{"id":83,"date":"2006-01-24T20:53:43","date_gmt":"2006-01-24T18:53:43","guid":{"rendered":"http:\/\/www.muratakyildiz.com\/wordpress\/?p=83"},"modified":"2006-01-24T20:53:43","modified_gmt":"2006-01-24T18:53:43","slug":"regresyon-analizi","status":"publish","type":"post","link":"https:\/\/www.istatistik.gen.tr\/?p=83","title":{"rendered":"Regresyon Analizi"},"content":{"rendered":"

\"Image\"Basit ve \u00e7oklu regresyon ile korelasyon konusunda Devrim Erdem, M\u00fccahit Ka\u011fan ve Fuat Tanhan’\u0131n haz\u0131rlam\u0131\u015f olduklar\u0131 \u00e7al\u0131\u015fmay\u0131 yaz\u0131n\u0131n devam\u0131nda okuyabilirsiniz. REGRESYON ANAL\u0130Z\u0130 <\/p>\n

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<\/span> Yazarlar (soyad\u0131 s\u0131ras\u0131na g\u00f6re): <\/span>Devrim Erdem, M\u00fccahit Kaan, Fuat Tanhan <\/strong><\/span><\/h1>\n

G\u0131R\u0131\u015f<\/span><\/h1>\n

<\/span>Bu \u00e7al\u0131\u015fma kapsam\u0131nda, regresyon analizi \u00fc\u00e7 b\u00f6l\u00fcmde incelenmi\u015ftir. Birinci b\u00f6l\u00fcmde regresyon analizinin temel mant\u0131\u011f\u0131na ili\u015fkin temel kavramlar ele al\u0131nm\u0131\u015ft\u0131r. S\u00f6zkonusu kavramlar tart\u0131\u015f\u0131larak ve birbiriyle ili\u015fkili bir bi\u00e7imde irdelenmeye \u00e7al\u0131\u015f\u0131lm\u0131\u015ft\u0131r. \u0131kinci b\u00f6l\u00fcmde ise basit do\u011frusal regresyon analizi a\u00e7\u0131klanm\u0131\u015ft\u0131r. \u00dc\u00e7\u00fcnc\u00fc b\u00f6l\u00fcmde ise \u00e7oklu do\u011frusal regresyon ele al\u0131nm\u0131\u015ft\u0131r. <\/span><\/h1>\n

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B\u00d6L\u00dcM I<\/span><\/strong><\/p>\n

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TEMEL KAVRAMLAR<\/span><\/strong><\/p>\n

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Neden Regresyon ve Korelasyon analizine ihtiya\u00e7 vard\u0131r?<\/span><\/em><\/strong><\/p>\n

Daha \u00f6nce de g\u00f6r\u00fcd\u00fc\u011f\u00fcm\u00fcz \u00fczere, t-testi<\/a>, varyans analizi <\/a>gibi ortalama farklar\u0131 ile ilgili hipotez testleri de\u011fi\u015fkenler aras\u0131ndaki ili\u015fkiye dair herhangi bir bilgi vermemektedir. Oysa serpilme diyagramlar\u0131na bak\u0131ld\u0131\u011f\u0131nda de\u011fi\u015fkenler anras\u0131nda bir ili\u015fki olabilece\u011fi hissedilebilmekte fakat bu t\u00fcr analizlerle bu ili\u015fkiler ortaya koyulamamaktad\u0131r. Dolay\u0131s\u0131yla de\u011fi\u015fkenler aras\u0131ndaki ili\u015fkinin \u015feklini, y\u00f6n\u00fcn\u00fc ve kuvvetini<\/em> belirleyebilmemiz i\u00e7in<\/u> yeni metotlara ihtiya\u00e7 vard\u0131r. Bu metotlar ise genel olarak regresyon (e\u011fri uydurma) ve korelasyon analizi olarak adland\u0131r\u0131l\u0131r. <\/span><\/p>\n

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De\u011fi\u015fkenlerin birlikte de\u011fi\u015fim \u00f6l\u00e7\u00fclerinin istatistiksel a\u00e7\u0131dan anlaml\u0131 olmas\u0131, bu de\u011fi\u015fkenler aras\u0131nda bir neden-sonu\u00e7 ili\u015fkisi bulunaca\u011f\u0131 anlam\u0131na gelir mi? <\/span><\/em><\/strong><\/p>\n

\u0131ki de\u011fi\u015fkene ait de\u011ferlerin birlikte azal\u0131p- \u00e7o\u011falmalar\u0131 istatistiksel a\u00e7\u0131dan anlaml\u0131 olsa bile bu de\u011fi\u015fkenler aras\u0131nda bir neden-sonu\u00e7 ili\u015fkisi bulunmayabilir. \u00d6rne\u011fin, bir \u00fclkede giyim e\u015fyalar\u0131n\u0131n ve g\u0131da maddelerinin imalat\u0131 bir arada ve ayn\u0131 s\u00fcre i\u00e7inde art\u0131\u015f g\u00f6sterebilir. Fakat bu, s\u00f6z konusu de\u011fi\u015fkenler aras\u0131nda sebep-sonu\u00e7 ili\u015fkisinin bulundu\u011fundan de\u011fil, ekonominin genel durumundan veya milli gelir ve n\u00fcfus art\u0131\u015f\u0131ndan ileri geliyor olabilir. Bir ba\u015fka deyi\u015fle, de\u011fi\u015fkenleri etkileyen ba\u015fka fakt\u00f6rler mevcuttur.<\/span><\/p>\n

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Peki, s\u00f6zgelimi iki de\u011fi\u015fken aras\u0131nda bir ili\u015fki oldu\u011fu tahmin edilebiliyor ise, bu ili\u015fki matematiksel bir denklem yard\u0131m\u0131yla ifade edilebilir mi?<\/span><\/em><\/strong><\/p>\n

E\u011fer iki de\u011fi\u015fken aras\u0131nda bir ili\u015fki oldu\u011fu tahmin ediliyor ise bu ili\u015fki en iyi \u015fekilde matematiksel bir denklem yard\u0131m\u0131yla ifade edilebilir. <\/span><\/p>\n

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S\u00f6zkonusu de\u011fi\u015fkenler aras\u0131 ili\u015fkiyi en iyi belirten bir matematiksel denklem\/fonksiyon elde etmek ara\u015ft\u0131rmac\u0131ya ne t\u00fcr faydalar sa\u011flar?<\/span><\/em><\/strong><\/p>\n

De\u011fi\u015fkenler aras\u0131ndaki ili\u015fkiyi ifade edebilen bir denklemin ortaya \u00e7\u0131kar\u0131ld\u0131\u011f\u0131nda;<\/span><\/p>\n

\u00b7 <\/span><\/span><\/span>\u0131li\u015fkinin y\u00f6n\u00fc belirlenebilir<\/span><\/p>\n

\u00b7 <\/span><\/span><\/span>\u0131li\u015fkinin \u015fekli belirlenebilir<\/span><\/p>\n

\u00b7 <\/span><\/span><\/span>Bilinmeyen de\u011ferlere dair yordamalar\/tahminler yap\u0131labilir<\/span><\/p>\n

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Peki, de\u011fi\u015fkenler aras\u0131ndaki ili\u015fkiler, ne t\u00fcr ili\u015fkilerdir?<\/span><\/em><\/strong><\/p>\n

\u0131li\u015fkiyi en iyi bi\u00e7imde belirten matematiksel fonksiyon iki parametreli do\u011frusal bir denklem olabilece\u011fi gibi, iki veya daha \u00e7ok say\u0131da parametre i\u00e7eren e\u011fri fonksiyonlar da olabilir. Hangi t\u00fcr fonksiyonun daha uygun olabilece\u011fi <\/span>meydana getirilecek <\/span>bir serpilme diyagram\u0131<\/em><\/strong>ndaki noktalar\u0131n durumundan anla\u015f\u0131labilir. \u00d6rne\u011fin, noktalar bir do\u011fru etraf\u0131nda toplanm\u0131\u015f ise do\u011frusal, b\u00fck\u00fclme noktalar\u0131 se\u00e7ilebiliyorsa e\u011frisel bir fonksiyonun kullan\u0131lmas\u0131 daha uygun olacakt\u0131r. Ayr\u0131ca b\u00fck\u00fclme noktalar\u0131n\u0131n say\u0131s\u0131na g\u00f6re fonksiyonlar\u0131n ka\u00e7\u0131nc\u0131 dereceden olduklar\u0131 da belirlenebilir. (Bir b\u00fck\u00fclme noktas\u0131 ikinci dereceden, iki b\u00fck\u00fclme noktas\u0131 ise \u00fc\u00e7\u00fcnc\u00fc dereceden bir fonksiyonun kullan\u0131lmas\u0131n\u0131 gerektirir.) <\/span><\/span><\/p>\n

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D\u00fcz do\u011fru<\/span><\/strong><\/p>\n<\/td>\n

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Y= ao<\/sub>+ a1<\/sub>X<\/span><\/p>\n<\/td>\n

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<\/span><\/p>\n<\/td>\n<\/tr>\n

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Parabol<\/span><\/strong><\/p>\n<\/td>\n

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Y= ao<\/sub>+ a1<\/sub>X + a2<\/sub>X2<\/sup><\/span><\/p>\n<\/td>\n

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<\/span><\/p>\n<\/td>\n<\/tr>\n

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K\u00fcbik e\u011fri<\/span><\/strong><\/p>\n<\/td>\n

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Y= ao<\/sub>+ a1<\/sub>X + a2<\/sub>X2 <\/span><\/sup> <\/span>+ a3<\/sub>X3<\/sup><\/span><\/p>\n<\/td>\n

