Chi-square - 136 Chapter 8 Crosstabuiation and Chi—Square Analyses THE PURPOSE of crossrabulaTion is $0 show in Tabular formaT The relaTionship

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Unformatted text preview: 136 Chapter 8 / Crosstabuiation and Chi—Square Analyses THE PURPOSE of crossrabulaTion is $0 show in Tabular formaT The relaTionship beTween Two or more caTegorical variables. Caregorical variables include Those in which disTincT caTegories exisT such as gender (female, male), eThniciTy (Asian, WhiTes, Hispanic), place of residence (ur- ban, suburban, rural), responses (yes, no), grade (A, B, C, D, F), and many more. CrossTabu— laTion can be used wiTh conTinuous daTa only if such daTa are divided inTo separaTe caTegories, such as age (0-) 9 years, 20—39 years, 4069 years, 60-79 years, 80-99 years), ToTal poinTs (O- 99, TOO—T49, l50wl99, 200-250), and so on. While iT is accepTable To perform crossiabula- Tion wiTh confinuous daTa ThaT has been caTegorized, if is rare To perform chi-square analyses wiTh conTiauous daTa because a greaT deal of useful informarion abouT The disTribuTion is losT by The process of caTegorizaTion. For insTance, in The ToTal poinTs disTribuTion (above), Two persons who scored 99 and TOO poipfs, respecTively, would be in The firsT and second cafegories and would be considered idenTical To Two persons who scored 0 and T49 poiaTs, respecTively. Nonerheless, crossfabuiarioa wiTh conTinuous daTa is ofTen used for purposes of daTa descrip- Tion and display. The SPSS command Qrosstabs and The subcommaeds grails and §tatistics are used To access all necessary informaTion abouT comparisons of frequency daTa. CROSSTABULATION While The frequencies command can Tell us There are 5 NaTives, 2O Asians, 24 Blacks, 45 Whires, and Ti Hispanics (and ThaT There are 64 females and 4i males) in our gradessav file, iT cannoT Tell us how many female Asians or male Whires There are. This is The funcTion of The Qrosstabs command. if would be appropriaTe To “cross” Two variables (ethnic by sex) To an- swer The ques’rions posed above. This wouid produce a Table of )0 differenr cells wiTh associ- aTed frequencies inserfed in each cell by crossing Two (2) leveis of gender (sex) wiTh five (5) lev-... els of eghnicify (ethnic). IT is possible To cross Three or more variables, alThough only wiTh a very . large daTa seT would a researcher be lil<eiy To perform a crossTabulaTion wiTh Three variables: because There would be many low—counT and empTy cells if The number of subjecTs was noT sufw ficienT. For The presenT sample, an ethnic by sex by grade crossiabulaTéon would probably noi be recommended. This procedure would creaTe a 5 (ethnic) x 2 (sex) x 5 (grade) display of" frequencies—a ToTal of 50 cells To be filled wiTh only 105 subiecrs. A iarge number of low- counT or empTy celfs wouid be guaranTeed. If such a crossTabulaTion were creaTed wiTh a large ' N, SPSS would produce five differenT 5 x 2 Tabies To display These daTa. CHI-SQUARE ( 2) TESTS OF INDEPENDENCE X in addiTioo To frequencies (or The observed va/ues) wiThin each ceil, SPSS cam also compuTe The expecTed value for each cell. Expeded va/ue is based on The assumpTion ThaT The Two variables- are independenT of each oTher. A simple example demonsTraTes The derivaTion of expecTed value. Suppose There is a group of TOO persons in a room and ThaT 30 are male and 70 are female. If There are )0 Asians in The group, if would be anTicipaTed (expecTed)—if The Two vari-' ables are independenT of each oThermThaT among The )0 Asians There would be 3 males and 7 females (The same proporfion as is observed in The enTire group). However, wiTh The same group of TOO, if TO of Them were fooTball players we would nor expecT 3 male fooTball players and 7 Temaie fooTbali players. in American socieTy, mosT fooTball players are male, and The Two caregories (gender and fooTball players) are nof independenT of each oTher. if There were Chapter 8 / Crosstabulation and Chi-Square Analyses 107 0'" no addiiional informaiion given, iT would be expecTed ThaT all H) of The players would be male. ES The purpose of a chi-square TesT of independence is To deTermine wheTher The observed values il- Tar The cells deviaTe significanle from The corresponding expecTed values for Those cells. : The chi-square sTaTisTéc is compuTed by summirig The