Mktg Res Report

8/4/2019 Mktg Res Report

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et Ownership: ANew Market Marketing managers have a host of bases fo r s egmenting~gmentation Basis? their markets, including demographics, lifestyles, benefits,and product usage, not to mention a number of commercially

developed systems that are available. I n general, market segmentation holds that if agroup of consumers can be identified by some basis, then it will differ fr om ot her mar-ket segments on key factors such as purchases, beliefs, or some other aspect that isuseful to a marketer seeking to target that segment. To state the concept somewhat dif-Ferently, the members of a ma rket segment will be uniquely associated wit~ prcdicposi-~i0h:;, .::;;td , , : h . : :~ c i'larkt::~il-18liIUIU~:1 iJt:!,;lj~it:!!i n e m arker segmen t. n e or sne can usethose predispositons in a marketing strategy.

Seeking to break new ground, researchers investiqoted the marketing segmenta-tion potential of pet ownership, and, specifically dog~ers~s cot ownership. I They sur-veyed olorqe number of American adults regarding their attitudes, interests, and opin-ions across a number of topics, and they compared these respondents' answers to theirpet ownership (or nonownership) with a technique calle d cr oss -tabulation, which isdescribed in this chapter.

The researchers discovered that pet ownership (cat, dog, or both) was associ-ated or related to a certain lifestyle profile. Namely, relative to nonowners of pets,pet owners were found to:

z:

Determiningand InterpretingAssociationsAmong Variables

1 . ,: ~ ' r . ~ J , ,'1 ,~ ( ; ,

.~~ .

'~!

.1 . " : 1 1 : 0 ~,,;

!!I Beodventurousill Be independenta Enjoy lifeOn the' other hand, relative to pct ()WInonowners of pets were found to be: ConservativeII Fatalistic13 Health-consciousIII Concerned for the environmentTaking a closer look at the nature of IJo l ,\ ,ship, the researchers then investigatecJ I}II,relationships with cat versus dog OWI,.", ,traits and demographic factors cssocicuoclowning a dog were found to be:

i\~, Ii',Traits Demographics

Researchers have idcnufled market segments based on dog ownership,cat ownership. or not owning a pet.

-' ..,..-----_. .__ ........ . . . --- .. .

Co nserv ati sm 35-54 years oldTraditionalism Less educated

Married

Being a cot owner was associakd with the following treits:Traits Demographics

More educateddventurousHealthconscious Metropolitan dwellerConcerned fo r the enviranment

While dog-and-cat segmentation is obvious for a. pet food company suchWhiskas (for cats) or Alpo (for dogs), there are more subtle uses of these ossociolicFor example, if an advertiser shows a middle-aged morried- couple worrying alIinonciol matters, it would lend more credence to the ad if there was a Family do"


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" Ih\jH. IV lo " \.:.1~JIIIIIHg ,111(,1J.lllCrprcung Assocrations ~nong Variables

sociative analyses determineecrher stable- relationshipssr between two variables,

ations!1ip is a consistentystcrnatir linkageecn the labels ornts for two var-iables.

th e a d. O n th e oth er h an d, if a n a ir p urific ation system wa s be ing advertise d, it w ouldb e a pp rop riate to show the fam ily cat reclin in g in fron t of the puri fication un it.

" : r h i ' chapter illustrates the usefulness of statistical m'y ,, , bey"",simple descriptive measures and stati st ical inference. Often, as we have described IIIour pet ownership segmentation example, marketers are interested in relationship,among variables. For example, Frito-Lay wants to know what kinds of people andunde r what circumstances these people choose to buy Dor it os, Fritos, and any of iheother items in the Frito-Lay line. The Pon ti ac Divi sion of General Motors wants to knowwha t t ype s of individuals would respond favorably to the various style chang~sproposed for the Firebird. An ewspaper wants to understand the lifestyle characteristicsof its prospective readers so that it is able to modify or change sections in tltenewspaper to better suit its audience. Furthermore, tl!e newspaper desires informationabout various types of subscribers so as to communicate this ill formation to itsadvertisers, helping them in copy design and advert isernem placement within tltevarious newspaper sections. For all of these cases, there arc statistical procedun-,available, termed "associative analyses." which determine answers to these questiou-.Associative analyses determine whether stable re la t ionslnps exist between t"."variables; they are the central topic of this chapter.

We begin the chapter by de sc ri bing the four different types of'relationships possiblebetween two variables. Then, we describe cross-tabulations and i ndicate how a cross-tabulation can be used to compute a chi-square value, which in turn, can be assessed todetermine whether o r not a st at is tically signiflcant association exists between the twovariables. For cross-tabulations, we move to a general discussion of correlationcoefficients, and we illustrate the use of Pearson product moment correlations. As in ourprevious analysis chapters, we show you SPSS steps to perform these analyses and there su lti ng output.

TYPES OF RELATIONSHIPS BETWEEN TWO VARIABLESIn order to describe a relationship between two variables, we must first remind you 01lhe ;Lale characteristic called "description" [hat we introduced 10 you ill Chapter 10.Every scale has unique descriptors, sometimes called "labels" or "amounts," that identifythe different labels of that scale. The term l e v e l s impl ie s t hat t he s cale is met ri c, namelyinterval or rattorwhile the term la be l s implies that the-scale IS not metric, typical ly nomi-nal. A simple labe l is a "yes" o r " no" one, for instance, if a respondent is labeled as a huyer(yes) or nonbuyer (no) of a particular product or service. Of course, if the researchermeasured how many t ime s a respondent bought a product, the amount would be thenumber of times, and the scale would be metric because this scale would satisfy theassumptions of a ratio scale.

A relationship is a consistent and systematic linkage between the lab els oramounts for two variables. This linkage is s tatistical, not necessarily causal. A causal link-age is one in which you ar e c ertain one variable affected the other one, but with a sta-tis tical linkage you cannot be certain because some other variable might have had some

Types of Re!a ti on sh ip s between Two Variablesmfluence. Nonetheless, statistical linkages o r r el at ionships often provide us with insightsIItJl lead to understanding even though they are not cause-and-effect relationships. Forl'X~l11ple, if we found a relationship that 9 out of I 0 bottled water buyers purchas ed f la -vored water, we understand that the flavorings are important to these buyers.

Associative analysis procedures are useful because they determine if there is a con-sistent and systematic relationship between the presence (label) or amount of one var i-.iblc and the presence (label) or amount of another va r iable. The re a re four bas ic types ofrclauonships between two variables: nonmonotonic , mono ton ic, linea r, and curvilinear.A discussion of each follows:INonmonoronic RelationshipsA norunonotonic relationship i sone in which the presence (or absence) of one vari-able i s systematically associared with the presence (or absence) of another vari able. Theterm nonmonotonicmeans e ssentially that there is no discernible direction to the relation-ship, but a relationship exists. For example, McDonald's knows fr om experience thatmorning customers typically purchase coffee, whereas noon customers typically pur-chase s oft drinks. The relationship is in no way exclusive-there is no guarantee that amorning customer will always or der a coffee o r th at an afternoon customer will alwaysor de r a s oft dr ink. In general, though, this relationship exists, as-can be seen in Figure18.1. The nonrnonotonic relationship is simply that the morning customer tends to pur-chase bre ak fas t f oods such as eggs, biscuits, and coffee, and the afternoon -customerstend to purchase lunch items such as burgers, fries, and soft drinks.

In other words, with a nonrnonoronic relationship, when you find the presence ofone label for a variable, you will tend to find the presence o r another specific label ofanother variable: breakfast diners typically order coffee. Here are some other examplesof nonmonotonic relationships: (I) People who live in apartments do not buy lawnmowers but homeowners do; (2) tourists in Daytona Beach, Florida, during "bikeweek" are likely to be motorcycle owners, not college s tudent s; and (3) Pl ay Stariongame players are typically children, no t adult s. Again each example reports that the pres-ence (absence) of one aspect of some object tends to be joined to the presence(absence) of an a spect o f some ot her obj ect. But the associa tion i s very general, and wemust state each one by spelling it our verbally. In other words, we know only the generalpattern of presence or nonpresence with a nonmonotonic relationship.

