All substances are toxic, but there is a dose level for each substance below which no adverse response will be observed. In order to find this level in humans, dose-response experiments on rodents have attempted to locate the no observed adverse effect level (NOAEL), this being the experimental dose level below the lowest experimental dose level with responses significantly different from the control level. This approach has been criticised, however, partly since it makes no use of any information about the relationship between dose level and response. An alternative approach is to define the change-point dosage as the largest experimental dosage that gives an estimated response no more harmful than the estimated response for the control group. With a suitable dose-response model the change-point dose level can be estimated.

West and Kodell (2005) model data involving the response of rats to daily doses of aconiazide. Fifty lab rats were split at random into five treatments groups. Over two weeks, the five groups of ten rats were respectively exposed to aconiazide at levels 0, 100, 200, 500, and 750 mg per kg of body weight per day $\left( X\right)$. The response recorded was weight gain over the fourteen days $\left( Y\right)$, a negative value indicating weight loss which can be considered harmful. Suppose a plot of the data recorded is as shown below:



West, R.W. and Kodell, R.L. (2005): Changepoint alternatives to the NOAEL. Journal of Agricultural, Biological, and Environmental Statistics 10, No. 2, 197--211.

Part a)
Suppose a model of the form



is fitted to the data, with $\epsilon$ an error term. Which of the following do you think would be a plot of the residuals against the dose level values?

??? A B C D



Part b)
If a model of the form



is fitted to the data via R, a summary of the model fit is given below:

$\text{Call:} \\ \text{lm(formula = WeightGain ~ Dose + I(Dose^2))} \\ \text{Residuals} \\ \begin{array}{c c c c c} \text{Min} & \text{1Q} & \text{Median} & \text{3Q} & \text{Max} \\ & & & & \\ \end{array} \\ \\ \text{Coefficients}\\ \begin{array}{l r r r r l} & \text{Estimate} & \text{Std. Error} & \text{t value} & \text{Pr(>|t|)} & \\ \text{(Intercept)} & & & & & \\ \text{Dose} & & & & & \\ \text{I(Dose^2)} & & & & & \\ \end{array} \\ \text{- }\text{- }\text{- } \\ \text{Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1} \\ \\ \text{Residual standard error: } \text{ on } \text{ degrees of freedom} \\ \text{Multiple R-squared: } \text{, Adjusted R-squared: } \\ \text{F-statistic: } \text{ on } \text{ and } \text{ DF, p-value: }$

Assuming the above model is a good fit for the data, estimate the mean weight gain (in g) of the rats in the control group (those that received no aconiazide). Give your answer to two decimal places.


Part c)
Assuming the above model is a good fit for the data, estimate the mean weight gain (in g) of a rat receiving 700mg of aconiazide per kg of bodyweight per day during the study. Give your answer to two decimal places.


Part d)
What is the estimated rate of change of weight with repect to increase in aconiazide dose at dose level 400mg per kg of bodyweight per day? Give your answer to three decimal places.


Part e)
What do these data suggest is the change-point dosage? In the event there is no change point, enter 0 as your answer