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<\/span><\/p>\n<\/td>\n<\/tr>\n

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\u00dcstel e\u011fri<\/span><\/strong><\/p>\n<\/td>\n

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Y= abX<\/sup> <\/span>veya logY= loga+ (logb)X = ao<\/sub>+ a1<\/sub>X<\/span><\/p>\n<\/td>\n

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De\u011fi\u015fkenler aras\u0131ndaki ili\u015fkiler, pekala e\u011friselsel de olabilmektedir. Peki, neden do\u011frusal ili\u015fkiler a\u011f\u0131ll\u0131kl\u0131 olarak incelenmektidir?<\/span><\/em><\/strong><\/p>\n

Birle\u015fik seriler aras\u0131ndaki ili\u015fkilerin belirlenmesinde uygulama sahas\u0131 en geni\u015f olanlar do\u011frusal denklemler ile <\/span>ikinci dereceden fonksiyonlar yani parabollerdir. Bunlar aras\u0131nda en \u00f6nemli yeri do\u011frusal denklemler tutmaktad\u0131r. S\u0131kl\u0131kla do\u011frusal denklemlerin kullan\u0131lmas\u0131n\u0131n sebepleri ise \u015funlard\u0131r:<\/span><\/p>\n

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\u00b7 <\/span><\/span><\/span>Bir \u00e7ok ili\u015fkinin \u015fekli do\u011frusald\u0131r.<\/span><\/p>\n

\u00b7 <\/span><\/span><\/span>Do\u011frusal denklemler, aksi halde matematik ifadesi \u00e7ok zor olan bir\u00e7ok ili\u015fkinin uygun birer tahmini olduklar\u0131ndan pratik bak\u0131mdan b\u00fcy\u00fck \u00f6nem ta\u015f\u0131makta <\/span>ve bundan dolay\u0131 ortaya \u00e7\u0131kacak hata da \u00f6nemsiz say\u0131labilecek kadar k\u00fc\u00e7\u00fck olmaktad\u0131r.<\/span><\/p>\n

\u00b7 <\/span><\/span><\/span>Do\u011frusal olmayan bir\u00e7ok ili\u015fki logaritmik ifadelerle do\u011frusal hale d\u00f6n\u00fc\u015ft\u00fcr\u00fclebilmektedir.<\/span><\/p>\n

<\/span> <\/span><\/p>\n

\u0131ki de\u011fi\u015fken aras\u0131ndaki ili\u015fkinin matematiksel bir denklemle ifade edilebilmesi hangi varsay\u0131m alt\u0131nda olabilmektedir?<\/span><\/em><\/strong><\/p>\n

\u0131ki seri aras\u0131ndaki ili\u015fkinin matematiksel bir fonksiyonla ifadesi de\u011fi\u015fkenlerden birinin \u201cba\u011f\u0131ml\u0131\u201d di\u011ferinin \u201cba\u011f\u0131ms\u0131z\u201d de\u011fi\u015fken olarak kabul edilmesi ile m\u00fcmk\u00fcnd\u00fcr. Bu \u015fekilde olu\u015fturulacak fonksiyon bir regresyon do\u011frusunun veya e\u011frisinin denklemi olacakt\u0131r. Daha \u00f6nce de belirtildi\u011fi gibi, ili\u015fkiyi ifade edebilen bir denklemin ortaya \u00e7\u0131kar\u0131lmas\u0131n\u0131n <\/span>ili\u015fkinin y\u00f6n\u00fcn\u00fc ve \u015feklini belirleyebilmenin yan\u0131 s\u0131ra <\/span>di\u011fer bir yararl\u0131 y\u00f6n\u00fc vard\u0131r ki bu, <\/span>Y= a+ bX <\/span>\u015feklinde <\/span>denklem ile , bir X de\u011feri verildi\u011fi takdirde <\/span>buna kar\u015f\u0131l\u0131k gelen Y de\u011ferinin de bulunabilmesi ve dolay\u0131s\u0131yla \u201ctahmin yapabilmenin<\/em>\u201d <\/span>m\u00fcmk\u00fcn olmas\u0131d\u0131r. Ancak, \u00f6zellikle ileriye y\u00f6nelik tahmin yaparken fonksiyonun \u015feklinin <\/span>de\u011fi\u015febilece\u011fi ihtimalini g\u00f6z \u00f6n\u00fcnde bulundurmak gerekir. <\/span><\/p>\n

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Regresyon analizi ile de\u011fi\u015fkenler aras\u0131ndaki ili\u015fkinin kuvveti ve derecesi hakk\u0131nda kesin bir bilgi sa\u011flanabilir mi?<\/span><\/em><\/strong><\/p>\n

Hay\u0131r. Regresyon analizi, iki de\u011fi\u015fkenden birini ba\u011f\u0131ml\u0131 di\u011ferini ba\u011f\u0131ms\u0131z de\u011fi\u015fken olarak kabul etti\u011fimiz durumlarda bize ili\u015fkiyi<\/u><\/em> ifade eden bir denklem vermektedir. Ancak regresyon analizi ili\u015fkinin kuvveti ve derecesi<\/u><\/em> hakk\u0131nda kesin bir bilgi sa\u011flayamamaktad\u0131r. Bu bilgiyi elde edebilmek i\u00e7in \u201ckorelasyon\u201d<\/em><\/strong> analizine ihtiya\u00e7 vard\u0131r. Gerek regresyon gerekse korelasyon analizinin \u00f6nemli bir varsay\u0131m\u0131, eldeki veriler yard\u0131m\u0131yla inceleme konusu de\u011fi\u015fkenler aras\u0131ndaki <\/span>\u201cger\u00e7ek\u201d ili\u015fkinin bir tahmininin<\/u> yap\u0131labilece\u011fidir. <\/span><\/p>\n

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De\u011fi\u015fkenler aras\u0131ndaki ili\u015fki sadece tek bir do\u011fruyla ifade edilebilir mi?<\/span><\/em><\/strong><\/p>\n

Hay\u0131r. Serpilme diyagram\u0131 yard\u0131m\u0131yla tespit edilen do\u011frusal bir ili\u015fki \u00e7ok say\u0131da<\/em> do\u011fru ile g\u00f6sterilebilir (\u015fekil 1.a). Ancak ili\u015fkiyi en iyi belirleyecek denklem en k\u00fc\u00e7\u00fck kareler y\u00f6ntemi<\/em><\/strong> ile belirlenecek olan denklemdir. Bu y\u00f6nteme g\u00f6re veri k\u00fcmesine en iyi uyan e\u011fri, bu e\u011friden serpilme diyagram\u0131ndaki noktalara olan dikey uzakl\u0131klar\u0131n karelerinin toplam\u0131n\u0131n en k\u00fc\u00e7\u00fck (minimum) oldu\u011fu e\u011fridir (\u015fekil 1.b).<\/span><\/p>\n

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<\/div>\n<\/td>\n
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\u015fekil 1. a<\/span><\/div>\n<\/td>\n
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\u015fekil 1. b <\/span> <\/span> <\/span> <\/span><\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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Peki neden kriter olarak uzakl\u0131klar\u0131n (hatalar\u0131n) toplam\u0131 de\u011fil de, bu uzakl\u0131klar\u0131n karelerinin toplam\u0131 kullan\u0131lmaktad\u0131r? <\/span><\/em><\/strong><\/p>\n

\u00c7\u00fcnk\u00fc, fark al\u0131nd\u0131\u011f\u0131nda baz\u0131 de\u011ferler pozitif baz\u0131 de\u011ferler de negatif olaca\u011f\u0131ndan <\/span>toplam s\u0131f\u0131ra yak\u0131n \u00e7\u0131kacak; bu ise ger\u00e7ek farklar\u0131 ortaya \u00e7\u0131karmayacakt\u0131r. Fakat kareler yard\u0131m\u0131yla <\/span>ger\u00e7ek farkl\u0131l\u0131klar \u00f6n plana \u00e7\u0131kmaktad\u0131r.<\/span><\/p>\n

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EN K\u00dc\u00c7\u00dcK KARELER Y\u00d6NTEM\u0131:<\/span><\/em><\/strong><\/p>\n<\/td>\n

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Veri noktalar\u0131n\u0131n (X1<\/sub>,Y1<\/sub>), (X2<\/sub>,Y2<\/sub>) , ………., (Xn<\/sub>,Yn<\/sub>) <\/span>olarak verildi\u011fini kabul edelim. Bu ikili de\u011ferleri bir do\u011fru denklemi ile ifade edebiliriz. Bu durumda her X de\u011ferine kar\u015f\u0131l\u0131k gelen bir g\u00f6zlenen bir de teorik olmak \u00fczere iki ayr\u0131 Y de\u011feri vard\u0131r. \u00d6rne\u011fin, X1 <\/sub>i\u00e7in g\u00f6zlemlerle bulunmu\u015f olan Y de\u011ferini Y1<\/sub> ile <\/span>ve denklem vas\u0131tas\u0131yla elde edilmi\u015f olan teorik Y de\u011ferini <\/span><\/em><\/p>\n

\u0176 = (a+bXi<\/sub>) ile g\u00f6sterelim.G\u00f6zlenen Yi<\/sub> ve teorik \u0176 = (a+bXi<\/sub>) aras\u0131ndaki farklar <\/span>sapma\/hata\/ ya da art\u0131k olarak tan\u0131mlanabilir.<\/span><\/em><\/p>\n

Yani;<\/span><\/em><\/p>\n

<\/span>ei<\/sub> = Yi <\/sub>– (a+bXi<\/sub>) <\/span>; i= 1,2,………..n<\/span><\/em><\/p>\n