IIjI

Monotonic RelationshipsMono tonic re lat ionships are ones in which the researcher can assign a general direc-tion to the associa tion between the two var iables . There are tworypes of monotonic rela-tionships: increasing and decreasing. Monotonic increasing relationships arc those inwhich one var iable increases as the other variable increases. As you would guess, monot-onic decreas ing r el ationships are those in which one variable increases as the other vari-able decreases. You should note that in neither case is there any indication of the exactamount of change in one variable as the other changes. "Monotonic" means that the

Breakfas t O rders Lu n ch O r de r s

A no nmonot onh 11'1111means two vari.lhl.,..., IIIdated, but only ill,\ \ 1'1eral sense.

A monotonic re lat iUI Ifr!:11imeans you know I,!W \"11direction of the n':I,IIIOI1between Lv~...rI,O"l~,

F IGURE 1 8 .1McDonald's Example"NornnonotonicRelationship for the 'f)Drink Ordered 1BrNand at Lunch


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\ 1I '!jll t'" 1 Ii: J)~ll'I'IIIJI"g1IlcJ interpreting Associauons Among Var iables

~Ionic relationships emI creasing or decreasing.

ar relationship means.,I() variables have agtu-Iinc" relationship.

R E 18.2ld's Control of His Ornoe Purchases:rtonic Increasingonship

.:

relationship can be described only in a general directional sense Beyond this, precisi(\llin the description is lacking. The following example should help to explain this concept.

The owner of a shoe store knows that older children tend to require larger shosizes than do younger children, but there is no way to equate a child's age with the rightshoe size. No universal rule exists as to the rate of growth of a child's foot or to the IInalshoe size he or she will attain. There is, .however, a monotonic increasing relationshipbetween a child's age and shoe size. At the same time, a monotonic decreasing rela(illil.ship exists between a child's age and the amount of involvement of his or her pareiu, inthe purchase of his or her shoes. As Figure 18,2 illustrates, very )'oung children ()ft~1lhave v irtually no input into the purchase decision, whereas older children tend to gainmore and more control over the purchase decision process until they ultimately beuJIl1eadults and have complete control over the decision. Once again, no universal rule "per.ares as to the amount of parental influence or the point in time at which the childbecomes independent and gains complete control over the decision-making prueess. Itis simply known that younger children have less influence in the decision-makingprocess, and older children have more innuence in the shoe purchase decision.The rela,tionship is therefore monotonic.

linear RelationshipsNow, we will turn to a more precise relationship. Certainly 'the easiest association tllenvision between two variables is a linear relationship. A linear relationship. is J"straight-line association" between two variables. Here, knowledge of the arnuu ru "I'one variable will automatically yield knowledge of the amount of' the other var iahlc ,l\ Jconsequence of applying the linear or straight-line formula that is known t" ni,tbetween them. In its general form, a straight-line formula is as follows:Formula for a Straight Line y = a + b x

where:y = the dependent variable being estimated or predicteda = 'the interceptb = the slopex = the independent variable used to predict the dependent variable

A ch ild ga in s conlrol 01 his /her shoe pu rcha ses w ith de velopm enl,bul lhe re la lionship is not p recise

Comp le le.. .

Non e

'2coUoc' "oE


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ell,t,ptcr I B: Determining and Interpreting Associations AmongYariables

::>rcsencc of a relationshipcell two variables is deter-d by a statistical test.

ion means that youif th!; relationship is

ve o r negative. while pat-ieans you know the gen-rure of the relationship.

t h me ans you know how:ent the relationship is.

curved relationship is used rather than the formula for a sr.raight line. lI[any curvillne ar patterns are possible. For example, the relationship may be an S-shape, a J-shap~,or some other curved-shape pattern. An example of a curvilinear relationship wilhwhich you should be familiar is the product life-cycle curve that describes the salc~pattern of a new product over time that grows slowly during its introduction and thenspurts upward rapidly during its growth stage and finally plateaus or sI0\\'5 down considerably as th e ma rket becomes saturated. Curvilinear relationships arc beyond tht,scope of this book; nonetheless, it is important to list them as a type of relationshltlthat can be investigated through the use of special-purpose statistical procedures. '

CHARACTERIZING RELATIONSHIPS BETWEENVARIABLESDepending on its type, a relationship can usually be characterized in three ways: by itspresence, direction, and strength of association. We need to describe these before taki ngup specific statistical analyses of associations between two variables.

PresenceP r es e nc e refers to the fl.nding that a systematic relationship exists between the two vari-ables of interest in the population. Presence is a statistical issue. By this staterncru , Wlmean that the marketing researcher relies on statistical significance tests to determinewhether there is sufficient evidence in the sample to support that a particular associationis present in the population. Chapter 17 on statistical inference introduced the conceptor a null hypothesis. With associative analysis, the null hypothesis states there is no asso-ciation present in the population and the appropriate statistical test is applied to test thishypothesis. If the test results reject the null hypothesis, then we can state tha t an a ssoci-ation is present in the population (at a certain level of confidence). We descri be the sta-tistical tests used in associative analysis later in this chapter.

Direction (or Pattern)You have seen that in the cases of monotonic and linear relationships, associations may bedescribed with regard to direction. As we indicated earlier, a monotonic relationship maybe increasing or decreasing. For a linear relationship, if b (slope) is positive, then the linearrelationship is increasing; and if b is negative, then the linear relationship is decreasing. Sothe direction of the relationship is straightforward with li~ear and monotonic relationshipsfor nonmonotonic relationships, positive or negative direction is inappropriate,because we can only describe the pattern verbally. 2 It will soon become clear to you thatthe scaling assumptions of variables having a nonmonotonic association negate the direc-tional aspects ofthe relationship. Nevertheless, we can-verbally describe the pattern of theassociation as w e have in our examples, and that statement substitutes f or direction. Finally,with curvilinear relationships, we can use a formula; however, the formula will d~fine dpattern such as an S-shape that we refer to in characterizing the nature of (he re lationship.

Strength of AssociationFinally, when present (that is, statistically Significant) the association between two vari-ables can be envisioned as to its s tr ength, commonly using words such as "strong,""moderate," '''weak,'' or some similar characterization; that is, when a consistent and

. 'IIl'rnatic association is found t o be p re sent between two variables, it is then up to the111,11kl'lillg researcher to ascertain the strength of the association. Strong associations areIIIII,.t' in which there is a high probability of the two variables' exhibiting a dependablelI'I,1l ronship, regardless of the type of relationship being analyzed. A low degree of asso-I 1 .1 1 rou, on the other hand, is one in which there is a low probability of the two vari-,rllll's' exhibiting a dependable relationship. The relationship exists between the va r i-,dll,'" but it is less evident.

There is an orderly procedure for determining presence, direction, and strength of,I n-lauonslup. First, you ,must decide what type of relationship can exist between theIWO variables of interest. The answer to this question depends oE., the scaling assump-lions of the variables; as we illustrated earlier, low-level (nominal) scales can embodyflilly imprecise, pattern-like, relationships, but high-level (interyal or ratio) scales canI ncorporare very precise and linear relationships. Once you identify the appropriate rela-uoriship type as either nonmonoronic, monotonic, or linear, the next step is to deter-mine whether that relationship actually exists in the population you are analyzing. This"l'P requires a statistical test, and, again, we describe the proper test for each of thesethree relationship types beginning with the next section of this chapter.

When you determine that a true relationship does exist in the population bymeans of the correct statistical test, you then establish its direction or pattern. Again,Ihe type of relationship dictates how you describe its direcrion'You might have toinspect the relationship in a table or graph, or you might need only to look for a pos-it ive or negative sign before the computed statistic. Finally, the strength of Ihe rela-tionship remains to be judged. Some associative analysis statistics indicate the>trcngth in aver)' stralghtfc)[\ovard manner-that is, just by their absolute size. vVithnominal-scaled va ri ahles , however. you must inspect the- pattern to judge thestrength. We describe this procedure next.

