Imiqondo Eyisisekelo ye-One-Way ANOVA
I-One-Way ANOVA iyindlela yezibalo esetshenziselwa ukuqhathanisa izindlela ezivela emaqenjini angaphezu kwamabili. Abantu abaningi bajwayelene nokuhlolwa kwe-t kokuqhathanisa izindlela ezimbili, kodwa uma inani lamaqembu lingaphezu kwamabili, ukusetshenziswa okuphindaphindiwe kokuhlolwa kwe-t empeleni kwandisa ingozi yokwenza izinqumo ezingalungile. Yilapho i-one-way ANOVA iba khona ebalulekile: inikeza indlela enembile nehlelekile yokuhlola ukuthi ngabe kukhona yini umehluko omkhulu ezindleleni phakathi kwamaqembu uma kuqhathaniswa ngokusekelwe esicini esisodwa (i-variable eyodwa yesigaba).
1. Kuyini i-One-Way ANOVA?
Igama elithi ANOVA livela ku-Analysis of Variance. Naphezu kwegama lalo, "ukuhlaziywa kokwehluka," inhloso yalo eyinhloko ukuhlola umehluko ngezindlela. Ukuqonda okuyisisekelo kwe-ANOVA yilokhu: uma izindlela zeqembu zihlukile ngempela, khona-ke ukwehluka phakathi kwamaqembu kuzobonakala kukhulu kunokwehluka ngaphakathi kwamaqembu.
Kubizwa ngokuthi “indlela eyodwa” ngoba kunesici esisodwa noma i-categorical independent variable eyodwa esetshenziswa ukwakha amaqembu. Isibonelo:
- Izindlela zokufunda (ngokuzimela, ngeqembu, ku-inthanethi) ngamamaki ezivivinyo.
– Uhlobo lomanyolo (A, B, C, D) ekuvuneni kwesivuno.
– Uhlobo lomuthi (umuthi 1, umuthi 2, i-placebo) othinta umfutho wegazi.
Esibonelweni esingenhla, "indlela yokufunda", "uhlobo lomanyolo", kanye "nohlobo lomuthi" yizinto ezihlukile ezinamazinga amaningi (izigaba).
2. Isetshenziswa nini i-One-Way ANOVA?
I-ANOVA yendlela eyodwa ivame ukusetshenziswa lapho:
1. I-variable exhomeke kuyo isesimweni sezinombolo/ubuningi (izibonelo: inani, isisindo, isikhathi, umfutho wegazi).
2. I-variable ezimele iyisici esisodwa esihlukaniswe ngamaqembu okungenani amathathu (k ≥ 3).
3. Abacwaningi bafuna ukwazi ukuthi ngabe kukhona okungenani iqembu elilodwa elinomehluko ohlukile kwamanye.
Uma kunamaqembu amabili kuphela, ukuhlolwa kwe-t kuvame ukwanele. Kodwa-ke, i-ANOVA ingasetshenziswa emaqenjini amabili futhi izokhipha iziphetho ezilingana nokuhlolwa kwe-t (ngaphansi kwezimo ezithile).
3. Umqondo Oyinhloko: Ukwehluka Phakathi Kweqembu vs. Phakathi Kweqembu
I-ANOVA ilinganisa imithombo emibili yokwehluka:
– Ukwehluka ngaphakathi kweqembu: ukuthi idatha ihluka kangakanani ngaphakathi kweqembu ngalinye. Isibonelo, noma iqembu linesilinganiso esithile, abantu balo bangase bahluke kakhulu kunesilinganiso.
– Ukwehluka phakathi kweqembu: ukuthi isilinganiso seqembu ngalinye sihluke kangakanani komunye nomunye.
Uma umehluko phakathi kwesilinganiso seqembu mkhulu, umehluko phakathi kwamaqembu uzoba mkhulu. Uma idatha ngaphakathi kwamaqembu isakazeke kakhulu, umehluko ngaphakathi kwamaqembu uzoba mkhulu. I-ANOVA iqhathanisa lokhu okubili isebenzisa isilinganiso esibizwa ngokuthi i-F statistics.