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En k\u00fc\u00e7\u00fck kareler y\u00f6ntemine g\u00f6re bu farklar\u0131n karelerinin toplam\u0131n\u0131n \u201cminimum<\/strong>\u201d olmas\u0131 gerekmektedir. <\/span><\/em><\/p>\n

Yani;<\/span><\/em><\/p>\n

<\/span>\u03a3 [Yi <\/sub>– (a+bXi<\/sub>)]2 <\/sup>= minimum<\/span><\/em><\/p>\n

<\/span>\u03a3 ei<\/sub>2 <\/sup>= minimum<\/span><\/em><\/p>\n

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\u03a3 [Yi <\/sub>– (a+bXi<\/sub>)]2 <\/sup>denklemin minimumunu bulabilmek i\u00e7in \u00f6nce a\u2019ya daha sonra da b\u2019ye g\u00f6re k\u0131smi t\u00fcrevler al\u0131narak 0 a e\u015fitlemek gerekir. Bu \u015fekilde ortaya \u00e7\u0131kacak denklemler de normal denklemler olarak adland\u0131r\u0131lmaktad\u0131r.<\/span><\/em><\/p>\n

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<\/span>\u03a3 Yi<\/sub> = Na + b \u03a3Xi<\/sub><\/span><\/em><\/p>\n

<\/span>\u03a3 Yi<\/sub>Xi<\/sub> = a \u03a3Xi<\/sub> + b \u03a3Xi<\/sub>2<\/sup><\/span><\/em><\/p>\n

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Ara\u015ft\u0131rma\/g\u00f6zlem ile elde edilen verilere dayanarak N, \u03a3 Yi, <\/sub>\u03a3Xi, <\/sub>\u03a3 Yi<\/sub>Xi <\/sub>ve \u03a3Xi<\/sub>2 <\/sup>de\u011ferlerini hesaplamak m\u00fcmk\u00fcnd\u00fcr. B\u00f6ylece a ve b katsay\u0131lar\u0131n\u0131n de\u011ferleri bulunabilir. <\/span><\/em><\/p>\n

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Bu \u015fekilde elde edece\u011fimiz do\u011fru denklemi X\u2019in ba\u011f\u0131ms\u0131z Y\u2019nin ise ba\u011f\u0131ml\u0131 de\u011fi\u015fken olarak kabul edilmesi halinde <\/span>ve en k\u00fc\u00e7\u00fck kareler y\u00f6ntemi kullan\u0131larak <\/span>ortaya \u00e7\u0131kacak do\u011frudur ve Y\u2019nin X\u2019e g\u00f6re regresyon do\u011frusu olarak tan\u0131mlanmaktad\u0131r.<\/span><\/em><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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B\u00d6L\u00dcM II<\/span><\/strong><\/p>\n

BAS\u0131T DO\u011fRUSAL REGRESYON<\/span><\/strong><\/p>\n

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Daha \u00f6nce de de\u011finildi\u011fi gibi, iki de\u011fi\u015fken aras\u0131ndaki ili\u015fkiler \u00e7e\u015fitli \u015fekillerde ortaya \u00e7\u0131kabilmektedir. Bu ili\u015fkiler aras\u0131nda en yayg\u0131n olarak kullan\u0131lan\u0131 ise de\u011fi\u015fkenler aras\u0131nda do\u011frusal ili\u015fkinin oldu\u011fu durumdur. Bir ba\u011f\u0131ml\u0131 ve bir ba\u011f\u0131ms\u0131z de\u011fi\u015fkenin oldu\u011fu do\u011frusal regresyon \u00e7\u00f6z\u00fcmlemesi basit do\u011frusal regresyon<\/em><\/strong> olarak adland\u0131r\u0131l\u0131r. <\/span><\/p>\n

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Basit Do\u011frusal Regresyon Modeli:<\/span><\/strong><\/p>\n

<\/span> <\/span>i=1, 2, …., n<\/em> <\/span>(1.1)<\/span><\/p>\n

<\/span><\/p>\n

<\/span>: Do\u011frunun y-eksenini kesti\u011fi nokta<\/span><\/p>\n

b<\/span><\/span>: <\/span>Do\u011frunun e\u011fimi veya regresyon katsay\u0131s\u0131<\/span><\/p>\n

e<\/span><\/span>: <\/span>\u015fansa ba\u011fl\u0131 hata de\u011feri<\/span><\/p>\n

<\/span><\/p>\n

(1.1) ile verilen denklemde <\/span>, regresyon do\u011frusunun y eksenini kesti\u011fi noktay\u0131 g\u00f6sterir ve sabit<\/em><\/strong> (constant) olarak adland\u0131r\u0131l\u0131r. <\/span>, x ba\u011f\u0131ms\u0131z de\u011fi\u015fkenin de\u011feri 0 iken , y de\u011fi\u015fkeninin alaca\u011f\u0131 ortalama de\u011feri verir. <\/span>b<\/span><\/span>, regresyon katsay\u0131s\u0131<\/em><\/strong> olarak adland\u0131r\u0131l\u0131r; ve ba\u011f\u0131ms\u0131z de\u011fi\u015fkende (x) bir birimlik de\u011fi\u015fme (artma veya azalma) oldu\u011funda, ba\u011f\u0131ml\u0131 de\u011fi\u015fkende (y) meydana gelecek ortalama de\u011fi\u015fkenlik miktar\u0131n\u0131 verir. <\/span>e<\/span><\/span>, hata terimidir; y\u2019deki de\u011fi\u015fimin, regresyon modeli (<\/span>+<\/span>b<\/span><\/span>x) ile a\u00e7\u0131klanamayan k\u0131sm\u0131n\u0131 g\u00f6sterir. Asl\u0131nda modeldeki <\/span><\/span> <\/span><\/sub>ve <\/span>b<\/span><\/span> <\/span><\/sub>de\u011ferleri, regresyon modelinin t\u00fcm evrendeki verileri kullan\u0131larak hesaplanan teorik de\u011ferlerdir. Ancak yine de dikkate al\u0131nmayan ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler olabilece\u011finden, verilerin rassal (\u015fansa ba\u011fl\u0131) de\u011fi\u015fimlerini g\u00f6steren hata de\u011feri <\/span><\/span>e<\/span><\/span> modele eklenmi\u015ftir.<\/span><\/p>\n

<\/span><\/p>\n

Bir regresyon denklemi, ba\u011f\u0131ml\u0131 de\u011fi\u015fken ile ba\u011f\u0131ms\u0131z de\u011fi\u015fken aras\u0131ndaki ili\u015fkiyi matematiksel bir model olarak belirtir. Ger\u00e7ek hayat uygulanamalar\u0131nda <\/span><\/span><\/span> <\/span><\/span><\/sub>ve <\/span>b<\/span><\/span> <\/span><\/sub>de\u011ferleri bilinmiyorsa, evrenden \u00f6rneklem al\u0131narak bunlar\u0131n <\/span>kestiricisi <\/span>olan a<\/em> ve b<\/em> kullan\u0131l\u0131r.<\/span><\/p>\n

<\/span><\/p>\n

<\/span>, <\/span>i=1, 2, …., n<\/em> <\/span>(1.2)<\/span><\/p>\n

<\/span>: <\/span>y<\/em>\u2019nin tahmini de\u011feri<\/span><\/p>\n

<\/span><\/p>\n

a ve b Katsay\u0131lar\u0131n\u0131n Bulunmas\u0131:<\/span><\/strong><\/p>\n

Regresyon denkleminin bilinmeyenleri olan <\/span> <\/span>ve <\/span>b<\/span><\/span> de\u011ferlerinin \u00f6rneklemden elde edilen kestiricileri olan a ve b katsay\u0131lar\u0131 en k\u00fc\u00e7\u00fck kareler y\u00f6ntemi<\/em><\/strong> (Least Squares Method) kullan\u0131larak bulunur. Burada esas, serpilme diagram\u0131nda (scatter diagram) g\u00f6r\u00fclen t\u00fcm noktalar i\u00e7in <\/span>do\u011fruya uzakl\u0131klar\u0131n\u0131n bulunmas\u0131 ve bunlar\u0131n toplam\u0131n\u0131n minimize edilmesidir. Ancak regresyon analizinde bu toplam fonksiyonu daima s\u0131f\u0131r olaca\u011f\u0131ndan a<\/em> ve b<\/em> de\u011ferlerini bulmada kullan\u0131lamaz.<\/span><\/p>\n

<\/span><\/span><\/p>\n

<\/span><\/p>\n

Bu durumda hatalar\u0131n (regresyon denkleminden sapmalar\u0131n) karelerinin toplam\u0131 bulunarak yeni bir fonksiyon olu\u015fturulur. <\/span><\/p>\n

<\/span><\/span><\/p>\n

<\/span><\/p>\n

Bu yeni fonksiyonu minimize eden optimal a<\/em> ve b<\/em>, <\/span> <\/span><\/sub>ve <\/span>b<\/span><\/span>\u2019\u0131n tahmini de\u011ferleri olacakt\u0131r. Yukar\u0131daki fonksiyon i\u00e7b\u00fckey (convex) oldu\u011fundan, fonksiyonu minimize eden optimal de\u011ferleri bulmak i\u00e7in fonsiyonun a<\/em> ve b<\/em> ye g\u00f6re k\u0131smi t\u00fcrevlerini s\u0131f\u0131r yapan de\u011ferleri <\/span>almak yeterli olacakt\u0131r.<\/span><\/p>\n