.CROSS-TABULATIONSCross-tabulation and the a ss ociated chi-square value that we are about to explain areused to assess if a nonmonotonic relationship exists between two nominal-scaled vari-ables. Remember that nonmonoronic relationships are those in which the presence ofone variable coincides with the presence of another variable, such as lunch buyersordering soft drinks with their meals.

Ii1;ip~Ltii1gtl,e Itc:lationship -v,.-ith a n(ii' ChartA handy graphical tool that illustrates a nonmonotonic relationship is a stacked barchart. With a stacked bar chart, two variables are accommodated simultaneously in thesame bar graph Each bar in the stacked bar chart stands for 100%, and it i sdivided pro-portionately by the amount of relationship that one variable shares with the other vari-ables. For instance, you can see in Figure 18.3 that there are two variables: buyer typeand occupational category The two bars are made up of tWO types of individuals: buyersof Michelob Beer and nonbuyers of Michelob. There are two types of occupations: pro-fessional workers, who might be called "white collar" employees, and manual workers,who are sometimes referred to as "blue collar" workers. With the buyers stacked bar:you can see that a large percent of the white collar stacked bar is accounted for by theMichelob buyers, while a smallerpercent of Michelob buyers is apparent on the bluecollar workers bar graph.

Cross- Tabulati ous

Based on sc;\ling ,,~,'111111HIIIIfirst determine Ih..: 1)/1'11 Irelationship, "rid iheu lUllform the appr-op-lau- ~ 1 . 11 1 ,o, j 1cal test.

Bar charts can be used to "t-tta nonmonoronic rclatiousbl]


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28 Chapter 18: Determining and Interpreting Associa ti on s Among Va riables

:IGURE 18.3-.1ichelob Light Purchases.rid Occupational Status

cross-tabulation consis ts ofows and columns defined by::: J . C categories classifying eachar iable.

-


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('11.1pICr 18: Determining and Interpret ing Assoc ia tions Among Var ia bles

entages are cellies d iv id ed byl total.

::msbetween product! preferences andJhic characteristics;eters identify their..kets.

"grand total." Just above it are the totals for the number of white-collar (160)blue-collar (40) occupati on respondents in the sample, Going to the left of the gra,total are the total s fo r Mi chelob L ight nonbuyers (34) and buyers (166) in the sanipie, The four cells are the totals for the inter se ction points: 15 2 white-coll ar M iche lohLight buyers, 8 white-collar non buyers, 14 blue-collar Michelob Light burers, and 26:~blue-collar nonbuyers. ,

These raw frequencie. can be converted to raw percentages br dh'iding each by ,the grand total. The second cross-tabulation table, the raw percentages table, con- ,tains the percc:ntages of the raw frequency numbers just discussed. The grand tOtal;location now has 100 percent (or 2001200) of the grand total. Above it are 80% and'20% for the raw percent'.ges o f white-collar occupational respondellts and blue-collar occupational respondents, respec tiv el y, in the sample. Divide a couple of the cel lsjustto verify that you understand how they are derived. For instance 152 -r - 200 :: 76percent.

Two additional cross-tabula tion t ables can be presented, and these are more valuablein revealing underlying relationships. The column percentages table divides the rawfrequency by its column total raw frequency. The formula is as follows:

Formula for aColumn Cell Percent

- Cell freq uencyColumn cell percent = Total of cells inthat column

For instance, it is apparent that of the nonbuyers, 24% were white-collar anrl 76%were blue-collar respondents. Note the reverse pattern for the buyers group: ,)2'~'" !Ifwhite-collar respondents were Michelob Light buyers and 8% '''vere blue-collar huycr.You are beginning to see the nonmonotonic relationship.

'~'5I . " ,." :?"

The row percentages table presents the data with the row totals as the 100 per-Il,ltl hase for each. That is, a row cell percentage is computed as follows:Jinl'Illu1a for a II C_e_ll_f_r_e-,q_u_e_n_c,-y_Row ce percent =I\oIV Cell Percent Total of cells in that row

Now, it is possible to see that, of the white-collar respondents. 95 1" were buyers,tllti S% were non buyers. ~\s you compare the RO\\' Percentages Table to the Columnl'I'I('~lItages Table, you should detect the relationship between Occupational Status andMiI'hclob L ight beer preference. Can you state it at this time)Unequal percentage concentratibns of individuals in a few celJ.s.,as we have in thisI'x,emple, i llustrates the possible presence of a non mor.otoni c association. If v.e h;:,dli.tlnt! ihrtt approximately 25% of t he sample had fallen in each ofrhe four cells, no rela-tlnll,hip would be found to exist-it would be equally probable for any person to be aMlchelob Light buyer or non buyer and a white- or a blue-collar worker. However, the1,lIgc concentrations of individuals in two particular cells here suggests that there is aIlIgh probab ilit y th at a buyer of Michelob Light beer is also a white-collar worker, andth ere is also a t endency for nonbu yers to work in blue-collar occupations. In otherwords, there is probably an association between occupational status and the beer-buyingbehavior of individuals in population represented by this sample. We must test the statis-tical Significance of the apparent relationship before we can say anything more about it.~CHI-SQUARE ANALYSISChi-square ( X 2 ) analysis is the examination of f[(~quencies lor two nominal-scaledvariables in a cross-tabulation table to determine whether the variables have a nO)]I11O-notonic relationship." The formal procedure for chi-square analysis begins when theresearcher formulates a statistical null hypothesis tbat the two variables under investiga-tion are notassociated in the population. Actually, it is not necessary for the researcher tostate this hypothesis in a formal sense, for chi-square analysis always explicitly takes thishypothesis into account. In other words, whenever we use chi-square analysis with across-tabulation, we always begin with the assumption that no association existsbetween the two nominal-scaled variables under analysis.'Observed and Expected FrequenciesThe statistical procedure is as follows. The first cross-tabulation table in Table 18. J con-tains observed frequencies, which are the actual cell counts in the cross-tabulationtabl,,, These observed freque nco-s are compared In expected f r~uencies, which arerl~B.Ded as the theoretica' frp~lJ,=ncics iha: are derived ::-:::;~nhis hyp()the~~i.s-ofno associ-ation between the two var iables. The degree to which the observed frequencies departfrom the expected frequencies is expressed in a single number called the "chi-squarestatistic." The computed chi-square statistic is then compared to a table chi-square value(at a chosen level of significance) to determine whether the computed value is s.ignifi-cantly different from zero.

Here's a simple example to help you understand what we just stated. Suppose youperform a blind taste test with 10 of your friends. First, you pour Diet Pepsi in J 0 papercups with no identification on the cup, Next, you assemble your 10 friends, and you leteach one try a taste frorn his or her paper cup. Then, you ask each friend to guesswhether it is Diet Pepsi or Diet Coke. If your friends guessed randomly, you wouldexpect five to guess Diet Pepsi and five to guess Diet Coke. This is you r null hypothesis:There is no relationship between the Diet Coke being tested and the guess. But you findIh,1 9 nf vour fripnrls r or rr-rtlv ~lle" "Diel Pe ris i " ;1l1r1 I inc orrt-ci lv \Jll("SS("S"DiPI

I,

Chi-Square Analysis 1;'11

Row (column) pcrCCnlll~!(\.iIar e row (co iumn) (;(,111'1""quencies divided by I he lOW(column) Iota I.

Chi-square analysis assessesnonmonoronic associations Incross-tabulat ion tables.

Observed frequencies are thecounts for each cell found inthe sample.Expected frequencies are cal-culated based on the nullhypothesis 'of no associationbetween the two variablesunder investigation.


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~Chapter 18: Determining and Interpreting Associations Among Variables

nputed chi-squarempares observed tod frequencies.

z:

-square statistic sum-how far away fromcted frequencies thed cell frequencies arc1be.

