Statistical tables are attached Two Hours UNIVERSITY OF MANCHESTER. May 2007 Final Draft

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1 Statistical tables are attached wo Hours UNIVERSIY OF MNHESER Ma 7 Final Draft Medical Statistics M377 Electronic calculators ma be used provided that the cannot store tet nswer LL si questions in SEION (5 Marks) nswer WO of the three questions in SEION ( 5 marks each) he total number of marks on the paper is SEION nswer LL si questions

2 SEION (i) (ii) In the contet of clinical trials briefl eplain what is meant b the term bias. Describe two possible sources of bias in clinical trials. [5 Marks]. In a trial comparing an cupuncture treatment () with a Homeopathic treatment (H) for patient suffering from chronic headaches, patients are allocated to treatment using deterministic minimization controlling for se and tpe of head ache (migraine, tension). he numbers of patients with each characteristic for each treatment are given in the table below after twent-five patients have entered the trial. Patient haracteristic Male Female Migraine ension reatment () (H) () (H) () (H) () (H) Number of Patients (i) How man patients have been allocated to each treatment? (ii) he characteristics of the net two patients to entering the trial are: 6 th (Male, Migraine) 7 th (Female, Migraine) Determine the treatment allocation of each patient. [5 marks]

3 3 randomised controlled equivalence trial is being planned to compare a new treatment () with a control treatment (). Suppose is the continuous and normall distributed outcome measure and suppose we consider the two treatments to be equivalent provided the (-α) % confidence interval for is in the range [- δ n δ,δ]. he power to demonstrate treatment equivalence is given b the epression Φ z α where σ σ is the within treatment group standard deviation, n is the sample size of both groups, and Φ is the cumulative distribution function of a standardised normal distribution. Suppose [-,] is to be used as the range of equivalence and the within treatment group standard deviation has been estimated to be 4. (i) Estimate the power of the stud 5 subjects in each treatment. (ii) Determine the sample size required to obtain 95% power. [ Marks]

4 4. researcher has carried out a randomized clinical trial to compare a new treatments () for chronic knee pain with a standard treatment (). he two treatments are randoml allocated to 89 patients including 6 with pain in both knees and 7 with pain in just one knee. fter three months follow-up patient s pain is measured using a mm visual analogue scale with higher scores representing greater pain. he researcher has carried out the following statistical analsis using a statistical software package, first analzing all patients and then patients with pain in just one knee and both knee separatel. ll subjects wo-sample t test with equal variances Obs Mean Std. Dev. [95% onf. Interval] diff diff = mean() - mean() -est of difference = (vs not =): -Value = -.43 P-Value =.57 DF = 87 One knee wo-sample t test with equal variances Obs Mean Std. Dev. [95% onf. Interval] diff diff = mean() - mean() -est of difference = (vs not =): -Value =.3 P-Value =.758 DF = 5 oth knees wo-sample t test with equal variances Obs Mean Std. Dev. [95% onf. Interval] diff diff = mean() - mean() -est of difference = (vs not =): -Value = -.44 P-Value =.36 DF = 6 Using a 5% significance level, the researcher concludes from the computer output that the new treatment () is more effective than the standard treatment () in patients with pain in both knees but no better in patients with pain in just one knee. (i) Derive a test statistic to compare the treatment effect in patients with pain in both knees with the treatment effect for patients with pain in just one knee.

5 (ii) ppl this statistical test to the data to compare the treatment effect in patients with pain in both knees with the treatment effect for patients with pain in just one knee. (iii) omment on the researcher s analsis and conclusions. Do ou agree with them? [4 Marks] 5 (i) stud measures the association between an eposure E and a disease D b comparing disease rates in eposed and uneposed subjects. It is claimed that this association is biased because of a confounding factor,. State two conditions which must be satisfied b for to be a confounder. (ii) riefl describe the principles underling two statistical approaches for removing bias due to a confounder. [6 marks] 6 Samples of eposed and uneposed subjects are chosen independentl and followed over time to monitor new cases of disease. Let and be random variables describing the number of new cases of disease seen in eposed and uneposed respectivel, and be the corresponding total person- time of observation in each group and let βλ and λ be the incidence densit rates in eposed and uneposed populations. (i) ssume that and have independent Poisson distributions with means βλ and λ respectivel. Show β that the conditional distribution of, given +=a+b, is inomial with parameters (a+b) and β + the fact that the sum of two Poisson variables also has a Poisson distribution., using (ii) Suppose that =.5. What is the condition distribution of given +=8 under H :β=? (iii) Suppose that a= 6, b=. est H :β= versus H :β> using a test of size.5. [ marks]

