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Unformatted text preview: 192 Chapter 16/ Multipie Regression Analysis MULTiPLE REGRESSlON is The natural extension at simple linear regression presented in
Chapter l5. In simple regression, we measured The amount at intiuence one variabie (The in—
dependent or predictor variable) had on a second variable (The dependent or criterion vari
able). We also compuTed The constant and coefﬁcient Tor a regression equation designed To
predicT The values of The dependent variable, based on The values of The independent variable.
While simple regression shows The .inTluence of one variable on another, multiple regression
analysis shows The intluence oi Two or more variables on a designated dependent variable. Another way To consider regression analysis (simple or multiple) is Tram The viewpoint oi a
slope—intercept torm oi an equation. When a simple correlation between Two variabies is com—
puted, The intercept and siope at the regression /ine (or line at best Tit) may be requested. This
line is based on the regression equation mentioned above with The y—inTercept determined by
The constant value and The slope determined by The coeTTicienT at the independent variable. We
describe here a simple regression equation as a vehicle for introducing multiple regression
analysis. To assist in This process we present a new example based on a iile called heip ing1.sav. This data Tile is reiated To a study of helping behavior; it is real data derived Tram a
sample at Si subiects. THE REGRESSION EQUATION in this chapter we introduce The new set at data mentioned above. Two at The variables used
to illustrate simple regression are zhelp (a measure at total amount 01‘ Time spent helping a
Triend with a probiem, measured in zscores) and sympathy (The amount at sympathy Telt by The
helper in response to The Triend’s problem, measured on a /i#ie(l) To mac/7(7) scale), Although
correlation is oTTen bidirectional, in This case zhelp is designated as the dependent variable
(that is, bne‘s sympathy influences how much help is given rather Than The amount of help inTlu—
encing one's sympathy). A signiﬁcant correlation (r = .46, p < .OOOl) was computed, demon
strating a substantiai relationship between the amount at sympathy one Teels and the amount at
help given. in addition, The intercept value (~l .892) and The slope (.498) were also calculated
for a regression equation showing The relationship between The (Two variables. From These
numbers we can create The Tormuia To determine The predicfed value at zheip: Zhelmpreaicted) = —‘1.892 + .498(sympathy) it a person measured 5.6 on The sympathy scale, The predicted value Tor That person tor zheip
would be .897. in other words, it a person measured Tairly high in sympathy (5.6 in this case),
it is anTicipaTed That he or she wouid give quite a lot at heip (a zscore at .897 indicates almost
one standard deviation more Than The average amount at help). The sequence below illus
Trates The computation at This number: Multipie regression analysis is similar but allows more than one independent variable To have
an influence on the dependent variable. ln This example, Two other measures were aiso signiﬁcantly correlaTed with zheip: anger (angry
or irritated emotions telt by the helper Toward the needy Triend, on a none(l) To mac/7(7) scale),
and efficacy (selTetticacy, or The helper's beliet That he or she had the resources To be oT help '
To The Triend, on a /i#/e(i) to mac/7(7) scale). The multiple regression anaiysis generated The
toilowing 8 values (The B can be roughly Translated as The slope or weighted consranrior the i’e Chapter 16 I Multiple Regression Analysis 193 variable oi inTeresT.): Blsympaihy) = .494), B(anger) = .2836, and BleTTicacy) a: .4125, and The
consTanT (iniercepi) 2 —4.3078. From These numbers a new equaTion may be generaTed To
deTermine The predicTed value Tor zhelp: Zhelpmredicted) = ~4.8078 + .4941(sympathy) + .2836(anger) + .4125(efficacy) lnserTing numbers Tram an acTual subiecT, subiecT 9 in This case: zheip(predgcted)=4.3078+.494’i(3.5)+.2836(1.0)+ .4125(2.9) : 4.09 This resuli suggesTs Thai This person who measured midrange on sympaihy (3.5), low in anger
(l .O), and low in selT—eTTicacy (2.9) would be expecTed To give liTTle help (a z score 0T MT .09 is
more Than one sTandard deviaTion below average). SubiecT 9‘s acfua/ zheip score was —.92.
