Estimation of the Parameters in One-Way Analysis of Covariance with the RAMML Method

Abstract

The main purpose of the experimental design is to determine whether there is a statistically significant difference among the means of the treatments defined as the levels of the factor of interest. In addition to the effect of the controllable factor on the response variable (y), in some cases, there may also be an effect of the uncontrolled covariate. In this case, the analysis should be performed by eliminating the effect of the uncontrolled covariate on the response variable in order to reduce the experimental error. For this purpose, the uncontrolled covariate (x) is included in the model. This method is called analysis of covariance (ANCOVA) in experimental design. ANCOVA, which is a combination of analysis of variance and regression analysis techniques, is sensitive to both y- and x-outliers. Although there are studies on ANCOVA models robust to y-direction outliers in the literature, to the best of our knowledge, there is no study in the context of ANCOVA based on robust adaptive modified maximum likelihood (RAMML) estimators that are also resistant to x-direction outliers. In this study, we address the presence of outliers not only in the response variable (y-direction outliers) but also in the covariate (i.e., x-direction outliers), and obtain RAMML estimators for the model parameters in one-way ANCOVA.

Keywords

• Covariance analysis
• x- and y-outliers
• RAMML method
• robustness

Bir-Yönlü Kovaryans Analizinde Parametrelerin RAMML Yöntemi ile Tahmini

Abstract

Deney tasarımında temel amaç, ilgilenilen faktörün düzeyleri olarak tanımlanan denemelerin ortalamaları arasında istatistiksel olarak anlamlı bir farklılık olup olmadığının belirlenmesidir. Yanıt değişkeni (y) üzerinde kontrol edilebilen faktörün etkisinin yanı sıra bazı durumlarda kontrol edilemeyen ortak değişkenin de etkisi söz konusu olabilir. Bu durumda, deneysel hatayı azaltmak amacıyla kontrol edilemeyen ortak değişkenin yanıt değişkeni üzerindeki etkisi arındırılarak analizin yapılması gerekir. Bu amaçla, kontrol edilemeyen ortak değişken (x) modele dahil edilir. Bu yöntem, deney tasarımında kovaryans analizi (analysis of covariance – ANCOVA) olarak adlandırılır. Varyans analizi ile regresyon analizi tekniklerinin bir birleşimi olarak ifade edilen ANCOVA hem y-yönlü hem de x-yönlü aykırı değerlere karşı duyarlıdır. Literatürde y-yönlü aykırı değerlere karşı dayanıklı ANCOVA ile ilgili çalışmalar bulunmasına rağmen bilindiği kadarıyla x-yönlü aykırı değerlere karşı dayanıklı adaptif uyarlanmış en çok olabilirlik (robust adaptive modified maximum likelihood – RAMML) tahmin edicilerine dayanan ANCOVA bağlamında herhangi bir çalışma bulunmamaktadır. Bu çalışmada, y-yönlü aykırı değerlere ek olarak ortak değişkende aykırı değerler olması, bir başka ifade ile x-yönlü aykırı değerler olması durumu ele alınmış ve bir-yönlü ANCOVA’da model parametrelerinin RAMML tahmin edicileri elde edilmiştir.

Keywords

• Kovaryans analizi
• x-yönlü ve y-yönlü aykırı değerler
• RAMML metodu
• dayanıklılık

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