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statistics principles and methods

Questions and Answers of Statistics Principles And Methods

Explain the relationship between a random effects model and a Bayesian analysis of a fixed eff’ects model.
Explain why an analysis of variance and an analysis of covariance can be handled analytically in the m e way.
13.1 Show that conditional distribution for y, given the other, kj and Y is Bernoulli. Describe how to implement a Gibbs sampler to explore the posterior distribution of models.
13.2 Consider the linear regression model, with Zellner's g-prior and the noninformative prior on 0'. p ( d ) = I/&.(a) Find the predictive distribution of Y at X = X, under model averaging in the
Let M denote the model Y = /?X + u and consider comparing &I to the niodel where M' : /I = 0.13.3 Using a noninfonnative prior distribution for 02, p(a2) = l / d , and Zeliner's g-prior for /?. p - N
13.4 Consider an improper prior distribution for jl and y given 02:p(By,y )proportional to I gX& lo?(a) What happens to the prior distribution of p, if g is multiplied by an arbitrary constant?(b)
14.1* Prove expression (14.2). with the associated defirution of a,, c,. (Recall that o2 and T? are assumed known. Hint: first derive the distributions of y I y and 8, I p, y and then integrate out p
14.2 Derive all of the conditional distributions required to implement a Gibbs sampler for the model (1 4.4).
I4.3* Consider the hierarchical model (14.5).(a) Draw a summary version of the hierarchical diagram as in Figure 14.1(a).(b) Assumc that L j , and k each have two leveIs and show the detailed diagram
14.4 Consider the collapsed hierarchical model (14.6).(a) Write out the matrices J, K. I,, as in Equation (14.3). Assume that i. j , and k each have two levels; make clear thc ordering of the
Each coin in a collection of bent coins has a different probability of coming up heads when tossed; the probability for each coin i s drawn h m a Beta distribution. Each coin is tossed several times
14.6 The genetic composition (genome) of a person is derived from that of the parents, but the way they are combined involves a random process. We will consider the process for a small group of'
Consider the model (14.10) and suppose that there is a single observation=y, from each Level 2 unit i, and thatb, = b is the same for every unit(i.e., the regression is a constant).(a) Derive the
Consider the random coefficient model (14.7).(a) Write the Level 1 conditional likelihood (given the variance components)for fit, i.e., the density of the part of the data that depends on /Il.(b)
14.9 Consider the following three-level hierarchical model Y y - N O , , var,o‘,>, 0, - “P,, T 2 h of - IG(v,, V Z ) , pt -” “1. CU2L T2 I ~ ( V I t 112h where j e l l , ..., n ) , j e { l .,
14.10 In a subject area that interests you, find a data set that has a hierarchical(clustered) structure and analyse it using any convenient software
15.1 Suppose you have data that consist of n = 10 independent and joint normally distributed random vectors representing multivariate data intensities.The data might correspond to the output of micro
Explain how you might assess the hyperpararrteters of the Bayesian factor analysis model when sample sizes are large.
What might be the advantages of standarkkg the data vectors (with respect to location and scale) before carrying out a Bayesian factor analysis?
Of what use might factor scores be in: marketing analysis; a political science voting context; an astronomy context; a sociology context; a biology context; and a medical context?
Suppose you carry out an exploratory Bayesian factor analysis (you had no idea in advance of collecting data of how many factors there might be) and you find there are four factors; but the four
Explain how you would go about fmding interval estimates of factor loadings in a Bayesian factor analysis containing a large number of data vectors.Why should you not be concerned about the possible
Suppose a Bayesian factor analysis is contemplated but some of the observable variables in the data vectors are jointly continuous, and others are jointly discrete. What ways might you explore to
16.1 Suppose there are two normal populations that are each two-dimensional with unknown parameters. We want to classify an observation z. Suppose the training samples give x = (2,3)' S = x = (3,1)'
16.2* Suppose there are three bivariate normal populations with training sample statistics as in Exercise 16.1, but, in addition, we know N = N = N = N = 10, and x = (0, 0), S = ( }). 3 Assuming the
16.3* For the classification problem in Section 16.1 adopt a natural conjugate prior family for (0,, A), instead of the vague prior family used there, and find: (a) The joint posterior density of 0,,
16.4* Explain the classification concept behind the notion of "second guessing undecided respondents," as discussed in Section 16.8.
16.5* In what types of situations would we be likely to encounter large fractions of "undecided" respondents, in the sense presented in Section 16.8? 16.7
16.6* While it is a general principle that it is better to have more information than less, why might it be better to have very few subsidiary questions in the undecided respondents problem? (Hint:
What are the arguments for and against merging observations already classified into say, , with its training sample, and using them to better estimate the parameters of r, in order to classify new
Suppose you would like to classify an unknown observation vector (px 1) into one of two populations or for which both ", and 16.8
16.9 2 are multinomial populations with probability mass functions /(x) and f2(x). How would you develop a Bayesian classification procedure for classifying z into ?