<\/span><\/p>\n

<\/span> <\/span> <\/span>(1.3)<\/span><\/p>\n

<\/span><\/p>\n

<\/span> <\/span> <\/span>(1.4)<\/span><\/p>\n

<\/span><\/p>\n

<\/span>xi<\/sub><\/span><\/em>: ba\u011f\u0131ms\u0131z de\u011fi\u015fkenin i<\/em>. g\u00f6zlemi, <\/span>i<\/em>=1,2,…,n.<\/span><\/p>\n

yi<\/sub><\/span><\/em>: <\/span>ba\u011f\u0131ml\u0131 de\u011fi\u015fkenin i<\/em>. g\u00f6zlemi, <\/span>i<\/em>=1,2,…,n.<\/span><\/p>\n

<\/span><\/span>: ba\u011f\u0131ms\u0131z de\u011fi\u015fkenin \u00f6rneklem ortalamas\u0131 (<\/span>)<\/span><\/p>\n

<\/span><\/span>: ba\u011f\u0131ml\u0131 de\u011fi\u015fkenin \u00f6rneklem ortalamas\u0131 (<\/span>)<\/span><\/p>\n

n: toplam g\u00f6zlem say\u0131s\u0131.<\/span><\/p>\n

<\/span><\/p>\n

(1.3) ve (1.4) e\u015fitlikleri yard\u0131m\u0131yla \u00f6rnekleme ili\u015fkin regresyon kestirim denklemi g\u00f6zlemler cinsinden ; <\/span><\/p>\n

<\/span><\/span><\/span><\/p>\n

olarak yaz\u0131l\u0131r. Bu denklemde her bir xi<\/sub> de\u011ferinin yerine konmas\u0131 ile elde edilen <\/span><\/span> <\/span>de\u011ferleri regresyon do\u011frusu \u00fczerinde yer al\u0131r. Di\u011fer bir deyi\u015fle, <\/span> <\/span>de\u011ferleri regresyon do\u011frusunu tan\u0131mlayan de\u011ferlerdir. G\u00f6zlenen y de\u011feri ile kestirilen y de\u011feri (<\/span>) aras\u0131ndaki farka at\u0131k<\/em><\/strong> (residual) denir ve <\/span> <\/span>e\u015fitli\u011fi ile ifade edilir. At\u0131k de\u011feri ne kadar k\u00fc\u00e7\u00fck ise modelin veriye o kadar uydu\u011fu s\u00f6ylenebilir. <\/span><\/p>\n

<\/span><\/p>\n

Varsay\u0131mlar:<\/span><\/strong><\/p>\n

    \n
  1. Regresyon \u00e7\u00f6z\u00fcmlemesinde ba\u011f\u0131ms\u0131z de\u011fi\u015fken de\u011ferleri hatas\u0131z \u00f6l\u00e7\u00fcl\u00fcr. Ancak, hi\u00e7 bir \u00f6l\u00e7\u00fcm m\u00fckemmel olmad\u0131\u011f\u0131ndan bu ifadenin anlam\u0131; ba\u011f\u0131ms\u0131z de\u011fi\u015fkendeki \u00f6l\u00e7\u00fcm hatalar\u0131n\u0131n ihmal edilebilece\u011fi \u015feklinde yorumlan\u0131r.<\/span><\/li>\n
  2. Her bir x de\u011feri i\u00e7in birden \u00e7ok y de\u011feri s\u00f6z konusudur. Her bir x de\u011ferine kar\u015f\u0131l\u0131k gelen y de\u011ferleri k\u00fcmesi normal da\u011f\u0131l\u0131m<\/em> g\u00f6sterir. Normality<\/em><\/strong><\/span><\/li>\n
  3. Her bir x de\u011fi\u015fkenine kar\u015f\u0131l\u0131k gelen y de\u011ferleri k\u00fcmelerine ili\u015fkin varyanslar homojen<\/em>dir. Homoscedasticity<\/em><\/strong><\/span><\/li>\n
  4. y de\u011ferleri istatistiksel olarak ba\u011f\u0131ms\u0131zd\u0131r. Yani, ba\u011f\u0131ms\u0131z de\u011fi\u015fkenin herhangi bir de\u011feri i\u00e7in elde edilen y de\u011feri, x in bir di\u011fer de\u011feri i\u00e7in elde edilen y de\u011ferinden etkilenmez. <\/span><\/li>\n
  5. y alt k\u00fcmelerinin olu\u015fturdu\u011fu da\u011f\u0131l\u0131mlara ili\u015fkin ortalamalar bir do\u011fru \u00fczerindedir. Linearity<\/em><\/strong><\/span><\/li>\n<\/ol>\n

    <\/span><\/em><\/strong><\/p>\n

    <\/span><\/p>\n

    <\/span><\/span><\/p>\n

    Grafik 1. Basit Do\u011frusal Regresyon Modelinin G\u00f6sterimi<\/span><\/em><\/strong><\/p>\n

    <\/span><\/em><\/strong><\/p>\n

    Grafik <\/span>1 de g\u00f6r\u00fcld\u00fc\u011f\u00fc gibi, her bir x de\u011ferine kar\u015f\u0131l\u0131k gelen birden \u00e7ok y de\u011feri vard\u0131r ve bu y de\u011ferleri e\u015fit varyans ile normal da\u011f\u0131l\u0131m g\u00f6sterirler. Ayr\u0131ca y alt k\u00fcmelerinin ortalamalar\u0131 bir do\u011fru \u00fczerindedir.<\/span><\/p>\n

    <\/span><\/p>\n

    <\/span><\/p>\n

    Hipotez Testi:<\/span><\/strong><\/p>\n

    <\/span> <\/span>(<\/span> <\/span>denklemine uyum anlaml\u0131 de\u011fildir.)<\/span><\/p>\n

    <\/span> <\/span><\/span>(<\/span> <\/span>denklemine uyum anlaml\u0131d\u0131r.)<\/span><\/p>\n

    <\/span><\/p>\n

    Bir regresyonun anlaml\u0131 olmas\u0131 asl\u0131nda do\u011frunun e\u011fimi olan <\/span>b<\/span><\/span>\u2019in s\u0131f\u0131rdan farkl\u0131 olmas\u0131 ile e\u015fde\u011ferdir.<\/span><\/p>\n

    <\/span><\/span><\/p>\n

    Test \u0131statisti\u011fi:<\/span><\/strong><\/p>\n

    Yukar\u0131daki hipotezi test etmek i\u00e7in t-testi kullan\u0131l\u0131r:<\/span><\/p>\n

    <\/p>\n\n\n\n\n
    \u00a0<\/td>\n<\/tr>\n
    \u00a0<\/td>\n <\/p>\n\n\n\n
    <\/p>\n
    \n

    0<\/span><\/p>\n<\/p><\/div>\n

    <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    <\/span> <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    <\/span> <\/span><\/p>\n

    <\/p>\n

    <\/span><\/span>Test istatisti\u011fi: <\/span><\/span><\/span><\/p>\n

    <\/span> <\/span>Kritik de\u011fer : <\/span><\/span><\/p>\n

    <\/span><\/strong><\/p>\n

    Karar Verme:<\/span><\/strong> <\/span><\/span><\/p>\n

    E\u011fer <\/span> <\/span><\/span> <\/span>veya <\/span> <\/span><\/span> <\/span><\/span> <\/span><\/span> <\/span>reddedilir.<\/span><\/p>\n

    Ya da,<\/span><\/em><\/p>\n

    E\u011fer <\/span> <\/span> <\/span> <\/span>p \u2013 de\u011feri <\/span><\/span> <\/span><\/span> <\/span><\/span> <\/span>reddedilir.<\/span><\/p>\n

    <\/span><\/p>\n

    Yorum:<\/span><\/strong><\/p>\n

    <\/span> <\/span>reddedilirse, <\/span>a<\/span><\/span> yan\u0131lma olas\u0131l\u0131\u011f\u0131 ile <\/span>\u2019d\u0131r; yani y<\/em> de\u011ferleri x<\/em>\u2019e ba\u011fl\u0131 olarak de\u011fi\u015fim g\u00f6sterirler ve bu durumda regresyon anlaml\u0131d\u0131r.<\/span><\/p>\n

    <\/span><\/p>\n

    Regresyon katsay\u0131s\u0131n\u0131n (<\/span><\/em><\/strong>b<\/span><\/span><\/em><\/strong>) anlaml\u0131l\u0131\u011f\u0131 varyans analizi (ANOVA) ile de test edilebilir. <\/span><\/span><\/em><\/strong><\/p>\n

    Bu durumda F istatisti\u011fi;<\/span><\/p>\n

    <\/span><\/span> <\/span>(1.5)<\/span><\/p>\n

    <\/span><\/p>\n

    <\/span>RKO: Regresyon kareler ortalamas\u0131<\/span><\/p>\n

    <\/span>AKO: At\u0131k kareler ortalamas\u0131 <\/span><\/p>\n

    <\/span><\/p>\n

    \u0131statistiksel karar i\u00e7in, bulunan F istatisti\u011fi, se\u00e7ilen <\/span><\/span>a<\/span><\/span> <\/span>yan\u0131lma d\u00fczeyinde regresyon ve regresyondan ayr\u0131l\u0131\u015f serbestlik dereceli F tablo istatisti\u011fi ile kar\u015f\u0131la\u015ft\u0131r\u0131l\u0131r. <\/span><\/p>\n

    <\/span><\/p>\n

    Unutulmamal\u0131d\u0131r ki, sadece bir ba\u011f\u0131ms\u0131z de\u011fi\u015fken oldu\u011funda, <\/span>F-testi<\/em> ve <\/span>t \u2013 testi <\/em>kullan\u0131larak yap\u0131lan hipotez testleri ayn\u0131 p \u2013 de\u011ferini vereceklerdir.<\/span><\/p>\n