squared devia‘rions [observed value (7%) minus expecTed value (Q) divided by The expecTed value for each cell: :— X2 flzllfa - fez/fa] es - 3y As you can see, i? There is a large discrepancy beTween The observed values and The expecTed as values, The x2 sTaTisTic would be large, suggesTing a significanT difference beiween observed id and expecTed values. Along wiTh This sTaTisTic, a probabiliTy value is compuTed. WiTh ,a < .65, iT is commonly accepied ThaT The observed values differ significanle from The expecTed values and ThaT The Two variables are NOTindependenT of each oTher. More compleie descripfions :3 and delini‘rlons will be included in The OquaT secTion of This chapTer. An addiTional concern addresses The facT Thai a chi-square sTaTisTic is ofTen ThoughT of as a TesT of associaTion (The opposife of independence) be’rween variables. This invalid assumpTion can creaTe difficulTy because a chi-square value is largely dependenf on The number of dimensions and sample size, and Thus comparisons of one chi—square value wiTh anoTher are oTTen mis— leading. To conTrol for This difficulTy, Pearson suggeséed The phi (a) sTaTisTic, which divides The 5 : chimsquare value by Nand Then Takes The posiTive sqaare rooT of The resulT. The purpose was To 3: sTandardee a measure of associafion To values beTween O and i (wiTh O indicaTing compleTeiy ‘3 independenT variables and a value close To l indicaTing a sTrong associaTion belween vari- l- ables). However, if one of The dimensions of The crossfabulaTion is larger Than 2, (i) may aTTain 1- a value larger Than 1.0. To cenTrol for This, CramEer’s V was inlroduced (The posiTive square '- rooT of xQT/[le—lfl, where /< is The smaller of The number of rows and columns). This measure 'y does vary beTween {J and if) and is a commonly used measure of The sTrengTh of associaTion ES beTween variables in a chinsauare analysis. f- )1 - The file we use To illusTraTe Crosstabs is our example described in The firsT chapTer. The daTa ,g file is called grades.sav and has an N I TOE. This analysis creales crossTabulaTéons and col— ,_ culaTes chi-square sTaTisTics for sex by ethnic. :r STEP BY STEP Crass Tabulation and Chi-Square Tests of Independence a . .. . .. . Stew- 3 Crear‘e and name a dare fi/e or ecu/fr fl'fnecessary) an a/reaaly exfsh'ng h'/e {see Chapfer 3/ T 3 To enfer SPSS, a dick on Start in The res/rear {bofiam ofscreenj acfr'vafes The sfan‘ menu: --,Eregrems_-.'-'._.;'-':,_==-'-_'f _) e’, gigg=~fiPSSiiigflffifgwlllfiplfifi 3 Affer dickan The 5/055 program icon, Screen :7 appears on The monh‘or. Screens I and 2 (er9- 3 played on The inside from cover) a//0w you To access The dafa file used in conduch'ng The aria/y- ? sis of infers-sf. The fol/owing sequence accesses The gradessav fife for furfher ana/yses: ' 108 Chapter 8 / Crosstabuiation and Chi—Square Analyses insgréén .. .. .. . . __ . . .. . ’ 3*; Eile m} Qpen —> ] grades . sav --> OK [or ‘2‘ gradessav ] Wnefner firsi‘ enrerfng SPSS or refurnmg from ear/fer operaffons The si‘andard menu orr com- mands across The Top is required {shown below). As long as if 1's w'si'b/e you may perform any analyses. If is no“ necessary for me dafa window ro be w'sib/e. i ' 2W _ _. is This menu oi commands disappears or modifies when using pivoT Tables or ediTing graphs. To uncover The sTandard menu of commands simply click on The or The icon. Afrer comp/error? of 5fep 3 a screen wir/v fne desired menu bar appears. When you click a command {from The menu bar}, a series of opfi'ons will appear {usual/y) be/ow fne se/ecfed ._ command. Wirh each new sei‘ of opffons, click fne desired i'rem. The sequence To access chr— I square srarisfics begins or any screen wr'fn f/Te menu of commands visfb/e: in Screen _ DaThis _ Anaiyze ---> Descriptive Statistics --> Crossta l _ ,_ bs A new winldow now appears (Screen 8.l , beiow) Thai provides The framework Tor concluding a crossiabs analysis. The procedure is To click on The desired variable in The iisT To The leTT (sex in This exampie), Then click The uppermosi of The righT arrows ( ) To indicaie Thai we wish gen— der To be The row variable. Then click a second variable (ethnic in This example) and click The middle righT arrow (To indicaie ThaT we wish eThniciTy To be The column variable). This is all Thai is necessary To creaTe a cross TabulaTion of Two variables. This will creoTe a 2 (sex) by 5 (eth- . nic) Table Thai conTains 10 cells. - -. Tress " The Crossfabs Window ' lestname E; firstnem ' Q) sex 52-1. 