4. I-hypothesis ku-ANOVA
Ngendlela eyodwa i-ANOVA, i-hypothesis yakhiwe kanje:
– H0 (i-null hypothesis): zonke izindlela zeqembu labantu ziyafana.
\[
\mu_1 = \mu_2 = \mu_3 = \amachashazi = \mu_k
\]
– H1 (inkolelo-mbono ehlukile): okungenani kukhona isilinganiso seqembu elilodwa elihlukile.
Okusho ukuthi, akuzona zonke \(\mu\) ezifanayo.
Kubalulekile ukuqonda ukuthi i-ANOVA ikutshela kuphela ukuthi kukhona umehluko noma cha iyonke. Uma imiphumela ibalulekile, kudingeka ukuhlolwa okwengeziwe ukuze kutholakale ukuthi yimaphi amaqembu ahlukene.
5. Izibalo Zokuhlola: Isilinganiso se-F
Isibalo esiyinhloko sokuhlolwa ku-ANOVA yi-F:
\[
F = \frac{\text{Between-group variance (MSB)}}{\text{Within-group variance (MSW)}}
\]
Lapha:
– I-MSB (Isilinganiso Sesikwele Phakathi) = isilinganiso sezikwele phakathi kwamaqembu, sichaza umehluko phakathi kwezindlela zeqembu.
– I-MSW (Isilinganiso Sesikwele Ngaphakathi) = isilinganiso sezikwele ngaphakathi kweqembu, sichaza ukwehluka ngaphakathi kweqembu.
Ingqondo:
– Uma izindlela zawo wonke amaqembu zifana, i-MSB incane ngakho u-F useduze no-1.
– Uma kukhona umehluko ocacile endleleni, i-MSB iyanda ukuze u-F abe mkhulu kuno-1.
– Inani elikhulu ngokwanele lika-F (uma liqhathaniswa nenani lika-F elibalulekile ngezinga elithile lenkululeko) lisenza senqabe i-H0.
6. Izingxenye Zokubala: i-SST, i-SSB, kanye ne-SSW
Ku-ANOVA, ukuhlukahluka okuphelele kwedatha kuhlukaniswe izingxenye ezimbili:
1. I-SST (Isamba Sezikwele Zonke): isamba sezikwele, ichaza ukuhlukahluka okuphelele kwayo yonke idatha uma kuqhathaniswa nesilinganiso esiphelele.
2. I-SSB (Isamba Sezikwele Phakathi): isamba sezikwele phakathi kwamaqembu, ukuhlukahluka ngenxa yokwehluka kwezindlela zeqembu.
3. I-SSW (Isamba Sezikwele Ngaphakathi): isamba sezikwele ngaphakathi kweqembu, ukuhlukahluka ngenxa yokwehluka komuntu ngamunye ngaphakathi kweqembu.
Ubudlelwano obuyisisekelo:
\[
I-SST = i-SSB + i-SSW
\]
Bese ngalinye lihlukaniswa ngamazinga enkululeko yokukhiqiza i-MSB kanye ne-MSW.
7. Izinga Lenkululeko
Amadigri enkululeko (df) ngendlela eyodwa i-ANOVA:
– df phakathi kwamaqembu: \(k – 1\)
(k = inani lamaqembu)
– df eqenjini: \(N – k\)
(N = isamba sayo yonke into ebonwe)
– ingqikithi ye-df: \(N – 1\)
Amazinga enkululeko abalulekile ngoba anquma ukuma kokusatshalaliswa kuka-F okusetshenziselwa ukuhlola ukubaluleka.
8. Izinkolelo ze-One-Way ANOVA
Ukuze imiphumela ye-ANOVA isebenze, ngokuvamile kudingeka imibono eminingana:
1. Ukuzimela: idatha phakathi kwabantu/okubonwayo izimele (ayithintani).
2. Ukufaneleka: idatha eqenjini ngalinye ivame ukusatshalaliswa (noma okungenani izinsalela ziseduze nokuvamile).
3. Ukulingana kokwehluka (i-homoscedasticity): ukwehluka phakathi kwamaqembu kuyafana.
Empeleni, i-ANOVA "iqinile" kakhulu ekwephulweni kokujwayelekile uma osayizi besampula bukhulu ngokwanele futhi bulinganisiwe. Kodwa-ke, ukwephulwa kokulingana kokuhlukahluka kungaba yinkinga enkulu, ikakhulukazi lapho osayizi besampula beqembu ngalinye bengalingani. Ukuhlolwa okufana nokwakwaLevene noma okwakwaBartlett kuvame ukusetshenziselwa ukuqinisekisa ukucabanga kokulingana kokuhlukahluka.