    <\/span><\/p>\n

    Regresyon Modelinin Performans\u0131 (<\/span><\/strong> r\u00b2<\/strong> ) :<\/strong><\/span><\/p>\n

    <\/span><\/strong><\/p>\n

    Regresyon analizinde ara\u015ft\u0131rmac\u0131n\u0131n temel ilgi oda\u011f\u0131, yap\u0131lan tahminin ne kadar g\u00fc\u00e7l\u00fc oldu\u011fu noktas\u0131ndad\u0131r. Yani, ula\u015f\u0131lan regresyon modeli, Y\u2019deki de\u011fi\u015fmelerin ne kadar\u0131n\u0131 a\u00e7\u0131klamaktad\u0131r? Regresyon analizinde, regresyonun sundu\u011fu tahminin ne kadar g\u00fc\u00e7l\u00fc oldu\u011funu belirten istatistik determinasyon katsay\u0131s\u0131<\/em> <\/strong>olarak adland\u0131r\u0131l\u0131r. Regresyon analizinde determinasyon katsay\u0131s\u0131 (r2<\/sup>), X\u2019in Y\u2019de a\u00e7\u0131klad\u0131\u011f\u0131 de\u011fi\u015fme oran\u0131n\u0131 g\u00f6sterir. <\/span><\/p>\n

    Pearson Momentler \u00c7arp\u0131m\u0131 Korelasyon Katsay\u0131s\u0131 hesapland\u0131\u011f\u0131nda;<\/span><\/p>\n

    r =<\/span><\/span><\/p>\n

    <\/span><\/span><\/p>\n

    \u0131ki ve daha fazla tahmin edici de\u011fi\u015fkenin oldu\u011fu \u00e7oklu regresyon analizinde korelasyon katsay\u0131s\u0131 R ve determinasyon katsay\u0131s\u0131 <\/span> <\/span>ile g\u00f6sterilir. <\/span><\/p>\n

    B\u00d6L\u00dcM III<\/span><\/h3>\n

    \u00c7OKLU DO\u011fRUSAL REGRESYON<\/span><\/h3>\n

    <\/span><\/p>\n

    AMACI<\/span><\/h4>\n

    Ba\u011f\u0131ms\u0131z de\u011fi\u015fken say\u0131s\u0131n\u0131n birden fazla oldu\u011fu regresyon modellerine \u00e7oklu regresyon modelleri denir. \u00c7oklu regresyon \u00e7\u00f6z\u00fcmlemesinde, bir ba\u011f\u0131ml\u0131 de\u011fi\u015fken ve bu ba\u011f\u0131ml\u0131 de\u011fi\u015fkeni etkiledi\u011fi d\u00fc\u015f\u00fcn\u00fclen birden daha \u00e7ok ba\u011f\u0131ms\u0131z de\u011fi\u015fken s\u00f6z konusudur. K\u0131saca, \u00e7oklu regresyon analizi, basit do\u011frusal regresyonun geni\u015fletilmi\u015f \u015fekli olarak d\u00fc\u015f\u00fcn\u00fclebilir. Buna ek olarak ise k\u0131smi korelasyon analizini kullan\u0131r.<\/span><\/p>\n

    \u00c7oklu reggresyon analizini kullanan bir ara\u015ft\u0131rmac\u0131n\u0131n iki genel amac\u0131 olabilece\u011finden bahsedilebilir (Alpar, 2001):<\/span><\/p>\n

    \u00b7 <\/span><\/span><\/span>Kolay elde edilebilir ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler yard\u0131m\u0131yla ba\u011f\u0131ml\u0131 de\u011fi\u015fken de\u011ferini kestirmek,<\/span><\/p>\n

    \u00b7 <\/span><\/span><\/span>Ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerden hangisi veya hangilerinin ba\u011f\u0131ml\u0131 de\u011fi\u015fkeni daha \u00e7ok etkiledi\u011fini belirlemek.<\/span><\/p>\n

    Model:<\/span><\/strong><\/p>\n

    <\/span> <\/span>, k=1,2,….<\/span><\/p>\n

    <\/span><\/p>\n

    Hipotez Testi:<\/span><\/strong> <\/span>Hipotez testinde ba\u011f\u0131ms\u0131z de\u011fi\u015fkenin ba\u011f\u0131ml\u0131 de\u011fi\u015fkeni yordamada manidar olup olmad\u0131\u011f\u0131 test edilir.<\/span><\/p>\n

    <\/span><\/span><\/p>\n

    <\/span> <\/span>de\u011ferlerinden en az biri 0 de\u011fil.<\/span><\/p>\n

    <\/span><\/p>\n

    Test istatisti\u011fi :<\/span><\/strong> <\/span><\/span><\/p>\n

    <\/span><\/p>\n

    Karar Kural\u0131:<\/span><\/strong> <\/span><\/span> <\/span> <\/span><\/span>\u015f<\/span><\/span> <\/span><\/span> <\/span>hipotezi reddedilir<\/span><\/p>\n

    F de\u011ferine kar\u015f\u0131l\u0131k gelen <\/span>p\u2013de\u011feri<\/span><\/em> <\/span><\/span><\/span>\u015f<\/span><\/span> <\/span> <\/span>hipotezi reddedilir.<\/span><\/p>\n

    <\/span><\/p>\n

    &lt;b style=’mso-bidi-font-weight:normal’>Yorum:<\/span> <\/span>p-de\u011feri <\/span> <\/span>ise <\/span> <\/span>hipotezi reddedilir. <\/span>Bu da, ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerle ba\u011f\u0131ml\u0131 de\u011fi\u015fken y<\/span> aras\u0131nda belirgin nitelikte bir ili\u015fki oldu\u011funu g\u00f6sterir. Yani ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler, ba\u011f\u0131ml\u0131 de\u011fi\u015fkeni etkileyen fakt\u00f6rlerdir. Testin reddedilmemesi (<\/span>\u2019\u0131n kabul edilmesi) durumunda, ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerin ba\u011f\u0131ml\u0131 de\u011fi\u015fkeni a\u00e7\u0131klayamad\u0131\u011f\u0131 sonucu do\u011far. <\/em>Bu durumda ya \u00f6rnek say\u0131s\u0131 artt\u0131r\u0131labilir <\/span>veya ba\u011f\u0131ml\u0131 de\u011fi\u015fkeni etkileyebilecek yeni fakt\u00f6rler (ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler aran\u0131r.)<\/span><\/p>\n

    <\/span><\/p>\n

    REGRESYON \u00c7\u00d6Z\u00dcMLEMES\u0131 SIRASINDA D\u0131KKAT ED\u0131LECEK \u00d6NEML\u0131 NOKTALAR<\/span><\/strong><\/p>\n

    <\/span><\/strong><\/p>\n

      \n
    • Ba\u011f\u0131ml\u0131 de\u011fi\u015fken s\u00fcrekli veya kesikli say\u0131sal veriler olmal\u0131d\u0131r; kategorik olmamal\u0131d\u0131r.<\/span><\/li>\n<\/ul>\n

      <\/span><\/p>\n

        \n
      • Ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler kategorik de olabilir. B\u00f6yle durumlarda bu ba\u011f\u0131ms\u0131z de\u011fi\u015fken s\u00f6zde\/yapay (dummy) de\u011fi\u015fken olarak adland\u0131r\u0131l\u0131r.<\/span><\/li>\n<\/ul>\n

        <\/span><\/p>\n

          \n
        • Ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler aras\u0131nda \u201c\u00e7okluba\u011f\u0131nt\u0131<\/strong>\u201d (multicollinearity) g\u00f6r\u00fclmemelidir. \u00c7okluba\u011f\u0131nt\u0131, ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerin birbirleriyle ili\u015fkili oldu\u011funu g\u00f6sterir (Tacq, 1997). Ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler aras\u0131nda b\u00f6yle bir ili\u015fkinin olmas\u0131, de\u011fi\u015fkenlerden birinin modele ek bir katk\u0131 getirmedi\u011fine dikkat \u00e7eker. Bu durum, ba\u011f\u0131ms\u0131z (yorday\u0131c\u0131) de\u011fi\u015fkenler aras\u0131ndaki korelasyon katsay\u0131lar\u0131n\u0131n mutlak de\u011ferinin .80 den b\u00fcy\u00fck bulunmas\u0131yla belirlenebilir. Multicollinearity olup olmad\u0131\u011f\u0131n\u0131 anlayabilmek i\u00e7in VIF de\u011ferlerine bak\u0131lmal\u0131d\u0131r (Alpar, 2001). \u00c7oklu ba\u011f\u0131nt\u0131 (veya \u00e7oklu birlikte do\u011frusall\u0131k) belirlendikten sonra bu sorunu a\u015fabilmek i\u00e7in ara\u015ft\u0131rmac\u0131, ara\u015ft\u0131rman\u0131n kuramsal temellerini dikkate alarak bu de\u011fi\u015fkenlerden sadece birini analize dahil edip, di\u011ferini\/di\u011ferlerini analiz d\u0131\u015f\u0131nda tutabilir. <\/span><\/span><\/li>\n<\/ul>\n

