT analysis, keep The aiready—selecTed Linear opfion, and also selecT Quadratic. The
sTeps now Tallow, beginning wéih Screen l5.4. Beg/mth wri‘h a dependenf variable of exam and checking for Wear and curvilinear Trends in The independenr yahoo/e (anxiety), perform The fol/owing sequence of sfeps. Perform The sfep
4a sequence ﬁeage 784/ To arrive or Screen 75.2, resU/i‘s of This aha/yer are in The lnfroc/uch'on. ) In screen ' Do This; . _ .. . .. . .. r. _ . $5195? anxiety ) midd/e "'9‘ @Quadmtic ' WhaT acTuaTTy Takes place when you Tormulaie The regression eqeaTion ThaT includes The inﬂu
ence 0T 0 quadra’ric Term, is The creaTion of a new variable ThaT is The square oT The independeni
variable (anxiety), or anxiety? You aTl vividiy remember Trom your Algebra T days ThaT The in—
clusion of a variable squared produces a parabola, Thar is, a curve wiTh a single change of di
recTion ThaT opens eiTher upward (ii The coeTTicienT is posiTive) or downward UT The coeTTicienT is
negaTive). NoTice in The equa’rion on page T70 Thai The coeTTicienT oT The squared Term is
negaiive [—l .52(anxiety)2], and ThoT The parabola opens downward. Also noTe The influence UT
The linear Trend (a Tendency oT daTa poinTs To go from lower leTT To upper righT) is also reTlecTed
in The curve. The leTT end oT The curve is lower Than The righT end oi The curve. Thus boTh linear
and curvilinear (quadraTic) Trends are reTiecTed in The graph. 136 Chapter 15 / Simpie Linear Regression To access ihel regression precedare that produces both linear and cerviiineor elements it is
necessary to create a new variable, the square of anxiety, assigned a variable name of anxi
ety2. Then when we begin the regression procedure iwo variables will be included as inde
pendent variables, anxiety and anxietyz. The me’rhod of creating new variables is described in
Chapter 4, and we will stepby—step you through the sequence but wiii not reproduce the screen
here (Screen 4.3 is on page 5T it you wish a visual reference). We do access one window that
has not been displayed previously, that is the window “that appears when you click on the Type
& wl=aibel button displayed in Screen 4.3. The use oi the box is selt~explanatory and is repro—
duced (below) tor visuai reterence only. The Compute Variable: Type and Label  .. 557'. .
Window The step—bystep sequence at creating a new variable, anxietyZ, and including it along with
anxiety as independent variables that intluence the dependent variable exam now toliows.
This is now actuaiiy mu/rrp/e regression because more than one independent variable is'inw cluded. § To run tire regression procedure wiih a dependeni variable oiexam and independent variables ' of anxiety and (anxiety)2 perform ine fo/iowing sequence of steps, Begin or the Dara Editor ; screen wifn fire menu of commands snowing. The procedure begins by crean'ng five new vari— ab/e anxietyZ. Resu/is from me ana/ysis are inc/uded in five Oufpui secfion, I H Screen" .. . .. . . . . . . Iransform —> Qompute i ' anxietyZ > Type & Label I I inside the Label box —) square of anxiety —> Continue Do This anxiety ~—> type is: ~——>' type; > “rem Analyze —) Regression —> @Linear
.3 exam —) Upper —> anxiety “') midd/e anXietyz T.) l 1: midd/e , eﬁw Upon compietion at step 5, 50, or 5b, the outpet screen will appear (Screen i, inside bac‘is
cover). All results from ihe (estcompleted anaiysis are included in the Ouiput Navigator;
Make use at the scroll bar arrows ( to view the results. Even when viewing outpuii
the standard menu at commands is still listed across the top at the window. Further analyses may be conducted without returning to the data screen. ~_ Chapter 15 / Simpie Linear Regression 187 El: PRINTING RESULTS 8” Results at the analysis (or analyses) that have iust been conducted requires a window that dis—
m plays the standard commands (Eite Edit Qata Iransfarm Analyze . . .) across the top. A
an typical print procedure is shown below beginning with the standard outpUt screen (Screen l,
Jm inside back cover). to To pn’ni‘ resu/i‘s, from the Omjour screen perform the {of/owing sequence orsfeps: 3_ . . . . . '. inissréen' ' D'o'ihi'é I . . . . .. . . .. .. . ... .