Coke." In other words, you have found a departure in your observed frequencies fromthe expected frequencies. It looks like your f ri ends can correctly identify Diet Pepsiabout 9 0% of the time. There seems to be a relationship, but we are not certain of its S ta.tistical Significance, because we have not done any Significance tests. The chi-square sta-tistic is used to perform such a test. We will describe the chi-square test and then aprlyit to your bli nd taste test using Diet Pepsi.

The expected frequencies are those that would be found if there were no assolia[ionbetween the two variables. Remember, this is the null hypothesis. About [he only "cHIl-cult" part of chi-square analysis i, in the computation of the expected freqllencie~. Thecomputation is accomplished using the following equation:Formula for anExpected CellFrequency

Expected celJ frequency = Cell column total x Cell row totalGrand total

The application of this equation generates a number for each cell that would kn'coccurred if the stud)" had taken place and no associations existed. Returning to ourMichelob Light beer example, you were told that 160 white-collar and +0 blue-collarconsumers had been sampled, and it was found that there 'were 166 buyers and 3+ non-buyers of Michelob Light. The expected frequency for each cell, assuming no association, calculated with the expected cell frequency is as follows:

Calculations of Expected CellFrequencies Using the MichelobBeer Example

16 0 x 166 . .White-collar buyer = = = 132. H200160 x 34Whire-collar nonbuyer = ---- = 27. 220040 x 166Blue-collar buyer = = 33.2200. 40 X 34Blue-collar nonbuyer = --- = 6 .8200

The Computed 1 .2 ValueNex t, c ompare the observed frequencies to these expected frequencies. The chi-squareformula is as follows:Chi-Square Formula n 2X 2 =L (Observed; - Expected;)

;-1 Expected;where

Observed; = observed frequency in cell iExpected = expected frequency in cell i

n = number of cellsApplied to our Michelob beer example,

Calculation o f Chi-Square Value (Michelob Example)2 (152 - 132-8)2 (8 - 272)2 (14 - 33.2/ (26 - 68)2X = + ---- + - + -'----"-132.8 27.2 33.2 6.8 = 81.64

You can see from the equat ion that each expected frequency is compared to theobserved freo uencv ann snuared to ad ju st for an" IW9a1i"E'values an d 10 avoid [he

I ,11I\('llation effect. This value is divided by the expected frequency to adjust for cell,, 1 / ( ' differences, and these amounts are summed across all of the cells. If there are[II,[I[}' large deviations of observed f requenc ie s from the expected frequencies, the(I unputed chi-square value will increase; but if there are only a few slight deviations1 '1(1 111 the expected frequencies, the computed chi-square number will be small. InIIIlit'!' words, the computed chi-square value is really a summary indication of how1 , ,, ,1IVayfrom the expected frequencies theobserved frequencies are found to be. As"It h, it expresses the departure of the sample findings from the null hypothesis of[HI,lSsociation.

Some researchers think of an expected-to-observed cornpansojj analysis as a "good-[11 '$ $ of fi t" t est. It assesses how closely rhe actual frequencies fit t ire pattern of expectedIItl j uencies. We have provided Marketing Research Insight 18 .1 .as,an illustration of thegoodness-of-fit notion."

Let us apply this equation to the example of your 10 friends guessing about DietP('psi or Diet Coke. We already agre ed that if they guessed randomly, you would find fiveguessing for each brand, or a S O -50 split. But if we found an 9 0-10 vote for Diet Pepsi,you would be inclined to conclude that they could recognize Diet Pepsi; that is, mostrecognized the cola taste, so they gave the name, Diet Pepsi, that is related to it. Let's usethe chi-square formula w ith observed and expected frequencies to see if the relationshipis statistically Significant. . -

~M~i\~,[*fEliZltN'i'Gtil-"'l~:.-;' v~-~t,1 i I I"- 't..-< 0 : ~ : l' ' '. ~_ ~"... .1 8 . 1 ('Zeroing in"on Goodness-of-Fit

.Can you gue ss t he next number based onthe apparent pat te rn of 1, 3, 5? Okay, what obout thisseries: 1 , 6, 1 1 , 1 6?

In the first series, you realize that 2 was added todetermine the next number (1, 3, 5, 7, 9, and so on).You looked at the series and noticed the equal intervalsof 2. You then erected a mental expectation of the seriesbased on your suspected pattern.

Letus toke the second series because it is a bit more difficult. Suppose your f ir st intuition was to odd 0 3 to theprevious number. Here is your expected series and' theactual one compared:

Expected 1 4 7 10Actual I 6 1 1 16Difference Q 2 4 6

Oops, not much of a match here. So let', try a 4.

Chi-Square Analysis 'III

Chi-square analysis i l osometimes referred I0 11~"goodness-of-fit" ICSt.

Expected 1 5 9 13Actual 1 2 .u l~Difference 0 1 2 3

Getting closer, but still not there. Now try a 5.

Expected 1 6 11 16Actual l ~ 11 . . 1 . 2Difference 0 0 0 0

You have been performing "goodnessof.fit" tests,Notice that the differences become smeller os youzeroed in an -the true pattern. (Catch the pun2) In otherwords, when the actual numbers are equo! to rhoexpected numbers, there is no difference, and the fil isperfect. This is t he concept used in chi-square onalysis,When the differences are small, you have a good fit tothe expected values. Whe~ the differences are larger,you have a poor fit, and your hypothesis (the expectednumber sequence) is incorrect.


8/17

Chapter i8: Determining and Interpreting Associations Among Variables

To determine the chi-square value, we calculate as follows:

Calculation ofChi-Squar e Val ueUsing Diet Peps i Tas te Test

n 2X 2 = IObserved; - Expected;);-1 Expected,(9-5/ (1-5)2=----+---5 5

= 6.4IRemember, you need to use the frequencies, not the percentages.

The Chi-Square Distribution

chi-square d istr-Ibur.ion's-echanges depending 011umber ofdegrees o f 'orn.

Now that you've learned how to calculate a chi-square value, you need to know if i t h sta.tistically Signi ficant. In Chapter 17, we described how the norma l cur ve, or L dislributi'Jn,the Fdistribution and Student's t d istribution, all ofwhich exist in tables, a re used by a UJIll.puter statistical program to de termine level of Significance. Chi-squa re analy sis requires theuse o f a d if ferent distri burien. The chi-square distribution i s s kewed to the right and therejection region is a lways at the right-hand tail of the flistribution. It differs from the nor.mal and t distributions in that it changes its shape depending on the situation at h and, hu t i tdoes not have negative values. Figure 18 ,4 shows examples of two chi-square distribuiion-; The chi-sq uare dis tr ibut ion ' s shape is determined by the number of degrees (l lrl'l'dom. The flgure shows that the more the degrees of freedom, the more the curve's i,li I ispulled to the right. In orher words, the more tbe degrees of f reedom, the larger the' chisquare value must be to fall i n th e re jec tion region for the null hypothesis.

It is a simple matter to determine the number of degrees of freedom. In a UtlStabulation table, the degrees of freedom are found through the formula below:Formula for Chi-SquareDegrees of Freedom Degrees of freedom = (r - I)(c - I)

wherer is the number of rows andcis the number of columns.

JRE 18.4 C hi - S q u a r e C u r v e f o r~ . 4 D e g r e e s o f Fr e e d o mi-Square Curve'se Depends on Itsees of Freedom-

C h i-S q ua r e C u rv e lo r/' 6 D e g re e s 01 F r e e d o m

a R e j e c t io n R e g io n is t h eR ig h t-H a n d E nd 0 1 C u r v e

'!;" ~..;'J[l~

Chi-Squnre AII;)t)'!I,II table of chi-square values contains critical points that determine the break

III'tween acceptance and rejection regions at various levels of significance, It alsot,tk('s into account the numbers of degrees of freedom associated with each curve;11 1 . \ 1 is, a computed chi-square value says nothing by itself-you must consider the1I111tlberof degre es o f freedom in the cross-tabulation table because more degrees ofI"'l'dom are indicative of higher critical chi-square table values for the same level of.,Igtlincance. The logic of this sttuation stems from the number of cells. vvith moreI r - l ls , there is more opportunity for departure from the expected values. The higher1,t1ilc values adjust for potential inflauori due to chance alone. After all, we want to. ,.h-iccr real nonmonotonic relationships, not phantom ones.

