Influence of Extremities Factor

The effect size m can be considered unstable when much of the pushing or pulling of m come from the extremities of data. That is when a collection of extreme data points have a high influence on m in an equation such as y=mx+b for that effect size is Unstable.

The Influence of Extremities Factor has been devised to measure the instability of the effect size m.

In Yi = mXi + b, Yi and Xi are vectors of data with Real Entries.

m is the regression coefficient and is calculated via least squares. This is termed as simple linar regression.

(x,y)i denotes the ith point of the data.

\displaystyle inf_i=m_U -m_{U-i}

In equation (5.1) infi is the influence of the point i in calculating m. This is defined as the following:

mU is m with all points in the dataset, where U denotes the entire dataset.

mU-i is m calculated without the ith data point.

We then sort the data by values of descending infi. The set of data comprising of the top 10% of infi is labelled as “UL” for the upper limit and the bottom 10% of infi is labelled as “LL” or lower limit.

By removing the UL set of data we calculate m{U-UL}.

By removing the LL set of data we calculate m{U-LL}.

\displaystyle m_{U-UL(10\%)} \leq m_U \leq m_{U-LL(10\%)}

 

 

By removing data that is pushing up “m” (i.e. high infi) we get m{U-UL(10%)} which is lower than mU.

 

By removing data that is pulling down “m” (i.e.low infi) we get m{U-LL(10%)} which is higher than mU.

 

Influence of Extremities Factor at 10% intervals would be:

\displaystyle IEF_{(n=10\%)}=\frac{(m_U-m_{U-UL(10\%)})+({m_{U-LL(10\%)}-m_U})}{m_U}

 

\displaystyle IEF_{(n\%)}=\frac{m_{U-LL(n\%)}+m_{U-UL(n\%)}}{m_U}

A low IEF indicates a stable effect size m. A high IEF indicates an unstable effect size m. By stable, the value of m is not overly influenced by a few data points and by unstable, the value of m is overly influenced by a few data points. Datasets that are stable are more representative of the entire cohort and datasets that are unstable are likely not representative of the entire cohort.

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