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Unformatted text preview: 124 Chapter 10 I Correlations CORRELATICJNS MAY be computed by making use at the SPSS command gorretate. Correla
tions are designated by thelower case letter r, and range in value tram a] to +l. A correla—
tion is otten called a Kari/airfare correlation to designate a simple correlation between two vari—
ables, as opposed to relationships among more than two variables, as frequently observed in
multiple regression analyses or structural equation modeling. A correlation is also trequently 
called the Pearson product~rnoment correlation or the Pearson r. Karl 5. Pearson is credited
with the formula from which these correlations are computed. Although the PearsOn ris predi
cated on the assumption that the two variables involved are approximately normally distributed,
the tormula often perlorms welt even when assumptions oi normality are violated or when one '
ot the variables is discrete. ideally, when variables are not normally distributed, the Spearman .
correlation (a value based on the rank order of values) is more appropriate. Both Pearson and .f '
Spearman correlations are available using the gorrelate command. There are other tormulas '
from which correlations are derived that reflect characteristics at different types at data, but a
discussion at these goes beyond the scope at this book. See the SPSS for l/l/fna’ows Base 5ys
rem User’s Guide tor additional information. WHAT IS A CORRELATION? Perfect positive (r= 1) correlation: A correlation at +1 designates a perfect, positive correla _
tion. Perfect indicates that one variable is precisely predictable tram the other variable. Posi '3
the means that as one variable increases in value, the other variable also increases in value (a I
conversely, as one variable decreases, the other variable also decreases). A scatter plot between two variables demonstrating a perfect correlation
{ r: 1.0) Received Number of
Hours Worked Perlect correlations are essentially never tound in the social sciences and exist only in mathe_'
matical tormulas and direct physical or numerical relations. An example would be the relation—"i
ship between the number at hours worked and the amount at pay received. As one number?
increases, so does the other. Given one at the values, it is possible to precisely determine the}
other value. = Positive (8 < r < 1) correlation: A positive (but not perlect) correlation indicates that as the.
value at one variable increases, the value of the other variabie aiso tends to increase. The"
closer the correlation value is to l, the stronger is that tendency; and the closer the correlation:
value is to O, the weaker is that tendency. Example of a positive correlation {G<r<l) Empathy Level ~. ‘MMLLJ(D Chapter 10 I Correlations 125 An example of a strong positive correlation is the relation between height and weight in adult
humans (r = .83). Tall people are usually heavier than short people. An example oi a weak . positive correlation is the relation between a measure of empathic tendency and amount of help given to a needy person (r = .l2). Persons with higher empathic tendency scores give
more help than persons who score lower, but the relationship is weak. No {r= G} correiation: A correlation of 0 indicates no relation between the two variables. For
example, we would not expect lQ and height in inches to be correlated. Exam pic of a zero correlation
(r r“ 0) Negative (1 < r< 0) correlation: A negative (but not perfect] correlation indicates a relation
in which as one variable increases the other variable has a tendency to decrease. The closer
the correlation value is to ml , the stranger is that tendency. The closer the correlation value is
to O, the weaker is that tendency. Example of a negative correlation Anxiety (0 < r < 1) Emotional
Stability An example ot a strong negative correlation is the relation between anxiety and emotional sta
bility (r e m.73). Persons who score higher in anxiety tend to score lower in emotional stability.
Persons who score lower in anxiety tend to score higher in emotional stability. A weak negative
correlation'is demonsirated in the relation between a person's anger toward a triend suflering a
problem and the quality of help given to that friend (r = —.l 3). lt a person's anger is less the
quality oi help given is more, but the relationship is weak. Perfect negative (r = —’t) correlation: Once again, periect correlations {positive or negative)
exist only in mathematical tormulas and direct physical or numerical relations. An example at a
perlect negative correlation is based on the formula distance = rate x time. When driving tram
point A to point B, if you drive twice as test you will take fie/{as long. Example of a perfect negative correlation (r 2 ~ 1.0) 126 Chapter 10 / Correlations ADDITIONAL CONSlDERATIONS Linear versus Curvilinear lti iS important to understand that the Correlate command measures only hear relationships.