SPSS and virtually all computer statistical analysis progums have chi-squarel,tIJles in memory and print out the probability of the null hwothesis. Let us repeatrhls point: The program itself will take into account the number of degrees of free-dOIl1 and determine the probability of support for the null hypothesis. This probabilII Y is the percentage of the area under the chi-square curve that lies to the right ofthe computed chi-square value. When rejection of the null hypothesis occurs, wehave found that a statistically Significant nonmonotonic association exis ts betweenIhe two variables.

As an example of the use of cross-tabulations and chi-square, we have preparedMarketing Research Insight 18.2, which illustrates how cross-tabulation can be usedwith qualitative data. In this case, the researchers judged the models and the wording orSeventeenmagazine advertisements and use a nominal classification system. Thus.the onlyway to analyze the data is with cross-tabulations .

COMPUTE CHI-SQUARE VALUESWe have described the concepts of observed frequencies. expected frequencies. andcomputed chi -square value. Plus. we have provided formulas for the latter two con-cepts . Your task in this Active Learning exercise will be to compute the expected fre-quencies and chi-square values for two different cross-tabulation tables. MarketingResearch Insight 18.2 has a cross-tabulation for Visual and a cross-tabulation for Verbaljudged "gi rl ish" advertisements i n Sevent eenmagazine. Compute the expected frequen-cies and chi-square values for each.

How to Interpret a Chi-Square ResultHow does one interpret a chi-square result? Chi-square analysis yields the amount ofsupport for the null hypothesis if the researcher repeated ule study many, manytimes wi th independent samples. By now, you should be well acquainted with theconcept of many, many independent samples. For example, if the chi-square analysisyielded a 0.02 Significance level for the null hypothesis, the researcher would con-clude that only 2% of the time he or she would find evidence to support the nullhypothesis. Since the null hypothesis is not supported, this means there- is a s ignifl-cant association.

It must be pointed out that chi-square analysis is simply a method to determinewhether a nonmonotonic association exists between two variables. Chi-square does notindicate the nature of the association, and it indicates only roughly the strength of asso-ciation by its size. It is best interpreted as a prerequisite to looking more closely at thetwo variables to discern the nature of the association that exists between them. That is,the chi-square test is another one of our "fl ags" that tell us whether or not it is worth-while to inspect all those row andcolumn percentages.

,

The cornput ed \ III ' i c I lilt ~v a l u e i s C'HlIMIt'd III.i 1 . 11 1 1 1 'val-ue to dl!II.. I nlnt ,\/;1.11111111Isigui ficnucc.

Computer stali~dl,' I p lt'H ' ,IIlook up table CII ~ql.U" \ Ii Iues and print 0111 tilt pltillo'bil i tv of sup port I i)! 1 1 1 1 ' 111\11hypothesis.


9/17

Chapter 18: Determining and interpreting Associations Among Variables--'v \~~ rR1rc~ "E~ f1f1N '@ i!il! E l h W I i @ n & 4 4 H M I I_ ' M r-~--8 2 Cross Tabulations RevealCultural Differences betweenJapanese and American Seventeen Magazine Adsjccasio"aliya researcher must work wilt, purely quai iro-e data, and this Marketing Research Insight describes,is situation. Researchers .were interested in comparing

le way Seventeen magaZIne portrayed teen-ope girls inifferent cultures? Specifically, the Japanese culturemphasizes shored identity and deemphasizes individu-lity while the American culture emphasizes individuality

VtSUAl IPtCTURESI

and even rebellion. Since consumers' selfidentitie, areshaped in port by mass media and advertising,researchers examined all of the relevant ods in four sue.cessive issues of the English (American) version ond theJapanese version of Seventeen magazine. They used ajudging system to classify the visual or pictured modeisand another judging system to classify the words (verbal)in the ads. In cla'ssifying the visual and verbal aspects ofthe advertisements, the judges decided whether or notthe advertisement's components were "girlish" meaningthat they were indicative of child-like norms. The cross.tabulation tables that resulted follow.

VERBAL IWORDS)JAPANESE U,S, JAPANESE U.S.SEVENTEEN SE VE NT EE N SEVE NT EE N SE VENT EEN

Gidish 73 64 Girlish 45 39Not girlish 31 95 Not girlish 59 120

In both cases , the computed chi-squor e values wererge and stalislicolly significant, meaning Ihatthere was

o relationship between the country's culture and the por-Irayal of teenaged females in the Seventeen magazine6dvertisements, The following column percentage tables

vividly depict the nalure of how Ihe odvertisements inSeventeen magazine communicate Japanese culturalnorms to Japanese teenaged females, while the Americanversion of Seventeen strongly communicates Americancultural norms to American teenaged female readers.

ViSUAlIPICTURES) VERBAL IWORDS)JAPANESE U,S, JAPANESE U,S.S E V ENT EEN SEVEN TEE N SE VEN TEEN SEVENTEEN

Girlish, 70% 40% Girlish 43% 25%Not girlish 30% 60% Not giriish 57% 75%

100% 100% 100% 100%r:

~iiic.lnt chi-square means~c2rchcr should look at[oss...:tabulation row and.npercentages. to "see"sociation pattern.

When the computed chi-square value is small, the null hypothesis or thehypothesis of independence between the two variables is generally assumed to betrue. It is not worth the marketing researcher's time to focus on associations,because the)' are more a function of sampling error than the)' are of meaningfulrelationships between the two variables. However, when chi-square analysis idenu-

II" ,I u-lauonship with a significince level of .05 or less (the flag is waving), the,,11.," her can be assured that he or she is not wasting time and is actually pursuing, I 1",11association , a relauonship that truly exists between the two variables in thel'''IIIII,llion. In our Diet Pepsi blind taste test, the chi-square table value for the 9S ~ oit'YI1 o r signifiG>.Ilce was 3.8, and the computed value was 6.4, so the computed\,11111 s larger than the critical value. If we used SPSS, the Significance level would be11''1 Iv e l as .000 I, indicating that the relationship is staristically Significant

~-~8:E:We are going to use our the Hobbit's ChoiceRestaurant survey data to demonstrate how toperform and interpret cross-tabulation analysiswith SPSS. You may recall that we used subscrip-tion to CityMagazine as a grouping variable and per-formed an independent-samples t test in Chapter16, We found that City Magazi1resubscribers were

1110re likely to intend to patronize the Hobbir's Choice Restaurant. We can use cross-tabulation analysis to get a better picture of the effectiveness of CityMagazineas an adver tis-ing medium. Subscription to City Magazine is a nominal variable because respondentsindicated "yes" or "no."We can categorize the respondents based on how likely. they are topatronize the Hobbit's Choice. By taking those who are "very likely" or are "somewhatlikely" and creating a "probable patron of the Hobbit's Choice Restaurant" variable inwhich respondents are either "probable patron:' or "not probable patron:'

The clickstream command sequence to perform a chi-square test with SPSS is ANALYZE-DESCRTPTtVE STATtSTtCS-CROSSTABS, which leads to a dialog box in which you can select the vari-ables for chi-square analysis. In our example in Figure I 8, S , we have selected Subscribe toCityMagazine as the row variable, and Probable patron of the Hobbit's Choice Restaurant asthe column variable. There are three options buttons at the bottom of the box. The Cells, , .option leads to the specification of observed frequencies, expected frequencies, row per-centages, column percentages, and so forth, We have opted for just the observed frequencies(raw counts) and the column percents. The Statistics .. , button opens up a menu of statis-tics that can be computed from cross-tabulation tables, Of course, the oilly one we want isthe chi-square option.

