# The tail(s) of p value

For any given $H_0$ vs $H_1$, the p value of any given point x is $\underset{\theta\in H_{0}}{sup}P\left(\left\{ z|L\left(z\right)\ge L\left(x\right)\right\} |\theta\right)$, Where $L\left(x\right)\equiv\frac{\underset{\theta\in H_{1}}{sup}\left[f\left(x|\theta\right)\right]}{\underset{\theta\in H_{0}}{sup}\left[f\left(x|\theta\right)\right]}$

-- See R. Weber's Statistics Note (Chap 6.2 & 7.1)

I made some wrong comment on the pdf Null Ritual (Gigerenzer, Krauss, & Vitouch, 2004) Where three types of significance level (rather than p value) were discussed. I had written the comment to note that the chapter had ignored the role of $H_1$ in definition of p value. In almost every textbook, the two-tail p vs single-tail p are differentiated. Usually, the two-tail p is defined by $H_1$ like $\mu\neq0$.

Here I demonstrate a three-tail p value case on R platform.

 z=(-1000:1000)*0.02; f=0.5 * dchisq(abs(z),df=5); h=dchisq(10,df=5)*.5; plot(z,f,type="h",col=c("black","grey")[1+(f>h)]); lines(c(-20,20),c(h,h)); ## $H_0$ is $\chi^2(5)$ * binomial(-1 vs 1) ##

Do you agree the region nearby zero under the "V" curve (which is below the horizontal line) should be the 3rd tail? I think so, if only $H_1$ includes all other possible distributions in the same shape.

You'll also agree there will be two asymmetrical tails if $H_1$ includes just two asymmetrical curves, for example, $\mu=-2$ and $\mu=1$ ($\sigma^2\equiv1$) while $H_0$ is the standardized normal distribution.

## One thought on “The tail(s) of p value”

1. lixiaoxu says:

However, the 3rd tail is still a freak even if there were two peaks in the population density.In psychometrics, judgment is rarely based just one single sample. With a realistic sample size, the density of calculated statistics like mean and variance would easily average into a single peak.