          <\/span><\/p>\n

            \n
          • Ba\u011f\u0131ms\u0131z de\u011fi\u015fken say\u0131s\u0131 fazla oldu\u011funda \u00e7e\u015fitli y\u00f6ntemler yard\u0131m\u0131yla modele katk\u0131s\u0131 en fazla<\/em><\/strong> olan daha az say\u0131daki de\u011fi\u015fken veya de\u011fi\u015fkenler<\/em><\/strong> belirlenebilir. S\u00f6z konusu y\u00f6ntemler aras\u0131nda; ad\u0131m ad\u0131m regresyon y\u00f6ntemi (stepwise), ileriye do\u011fru se\u00e7im (foward selection), geriye do\u011fru \u00e7\u0131karma (backward elemination) gibi y\u00f6\u00f6ntemler say\u0131labilir.<\/span><\/li>\n<\/ul>\n

            <\/span><\/p>\n

              \n
            • G\u00f6zlem say\u0131s\u0131 (n), ba\u011f\u0131ms\u0131z de\u011fi\u015fken say\u0131s\u0131n\u0131n en az 5 kat\u0131 kadar olmal\u0131d\u0131r. \u0131deali ise, g\u00f6zlem say\u0131s\u0131n\u0131n ba\u011f\u0131ms\u0131z de\u011fi\u015fken say\u0131s\u0131n\u0131n 20 kat\u0131 kadar olmas\u0131d\u0131r. Yordamaya y\u00f6nelik \u00e7al\u0131\u015fmalarda bu say\u0131n\u0131n daha da artt\u0131r\u0131lmas\u0131 ve evrene genelleme yapabilmek i\u00e7in \u00e7al\u0131\u015f\u0131lacak \u00f6rneklemin genelleme yap\u0131labilecek \u015fekilde, uygun \u00f6rneklem alma y\u00f6ntemlerinin kullan\u0131larak se\u00e7ilmesi gerekir.<\/span><\/li>\n<\/ul>\n

              <\/span><\/p>\n

                \n
              • Regresyon \u00e7\u00f6z\u00fcmlemelerinde ayk\u0131r\u0131 de\u011ferlere (outlier) dikkat edilmelidir. \u00c7oklu regresyon analizinde do\u011frusall\u0131k ve normallik varsay\u0131mlar\u0131n\u0131n kar\u015f\u0131lanmas\u0131n\u0131 g\u00fc\u00e7le\u015ftiren u\u00e7 de\u011ferlerin olup olmad\u0131\u011f\u0131, at\u0131k de\u011ferler (residual) \u00fczerine kurulu grafiklerle incelenebilir. Bunun yan\u0131 s\u0131ra Mahalanobis uzakl\u0131k de\u011ferleri kullan\u0131larak da belirlenebilir (B\u00fcy\u00fck\u00f6zt\u00fcrk, 2004). <\/span><\/li>\n<\/ul>\n

                <\/span><\/p>\n

                \u00c7oklu Regresyon \u0131statistikleri<\/span><\/h3>\n

                <\/span><\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
                \n

                bi<\/sub> :<\/span><\/strong><\/p>\n<\/td>\n

                		\n

                Regresyon katsay\u0131lar\u0131. <\/span><\/em>Di\u011fer de\u011fi\u015fkenler sabit tutuldu\u011funda, s\u00f6zkonusu yorday\u0131c\u0131 de\u011fi\u015fkendeki birim art\u0131\u015fa kar\u015f\u0131l\u0131k yardanan (ba\u011f\u0131ml\u0131) de\u011fi\u015fkendeki de\u011fi\u015fim miktar\u0131n\u0131 g\u00f6sterir. Ayn\u0131 zamanda k\u0131smi e\u011fim<\/em> ya da k\u0131sm\u0131 regresyon katsay\u0131s\u0131<\/em> olarak da isimlendirilir. <\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

                \n

                S(bi<\/sub>) :<\/span><\/strong><\/p>\n<\/td>\n

                		\n

                Regresyon katsay\u0131lar\u0131n\u0131n standart hatalar\u0131<\/span><\/em><\/p>\n<\/td>\n<\/tr>\n

                \n

                BETA :<\/span><\/strong><\/p>\n<\/td>\n

                		\n

                Standartla\u015ft\u0131r\u0131lm\u0131\u015f regresyon katsay\u0131lar\u0131.<\/span><\/em> Modele katk\u0131s\u0131 daha fazla olan (y\u2019yi daha fazla a\u00e7\u0131klayan) de\u011fi\u015fkenin BETA katsay\u0131s\u0131 daha b\u00fcy\u00fckt\u00fcr.<\/span><\/p>\n<\/td>\n<\/tr>\n

                \n

                VIF :<\/span><\/strong><\/p>\n<\/td>\n

                		\n

                Varyans \u015fi\u015fme De\u011feri.<\/span><\/em> Ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler aras\u0131nda bir ba\u011f\u0131nt\u0131 olup olmad\u0131\u011f\u0131n\u0131 g\u00f6sterir. VIF de\u011ferlerinin 10 un \u00fczerinde olmamas\u0131 istenir. VIF>10 oldu\u011fu durumlarda \u00e7oklu ba\u011f\u0131nt\u0131n\u0131n varl\u0131\u011f\u0131 (multicollinearity) s\u00f6zkonusudur.<\/span><\/p>\n<\/td>\n<\/tr>\n

                \n

                t :<\/span><\/strong><\/p>\n<\/td>\n

                		\n

                Regresyon katsay\u0131lar\u0131n\u0131n anlaml\u0131l\u0131\u011f\u0131na ili\u015fkin t istatisti\u011fi.<\/span><\/em> t de\u011ferleri regresyon katsay\u0131lar\u0131n\u0131n standart hatalara b\u00f6l\u00fcnmesi ile bulunur.<\/span><\/p>\n<\/td>\n<\/tr>\n

                \n

                p :<\/span><\/strong><\/p>\n<\/td>\n

                		\n

                Regresyon katsay\u0131lar\u0131n\u0131n anlaml\u0131l\u0131\u011f\u0131na ili\u015fkin p olas\u0131l\u0131\u011f\u0131.<\/span><\/em> <\/span>\u00d6rne\u011fin, alfa (<\/span>) yan\u0131lma d\u00fczeyi .05 al\u0131n\u0131rsa, p<.05 ko\u015fulunu sa\u011flayan p de\u011ferlerine sahip katsay\u0131lar\u0131n modele katk\u0131s\u0131n\u0131n anlaml\u0131 oldu\u011fu s\u00f6ylenebilir.<\/span><\/p>\n<\/td>\n<\/tr>\n

                \n

                s :<\/span><\/strong><\/p>\n<\/td>\n

                		\n

                Regresyon denkleminin standart hatas\u0131.<\/span><\/em><\/p>\n<\/td>\n<\/tr>\n

                \n

                R :<\/span><\/strong><\/p>\n<\/td>\n

                		\n

                \u00c7oklu korelasyon katsay\u0131s\u0131 (Multiple R).<\/span><\/em> Ba\u011f\u0131ml\u0131 de\u011fi\u015fken ile ba\u011f\u0131ms\u0131z de\u011fi\u015fken aras\u0131ndaki ili\u015fkinin derecesini verir. <\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

                \n

                R2<\/sup> :<\/span><\/strong><\/p>\n<\/td>\n

                		\n

                \u00c7oklu a\u00e7\u0131klay\u0131c\u0131l\u0131k katsay\u0131s\u0131 (R2<\/sup>).<\/span><\/em> \u00c7oklu korelasyon katsay\u0131s\u0131n\u0131n karesidir. Ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerin ba\u011f\u0131ml\u0131 de\u011fi\u015fkeni ne oranda a\u00e7\u0131klad\u0131\u011f\u0131n\u0131 g\u00f6sterir.<\/span><\/p>\n

                <\/span>.<\/span><\/p>\n

                \u00d6rn;<\/span><\/strong> R2<\/sup>= .81, verilerin % 81 i olu\u015fturulan regresyon do\u011frusuyla a\u00e7\u0131klanabilir.<\/span><\/p>\n<\/td>\n<\/tr>\n

                \n

                F :<\/span><\/strong><\/p>\n<\/td>\n

                		\n

                Regresyon katsay\u0131lar\u0131n\u0131n s\u0131f\u0131ra e\u015fit olup olmad\u0131\u011f\u0131na ili\u015fkin t\u00fcmel F de\u011feri olup regresyona ili\u015fkin olarak yap\u0131lan varyans analizi sonucunda elde edilir. F de\u011feri anlaml\u0131 ise en az\u0131ndan bir regresyon katsay\u0131s\u0131 s\u0131f\u0131rdan farkl\u0131d\u0131r; ba\u015fka bir deyi\u015fle bu de\u011fi\u015fkenin modele katk\u0131s\u0131 anlaml\u0131d\u0131r. Bazen, regresyon katsay\u0131lar\u0131n\u0131n t\u00fcm\u00fc anlams\u0131z iken, F de\u011feri anlaml\u0131 \u00e7\u0131kabilir. Bu durumda da verilerde \u00e7oklu ba\u011f\u0131nt\u0131 sorunu vard\u0131r. <\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                <\/span><\/p>\n

                <\/span><\/strong><\/p>\n

                <\/span><\/span><\/u><\/strong><\/p>\n

                <\/span><\/span><\/u><\/strong><\/p>\n

                <\/span><\/span><\/u><\/strong><\/p>\n

                <\/span><\/span><\/u><\/strong><\/p>\n

                <\/span><\/span><\/u><\/strong><\/p>\n

                <\/span><\/span><\/u><\/strong><\/p>\n

                <\/span><\/span><\/u><\/strong><\/p>\n

                \u00c7OKLU REGRESYON ANAL\u0131ZDE KULLANILAN Y\u00d6NTEMLER<\/span><\/strong><\/u><\/strong><\/p>\n