The resulting output is found in Figure 18.6, In the first table, y,pu can see that we havevariable and value labels, and the table contains the raw frequency rr-the first entry in eachceil. Also, the row percentages are reported along with each row and column total. In the sec-ond table, there is information on the chi-square analysis result. For our purposes, the onlyrelevant statistic is the Pearson chi-square, which has been computed to be I I 2.878, The dfcolumn pertains to the number of degrees of freedom, which is 1; and the Asyrnp. Sig. corre-sponds to the probability of suppOrt for the nul l hypothesis, Significance in thi~ example is,000, which means that there is practically no support for the hypothesis that subscription toCityMagazineand Probably patronage of the Hobbit's Choice Restaurant are: not associated, Inother words, they are related. (Actually, the probability is not exactly equal to zero becauseSPSS reports only three decimal places, If it reported, say, 10 places, you would see a numbersomewhere past the third decimal place.)

. So, SPSS has effected the first step in determining a nonmonotonic association.Through chi-square analysis it has signaled that a statistically significant association

The Hobbit's ChoiceRestaurant Survey:Analyzing Cross-Tabulationsfor Significant Associationsby Performing ChiSquareAnalysis with SPSS

Chi-Sgll,ln' )\1111)',1,

~

With SPSS. chi-square is .'11option under the "Crossr.rbv"analysis routine.


10/17

Chi-square analysis.et the SPSS significanceasthe aTIIQUntof supporta~ciation between ther iables being analyzed.

\ 11''III\.:1Its: uelcrnuIllug anc11nterpretln.g Associa tions Among Variables

'!!II._-.. ). , .TrSf~ ' . - ij ; T . G " ! : ~ . 'I" '~~~-:; ._-.";: : :_~. - :7 " !.; c~eMe1S

I' S. Mer selecting the Variahb,Celli; a ; : : ; S ~ a n ~ ~ : n OK t o"~if&BWlJP.H._ '",. _. ,,~~~;

C DCoIu rnt -(s}o [ ;' > 1 1 0 0 " '< p, , ,,o d H d c .,

H ot..,""t... __ ~__. .~_______ _: i 2. In the Crosstabswidcw, ~, Row Variable a nd the Cobu.o

Variable t o he aJfa};ned~~~""""-';~rEta

. ResiI:iJMI " U r n ta n da r dizOO

. P CoIum F" 5 tancfatdized----~I 'rToted F" .AriI.l-,d,.,rl..d~riNli ..'._..... '3. Cl ic k on Cells ... &lectth.C6unlsw, Ncnn tege r Weq.!J ... ' and Percentages des ired inthe :~

r . Ro~dctJlcOlnts c.t;'.~~~.~~~._~4~o.~!!.'!.u_~._. :~;.r tuocere cel ccc+s " ~~~~~~~~

rCoch ran 's;

F IG U RE 1 8.5The SPSSClickstream to C reate Cross-Tabulation with Chi-Square Analysis

Rle Edt View [ ).,to Trensform_~~_~~ __ ,/yle_~r~~ __~~~~~.:..~~_j Th e Crosstabulasions table has raw ~'-::I~I~~~II EilllI'=11iI101'1!1I ' counts and column percerus as ~r ;w . . : ." .~c.n.n. . .,- ~ :% : f ~ m a h S menu ,:_I Doyou~Ubs:crlbetoCIyMolQ.

1 8 . 3 Use of Cross-Tabulationsto Test and GraphicalPresentations to Show Cross-Tabulation Relationshipsfor Online versus NononlineShoppersThe frequencies found in cross-tobulotions. when convertedto percenloge tobles, ore quite amenable to graphical pre-sentation s t ha t ore very useful in depicting the nature of therelationships found in a survey. With th is Mar ketingResearch Ins ight, we are using the cross-tabulationsreported in a vveb-bosed survey that compared onlineshoppers with individuals who had never made an onlinepurchose.f In the survey, these two types of purchasers

were measured by a number of demographic chorccterist ics such as gender, age, education, ethnicity, maritol slCItus', and income. Ih ey were also measured on sell-reportsof computer competency and how they prefer :0 search forinformation about morketplace allernotives. The followingthree relationships were found to be statistically significanl,meaning that the relationships exist in the population. Thnllfinding of significance with a cros s-tcouloticn allows thresearcher to examine and describe the relulionship, 05 ilis a nonmontonic pattern and not one rhot can be choroc-terized by direction or strength fr om t he chi-square resultsa lone. We hove included a s lo tement o f the apparenl relo-tionships in each graph. This graphical approach is onappropriate one for nonmontoni c r el otionships.

(continued)


11/17

605040 ' '..~\- iiiO n l in e--~; .

~ 30 I~ S h o p p e r


12/17

\ 1I .1I '1t -1 I I~ . IJI I\ I I II IU I IIH 1111 (1 1111 (11)1IIIIH ASM)(,.I.lI il)1I1'o A II HlIg V. II 'I,\ I)lc!


13/17

\ iil l H I I III In i l 1111111111)1 111111 11111 '1"1 I III)! I\~ltjl II 11111111 ''' 1\1111111)1 Y,III,WIl ~

THE PEARSON PRODUCT MOMENT CORRELATIONCOEFFICIENTThe Pearson product moment correlation measures the linear relationship h\t II VI IItwo interval- and/or ratio-scaled variables such as those depicted conceptually b} '(,IIter diagrams. The correlation coefficient that can he computed between the t\\'o vanilhll'"is a measure of the "tightness" of the scatter points to the straight line. You already kllli\\that in a case i n which all of the points fall exactly on the straight line, the corr~lillllilicoefficient indicates this as a +Ior a -I.n the c ase in which it was impossible to d"cern an ellipse such as in scatter diagram Figure IS.S(a), the correlation coefTicl\'lliapproximates Zero. O course, it is extremely unlikely that you will find perfect 1.1) you see how the concepts we [ust discussed fit in. In ficient for' the re la tionship . Thi s val ue reveals thai Ihestotisticion's terminology, the numerator represents the greater the number of citizens living in a county, the I:55-products sum and indicates the covariation or greater the county's retail sales. - - - - 1

z:nsxsy25,154

10 x 7.8 x 384.4=~5,15429,975.4.84

\.-1/1'1'

The Pearson Product MOIIICIIIon'd,llloil ('1111111I "IIIH' rill'mula for calculating a Pearson product moment correlation is couipltcatcd.

111,11I"I'.I1L'hers never compute it by hand. as they invariably find these all computer,,1111"11 lowcver, some instructors believe that students should understand the worki ngs"I 1111'orrclauon coefficient formula. We have described this formula and provided anI 11I1/11"II Marketing Research Insight 18.4.

1"',INlI1 product moment correlation and other linear association correlation coeffl-, lilli, IlIciicarF not only the degree of association but the direction as well, because as we11',1IlllI'd in our introductory camments on correlations, the sign of the correlation\ "I lIit 11'111ndic ates the direction of the relationship. Negative correlation coefficients1I\'I,.tl'lliilL the relationship is opposite: As one' variable increases, the other variable'/1'1II',;,CS. Positive correlation coefficients reveal that the relati'9'"nship is increasing:III >:"1quantities of one variable are associareci with larger quantities of another variable.II I', uuportant to note that the angle or the slope of the ellipse hasnothing to do withIIII' ',I/.c of correlation coefflcient. Everything hinges on the width of the ellipse. (The,IIIPI' will be considered in Chapter 19 on regression analysis.)UATE.NET: MALE USERS CHAT-ROOM PHOBIA11.lIt,net is an onl ine meet ing service. Its purpose is to operate a vir tu ra l meeting place for111(11 seeking women and women seeking men. Internal analysi s has revealed that femaler h.u-room users greatly outnumber male chat-room users. This is frustrating to Date.netprlucipals, as they know that the number of "men seeking women" is about the same as"women seeking men." Men seem to have a c hat-room phobia.

