Solution of Behrens-Fisher Problem For High-Dimensional Data and Comparison of Proposed Test Statistics

Abstract

Comparison of multivariate mean of more than two groups is generally known as multivariate variance analysis (MANOVA) problem. However, classical test statistics used to solve MANOVA problems are highly affected by assumption violations. In addition, multivariate test statistics are insufficient in cases where the number of dependent variables is greater than the number of observations. This study aims to compare some proposed test statistics to determine whether mean vectors are equal in high-dimensional Behrens-Fisher problems. For this purpose, a simulation analysis was carried out for four different test statistics in cases where variance-covariance matrices are heterogeneous. Three different variance-covariance models were used in the study. In addition, different experimental conditions were taken into account for the number of dependent variables and the number of observations. The results of the study showed that the performances of the test statistics were generally comparable. However, it has been observed that the performance of test statistics varies according to experimental conditions.

Keywords

• Multivariate variance analysis (MANOVA)
• High dimensional data
• Behrens-Fisher problem
• Monte Carlo simulation

Yüksek Boyutlu Veriler İçin Behrens-Fisher Probleminin Çözümü ve Önerilen Test İstatistiklerinin Karşılaştırılması

Abstract

Çok değişkenli ikiden fazla grup ortalama vektörünün karşılaştırılması genellikle Çok Değişkenli Varyans Analizi (MANOVA) problemi olarak bilinmektedir. Ancak MANOVA problemlerinin çözümü için kullanılan klasik test istatistikleri, varsayım ihlallerinden oldukça fazla etkilenmektedir. Ayrıca bağımlı değişken sayısının gözlem sayısından daha büyük olduğu durumlarda çok değişkenli test istatistikleri yetersiz kalmaktadır. Bu çalışmanın amacı, yüksek boyutlu Behrens-Fisher problemlerinde ortalama vektörlerin eşit olup olmadığını belirlemek için önerilen bazı test istatistiklerini karşılaştırmaktır. Bu amaçla varyans-kovaryans matrislerinin heterojen olduğu durumlarda dört farklı test istatistiği için bir simülasyon analizi gerçekleştirilmiştir. Çalışmada üç farklı varyans-kovaryans modeli kullanılmıştır. Ayrıca bağımlı değişken sayısı ve gözlem sayısı için farklı deneysel koşullar dikkate alınmıştır. Çalışmanın sonuçları, test istatistiklerinin performanslarının genel olarak karşılaştırılabilir olduğunu göstermiştir. Ancak deneysel koşullara göre test istatistiklerinin performansının değiştiği gözlemlenmiştir.

Keywords

• Çok değişkenli varyans analizi (MANOVA)
• Yüksek boyutlu veri
• Behrens-Fisher problemi
• Monte Carlo simülasyonu

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