There are many relations that are not li inear. Take the example of nervousness before a major
exam. Too much or too little nervousness generally hurts performance while a moderate .
amount of nervousness typically aids performance. The relation. .on a scatter piot would look
like an inverted U, but computing 0 Pearson correlation would yield no relation or a weak rela I
tion. The chapters on simple regression and multiple regression analysis (Chapters 15 and lo)
will consider curvilinear relationships in some detail. it is often a good idea to create a scatter
plot of your data before computing correlations, to see if the relationship between two variables '
is linear. if it is linear, the scatter plot will more or less resemble a straight line. While a scatter
plot can aid in detecting linear or curvilinear relationships, it is often true that significant corre
lations may exist even though they can not be detected by visual analysis alone. Significance As with most other statistical procedures, a significance or probability is competed to determine
the likelihood that a particular correlation could occur by chance. The significance (or ,0 value)
represents the degree ofrarr'ry of a certain result. A significance less than .05 (p < .05) means
that there is less than a 5% chance that this relationship occurred by chance. SPSS has two
different significance measures, one tailed significance and two tailed significance. To deter
mine which to use, the rule of thumb generally followed is to use two— tailed signiifcance when
you compute a table of correlations in which you have little idea as to the direction of the cor
relations. lf, however, you have prior expectations about the direction of correlations (positive
or negative), then the statistic for onewtailed significance is generally used. Causality Correlation does not necessarily indicate causation. Sometimes causation is clear. if height;
and weight are correlated, it is clear that additional height causes additional weight. Gaining":
weight is not known to increase one’s height. Also the relationship between gender and em—'
pathy shows that women tend to be more empathic than men. if a man becomes more emi' pathic this is unlikely to change his gender. Once again, the direction of causality is clear: _
gender influences empathy, not the other way around I There are other settings where direction of causality Is likeiy but Open to auesti.on For instance
self— efficacy (the belief of ability) is strongiy correlated with helping. it would generally be
thought that belief of ability will influence how much one helps, but one could argue that one:
who helps more may increase that ii” belief of self efficacy as a result of their acti.ons The former
answer seems more likely but both may be partially valid. Thi rdly, sometimes iii is difficult to have any idea of which causes which. Emotional stability and.
anxiety are strongly related (more emotionally stable people are less anxious). Does greater.
emotional stability result in less anxiety, or does greater anxiety result in lower emotional stabil—'
ity? The answer, of course, is yes. They both influence each other. Finally there is the third variable issue. it is reliably shown that ice cream sales and homicides
in New YorkCity are positively correlated. Does eating ice cream cease one to become haml—
cidal? Does committing murders give one a craving for ice cream? The answer is neither. :meDCD Chapter 10 T Correlations 127 BoTh ice cream sales and murders are correiaTed wiTh heaT. When The weaTher is hoT more
murders occur and more ice cream is sold. The same issue is aT play wiTh The reliable ﬁnding
ThaT across many ciTies The number CST churches is posiTively correlaTed wiTh The number of bars.
No, iT’s noT ThaT church going drives one To drink, nor is H ThaT heavy drinking gives one an
urge To aTTend church. There is again The Third variable: populaTion. Larger ciTies have more
bars andchurches while smaller ciTies have Tewer oi boih. Partial Correiation We menTion This issue because pariial correiarion is includedas an opTion wiThin The conTexT of
The gorreiate command. We menTion iT only brieﬂy here because ii is covered in some deTail
in ChapTer T4 in The discussion abouT covariance. Please refer To Thar Chap’rer Tor a more de
Tailed descripTion oi parTial correla‘rion‘ Pariial correlaTion is The process of Tinding The correla
Tion beTween Two variables ah‘erihe inﬂuence oT oTher variables has been conTrolled Tor. ii, for
instance, we compuTed a correlaTion beTween GPA and ToTal poinTs earned in a class, we could
include year as a covariaTe. We wouid anticipaTe Thai Tourih year sTudenTs would generally do
beTTer Than TirsT year sTudenTs. By compuTing The panial correlaTion, ThaT ”parTials ouT” The inilu
ence of year, we marhemaTically eiiminaie Theiniluence 0T years 0T schooling on The correloTion
beTween ToTal poinTs and GPA. WiTh The parTiai correlaTion opTion, you may include more Than
one variable as a covariaTe ii There is reason To do so. ' i The Tile we use To illusTraTe The gorreiate command is our example inTroduced in The TirsT chapTer. The Tile is called gradesrsav and has an N = 105. This analysis compuTes correla—
Tions beTween live variables in The Tile: gender (sex), previous GPA (gpa), The TirsT and Tihh quéz~
zes (quiz1, quizS), and The final exam (final). T STEP av STEP Correiations :— CreaTe and name a daTa Ti/e or ediT (ii‘necessaryj an a/reac/y exisTing ri/e {see Chapi‘er 3/ 3 To enTer 5/355, a dick on Start in The Tas/(har (haTTom ofscreenj acTivaTes The sTarT menu: ' Iii seesn. ' Do This: a f"
1 as 'i
Ah‘er c/icking The SPSS program icon, Screen i appears on The moniTor. Screens i and 2 {dis p/ayed on The inside TronT cover) a/low you To access The daTa fife used in conducTing The analy
sis of inTeresT. The To/lowing sequence accesses The grades.sav Ti/e for Tun‘her ana/yses: I .. Step 3 .. .“s't'eab 2. ‘A A: sizzle $1555 [3:1 iii Ea: seems Thisdimes]. Dorms .. . . .. .