4;!) ethnic .; eyes '2'; 65) iawup ® Section a spa : <55) ex‘trcrea @review -' . <3? mm - a}: auiZE ®quiz3 {1‘9 quiz4 '. @auizfi iseifislélyéiiéréd _ _ kLQ LM Chapter 8/ Crosstabuiation and Chi~Square Analyses 109 The lowesT box in The window allows for crosstabalafion of Three or more variobles. if, for in- sTance, we wanted To find The gender by ethnic breakdown for The Three sections, you would clicl< The section variable in The list of variables Then click The lowesT of The Three righT arrows. This would reselT in Three Tables: A gender by eThniciTy breakdown for The firsT secTion, a gender by eihnicéty breakdown for The second secTion, and a gender by eThniciTy breakdown for The Third secTion. The Previous and Next To The lefT and right of Layer 1 of ‘1 are used if you wanTed a gender by eihnic analysis for more Than one variable. For insTance, if you wanted This breakdown for ba‘Th section and year (year in school), you would click section, click The lowesT righ’r arrow, click Next, click year, Then click The lowest righT arrow again. This would produce Three 2 X 5 Tables for section and four 2 X 5 Tables for year. The sequence of steps below shows how To creaTe a sex by ethnic by section crassTabulaTion. The {OI/owing sequence begins from Screen 8. l . To arrive 07‘ This screen, perform whichever of sreps 7—4 finages 707-108) are necessary. Inscreén. Do'Tfifis . . . . . . . . . . . sex ‘9 upper .. *—) ethnic —) middle . .. "9 section -—> lowest . *5 f: UK For a simple sex by ethnic crossfabulaTian, omit The Two clicks Thai involve The section variable. H is rare for a researcher To want To compute only cell frequencies. in addition To frequencies, iT is possible To include wéihin each cell a number of addifional options. Those mos? frequenily used are listed below wiTh a brief definition of each. When you press The Qeils buTTon (Screen 8.l), a new screen appears (Screen 8.2, below) That allows you To selecT a number of options. The Qbserved count is selected by defaulT. The Expected counT (more frequenle referred To as The expecfed va/ue) is in mos? cases also desired. Inclusion of other values depends on The preference of The researcher. ‘ The Crossrahs: Ce// DrLsp/ay Window The actual number of subiecTs or cases wiThin each cell. The expected value for each cell (see page Too). The perceni of values in each cell for That row. The percenT of values in each ceil for ThaT column. Qbserved Count: Expected Count: Bow Percentages: goiumn Percentages: :otal Percentages: gnstandardized Residuals: Observed value minus expecTed value. The percent of values in each cell for The whole Table. QUBDGB 110 Chapter 8 / Crosstabuiation and Chi-Square Anaiyses To creai‘e a sex bi} ethnic crossiabulafion fliai includes observed couni‘, expecied couni‘, fofal perceni‘ages, and unsfandardized residuals wiibin eacb cell, ,aen‘arm The following sequence of sfeps. We begin 02‘ Screen 8. l ,- press asset il variables remain from a prior analysis. .flo'T-hi's' " _ _ sex rm} upper #5 ethnic -> middle #9 3 in. Screen; Total —> @Qnstandardized --> Continue l Noie: We don’T dick on Qbserveci, because This opiion is already selecied by delauli. Thus Tar we have only creoTed TabulaTions 0T numbers wiihin cells. Usuaily, along wiTh cross iabuloiion, a chiwsquare analysis is conducied. This requires a click on The §tatistics pushhui- 5: Ton (see Screen 8.1}. When This buTTon is clicked, a new window opens (Screen 8.3, below). 1..- Many diTTereni TesTs of independence or associaiion are lisTed here. Only Chi-square, and Bhi and Cramér’s V (see page 107) and Correlations (explained in Chaplez‘ 10) will be consid-‘ 5 ered here. As in The galls window, The procedure is To click in The small box To The leTT oT The 7_ desired sTaTisTic belore reiurning To The previous screen To conducT The analysis. See The SPSS .3 :- lor Windows Base Sysfen’i Users Guide Tor a descripTion ol The oTher sTaTisTics included in This chad. The Crossiabs: Sfalisfics Window 4‘ in The sequence below, we creare a sex by ethnic crassiabulai‘ian, requesi gbserved count, : Expected count, and w{J‘nstanciardized Residuals wifbi'n each cell, and include The Chi- square, and fihi and CramCJr’s V as