Exercises Sheet 1

(W) denotes workshop exercises; (A) denotes homework questions; (E) additional exercises for completion in your own time.

Workshop Exercises

1. (W) As professional statisticians we need to practice according to regulatory body guidance. For this course we are focusing upon the ICH guidelines available from: https://www.ema.europa.eu

   Visit the EMA homepage and download the ICH E9 guideline.

   1. Which section describes study blinding?

   2. When and only when should study blinding be broken?

   3. Which section describes randomisation?

   4. The guideline includes a glossary. What is a multi-centre trial? How should randomisation procedures be organised in this setting?

2. (W) Why is randomisation the method of choice for allocating treatments to patients? State two reasons. [Note that there is extra short reading on this topic on the Moodle page].

3. (W) Simple randomisation: $p=0.5$ is to be used to allocate $n$ subjects to two treatments labelled $A$ and $B$.

   1. What is the total number of possible allocations of the $n$ subjects?

   2. Express the probability of achieving perfect balance: $n_A$ = $n_B$ = $n/2$.

   3. Suppose that $n=60$, what is the probability that the treatment groups have equal numbers?

   4. Trialists are interested in the probability that the imbalance will exceed four. Compute this probability.

4. (W) Visit the website: https://www.randomisation.com. Assuming three treatment groups and a sample size of 96 use the generator to:

   1. Provide a treatment allocation sequence using a fixed block size $b:b\ne 1$. [Note: You will need to enter the treatment group labels (A,B,C, say) in the upper array (table) and then specify the block size and number of blocks in the lower array.] What is the maximum possible imbalance for your generated sequence?

   2. Repeat the procedure this time varying the block size. What is the purpose of varying the block size? What is the maximum possible imbalance for your specification?

5. (W) Researchers are planning a clinical trial in which patients will be randomised to receive either a new treatment or placebo. They decide to stratify on age (5 levels) and gender.

   1. What is the purpose of stratified randomisation?

   2. Suppose simple randomisation was used within the strata. Would this procedure yield the intended aims?

   3. Under what circumstances may it prove impractical to use a stratified randomisation procedure? What alternative method of treatment allocation could be used? Discuss briefly in the context of this trial.

6. (W) A clinical trial is to be conducted to compare two treatment group means. Assume that responses on treatment, group indexed by $T$, are: $X_{iT}\sim N({\mu }_T,{\sigma }^2)$ and responses on control, group indexed by $C$, are: $X_{iC}\sim N({\mu }_C,{\sigma }^2)$.

   1. What is the variance of the estimated treatment effect given by the difference in the sample means: $\overline{D}={\overline{X}}_T-{\overline{X}}_C$?

   2. Let $n=n_T+n_C$ and put $n_T=r.n$ and $n_C=(1-r).n$ where $r$ is the proportion allocated to the experimental treatment. What is the optimal value of $r$? (i.e. that which minimises the variance).

   3. Now assume that the variance for the two groups differ: $X_{iT}\sim N({\mu }_T,{\sigma }_T^2)$ and $X_{iC}\sim N({\mu }_C,{\sigma }_C^2)$. What is the optimal treatment allocation (value of $r$) in this setting?

   Homework Exercises

7. (H) The data below derive from a study conducted to compare the PRE and POST menopausal dietary intake (kJ) in 11 women. Perform appropriate analysis (clearly stating your research hypothesis) and interpret the results in context. State the assumptions underpinning your choice of analysis.

   SUBJECT   PRE   POST
   1         5260  3910
   2         5470  4220
   3         5640  3885
   4         6130  5150
   5         6390  5645
   6         6515  4680
   7         6805  5265
   8         7515  5975
   9         7515  6790
   10        8230  6900
   11        8770  7335
   

   Additional Exercises

8. (E) What is a placebo? When might it be unethical to use a placebo?

9. (E) Weiler et al. (2000) investigate the effects of drugs and alcohol on driving performance. Alcohol was given in form of a beverage. The placebo contained no alcohol but alcohol was swabbed around the rim. What do you think about this blinding procedure?

10. (E) Suppose a randomised trial is being planned to show the efficacy of acupuncture in patients with a certain disease. Can you think of a blinding procedure?
    

    Challenge Exercise

11. To promote balanced treatment allocation trialists decide to used randomised blocking within centres in a multi-centre trial. So the centre is a strata. Assuming $k$ centres and a fixed block size $b$ give an expression for:

    1. the maximum possible imbalance

    2. the probability of observing this maximum imbalance. You should assume a uniform stopping probability within a block.