They commissioned an online marketing research company to design a questionnaireIh.u was pos ted on. the date.net Web site for 15 days. The survey is a success. as over 5000d.ue.net users f ill it out in this time period. Date.net executives request a separate analysisof'''Jl1cn seeking women" user res pondent s to look into the chat- roam-related questions.The research company decided to report all correlations that are significant at the 0.01level. Here is a summary of the correlation analysis findings

FactorCorrelation withAmount of date.netChat-Room Use

Demographics: -.68-.76-.78+.57+.68-.90- & " 1

Ag eIncomeEducationNumber of years divorcedNumber of chlldrenYears at present addressYears at present jobRelationshipsJob/ careerPersonal appearanceLife in generalMinutes online dailyOnline purchasesOther chatting time/monthNumber of e-mail accounts

Sansfacnon w ith: -.16-.86-.72-.50

Online behavior: +.90-.65+.86+.77

Use of date.net (1 = not important and 5 = very important)Meet new peopleOnly way tv till to womenLooking for a li fe partnerNo t much else to do

+.38+.68-.72+.59

1,1

A I,oslll,,' 11I11I,111I11I1I.IWIIall ill\I"\,,,,IIIU IIIH 11II LIIIIIIIltip,"'hl'lljl.ljIIIlI~IHh lilli'b.liol \I H I I . d t - : II tiP! IJIII ! M filii


14/17

\ jj,IJiH1 10: IJCtC.J IIIIIUl1g :U1U InlCrprCLJng ASSQC1:lU0l1SAmong var ianres

..

SPSS,oorrelauons areouted wi th theELATE-BIVARIATE.re,

correlation analysis. each~ation willhave a uniqueicance level.

s:

tudent.Assistanr:tegrate.d .Case:ing with Correlations

For each fac tor, use your knowledge of correlations and provide a statement of how IIcharacterizes the typical date.net male chat room user. Given your findings, what tac : icS \10you recommend to Date.net to combat the male chat phobia problem?

ffV~~E':The Hobbit's ChoiceRestaurant Survey: Howto Obtain Pearson -ProductMoment 'Correlation(s)with SPSS

. With SPSS, i t t akes only a few cli cks to compute Cor.relation coefficients. Once again, w e will use thHobbit's Choice Restaurant survey case study becausyou are familiar with it. If you recall, we ha~e deter.mined from previous analysis, that a waterfront viewis generally preferred. Remembering this, you would

probably feel very confident about recommending this location to Jeff. But let's take a doserlook using correlation analysis. Correlation analysis can be used tof ind out what people wantwi th the waterfront view; that is, high positive correla tions would indicate tha t they wantedthe waterfront location and the items that are highly correlated with this location preferences.Conversely, high negative correlations would signal thacrhey did not want those items with thewaterfront location. Recall that there were several menu, decor, and atmosphere questionsbeing mulled over by Jeff Dean. Correlation analysis is very powerful, as it can reveal to whatextent people prefer (or do not prefer) these items as they prefer the waterfront view. We'llonly do a few of the items here, and you can do the rest in your SPSS integrated case analysiswork specified at the end of the chapter.

So,we need to perform correlation analysis with the waterfront locacon preference vari-able and the other factors that will determine the Hobbit's Choice Restaurant's "personality."The clickstrearn sequence is ANALYZE-CORRELATE-BIVARIATE, which leads, as can be seen in Figure18.9, to a selection box to specify which variables are to be correlated. Note that we haveselected the waterfront location and several other items related to decor, atmosphere, andmenu. Different types of correlations are optional, so we have selected Pearson's, and the two-tailed test of significance isthe default.