Eiie —>Qpen —9 Dgta Front2 type grades . eav —> “5:; OK Whei‘her TirsT enTering 5P55 or reforming from ear/ier operaiions The sTano/ard menu of com
mands across The Top is required {shown [De/ow). As /ong as if is w'sib/e you may perform any
ana/yses. TT is nor necessary for The dafa window To be visible. _ 123 Chapter 10 / Correiations MUIMHI _. This menu oi commands disappears or modiﬁes when using Ejoi Tables or ediiing graphs. To
uncover The siandard menu oT commands simply click on The or The ' Icon. Aﬂer comp/efion of Siep 3 a screen wir/v The desired menu bar appears. When you c/ick a _
command {from The menu bar), a series of aprions will appear {usual/y) be/ow The se/ecfed .
command. Wifb each new sef of opi‘ions, c/ick The desired irem. The sequence J‘o access ;
Corre/an'ons begins of any screen wif/i The menu ofcommands wisib/e: ' ' leisorgan Do This l Q Analyze *> Correlate —> {:9 Bivariate l”
Aiier clicking Eivariate, a new Window opens (Screen 10.1, below) Thai specifies a number oi 
opTions available wiTh The correiaiion procedure. FirsT, The box To The leTT iisTs all The numeric}
variabies in The file (noie The absence oi firstnam, lastname, and gradeali nonnumeric).
Moving variables Tram The lisT To ihe Mariablqs) box is similar To ihe procedures used in previ
ous chapTers. Click The desired variable in The lisT, click ., and Thai variable is pasied inio 
The Variable{s) box This process is repeaTed Tor each desired variable. Also, ii There are a
number oi consecuiive variables In The variables lisT you may ”click & drag” irom The TirsT To ihe lasi desired variable To seleci Them aii Then a single cii cl< oT  will pasie ail highligh’red vari
ables inTo The aciive box. " " Co'neieﬁo ' The Bi'voriare Corre/afions __ \
. 3: as sex
WHO/OW @eihnic
~®year
.5' @iowup
<3!) semian
e sea
0 extrcred
<e> review in The nexi box labeled Correlation Coefficients The Pearson r is selecTed by deiouli. IT yOUr
daia are nor normally disTribuTed Then seleci §pearman You may selec’r bo’rh opiions and 53?
how The values compare. ' Under Test of Signiﬁcance Twotaiied is seiecTed by defauli. Ciick on Onetaiied ii you have
clear knowiedge of The direcfion (posiiive or negative) oi your correiaii ons ﬁlag significant correlationsi is seiecTed by deiauli and places an asierisk (* ) or double asTerisi
(**) nexT To correlaiions Thai aTTain a pariicular level oi signiﬁcance (usualiy .05 and .Oi‘li
WheTher or noT signiTicanT values are ilagged, The correlaiion, The signiﬁcance accuraie To Three
decimals and The number of subiecis invoived in each correiaiion will be included. ' Chapter 10 / Correlations 129 a} For analyses demensTraTed in ihis chapTer we will sTicl< wiTh The Pearson correlaiion, The Iwo
' tailed TesT of significance, and also l<eep flag significant correlations. if you wish oTher op—
~wl Tions, simply click The desired procedure To seleci or deseleci beiore clicking The Tina! OK. For
lo sequences Thai Tallow, The sTan‘ing poinT is always Screen lO.l. ll you are already in The pro gram, perform whichever ol sTeps l—4 (pp. l27—i28) are necessary To arrive ai Thai screen. A
click 0T Reset may be necessary io clear Tormer variables. a
9d : To produce or come/anon mofn'x of sex, gpa, quiz1, quizS, and final, perform #75 fol/owing
.355 1 sequence ofsfeps: _1 ieSnrersn .B'nT‘hiS .. . . . . . ..
a ‘ sex m) Q. gpa —> quiz‘i —> @% quizS m)
j ‘ {3%ﬁnal ——> “4% OK .I of .‘ Addiiional procedures are available iT you click The thions buTTon in The lower righi corner oT
"ic " Screen lO.l. This window (Screen l0.2, below) allows you io select addiTional sTaTisTics To be
2). ‘ prinTeCl and To deal wiTh missing values in Two diTTerenT ways. Means and standard deviations
ii ‘ may be included by clicking The appropriaie opTion, as may grossuproduct deviations and
10 ‘ covariances. a ; 1e : The Bivariore Correlafions: Opiions
Who/OW 3‘ l To Exciude cases pairwise means Thai Tor a panicular correlaiion in The mairix, ii a subiecT
has one or Two missing values Tor ihai comparison, Then Thair subieci's inﬂuence will noT be in
cluded in Thai pariicular correlaiion. Thus correlaTions wiThin a maTrix may have diTTereni num
bers of subiecTs deTermining each correlaiion. To Exclude cases [istwise, means ihaT ii a
subjecT has any missing values, all daTa from Thai subieci will be eliminaied Tram any analyses.