forms of analysis. We do nol include carrelafions be— _ cause ibey nave no meaning wben fliere is no infrinsic order io ibe assaciaied variables. The . sianiing poinf is Screen 8. l. Comp/ere resulis from ibis analysis are included in The Ouipul seClion. in 'D'o‘lifis . .l . . . . .. sex ~——> upper --> ethnic —> ‘ giggxpected *5 @gastandardized -> Continue 1 Statistics 1 Chi-square *9 fihi and Cram [:1 r’s V —> Continue Tail The Chapter 8 / Crosstabulation and Chi—Square Analyses 111 3/ Chen we may wish To conducT a cross TabulaTion and chi-square analysis on a sabsef ol a cer— 3f Tain variable. For insTance, in The sex by ethnic crossTabulaTion described earlier, we may wish To deleTe "NaTive" caTegory lrom The analysis since There are only 5 ol Them and earlier analy- ol gender) X 4 (levels 01‘ eThniciTy alTer excluding The TirsT level) analysis. ATTer you have selecTed The variables lor crossTabulaTion, have chosen cell values and desired sTaTisTics, Then click on The Qata command in The Menu Bar aT The Top ol The screen. in The menu of opTions ThaT opens below, click on Select gases. AT This poinT, a new window will open (Screen 4.8 lrom Chapier 4). in This window, click The small circle To The righi of if gondition is satisfied (so a black doT lills iT), Then click The if buTTon )usi below. A new dialog box again opens (see Screen 4.3, also Tram ChapTer 4) TiTled Select Cases: If. This window provides access To a wide varieTy ol operaTions described in Chapier 4. For The presenT we are only concerned wiTh how To selecT levels 2, 3, 4, and 5 of The ethnic variable. FirsT sTep is To selecT ethnic lrom The variable lisT To The leTT, Then click The To pasTe The vari- able inTo The “acTive” box, Then clic§< ihe (on The small keyboard below The acTive box), Then : clicl< The 2. You have now indicai‘ed ThaT you wish To selecT all levels oi ethnic greaTer or equal 3 . To 2. Then click Continue, Screen 4.8 will reappear, click OK, The original screen will appear 5 (Screen 8.l) click The OK and your analysis will be compleTed wiTh only four levels oT elhniciTy. The sTep-by-siep sequence iollows. i ses have indicaied ThaT There is a problem wiTh low counT cells. This means creaTing a 2 (levels 5 l l l The sfan‘i'ng ,aai'nf for This sequence is again Screen 8. l . Review si‘eps ] ~4 To see how To access This screen. If may be necessary To press The Reset buffon. .Doi'rhisT _ _ . . . _ _ _ . sex ~—> ethnic —~> med/e —-> €3§eiis fikgxpected --> @Uflflstanaardized ~9®Cantinue ' I Statistics Chi-square —> Ehi and Cramér’s V % Data {or The Top of The screen in The menu bar} —) The 0 To The er? of if goedition is satisfied we 1f bar/usi‘ beneafh ethnic -> _ _ OK j. 5 Upon compleTion of sTep 5, 5a, 5b, or 5c, The ouipuT screen will appear (Screen l, inside back cover). All resalTs lrom The iusT-cornpleied analysis are included in The OquuT Navigaior. Make use ol The scroll bar arrows () To view The resulTs. Even when viewing oquuT, The sTandard menu oi commands is sTill lisTed across The Top oT The window. FurTher analyses may be conducTed wiThouT reiurning To The daTa screen. Confinae Select gases Continue PRINTING RESULTS ResulTs of The analysis (or analyses) ThaT have )usT been concluded requires a window Thai dis- plays The sTandaral commands (Elle gait Qata Iransform Analyze . . .) across The Top. A 112 Chapter 8 / Crosstabuiation and Chi-Square Analyses Typical prini procedure is shown below beginning Wlll‘l ihe siaodard ouipul screen (Screen i, inside back cover). To prfo resu/z‘s, from #79 Our/our screen perform fhe fo/lowr'ng sequence ofsfeps: 5 hi. Set-em. DoTiiis... 3 in. Screen . DOT-his" r; l @Ene ~—> Exit Noie: Afirer clicking Exit, ihere will lrequenily be small windows lhal appear asking ii you wish lo save or change any’ihing. Simply click each appropriaie response. OUTPUT Crossiabula’rion'and Chi~Square (X2) Analyses Whal follows is pariial ouipo? from sequence siep 5b, page 3 l0. SPSS for Windows: Crosstabulation and Chi-square analyses ETHNIC 1 Native 2 sian 3 Black "‘—5 Hisanic -' Female Coun Exp. Count Residual 2 Male - Count Exp. Count Residual Count Exp. Count Significance Pearson Chi-Square ' Likeiihood Ratio I Linearuby~Linear Association Phi I Cram [I] r's V ‘ N of valid cases 105 a. 3 cells (30%) have expected count less than 5. The minimum expected count is 195 