Statistics Definitions > Non Centrality Parameter (NCP)

## What is the Non Centrality Parameter (NCP)?

The non centrality parameter (λ) is a measure of “…the degree to which a null hypothesis is false” (Kirk, 2012). In other words, it tells you something about the statistical power of a test.

For example, an F-distribution with an NCP parameter of zero means that the F-distribution is a central F-distribution. As the Non Centrality Parameter increases, the shape of the F-distribution shifts to the right. This also means that a larger percentage of the curve moves to the right of the critical value for alpha. The result is an increase in statistical power.

The F-distribution, Chi-square distribution, Student T-distribution, and Beta distribution have central distributions, which all occur when the NCP is set to zero. The central distribution can be viewed as a special case (λ = 0) from the set of all possible non central distributions.

## Formula for F-Distribution NCP

The formula for the NCP is related to the F ratio:

F = (σ_{e}^{2} + σ_{Β}^{2} / σ_{e}^{2}). When the variance of the group means in the numerator increases, the F ratio gets larger and the F distribution stretches to the right.

As a formula, the NCP is the ratio of the ANOVA sum of square between to mean square within (Carlberg, 2004; Kirk, 2012):

Where:

- j indexes the groups,
- n is the number of observations per group,
- &Beta is the
*grand mean*– a group mean.

## Disambiguation

Some authors describe the Non Centrality Parameter as the* degree of misspecification of a model*. Others define it as a control for the shape of a distribution. Sometimes the NCP is simply defined as a “Difference in means.” One thing is clear: **The term “Non Centrality Parameter” has been defined inconsistently across texts for years **(Carlberg, 2014). The exact definition of the NCP depends largely on where you are reading it.

Jondeau et. al (2007) describe the NCP as a parameter that “controls the shape of the [T-distribution].”

Even the notation is up for debate: φ, δ, d, or λ? Roncoroni et. al. use the following formula for the NCP:

**The recent majority seem to be in agreement that the accepted notation is λ**(Carlberg, 2004). After some pretty hefty reading, I decided to skew this article towards the more commonly accepted definition of λ as it relates to the F Distribution and statistical power.

**There are other interpretations,**so you might find the following notes useful:

- φ has been used for decades to look up statistical power in charts (Carlberg suggests the September 1957 Journal of the American Statistical Association for an example of how φ is used). It is not the same as the NCP.
- δ is used by some fairly modern texts (like this one from 2007) and it has also been used by many prominent texts — some with the incorrect formula as well. Carlberg notes a “prominent older text” which changes over the years to not only use λ instead of δ, but loses a square root symbol as well:

.

**References:**

Carlberg, C. (2014). Statistical Analysis: Microsoft Excel 2013. Pearson Education.

Kirk, R. (2012). Experimental Design: Procedures for Behavioral Sciences. SAGE Publications.

Roncoroni A, et. al. (2015). Handbook of Multi-Commodity Markets and Products: Structuring, Trading and Risk Management. John Wiley & Sons.

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