I 5e/ecf desired auteur or edit {see pages 7 9  25) m} Eile ~) Erint Consider and se/ecf desired print options oﬁ’ereaﬂ then —> 70 exit you may begin from any screen that shows the File command of the top. .«rin' Screen . Da“Tﬁié i tissue “>1” Egit 5 Note: Atter clicking Eggit, there wilE trequently be smali windows that appear asking ii you wish to save or change anything. Simply click each appropriate response. m OUTPUT Simple Linear and Curvilinear Regression Analysis 95 Whai‘ {13/be5 is output from sequence step 5 on page 184.
or
W; Model Summary R . 5 Model 1
R square I Adjusted R square I . Dependent Veriabie: EXAM
Std Error of Estimate _ Predictors: (Constant, ANXIETY) ANOVA .' Sum ofquares SignifF
' Regression 1 3310.476 3310.475 l 22.186 .0000
Residual 71 10594209 149.214 l
b Coefficients
1 i Unstanderdized coefficients Std Coeﬁicints 
m i Std. Error Signifoft
' ' (Constant) =
3k : ANXIETY
1’. ; '
This output shows that there is a signiﬁcant linear relationship between exam pertormance and anxiety such that a higher level at anxiety results in higher scores. The speciﬁc meaning of this
output is summarized in the definitions of terms that iollow. 188 Chapter 15 / Simple Linear Regression 'f .befihitichipésictipﬁon. ' R SQUARE ADJUSTED R SQUARE STANDARD ERROR
REGRESSION
RESIDUAL DF SUM OF SQUARES MEAN SQUARE
F
SIGNIF F B
F STD ERROR
i‘ BETA t
SIGNEF t Since there is oniy one independent variable, this number is the bivariate correla»
tion (r) between exam and anxiety. The R SQUARE vaiue identifies the proportion of variance in exam accounted for by
ANXIETY. in this case 23.8% of the variance in exam is explained by anxiety. R SQUARE is an accurate value for the sample drawn but is considered an optimis~
tic estimate for the popuiation value, The ADJUSTED R SQUARE is considered a
better population estimate. The standard deviation of the expected vatues 1tor the dependent variable, exam.
Statistics reiating to the explained portion of the variance.
Statistics relating to the unexplained portion of the variance. Degrees of freedom: For regression, the number of independent variables (1 in i
this case). For the residual, the number of subjects {73) minus the number of in—
dependent variables {1), minus 1: (73 wt W1 "a 71). For regression this is the between—groups sum of squares; for residuai, the within
groups sum of squares. Note that in this case there is a larger portion of unexa
.plained variance than there is of explained variance, a reality also refiected in the E
R2 value. Sum of squares divided by degrees of freedom. Mean square regression divided by mean square residuai. Likeiihood that this result could occur by chance. Coefficient and constant for the linear regression equation:
exammred) a 64.24? + 2.818(anxiety). Standard error of B: A measure of the stability or sampling error of the Bvaiues. 5
it is the standard deviation of the Bavalues given a targe number of samples drawn '
from the same population. The standardized regression coefficients. This is the B—value for standardized
scores (zwscores) of the variable anxiety. This value wiil aiways vary between i 1.0 r
in iinear relationships. For curvilinear relationships it will sometimes extend out
side that range. 8 divided by the standard error of B. F
Likeiihood that this result could occur by chance. ‘ A REGRESSION ANALYSIS THAT TESTS FOR A CURVILINEAR TREND
What follows is output from sequence step 5b on page [86. Model Summary R
R square Adjusted R square
Std Error of Estimate Source of Variation _ _ Model 2 Dependent Variable: EXAM
Predictors: (Constant. ANXiETY) ANOVA Sum of Suares
8194.538 l Me Square
4457.269 Regression Residual 4990.146 ...
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