                <\/u><\/strong><\/span><\/p>\n

                \u00c7oklu regresyon analizinde kullan\u0131lan pek \u00e7ok y\u00f6ntem vard\u0131r. En yayg\u0131n olarak kullan\u0131lan anliz <\/span>y\u00f6ntemleri ise \u015funlard\u0131r: i.<\/em><\/strong> Stardart \u00c7oklu Regresyon (Standard multiple regression), <\/span>ii. <\/em><\/strong>Hiyerar\u015fik \u00c7oklu Regresyon (Squential \/ Hierarchical multiple regression), <\/span>iii.<\/em><\/strong> A\u015famal\u0131 veya \u0131statistiksel \u00c7oklu Regresyon (Stepwise \/ Statistical multiple regression)<\/u><\/strong><\/span><\/p>\n

                <\/u><\/strong><\/span><\/p>\n

                  <\/u><\/strong><\/p>\n
                1. Stardart \u00c7oklu Regresyon (Standard multiple regression):<\/span><\/u><\/strong> Bu regresyon y\u00f6nteminde, \u00f6ncelikle b\u00fct\u00fcn ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler denkleme girer. Ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerin herbiri, di\u011fer ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerin hepsi denkleme girdikten ssonra denkleme al\u0131nm\u0131\u015f gibi de\u011ferlendirilir. Her bir ba\u011f\u0131ms\u0131z de\u011fi\u015fken, ba\u011f\u0131ml\u0131 de\u011fi\u015fkeni yordamada, di\u011fer ba\u011fms\u0131z de\u011fi\u015fkenlerin t\u00fcm\u00fcnden farkl\u0131 olarak ne kadar katk\u0131da bulundu\u011fu a\u00e7\u0131s\u0131ndan de\u011ferlendirilir (Tabachnick & Fidell, 2001). Bu y\u00f6ntemde, t\u00fcm ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerin ba\u011f\u0131ml\u0131 de\u011fi\u015fkendeki ortak etkilerinin incelenmesi esast\u0131r. <\/span><\/span><\/u><\/strong><\/li>\n<\/ol>\n

                  <\/u><\/strong><\/span><\/p>\n

                    <\/u><\/strong><\/p>\n
                  1. Hiyerar\u015fik \u00c7oklu Regresyon (Squential \/ Hierarchical multiple regression):<\/span><\/u><\/strong> Hiyerar\u015fik regresyonda, ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler ara\u015ft\u0131rmac\u0131n\u0131n belirledi\u011fi s\u0131rada denkleme girer. Herbir ba\u011f\u0131ms\u0131z de\u011fi\u015fken, denkleme girdi\u011fi noktada e\u015fitli\u011fe ne kadar katk\u0131da bulundu\u011fu a\u00e7\u0131s\u0131ndan de\u011ferlendirilir. Ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerin denkleme giri\u015f s\u0131ras\u0131n\u0131 ara\u015ft\u0131rmac\u0131 mant\u0131ksal veya kuramsal bir yap\u0131ya g\u00f6re belirler. Fakat, ara\u015ft\u0131rmac\u0131 daha \u00f6nemli g\u00f6rd\u00fc\u011f\u00fc de\u011fi\u015fkenleri son a\u015famalarda denkleme almak; modele daha az katk\u0131s\u0131 olabilecek de\u011fi\u015fkenlere ise giri\u015fte \u00f6ncelik vermek gibi bir yol da izleyebilir. Ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler, modele katk\u0131da bulunmalar\u0131 bak\u0131m\u0131ndan de\u011ferlendirilir.<\/span><\/u><\/strong><\/li>\n<\/ol>\n

                    <\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/p>\n
                    1. \u0131statistiksel \u00c7oklu Regresyon (Statistical multiple regression):<\/span><\/u><\/strong> Bu y\u00f6ntem ayn\u0131 zamanda a\u015famal\u0131 (stepwise) <\/span>\u00e7oklu regresyon y\u00f6ntemi olarak da adland\u0131r\u0131l\u0131r. Bu y\u00f6ntemde, regresyon e\u015fitli\u011fine sadece ba\u011f\u0131ml\u0131 de\u011fi\u015fkenin manidar yorday\u0131c\u0131lar\u0131 olan de\u011fi\u015fkenler al\u0131n\u0131r; di\u011fer de\u011fi\u015fkenler e\u015fitlik d\u0131\u015f\u0131 b\u0131rak\u0131l\u0131r. A\u015famal\u0131 \u00e7oklu regresyon analiiznde, \u00f6ncelikle ba\u011f\u0131ml\u0131 de\u011fi\u015fken ile en y\u00fcksek korelasyonu veren, yani ba\u011f\u0131ml\u0131 de\u011fi\u015fkenin varyans\u0131na en y\u00fcksek katk\u0131y\u0131 sa\u011flayabilecek ba\u011f\u0131ms\u0131z de\u011fi\u015fken se\u00e7ilerek i\u015fleme ba\u015flan\u0131r. Daha sonra ba\u011f\u0131ml\u0131 de\u011fi\u015fkenin varyans\u0131na birinviyle birlikte en y\u00fcksek katk\u0131y\u0131 veren ikinci ba\u011f\u0131ms\u0131z de\u011fi\u015fken i\u015fleme al\u0131n\u0131r ve i\u015flem bu \u015fekilde s\u00fcrd\u00fcr\u00fcl\u00fcr. \u0131statistiksel \u00e7oklu regresyon analizi \u00fc\u00e7 farkl\u0131 yolla yap\u0131labilir:<\/span><\/u><\/strong><\/li>\n<\/ol>\n

                      i. <\/span><\/span><\/span><\/em><\/strong>\u0131leriye Do\u011fru Se\u00e7me (Foward Selection) \u2013 <\/span><\/em><\/strong>Her bir ba\u011f\u0131ms\u0131z de\u011fi\u015fkenle ba\u011f\u0131ml\u0131 de\u011fi\u015fken aras\u0131ndaki korelasyon hesaplan\u0131r ve \u00f6ncelikle ba\u011f\u0131ml\u0131 de\u011fi\u015fkenle en y\u00fcksek korelasyonu veren ba\u011f\u0131ml\u0131 de\u011fi\u015fken analize al\u0131n\u0131r. Bu de\u011fi\u015fkenin katk\u0131s\u0131 (R2<\/sup>) de\u011ferlendirilir. Daha sonra, ikinci olarak ba\u011f\u0131ml\u0131 de\u011fi\u015fkenle y\u00fcksek korelasyon veren yorday\u0131c\u0131 analize al\u0131narak ve a\u00e7\u0131klay\u0131c\u0131l\u0131k katsay\u0131s\u0131ndaki art\u0131\u015fa g\u00f6re s\u00f6zkonusu de\u011fi\u015fkenin modele katk\u0131s\u0131 incelenir. Bu i\u015flem, art\u0131k ba\u011f\u0131ms\u0131z (yorday\u0131c\u0131) de\u011fi\u015fkenlerin ba\u011f\u0131ml\u0131 de\u011fi\u015fkeni a\u00e7\u0131klamada manidar bir katk\u0131lar\u0131n\u0131n olmad\u0131\u011f\u0131 noktaya kadar devam eder. <\/span><\/span><\/u><\/strong><\/p>\n

                      ii. <\/span><\/span><\/span><\/em><\/strong>Ad\u0131m Ad\u0131m Regresyon (Stepwise) \u2013<\/span><\/em><\/strong> \u0131leriye do\u011fru se\u00e7me y\u00f6nteminin daha geli\u015fmi\u015fi olarak da d\u00fc\u015f\u00fcn\u00fclebilir yaln\u0131z her ad\u0131mda o an modelde bulunan t\u00fcm ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler sanki modele en son girmi\u015f gibi de\u011ferlendirilir. Bu \u015fekilde, her bir de\u011fi\u015fkenin modele girmesiyle yeniden t\u00fcm modelin de\u011ferlendirilmesi sayesinde ba\u015fta iyi bir yorday\u0131c\u0131 olarak g\u00f6r\u00fclen bir de\u011fi\u015fkenin daha sonra t\u00fcm model i\u00e7inde etkili bir katk\u0131s\u0131n\u0131n olmad\u0131\u011f\u0131 belirlenebilir (Pedhazur, 1982). <\/span><\/u><\/strong><\/p>\n

                      iii. <\/span><\/span><\/span><\/em><\/strong>Geriye Do\u011fru \u00c7\u0131karma (Backward Elemination) \u2013<\/span><\/em><\/strong> \u0131lk a\u015famada, b\u00fct\u00fcn yorday\u0131c\u0131 de\u011fi\u015fkenler analize dahil edilir. Daha sonra, her bir yorday\u0131c\u0131n\u0131n modele katk\u0131s\u0131n\u0131n manidarl\u0131\u011f\u0131n\u0131 belirlemek i\u00e7in, s\u00f6zkonusu de\u011fi\u015fken sanki en son modele giriyormu\u015f gibi k\u0131smi F testi yap\u0131l\u0131r. En k\u00fc\u00e7\u00fck F de\u011ferini veren de\u011fi\u015fken modelden \u00e7\u0131kar\u0131l\u0131r. Bu i\u015flem, modele manidar (significant) katk\u0131s\u0131 olan ba\u011f\u0131ms\u0131z (yorday\u0131c\u0131) de\u011fi\u015fkenler belirlene kadar devam eder.<\/span><\/u><\/strong><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      Bu \u00fc\u00e7 y\u00f6ntemden e\u011fer di\u011fer y\u00f6ntemlerin kullan\u0131lmas\u0131 i\u00e7in \u00f6nemli gerek\u00e7eler yoksa \u201cStandart \u00c7oklu Regresyon\u201d un kullan\u0131lmas\u0131 \u00f6nerilir (Tabachnick & Fidell, 2001). Hiyerar\u015fik ve \u0131statistiksel regresyon i\u00e7in kullan\u0131lan programlar ve elde edilen sonu\u00e7lar aras\u0131nda baz\u0131 benzerlikler olmas\u0131na ra\u011fmen, ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerin e\u015fitli\u011fe girmesi ve sonu\u00e7lar\u0131n yorumlanmas\u0131 konular\u0131nda bu iki y\u00f6ntem aras\u0131nda farkl\u0131l\u0131klar bulunmaktad\u0131r. Hiyerar\u015fik regresyonda ara\u015ft\u0131rmac\u0131 de\u011fi\u015fkenlerin giri\u015fini kontrol ederken, istatistiksel y\u00f6ntemde verilerden elde edilen istatistiksel de\u011fi\u015fken giri\u015fini kontrol eder. Bu y\u00f6ntemler, model test etme yerine model olu\u015fturma y\u00f6ntemleri olarak kabul edilmektedir. <\/span><\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/span><\/p>\n