The predicTion in This case was Tairly close To accuraTe. A posiﬁve value Tor one oT The B coeTTicienTs indicaies Thai 0 higher score on The associaied
variable will increase The value 0T The dependenT variable (i.e., more sympaihy yields more
help). A negarive coeTTicienT on a predicTor variable would decrease The value oT The depend
enT variable (The equaTion above does noT illusTraTe This; an example mighT be more cynicism
yields /ess help). The greaTer The 5’ value (absoluTe values), The greaTer The influence on The
value 0T The dependeai variable. The smaller The 8 value (absoluTe values) The less inﬂuence
Thai variable has on The dependeni variable. However, 8 values when cannoT be compared direcily because diTTerenT variables may be
measured on differenT scales, or have differeni mefn’cs. To resolve This diTTiculTy, sTaTisTicians
have genera‘red a sTandardized score called Befa ([3), which allows Tor direcT comparison CT The
relaTive sTrengThs oT relaTionships beTween variables. [3 varies beTween $1.0 and is a pan‘r’a/
come/affair. A parTial correlaTion is The correlaTion beTween Two variables in which The inﬂuence
oT all oTher variables in The equaTion have been pariialed ouT. in The conTexT of The preseni ex
ample, The Befa beTween anger and zhelp is The correlaiion beTween The Two variables aﬁer
sympathy and efficacy have been enTered and The variabiii’ry due To The subiecTs’ sympaThy
and eTTicacy have already been calcalaied. Thus Befa is The unique conTribuTion of one vari—
able To explain anoTher variable. The Safe weighT, oTTen called The sfandardized regression co
efﬁcient is noT only an imporianT concepT in regression analysis, buT is The consTrucT used in
sTrucTural equoTion modeling To show The magniTude and direcTion of The relaTionships beTween
all variables in a model. STrucTural equaTion modeling is becoming increasingly popular in so
cial science research buT requires The purchase oT an addiTional SPSS module. in The previous equaTion, we Tind, as expec‘red, Thai higher sympathy and higher efficacy 'scores correlaTed wiTh higher zheip scores. ConTrary To inTuiTion, we find Thar more anger also correlaTed posiiively wiTh zhelp. Why This is True is a maTTer 0T discussion and inTerpreTaTion of
The researcher. When an unexpecTed resul’r occurs, The researcher would be well advised To
recheck Their doTa To ensure ThaT variables were coded and enTered correchy. SPSS gives no
clue as To why analyses Tum our The way They do. REGRESSION AND R2: THE AMOUNT OF VARIANCE EXPLAINED ln multiple regression analysis, any number O'l variables may be used as predicTors, buT many
variables are noT necessarily The ideal. IT is imporTanT To find variables ThaT Sigm‘ﬁcani‘ly inTlu»
ence The dependenT variable. SPSS has procedures by which only Signiﬁcanr prediciors will be
enTered inTo The regreSSion equaTion. WiTh The ﬁerward enTry method a dependenT variable 1 94 Chapter 16 / Muttipte Regression Analysis and any number at predictor (independent) variables are designated. Regression wilt tirst
compute which predictor variable has the highest bivariate correiation with the dependent vari—
able. SPSS wilt then create a regression equation with this one seiected independent variabie.
This means that otter the tirst step, a regression equation has been calculated that includes the
designated dependent variable and on/y One independent variable. Then ﬁegression will en
ter the second variable, which explains the greatest amount at additiona/ variance. This sec
ond variable will be included oniy it it explains a Significant amount at additional variation.
Atter this second step, the regression equation has the same dependent variable but now has
two predictor variables. Then, it there is a third variable that significantly explains more at the
variance, it too will be inciuded in the regression equation. This process will continue until no
additional variables signiticantiy explain additional variance. By detault, Begression will cease
to add new variables when p value associated with the inclusion at an additional variable in
creases above the .05 level at signiticance. The researcher, however, has the option to desig
nate a ditterent level at significance as a criterion tor entry into the equation. The measure at the strength at relationship between the independent variables (note, piurai)
and the dependent variabie is designated with a capital Rand is usually reterred to as mu/rr'p/e :"j3
R This number squared (R9) yields a value that represents the proportion at variation in the j.
dependent variable that is explained by the independent variables. In the regression analysis
that produced the regression equation shown above, the value oi muitiple Rwas .616, and the I 3}
R2 was .380. This indicates that 38% at the variance in zhelp was accounted tor by sympathy, 3.” i=3
anger, and efficacy. See the Output section at this chapter for additional intormation about 5f 2”, multgpie R. CURVILINEAR TRENDS, MODEL BUILDING, ANB REFERENCES Regression, like correlation, measures tor linear relationships. in the previous chapter we de—_':
scribed the procedure to test tor a curvilinear relationship. The same process operates with "31'
muitiple regression. It there is theoretical or statisticai evidence that one or more at the pre I I
dictor variables demonstrates a cuwiiinear association with the dependent variable, then a
quadratic (the variabie squared) tactor may be added as a predictor. Piease refer to the previ~  
ous chapter tor an expianation at this process. A tinal critical item concerns model building, that is, conducting a regression analysis that is .
