# Noncentrality Parameter - what is it, what does it do, what would be a suggested value?

I have been trying to brush up on my stats knowledge, especially in relation to Sample size determination and Statistical Power Analysis. But it seems that the more I read the more I need to read.

Anyway I found a tool called G*Power which seems to do everything I need but I am having an issue understanding Noncentrality Parameter, what is it, what does it do, what would be a suggested value etc?

The information on wikipedia etc is either incomplete or I'm not doing a very good job understanding it.

I am conducting a series of two tailed z-tests if that is any help.

p.s. Can anyone add better tags to this question?

In power calculations, we calibrate tests using knowledge of what the sampling distribution of the test statistic would be under the null hypothesis. Usually, it follows a $\chi^2$ or normal distribution. This allows you to calculate "critical values" for which, values exceeding this are deemed to be too highly inconsistent with what would be expected if the null were true.

The power of a statistical test is calculated by specifying the probability model for the data generating process under an alternative hypothesis, and calculating the sampling distribution for the same test statistic. This now takes on a different distribution.

For test statistics having a $\chi^2$ distribution under the null, they take a non-central $\chi^2$ distribution under the alternative that you create. These are very complicated distributions, but standard software can calculate density, distribution, and quantiles for them easily. The trick is that they are a convolution of standard $\chi^2$ densities and Poisson densities. In R, the dchisq, pchisq, and rchisq functions all have an optional ncp argument which is, by default, 0.

If the test statistic has a standard normal distribution under the null hypothesis, it will have a nonzero mean normal distribution under the alternative. Here that mean is the noncentrality parameter. For a t-test under an equal variance assumption, the mean is given by:

$$\delta = \frac{\mu_1 -\mu_2}{\sigma_{pooled}/\sqrt{n}}$$

In either case, data generated according to an alternative hypothesis will have test statistics following some noncentral distribution with noncentrality parameter ($\delta$). The $\delta$ is a sometimes unknown, often complicated function of the other data generating parameters.

• I get why random sampling would lead to a normally distributed mean if the null hypothesis were true (your black line). But the web has given me conflicting descriptions of the distribution under the alternative (i.e., when $\mu_2$ is assumed to be different to $\mu_1$) - in yours it is also normal (red line) but e.g., real-statistics.com has shown it to be skewed (see image half-way down the page). Surely, I've missed a trick. Can you help clear things up for me? – Ben Apr 25 '18 at 12:14