                      \u00c7OKLU KORELASYON ANAL\u0131Z\u0131<\/u><\/strong><\/span><\/h3>\n

                      <\/u><\/strong><\/span><\/p>\n

                      \u00c7oklu regresyon fonksiyonu belirlendikten sonra X1 <\/span><\/sub>ve X2 <\/sub> <\/span>de\u011fi\u015fkenlerinin Y de\u011fi\u015fkenini a\u00e7\u0131klamadaki \u201c\u00f6nemlerini\u201d ve fonksiyonun \u201cuyum derecesini\u201d \u00f6l\u00e7mek i\u00e7in bir \u00f6l\u00e7\u00fcye ihtiya\u00e7 vard\u0131r. Her iki X1 <\/span><\/sub>ve X2 <\/sub> <\/span>de\u011fi\u015fkenlerinin katk\u0131s\u0131n\u0131 <\/u><\/em>bir arada dikkate alan \u00e7oklu korelasyon katsay\u0131s\u0131<\/em><\/strong> R, ve onun karesi olan \u00e7oklu determinasyon katsay\u0131s\u0131<\/em><\/strong> R2<\/sup> nin anlam\u0131 \u00fczerinde durulduktan sonra, herbir ba\u011f\u0131ms\u0131z de\u011fi\u015fkenin (di\u011fer ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler sabit tutuldu\u011fu durumda) <\/span>katsay\u0131s\u0131n\u0131 ayr\u0131 ayr\u0131 \u00f6l\u00e7en k\u0131smi korelasyon katsay\u0131lar\u0131 r ile , bunlar\u0131n kareleri olan k\u0131smi determinasyon (belirleme) katsay\u0131lar\u0131 r2<\/sup> de\u011ferlerinin anlam\u0131na de\u011finilecektir. <\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      \u00c7oklu Korelasyon Katsay\u0131s\u0131 (R) ve \u00c7oklu A\u00e7\u0131klay\u0131c\u0131l\u0131k Katsay\u0131s\u0131 (R2<\/sup>):<\/span><\/u><\/strong><\/u><\/strong><\/p>\n

                      R2 <\/sup> <\/span>= 0,9570 ve R=0,98<\/span><\/span><\/strong><\/u><\/strong><\/p>\n

                      Bu de\u011ferler her iki ba\u011f\u0131ms\u0131z de\u011fi\u015fken bir arada dikkate al\u0131nd\u0131\u011f\u0131nda<\/em> ili\u015fki ile a\u00e7\u0131klanabilen de\u011fi\u015fkenli\u011fin yakla\u015f\u0131k %95 oldu\u011funu (yani bu olu\u015fturulan regresyon d\u00fczlemi ile verilerin <\/span>%95 i a\u00e7\u0131klanabilir) ve korelasyon katsay\u0131s\u0131n\u0131n yakla\u015f\u0131k 0,98 <\/strong>oldu\u011funu g\u00f6stermektedir. \u0131li\u015fkinin kuvvetli g\u00f6r\u00fcnmesine kar\u015f\u0131l\u0131k <\/span>hangi ba\u011f\u0131ms\u0131z de\u011fi\u015fkenin<\/em> katsay\u0131s\u0131n\u0131n daha fazla oldu\u011funu anlamak m\u00fcmk\u00fcn de\u011fildir. Bunun i\u00e7in k\u0131smi korelasyon katsay\u0131lar\u0131n\u0131n incelenmesi gerekmektedir. <\/u><\/strong><\/span><\/p>\n

                      <\/span><\/span><\/u><\/u><\/strong><\/p>\n

                      K\u0131smi Korelasyon Katsay\u0131lar\u0131:<\/span><\/u><\/strong><\/u><\/strong><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      \u00c7oklu korelasyon katsay\u0131s\u0131 R ile ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerin (X1 <\/span><\/sub>ve X2<\/sub>) ba\u011f\u0131ml\u0131 de\u011fi\u015fkeni (Y) a\u00e7\u0131klamadaki \u00f6nemlerini yani X1 <\/span><\/sub>ve X2 <\/sub> <\/span>nin birarada katk\u0131lar\u0131n\u0131n kuvveti \u00f6l\u00e7\u00fclmektedir. K\u0131smi korelasyon katsay\u0131s\u0131 ile de sadece bir ba\u011f\u0131ms\u0131z de\u011fi\u015fkenin (\u00f6rne\u011fin sadece X1 <\/sub>in) di\u011fer ba\u011f\u0131ms\u0131z de\u011fi\u015fkenler sabit tutuldu\u011fu yani b\u00fct\u00fcn ba\u011f\u0131ms\u0131z de\u011fi\u015fkenlerin etkilerinin kald\u0131r\u0131ld\u0131\u011f\u0131 halde yapt\u0131\u011f\u0131 katk\u0131 \u00f6l\u00e7\u00fclmektedir.<\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      X1<\/sub> sabit tutuldu\u011funda, <\/span>X2 <\/sub>nin ili\u015fkiyi a\u00e7\u0131klamadaki katk\u0131s\u0131:<\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/u><\/strong><\/span><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      X2<\/sub> sabit tutuldu\u011funda, <\/span>X1 <\/sub>nin ili\u015fkiyi a\u00e7\u0131klamadaki katk\u0131s\u0131:<\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/u><\/strong><\/span> <\/u><\/strong><\/span><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      <\/span><\/strong><\/u><\/strong><\/p>\n

                      KAYNAK\u00c7A<\/span><\/strong><\/u><\/strong><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      ALPAR, Reha. Spor Bilimlerinde Uygulamal\u0131 \u0131statistik<\/em>. Nobel Yay\u0131n Da\u011f\u0131t\u0131m, Ankara, 2001.<\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      ARICI, H\u00fcsn\u00fc. \u0131statistik Y\u00f6ntemler ve Uygulamalar<\/em>. Meteksan A.\u015f., Ankara, 2001.<\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      B\u00fcy\u00fck\u00f6zt\u00fcrk<\/u><\/strong><\/span>, \u015f.. Sosyal Bilimler \u0131\u00e7in Veri Analizi El kitab\u0131<\/em>. Pegem Yay\u0131nc\u0131l\u0131k. Ankara, 2004.<\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      K\u00d6KSAL, B. A.. \u0131statistik analiz Metodlar\u0131. 5. Bask\u0131. \u00c7a\u011flayan Kitapevi. \u0131stanbul, 1998.<\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      <\/a>MERTLER, C. A. & VANNATTA<\/span>, R. A.. Advanced and Multivariate Statistical Methods: Practical Aplication and Interpretation.<\/em> (2th<\/sup> ed.). Pyrczak Publishing. Los Angeles, 2002.<\/span><\/u><\/strong><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      PEDHAZUR, E. J.. Multiple Regression in Behavioral Research: Explanation and Prediction. Fort Worth, TX: Holt, Rinehart, and Winston, 1982.<\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      TABACHNICK, B.G., & FIDELL, L. S.. Using Multivariate Statistics<\/em>. (4th ed.). Needham Heights, MA: Allyn & Bacon, 2001.<\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      Tacq<\/u><\/strong><\/span>, J.. Multivariate Analysis Techniques in Social Science Research<\/em>. Sage Publication, 1997.<\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      TATLID\u0131L, H\u00fcseyin. Uygulamal\u0131 \u00c7ok De\u011fi\u015fkenli \u0131statistiksel Analiz<\/em>. Cem Web Ofset L.\u015f., Ankara, 1996.<\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n

                      <\/u><\/strong><\/span><\/p>\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"

                      Basit ve \u00e7oklu regresyon ile korelasyon konusunda Devrim Erdem, M\u00fccahit Ka\u011fan ve Fuat Tanhan’\u0131n haz\u0131rlam\u0131\u015f olduklar\u0131 \u00e7al\u0131\u015fmay\u0131 yaz\u0131n\u0131n devam\u0131nda okuyabilirsiniz.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_kad_post_classname":"","footnotes":""},"categories":[],"tags":[],"class_list":["post-83","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"https:\/\/www.istatistik.gen.tr\/index.php?rest_route=\/wp\/v2\/posts\/83","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.istatistik.gen.tr\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.istatistik.gen.tr\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.istatistik.gen.tr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.istatistik.gen.tr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=83"}],"version-history":[{"count":0,"href":"https:\/\/www.istatistik.gen.tr\/index.php?rest_route=\/wp\/v2\/posts\/83\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.istatistik.gen.tr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=83"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.istatistik.gen.tr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=83"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.istatistik.gen.tr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=83"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}