## “Effect Size” — same data, different interpretations

Just a short R-script note to embody the 3-page-paper of Rosenthal & Rubin (1982).

Table 1. (p. 167) listed a simple set-up. There was a between-subject treatment. Control group includes 34 alive cases and 66 dead cases. Treatment group includes 66 alive cases and 34 dead cases. The question is what is the percentage of the variance explained by the nominal IV indicating the group?

The authors pointed out that one may interpret the data result as death rate was reduced by 32% while the other may interpret the same as 10.24% variance was explained. Let's demo it more dramatically to imagine just 4% explained variance would reduce death rate by 20%.

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Rosenthal, R. & Rubin, D. B. (1982). A simple, general purpose display of magnitude of experimental effect. Journal of Educational Psychology, 74, 166-169.

## 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.