In a toxicological study with six treatments T$1,...,$T$6($corresponding to an increasing

dose$),$ in pairwise comparisons the following differences are significant: T$1$ vs$.$ T$4,$ T$1$

vs$.$ T$5,$ T$1$ vs$.$ T$6,$ T$2$ vs$.$ T$5,$ T$2$ vs$.$ T$6,$ and T$3$ vs$.$ T$6.$ Perform the insert$-$and$-$absorb

algorithm by hand to derive the corresponding letter displayExercise $13$: Multiple comparison procedures points $(2+2+2+4)$

Load the file data$\_$gradedsheet. RData. It contains simulated response values for a toxicological

experiment measured for six increasing doses, in six replicates each. It is assumed that the first

dose corresponds to the control.

$($a$)$ Display the data via boxplots and describe the overall shape.

$($b$)$ Making use of the functions glht and cld from the package multcomp, calculate the

compact letter display $($cld$).$

$($c$)$ Explain, why the resulting cld is equivalent to the result from Exercise $10,$ i$.$e$.,$ to a resulting

cld with the letters 'abc' for treatment T$3$ and otherwise the same results as obtained here.

$($d$)$ Perform three types of multiple testing procedures, using the function glint from the

package multcomp: Dunnett, Williams, and extension of Dunnett and Williams including

downturn effects. Plot two$-$sided confidence intervals of the results. Compare and interpret

the results.