I would like to know if there is a test for the difference of to means m1 and m2 (continuous variables) if I have only information for mean, 2.5%- and 97.5%-quantiles. For example:
$m1 \ \ \ = 10.5$
$q1_{025}= 8.3$
$q1_{975}= 12.5$
$m2 \ \ \ \ = 15.5$
$q2_{025}= 12.7$
$q2_{975}= 17.3$
No further informations are available. One possible way to solve this is to assume that the quantiles range is 1.96*standard errors (se), to compute the se and to perform a t-test. Maybe there is another test which is more suitable to this kind of data (and preferably as R function). Any idea?
Updates to clarify question:
The original question above was motivated from a regression analysis with a linear (x) and quadratic term ($x^2$), interaction z (factor with 7 levels) with x, other fixed variables and a random variable
$$y
\tilde~b_1x+b_2x²+b_3z+b_4xz+b_5x²z+...
$$
The parameter of interest is the x value at maximum y ($x_m$). I computed 500 $x_m^{\ast}$ through (s=500 resamplings of the residuals $\epsilon=y-\hat y$ and 500 regressions). From this 500 $x_m^{\ast}$ I computed $m1$, $q1_{025}$,$q1_{975}$,$sd1$,...,$m7$ etc.
My questions:
(1) If I have only the quantiles of $x_m^{\ast}$ for z-levels=1...7 and no other informations how can I perform a test to compare $x_m$ of z-level 1 and 2? (This was my original question above).
(2) I want to compare z-levels 1 and 2 and I have all $x_m^{\ast}$ available: $$x_{{m_1}_1}^{\ast}$,$x_{{m_1}_2}^{\ast}$,...,$x_{{m_1}_{500}}^{\ast} $$ for level 1 and $$ x_{{m_2}_1}^{\ast}$,$x_{{m_2}_2}^{\ast}$,...,$x_{{m_2}_{500}}^{\ast} $$ for level 2.
(3) Additionally to question (2) I have the number of observation for level 1 and 2 of z: $n_1$ and $n_2$.