conceptually sound. Certain tundamental criteria are necessary tor creating a reiiable model: l. Your research must be thoughttully cratted and caretuily designed. The arithmetic ol
regression will not correct tor either meaningless relationships or serious design tiaws; ' _
Regrettably, many details at what constitutes research that is "thoughttuliy cratted and 
caretully designed“ extend well beyond the scope at this book. 2. The sample size shouid be large enough to create meaningtui correlations. There are.
no hard rates concerning acceptable sample size, but as Ndrops below 50, the vaiidityl.
at your resuits become increasingly questionable. Also there is the consideration at the
number at variabies in the regression equation. The more variables involved, the iargéi;
the sample size needs to be to produce meaningtui results. ' ' 3.» Your data should be examined caretuliy tor outliers or other abnormalities. 4. The predictor variabies shouid be approximately normaity distributed, ideaily with skeW‘f .I'
ness and i<urtosis values between it. HoWever, good resuits can otten be obtai'rtt‘iCE " U)
: 3* LU
I f'a is at 2’8 l6 Ei’ Chapter 16 / Multiple Regression Analysis 195 with an occasional deviation trom normality among the predictor variables, or even the
inclusion of a discrete variable (such as gender). A normal distribution tor the depend»
ent variable is also urged; but discriminant analysis (Chapter 22) uses a discrete meds»
are as the criterion variable in a regression procedure. 5. Be keenly aware at the issue at linear dependency between the predictor variables.
Never use two variables one ot which is partially or entirely dependent upon the other
(such as points on the tinal and total points in the class). Also avoid variables that are
conceptually very similar (such as worry and anxiety). To compute a matrix at correla~
tions between potential predictor variables is always wise. Variables that correlate
higher than r m .5 should be scrutinized carefully betore both are included in a regres—
sion analysis. The power and interpretability of results are substantially compromised
when variables that are linearly dependent are included in the analysis. Multiple regression analysis is not a simple procedure. Like analysis at variance, 0 number at
thick volumes have been written on the topic, and we are in no way attempting to duplicate
those ettorts. it is suggested that, betore you attempt to conduct multiple regression analysis,
you take a course in the subiect. A number at books are available that cover both simple and
multiple regression. Several that the authors teel are especially good include: Chatteriee and Price (i999), Gonick and Smith “993), Schulman (T998), Sen and Srivastava (i997), and
Weisberg ('l 985). Please see the reterence section tor more detailed intormation. The purpose at the previous tour pages has been to remind you at the rationale behind regres
sion it you have been away trorn it tor a while. The purpose ot the pages that tollow is to ex
plain step by step how to conduct multiple regression analysis with SPSS and how to interpret
the output. The data we use by way at example are tram the study already discussed on pages 192194. The tile name is helping1.sav, N I: ST. The tollowing variables will be used in the description; all variables except zhelp are measured on a lift/ell) to mac/7(7) scale. C3 zhelp: The dependent variable. The standardized score (2 score) tor the amount at
help given by a person to a triend in need on a —3 to +3 scale. sympathy: Feelings at sympathy aroused in the helper by the triend‘s need. anger: Feelings of anger or irritation aroused in the helper by the friends need.
efficacy: Selt—etticacy of the heiper in relation to the triend’s need. severity: Helper‘s rating at how severe the triend‘s problem was. empatend: Empathic tendency ot the heiper as measured by a personality test. E]
El
E]
Cl
C) STEP BY STEP Multiple Regression Anaiysis . .. . ... . .. . .. .. .. . .. Stein's}
Create and name a data 29/8 or edit {if necessary) an already existing fi/e (see Chapter 3/ To enter 5P55, a click on Start in the raskbar (bottom ofscreen) activates the start menu. .instteen' . . . . . . .. .. Steed?