The output generated by this command is provided in Figure 18.10. Whenever youinstruct SPSSto compute correlations, its output is a symmetric correlation matrix composedof rows and columns tha t pertain to each of the variables. Each cell in the matrix contains threeitems: (1) the correlation coefficient, (2) the significance level, and (3) the sample s ize. Asyoucan see in .Figure 18.10, the computed correlations between "prefer waterf ront locat ion" andthree of Jeff's questions-Simple decor? Prefer unusual entrees? Prefer unusual desserts?-are+.780, -.782, and -.81 0, respectively. They all have a "Sig" value of .000, which t ranslates intoa .001 or less probability that the nul l hypothesis of zero correlation is supported. if you lookan our correlat ion printout, you wil l al so notice that a correlation of 1.000 is reported, inwhich a vari ab le i scor related with itself. Thi s report ing may seem strange, but it serves thepurpose of reminding you that the correlation matr ix rhat is generated with thi s procedure iss~etric. I~ other words, the correlations in the matrix above the diagonal 1s are identical tothose correlations below the diagonal. With only a few variables, this fact is obvious; however,sometimes several variables are compared in a s ingle run, and the I s on the diagonal are handyreference points.. Since Wenow know that the correlations are statistically significant, or significantly dif-fe~ent from zero, Wecan assess their strengths. They hover around .80 which, according to ourrules of thlllnb 0 I'" di d I . . th d., n corre anon SIZe I n cates a mo erate y strong aSSOCIatIon. In 0 er war s,we have some rei" bl s: I . .":,: the - . tionships tha t are sta e and rair y s trong. Last, we can use the SIgnsto mter-pret '.. eassoci .t ions. What isyour interpretation? ...

~~~~.~~ ..,,~-;r~=~,..:r~-?::,,_.,.-:.

The Pearson Product Mornelll ("orrl,I,lIlo,II"I"'II, 1 " " 1

1 . 1 ~ C '" I : i ; ! j m - ~ - . : o m u u . _ u u u _ . . _ . . _ _1 1 $2~! ' ., . fy . , 1 C>"" ~ 3. A/tersel=tinz tlJe Variables; 1I $11\ o~o_~e~~~_ ... _.. ~ _ ' - r o o _ - = - - dic/;onOKtqp~nrmtJst~sU II. U.f8 Correlate> Blvariase.: to 1 ; ; : F . ~ ~ ~ ~ ~ ~ ~ I j i l i - i - - i - i - i - ; - ; i - i - i i i i - i i - - i j - - i ' i - ; ~ j l ~ .1 1 1 " " up t he Bi .ariase Correlauons :

window ; ~- mm __ ._ I~WNchsecionr:Jth~. V

~~Wnc.jl~. .- - - - - . . - - . - . . - .. - - - 1i 1. Select tlJe VarialJle(s) to b.I "",*",d and by tkfaJ.tUSPSS .ll !

I c()1npute pi!lJNOlJ coefficient.:J 1i .-T;;;is~~--.-.----: r. Tw o-tM e.d r One-tli ied i _ _ 1 0 ,, ,, ", ... Ip~~.ant correlatiom

F IG UR E 1 8.9The SPSSClickstream to Obtain Correlations

Ale Edt VIew 'Dot/! Tr.:nsform Ins er t Ftm4 lt PIldyze Gr/!Dhs Wt es W rld ow Hetlii ! II J i 3 T t !3 .,

~l~!J~J0./l it J / ~/ jgi j/ t;, . L i; L / < ?I 1 / 1 f ; J _ 1 . J'"

Correlations r - . . 17 .- ;C ; ; ;; ;; i ; ; ; ;; ;; ' ; - ; ;W k '; ; '; .. - . . l l!cotre14JUJns, :rzni/C4lJCe lnel..; and


15/17

I. 11.111 I I ni IJCLC1"!!UUUg anu Ullerprcung Assocauons AIllong var rauies

!'"' l il l ll l' " " , 1 11 , I,.)"'ill I' 1 \' 1 1 1 1 I~"" 1 1 i( ' < l1 . g o -I

,\ ,., ivc tllttl

hen dining on seafood.rauran t p at ro ns o ft en ' l ikekeep i t s imple .

Here is what we have found. People who prefer to ea t at a r es ta urant with a wat "',. I de I I C Ifront view also prefer a sImp e ecor. At t re same time, t iey do not want unusual'entrees or unusual desser ts . Apparently, when folks go to a waterfront restaurant theiwant to kick back, be comfortable, and not be bothered with choosmg from a varlCl1of curious dishes or an array of exotic desserts. They probably want seafood. Anupscale Hobbie's Choice Restaurant with unusual entrees and desserts would d~n.nitely not fit the prefe rence s o f th ese peop le . ~o , now how do you feel about your pre-vious recommendation to locate the Hobbit s ChOICe Restaurant on some expensivwaterfront property)

USING SPSS TO COMPUTE CORRELATIONSYou have jus t seen the cor re la tion analysi s f indings for one set ofpreferences for a restau,rant. Now let 's t ake a speci fi c a spect o f th e r es taurant, n ame ly , th e fa ct th at it could bewithin a 3D-minute drive f rom pat ron' s homes. Use SPSSto determine the cor relation ofth e pr efe rence fo r th is lo ca tion wit h p re fer ences f or st ri ng qua rt et music , ja zz combomus ic, f orma l wai ts ta ff wea ri ng tuxedos , and unusual en tre es on the menu. When youinspect the cor re la tion mat rix tha t resul ts , what have you discovered about the combina-t ion of res taurant a tt ributes tha t these pat rons prefer for a res taurant that is within a 30-minute drive?

Special Considerations in Linear Correlation ProceduresWe have p repa red Table 18. 3 to summarize and remind you of four considerations 10keep in mind when working with correlations. We will discuss each of these in turn. Tobegin', the scaling assumptions underlying linear correlation should be apparent 10 you,

~-,,~""""" i* .v.F . ~~~.!t~:-~"~~~""~~'>::':'~ll'7!"';'~~_~_._..._... ..

The PearsoII P r or l l l " 1 MOII"''' CIi'""!."I"1i ,, " 1 1 1 11 , ,1 1

;!~ :~ ~ ~ l~ ~ ~ ~ J:6Y ~ ~ hJ~ g b ~ f~ 1~ ;~ ':'Ih,~ I (II relations only for marie variables (interval or ratio scaling).

t III w l.ll lon assumes that only the two variables involved ar e r el evant: all other variables andLUlor .arc considered [Q be constant.

t ru n-l.uion does not indicate cause-and-efTect, only covariance between the two variables is1 , , 'lilfI,lnalyzed.I'(illl'!ation expresses only the Iinearrelauonship between two variables.

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en the null hypothesis isecred, the researcher may:; "~a rr:.aIi .d .ge riall) , important -ationship to share with the..flager. _ =

~;S Student Assistant Online:ar Integrated Caseie in the Bottle: SPSSeistics Coach

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Regression Analysis ill MarKeting Research

.:

ConceptMultipleregressionanalysisAddit ivi ty

IndependenceassumptionM u l ti p le R

Mult icol l ineari ty

Variancei n f l at i on fa c to r(V lF )Trimming

Standardiz edbet a c oefficientsDummyindependentvariableStepwisemult ipleregression

ExplanationA powe rful form of regress io n where more than one xva ri ab le i s in the regressiont"f~uc.ti0n

A statistica l a ssumpt ion tha t a llows the use of more than one x variable in amultiple regression equation: y = a + blXl + b Ix 2 . + b r nx mA statist ic al r equi rement tha t when more than one x variable is used , n o p ai r ofxvariables hasa high correlationAlso called the coefficient of determination, a number that ranges from 0 to !.Othat indicates the strength of th e overall linear relationship in amult ip leregression, the higher the betterThe term use d t o d enot e a v iola ti on o f t he independence assumption that causesregres sion resul ts to be in errorA statistical value that identif ies what x variable(s) contribute to multicollinearitya nd shou ld b e removed f rom t he analysis to eliminate multicollinea ri ty . Anyva r ia bl e w ith a VIF of 10 or greater sh ou ld be rem ove dRemoving an x variable in multiple regression because it is nor statisticallysignificant, rerun ning the regression, and repeating until all rema in ing xva ri ablesa re SignificantS lopes (~ values) t ha t a re no~m aliz ed so they can be compa red dir ec tl y 10determi ne t heir relative importance in y's predicuouUse of an x variab le tha t has a 0,1 or similar coding, used spa ring when nominalvariables must be in the independent variables set

A specialized mu lt ip le regressi on t hat i s a ppropriate when there is a large numberof x variables that need to be trimmed clown to a small, signiHcani . set and theresearcher wishes the statistical program to do this automatically

. Predicti ..e analyses are methods used to forecast the levels of a var iable such as sales.Model building and extrapolation are two general options available to lua,~,c:~researchers. In ei ther case, it i s impor tant to as ses s th e goodness of the prediction. Thisassessment is typic al ly pe rfo rmed by compar ing t he p red ictions against the actual datawith procedures called "residual s analyses." --Market researchers use regression analysis to make predictions. Till': ba sis of thistechnique is an assumed st ra ight -li ne r ela tionship exi st ing between the variables. Withbivariate reg res sion, one i ndependent variable, x, is used to predict the dependent vari-able, y , using the s traight-line formula of y = a + bx. A high R squa re and a statisticallySignifican t s lope indic ate that the linear model is a good fit. With multiple regressi0n,t he unde rly ing concep tua l mode l speci fi cs th at several mdependcnr variables are to beused. and it is necessary to determine which ones are Significant. By systematicallye liminating the nonsignif icant independent var iables in an iterative manner, a processcalled "trimming," a res ea rcher wil l u ltimate ly der ive a set of Significant independent

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I\niI W 1j1l11\11I111\,,111,11iI'S that yield J significam predictive model. The stand.ml ('11111111tIII "~lllilI111 I1I'.I,d 111compute a confidence interval range for a regression. pl'"dlulltil

~;t\l',()t1ed researchers Illay opt to use stepwise ruuhiple n:gl ~'"IIIII 11 1.11'Ii \l'1I11.tl , u f:I' number of candidate independent variables such as several dellltllP il!illll , Illnl I'll,11111uycr behavior characteristics. With stepwise multiple regress,iulI, intil'IH'lIdl'lli 1.11111111"ll't' entered into the multiple regression equation containing only 't"llI~lli ,tll} "IfIItrl lt ,1111ndependent variables. .,,'

I'mliction (p, 560)l.xtrapolation (p. 561)l'u-dictive mode l ( p. 561)Analysis of residuals (p. 562)I\ivariate regress ion analysis (p. 563)Intercept (p. 563)Slope (p . 563)Ikpendent variable (p. 564):.idcpendent var iable (p. 564)Least squares criterion (p. 564)Standard error of the estimate (p. 570)Outlier (p. 572)

General conceptual model (p, 573)Mu lti pl e r eg res si on analysis (p. 57 S )Regress ion pl ane (p. 575)Addit iv ity (p. 576)Coefficient of determination (p. 577)Independence assumption (p. 577)Multicollinearity (p. 577)Variance inflation factor (VIF) (p. 5J 7)Dummy independent variable (p. S8 I)Standardized beta coefficient (p. 582)Screening device (p. 583)Stepwise multiple regression (p. 585)

1. Construc t and explain a reasonab ly simp le p redic ti ve mode l fo r each of the follow-ing cases:a. What is the relationship between gasoline prices and distance traveled for fam-

ily automobile touring vacations? .b. How do hurricane warnings rel at e t o pu rchases of flash light batteries in the

expected landfall area?c. What do florists do with regard to thei r i nventory of flowers for the week prior

to and the week foilowing Mo the r's Day? ~2. indicate what the scatter diag ram and p robab le regression line would look like fortwo va riable s t hat a re correlated in each of the following ways (in each instance,

assume a negat ive intercept): (a) -0,89, (b) +0.48, and (c) -0.103. Circle K runs a contest, invit ing cus tomers to fill out a registration card. In

exchange, they are eligible for a grand prize drawing of a trip to Alaska. The carelasks for the cus tomer's age, education, gender, estimated weekly purchases (in dol-lars) a t t ha t C ir cl e K, and app rox ima te distance the Circle K is from his 0:her horne ..I denti fy e ach of the following if a multiple regress ion analy sis were to be per-formed: (a) independent variable, (b) dependent variable, and (e) dummy variable.

4. Exp lai n wha t i smeant by the i ndependence a ssump tion i n mu lt ip le regre ssi on. Howcan you examine your data for independence, and what s ta ti stic is issued oy most sta-tistical analysis programs? How is this statistic interpreted? In other words, what wouldindicate the presence of multicollinearirv, and what would you do to eliminate it?

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