9. Ukuhunyushwa Kwemiphumela: inani le-p kanye nesinqumo
Imiphumela ye-ANOVA ivame ukwethulwa kuthebula le-ANOVA eliqukethe i-SSB, i-SSW, i-df, i-MSB, i-MSW, inani le-F, kanye nenani le-p.
– Uma inani le-p lingu-≤ α (isb. α = 0,05), yenqaba i-H0: kunomehluko omkhulu esilinganisweni phakathi kwamaqembu.
– Uma inani le-p > α , khona-ke yehluleka ukwenqaba i-H0 : abukho ubufakazi obanele bokuthi izindlela zihlukile.
Kodwa-ke, "ukwehluleka ukwenqaba i-H0" akusho ukuthi izindlela ziyafana ngempela; kusho ukuthi idatha ayinamandla ngokwanele ukufakazela umehluko.
10. Ukuhlolwa Kwangemva Kwe-Hoc Ngemva kwe-ANOVA
Uma i-ANOVA ibalulekile, isinyathelo esilandelayo ukuthola ukuthi yimaphi amaqembu ahlukile. Lokhu kwenziwa ngokuhlolwa kwe-post hoc, isibonelo:
– I-Tukey HSD (evame ukusetshenziswa ukuqhathanisa wonke amabhangqa).
– Bonferroni (ogcina imithetho).
– Scheffé (iyakwazi ukushintshashintsha uma kukhulunywa ngokuhlukahluka okuhlukahlukene).
– Imidlalo-Howell (ifaneleka kakhulu uma umehluko ungalingani).
Ngaphandle kokuhlolwa okwengeziwe, siyazi kuphela ukuthi “kukhona umehluko,” kodwa asazi ukuthi umehluko ukhona kuphi.
11. Usayizi Womphumela
Ngaphezu kokubaluleka, kubalulekile futhi ukubika ukuthi i-factor inathonya elingakanani ku-dependent variable. Osayizi bemiphumela evamile ku-ANOVA yile:
– I-Eta eyisikwele (\(\eta^2\)): isilinganiso sokwehluka okuphelele okuchazwe ngomehluko weqembu.
– I-Omega enesikwele (\(\omega^2\)): inguqulo engachemile kangako, ikakhulukazi kumasampula amancane.
Osayizi bemiphumela basiza ekuhloleni ukubaluleka okungokoqobo, hhayi nje ukubaluleka kwezibalo.
Isiphetho
I-ANOVA yendlela eyodwa iyithuluzi eliyisisekelo lezibalo lokuqhathanisa izindlela zamaqembu angaphezu kwamabili ngokusekelwe esicini esisodwa. Umqondo oyisisekelo ukuqhathanisa umehluko phakathi kwamaqembu nomehluko ngaphakathi kwamaqembu kusetshenziswa izibalo ze-F. Ukusetshenziswa kwayo kudinga ukucabanga kokuzimela, ukujwayelekile, kanye nokulingana kokuhlukahluka ukuqinisekisa iziphetho ezinokwethenjelwa. Uma imiphumela ye-ANOVA ikhombisa umehluko omkhulu, ukuhlaziywa kuyaqhubeka nokuhlolwa kwe-post hoc ukuhlonza amaqembu ahlukene nokubika osayizi bomphumela ukuhlola amandla ethonya cishe.
Uma ufuna, ngingangeza isibonelo esiphelele (idatha encane), izinyathelo ezilula zokubala ngesandla, noma isibonelo sokukhishwa kwe-ANOVA okuvela ku-SPSS/R/Excel kanye nendlela yokukufunda.