_ 1 f" " _ n. $5p55113rsi1et_aaser'' _ r  196 Chapter 16 1 Multiple Regression Analysis Ah‘er clicking i‘he SPSS program icon, Screen l appears on fhe moniiar. Screens l and 2 {dis
played on ihe inside fronr cover) allow you io access the daia file used in conducimg fhe analy
sis ofinieresi. The following sequence accesses rhe helping1.sav file for iun‘her analyses: iisseen D'QTHT'S . fees: 3. i  @Eile > Qpen —9 “Qt Data  Ltype helpingleav —> [oral]
[or %% helping'isav ] Wheiher firsr eniering 5P55' or rerurning from earlier operaiions ihe siandard menu of com ___3:
mands across ihe rap is required {shown below}. As long as if is visible you may perform any ; ihule. ’ :w :‘m "T .ssp v « ““73“. "F'T'zr‘x'I‘z’. This menu of commands disappears or modifies when using pivoT Tables or ediiing graphs. To
uncover The sTandarol menu of commands simply click on The W or The icon. Afrer comp/error? of Siep 3 a screen wiih ihe desired menu bar appears. When you click a _
command {from ihe menu bar], a series of opiians will appear (usually) below The selecied
command. 144% each new sei of opiions, click rhe desired iiem. The sequence fa access mul—
rip/e regression analysis begins of any screen wiih ihe menu of commands visible: 'DoiThis ' . ii: Sc'reen' . . . . . .
$4: Analyze —> gegression e {a Linear Li‘LJ ! The screen Thai appears aT This poinT (Screen lei, below) is ideniical To The dialog box shown I
(Screen l5.l) from The previous chapier, We reproduce ii here Tor sake ol visual reference and
so show The new lisT of variables in The helping1.sev file. l": 16.1 Rear Regression The Linear Regression Window <3!) Ensihelp
@insiee
(it?! infhelp
(is) inise e tomes .. AT The Top ol The window is The erendent box ThaT allows room for a single dependeni vari:
able. la The middle of The screen is The independenﬂs) box where one or more carefully se ri—
;e~ Chapter 16 / Multiple Regression Analysis 197 lected variables will be pasted. There are no SPSS restrictions To the number of independent
variables you may enter; There are common sense restrictions described in the introduction.
The Block 1 0le option allows you To create and conduct more than one regression analysis in
a single session. Atter you have chosen The dependent variabie, independent variables, and
selected all options That you desire related To an analysis, then click The Next peshbutton To the
right of Block ‘T of ‘i. The dependent variabie will remain, all selected options and specifica
tions will remain, but the independent variables wiil revert to The variable list. Then you may
select specifications for a second regression analysis. You may specify as many different analy
ses in a session as you like; a click of The OK pashbutton will run them in the some order as
They were created. Several additional options are described in The paragraphs That Toliow. The menu box labeled Method is an important one. It identifies five different methods 0T en
tering variables into The regression equation. With a click of The ”V, the five options appear. Enter is the default method of entering variables. Each one is described below: :3 Enter: This is The forced entry option. SPSS will enter at one Time all specified variables
regardless of significance levels. :3 Forward: This method will enter variables one at a Time, based on The designated sig—
nificance value To enter. The process ceases when there are no additionai variables That
explain a signiﬁcant portion OT additional variance. E3 Backward: This method enters al/independent variables at one time and Then removes
variables one at a time based on a preset significance value to remove. The default value to remove a variable is p 2 .TO. When There are no more variables That meet the
requirement for removal, the process ceases. E'J Stepwise: This method combines both Forward and Backward procedures. Due to
The complexity of intercorreiations, The variance explained by certain variables will
change when new variables enter the equation. Sometimes a variable That qualified to
enter ioses some of its predictive validity when other variables enter. If this takes place,
The Stepwise method will remove The ”weakened" variable. Stepwise is probably The
most frequently used of the regression methods. C] Remove: This is the forced removal option. lT requires an initial regression analysis
using The Enter procedure. In The next block (Stock 1 of 1) you may specify one or
more variabies To remove. SPSS wiTl Then remove The specified variables and run The
analysis again. it is also possible To remove variables one at a Time for several blocks. The ﬂLS» pushbuTTon (Weighted Least Squares) ailows you to select a single variable (not
already designated as a predictor variable) That weights The variables prior To analysis. This is
an infrequently used option. The Plots opTion deals with plots of residuals only and will be considered in The final chapter of
the book (Chapter 28). A click on ﬁtatistics opens a small dialog box (Screen 16.2). Two options are selected by de
tault. Estimates will produce the 8 values (used as coefficients for the regression equation),
Betas (The standardized regression coefficients) and associated standard errors, fvalues, and
significance values. The Model fit produces The Multiple R, R2, an ANOVA Table, and associ— ated Fratios and significance values. These Two options represent The essence 0T multiple re
gression output. 198 Chapter 16 / Multiple Regression Analysis The Linear Regression: Statlstics
Window ”Linear integrassiani Statistics Other treauently used options include:
Cl Cogﬁdence intervals: Produces 95% contidence intervals tor the Bval...
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