Log-logistic probability plot ($\log \frac{F(x)}{1-F(x)}=(\beta -1)\log x-(\beta -1)\log \alpha$)

For $F(x)=1-[1+(x/\alpha {)}^{\beta -1}{]}^{-1}$, plotting the logits of $p_i=i/(n+1)$ against sorted log observations gives an approximately straight line under the model. The probability integral transform makes $F(X_{(i)})$ a uniform order statistic with expectation $i/(n+1)$, motivating these plotting positions. With those axes the slope is $\beta -1$ and the intercept is $-(\beta -1)\log \alpha$.