Chapter 8 / Crosstabulation and Chi—Square Analyses 113 The first step in interpreting a crosstabulation or chi-square analysis is to observe the actual values and the expected values within each cell. Preliminary observation indicates that ob— served values and expected values are quite sirniiar. "the greatest discrepancy is for Whites (females, 26 actual count, 27.4 expected; and males )9 actual count, 37.6 expected). Note also that the residual value (the number beneath the other two) is simply the observed value minus the expected value. Even without iooking at the chi-square statistics you would antici— pate that observed values and expected values do not ditler signiticantly (that is, gender and ethnicity in this sampie are independent of each other). The results support this observation with a low Chi—square value (i .l9288) and a significance greater than 0.8 (87927). Notice that measures of association are aiso small and do not approach significance. As suggested in the Step by Step section, low-count cells is a problem. Three at )0 have an expected value iess than 5. The usual response would be to redo the analysis after deleting the “Native” category. To further assist understanding, definitions of output terms toiiow. " '_ isan;summarises“- The top number in each of the 10 cell (4, $3, 14, . , .), ilcatesthe number of subjects COUNT . in each category. 1 EXP COUNT The second number in each of the 10 celis (3.0, t2.2, 34.6, . . .), indicates the number ‘ that would appear there if the two variables were perfectly independent of each other. RESiDUAL The observed value minus the expected vaiue. Row TOTAL The total number of subjects for each row (64 females, 41 males). COLUMN The total number of subjects in each category for each column (5 American Indians, 20 TOTAL Asians, 24 Blacks, 45 Whites, t1 Hispanics). CHI SQUARE: Two different methods for computing chi—square statistics. With a large N, these two val~ PEARSON and ues will be ciose to equal. The formula for the Pearson chi—square is: LiKELIi—IOOQ RATED ‘ X2 = Eilfo ” felzl-fe] For PEARSON and MAXIMUM LIKELIHOOD methods, as the test—statistic value gets larger the VALUE 3 likelihood that the two variables are not independent (e.g., are dependent) also in— creases. The values close to 1 (1.193, 1.268) suggest that gender balance is not de— pendent an which ethnic groups are involved, DEGREES OF Degrees of freedom is the number of levels in the first variable minus 1 (2 - 1 = 1) times FREEDOM the number of levels in the second variable minus 1 (5 — 1 = 4); 1 x 4 = 4. SIGNIFICANCE The likelihood that these resutts couid happen by chance. The large p value here indi— cates that observed values do not differ significantly from expected values. This statistic tests whether the two variables correiate with each other. This measure is often meaningless because there is no logical or numeric relation to the order of the vari- LlNEAR-BY- . . . . . UNEAR ables. For instance, there is no iogicai order (from a tow value to a high value) of ethnic— ity. Therefore, the correlation between gender and ethnicity is meaningless. it, however, ASSOCIATiON . . . . . , the second variable were income, ordered from low to high, a valid correlation could rem suit. MiNiMUM The minimum expected count is for the first ceil of the second row (male American in- EXPECTED dian). The expected value there is rounded off to the nearest tenth (2.0). The value ac— a COUNT curate to two decimals is 1.95. A measure of the strength of association between two categorical variables. A value of _ PH] .10659 represents a very weak association between gender and ethnicity. The equation: a = .ng / N CELLS With Three at the 10 cells have an expected frequency less than 5. it you have many low— EXPECTED count cells (more than 25% is one accepted criteria), the overall chi—square value is less COUNT < 5 likely to be vaiid. ...
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This note was uploaded on 11/09/2011 for the course SYA 4300 taught by Professor Staff during the Fall '08 term at University of Florida.

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Chi-square - 136 Chapter 8 Crosstabuiation and Chi—Square Analyses THE PURPOSE of crossrabulaTion is $0 show in Tabular formaT The relaTionship

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