Missing values is a Thomy problem in daTa analysis and should be deali wiih before you gei To
The analysis siage. See ChapTer 4 Tor a more compleTe discussion ol This issue. The {oi/owing procedure, in addifion fo producing a come/anon mornx similar 2‘0 #701‘ creofed in
sequence Srep 5, wi/l prini‘ means, grandma deviorions, crossproducr deviorions, and covari—
ances in Mo fob/es prior ro priming rhe corre/ofion moin'x: fer I. in Spree nf Did fhis. i
k I 102 grossproduct deviations and co l
i _ ‘
le .. . .. i . 130 Chapter 10 / Correlations What we have iltustroted thus far is the creation of a correlation matrix in which there are equal
number of rows and columns. Often a researcher wishes to create correlations between one
set of variables and another set of variables. For instance she may have created a 12 X 12
correlation matrix but wishes to compute correlations between 2 new variables and the original
12. The windows format does not aliow this option and it is necessary to create a ”command
file”, something familiar to users of the PC or mainframe versions of SPSS. If you attempt any
thing more complex than the sequence shown below, you will probably need to acquire the
SPSS Base System Syntax Reference Guide. 7 _
The SPSS Syntax Editor
Window By clicking Eile, then ﬂew, then §yntax, a new screen opens that allows you to type in a com—
mand file to accomplish the procedure described above. Type in the words exactly as shoWn in 2
sequence 5b (exchanging only the variable names you desire). Then highlight the text you have _
lust typﬁed and click the i3 icon at the top of the screen to run the program. Note that the text .I
is already typed in the Screen 10.3 display. All elements of the command tine are necessary
for it to run. If you omit equal signs, periods, or misspell words, an error message will tlash { and your program will not run. You may begin this sequence from any screen that shows the standard menu at commands ;
across the top of the screen. Be/ow we create a 2 by 5 matrix of Pearson correlations cam " paring sex and total with year, gpa, quiz1, quiz5, and final.
13 Db‘Th':iS'_ i... . i. . . ' ' ' ' '
(53$ Eiie m) blew "6 {E ﬁyntax
EVE] correlations variables x sex total with year
gpa quiz}. quizS final. —> {to high/tight this texﬂ@ Home
key ——) lﬂ the Down Cursor key once whi/e holding down the Shift they —> "i {at top ofscreenj  When you use this format, simply replace the variables shown here by the variables you desire. You
may have assmany variables as yoa desire both before and otter the ”with”. it you get an error mes
sage, you probably speiled a variable name wrong or left off the period. Upon completion of step 5, 5a, or 5b, the output screen will appear [Screen 3, inside back
cover). All results from the gustcompleted analysis are included in the Output Navigator; Make use of the scroll bar arrows l... to view the results. Even when viewing output,
the standard menu of commands is still listed across the top of the window. Further analyses may be conducted without returning to the data screen. __.—v IV LU .2... \Jw__ a
a Chapter 10 / Correlations 131 PRINTING RESULTS Results of the analysis (or analyses) that have lost been conducted requires a window that dis~
plays the standard commands (file Edit Qata Iransform Analyze . . .) across the top. A
typical print procedure is shown below beginning with the standard output screen (Screen l,
inslo‘e back cover). To prfnf resu/i‘s, from #26 Output screen pen’orm the fo//owr’ng sequence ofsr‘eps: 'tn'Slcreén DoTh'iis ' .. .. .. . ' . . 'é'ish 6]  Se/ecf desired output or edif {see pages I? ~ 25/ m) {E Eiie —E> % Erint IlaScreen“ beﬁts .. . .. .. . . . .
l seesaw Ezsit Note: After clicking Egit, there will frequently be small windows that appear asking if you wish to save
or change anything. Simply click each appropriate response. OUTPUT Correlations This oulpo‘t is from sequence step 5 (page 129) with twotailed Significance selectea ariaI srg
nificant correlations flagged.
f Correlations _ ' . i
I Pearson Correlation Sig. (2—tailed) N Pearson Correlation Sig. (2tailed) N Pearson Correlation Sig. (2—taiiecl) N Pearson Correlation Sig. (Z—tailed) N Pearson Correlation Sig. (2—talied) N ' _
* Correlation is significant at the 0.05 level (2—tailed)
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