6 SEION 7 For a parallel group trial comparing a control treatment () with a new treatment () suppose is a continuous, normall distributed outcome variable and is the value of the same variable recorded at baseline prior to randomisation. Suppose that δ is the treatment effect such that: = + ε and = + ε for treatment δ ε = + + and ε = + for treatment. with E ε = E ε =, Var ε = σ, Var ε = σ, and ov ε, ε = σ Let d = with d and d the sample means of treatment and treatment respectivel. ˆ δ θ = θ θ. Define ( ) ( ) ( ) (i) Show that ) E δ ( θ) = δ.[4 marks] ) (ii) Show that Var δ ( θ) = λ ( σ + θ σ θσ) where λ = +, n is the numbers of patient n n allocated to the new treatment and n is the number allocated to the control treatment. [7 marks] (iv) Show that this has a minimum when θ = σ σ.. [5 marks] (iv) omment on the implications of these results for the choice between an unadjusted comparison of the two treatments based on, and analsis using the change score - and an adjusted analsis using a linear model of with treatment group and as covariates. [5 marks] (v) Suppose and are the mean for each treatment as baseline and outcome. One tpe of statistical analsis seen within the medical literature is to test the hpothesis H = and H = using paired t-test. Rejection of the hpothesis for one treatment but not the other is interpreted as a difference between the two treatment. Give two reasons wh this tpe of analsis is flawed. [4 marks]

7 8. For an / crossover trial a model for a continuous outcome ij of the i th patient in the j th period can be written as i δ ξi ε i = for a patient in sequence in period, i φ ξi ε i = for a patient in sequence in period, = + ξ + ε for a patient in sequence in period, i i i i δ φ ξi εi = for a patient in sequence in period. where is the mean for the sequence in period, δ is the treatment effect of relative to, ξ i is a random variable representing patient i with mean zero and varianceσ, and ε ij is the error term for patient i in period j d assumed to be normall distributed with mean zero and variance σ ε. Defining di = i i let d,, d d and be the sample and population means of these for sequences and respectivel. (i) Show that a test of the null hpothesis : d H = is the same as a test of no treatment effect, d (ii) H δ = [3 marks] : wo anti-cholesterol lowering drugs were compared in a randomised controlled crossover trial. en patients were randoml allocated to sequence drug then drug and eight patients were randoml allocated to sequence drug then drug. he table below summarizes the sample mean and standard deviation for each sequence and period. interval of the treatment effect. est the hpothesis H : δ = and compute the 95% confidence interval. [7 marks] (iii) Period Period Period Period Sequence mean s.d. mean s.d. mean s.d n Define c i = i i for sequence and c i = i i for sequence. Let c, c c and c be the population and sample means of these for sequences and respectivel. Show that a test of the null hpothesis : c c H = is the same as a test of the period effect, H : φ =. [3 marks] (iv) For data in the table above test the null hpothesis H : φ =. [5 marks] (v) (vi) riefl comment on the result of the trial [ marks] It is sometimes suggested that the treatment effect δ can be estimated b the overall sample mean of the differences c i sa c = N i= be a biased estimate of δ. [5 marks] N c i, where N is the total number of subjects in the trial. Show that c ma

8 9 (i) o investigate whether an eposure, E, is a cause of a disease D, random samples of eposed and uneposed men, of size n and n respectivel, are chosen from a population. he cumulative incidence of disease in each group over a ear period is measured. ssume that the numbers of subjects in each group who develop the disease have independent inomial distributions with probabilit parameters π and π π ( π ) respectivel and consider the odds ratio, γ =. ( π ) π d d Using the approimate relationship Var[ ] Var[] Var [ ln ˆ γ ] n π + n + ( π ) n π + n = E[ ] ( π ), show that (ii) he following data are obtained: Disease No Disease Sub-total Eposed 9 Uneposed 38 5 Use this formula to calculate an approimate 95% onfidence Interval for γ. You ma assume that approimatel Normall distributed with mean ln(γ). ln( ˆ γ ) is (iii) onsider a different population with the same disease probabilities, π and π. proportion, q, of this population is eposed to E. Give epressions for the proportions in the population in the four cells formed b qπ cross-classifing E and D. Use these to show that ν = Pr{ E D} = and derive a similar qπ + ( q) π epression for ν( ν ) ν = Pr{ E nod}. Hence show that ( ν ) ν = γ. (iv) On the basis of the result in (iii) suggest an alternative stud design to that in (i) for estimating γ.