You would like to use Bayesian classification methods to classify an unknown observation vector z: (px 1) into o , where ~ N(0,2),i=1,2. You have training data with N, observation vectors believed to
Consider a survey of nurses’ opinions of their working conditions. What type of variables are: (i) length of service (ii) staff grade (iii) age (iv) salary(v) number of patients seen in a day (vi)
What differences do you think are there in a discrete measurement such as shoe size, and a discrete measurement such as family size?
Many continuous variables are dichotomised to make them easier to understand e.g. obesity (body mass index >30 kg/m2) and anaemia(haemoglobin level
If a relative risk is quoted, what is the absolute risk difference? Is this a very small number? Beware of reports that only quote relative risks and give no hint of the absolute risk!
If an odds ratio is quoted, is it a reasonable approximation to the relative risk? (Ask what the size of the risk in the two groups are.)
The blood group of 55 women diagnosed as suffering from thromboembolic disease and 145 healthy women are displayed in Table 2.7.Table 2.7 Blood group distribution for healthy women and those with
Ninety-nine pregnant women, with dystocia (diffi cult childbirth or labour), were allocated at random to receive immersion in water in a birth pool(Intervention group: Labour in water 49 women) or
A newspaper headline states that a new drug for early stage breast cancer reduces the risk of recurrence of the disease by 50%. What other information would you like before deciding to take the drug?
Is the number of subjects involved clearly stated?
Are appropriate measures of location and variation used in the paper?For example, if the distribution of the data is skewed, then has the median rather the mean been quoted? Is it sensible to quote a
On graphs, are appropriate axes clearly labelled and scales indicated?
Do the titles adequately describe the contents of the tables and graphs?
Do the graphs indicate the relevant variability? For example, if the main object of the study is a within-subject comparison, has within-subject variability been illustrated?
Does the method of display convey all the relevant information in a study?For example if the data are paired, is the pairing shown? Can one assess the distribution of the data from the information
The age (in years) of a sample of 20 motor cyclists killed in road traffi c accidents is given below.18 41 24 28 71 52 15 20 21 31 16 24 33 44 20 24 16 64 24 32(i) Draw a dot plot and histogram. Is
The table below shows the height of 12 fathers and their fully-grown sons.Father’s height Son’s height(cm) (cm)190 189 184 186 183 180 182 179 179 187 178 184 175 183 174 171 170 170 168 178 165
To whom has the diagnostic test been applied? It is possible that characteristics of the patients or stage and severity of the disease can infl uence the sensitivity of the test. For example, it is
How has the group of patients used in the analysis been selected, and in particular how has the decision to verify the test by the gold standard been made? A common error is to select patients in
How have the investigators coped with specimens they were not able to interpret? If the reason for failure to interpret is essentially random, and is unrelated to disease status, then the test
Did the investigator who provided the diagnostic test result know other clinical results about the patients? Diagnostic tests are usually carried out during or, in conjunction with, the clinical
Was the reproducibility of the test result determined? This could be done by repeating the test with different operators, or at different times, or with different machines, depending on the
Did the patients who had the test actually benefi t as a consequence of the test?
How good is the gold standard? An ideal gold standard either may not exist or be very expensive or invasive and therefore not carried out. In this case, the test used as the gold standard may be
In a group of patients presenting to a hospital casualty department with abdominal pain, 30% of patients have acute appendicitis. Seventy per cent of patients with appendicitis have a temperature
A new laboratory test is developed for the diagnosis of rectal cancer.(a) A sensitivity of 85% implies that 15% of patients with rectal cancer will give negative fi ndings when tested.(b) A specifi
Three tests (A, B and C) for the diagnosis of breast cancer in premenopausal women were assessed against a ‘gold standard’ taken to be 100%accurate. Their sensitivities were A 90%, B 85%, C 80%.
What is the population from which the sample was taken? Are there any possible sources of bias that may affect the estimates of the population parameters?
Have reference ranges been calculated on a random sample of healthy volunteers? If not, how does this affect your interpretation? Is there any good reason why a random sample was not taken?
For any continuous variable, are the variables correctly assumed to have a Normal distribution? If not, how do the investigators take account of this?
A GP estimates that about 50% of her patients have ‘trivial’ problems.What is the probability that four out of fi ve patients in one evening’s surgery have trivial problems?
Suppose a hospital Accident and Emergency department has an average of 10 new emergency cases per hour. Calculate the probability of observing exactly 10 new emergency cases in any given hour.
The systolic blood pressure of 16 middle age men before exercise has a Normal distribution with a mean of 141.1 mmHg and a standard deviation of 13.62 mmHg. What is the probability having a systolic
The diastolic blood pressures (DBP) of a group of young men are Normally distributed with mean 70 mmHg and a standard deviation of 10 mmHg.Decide whether the following statements are true or
Given the sample described in question 4.(i) What is the probability of a young man having a DBP of 95 mmHg or above?(ii) What is the probability of a young man having a DBP of 55 mmHg or less?(iii)
A GP and partners have 6000 patients and refer 27 patients to neurology in one year. In the health authority region, there are 1400 neurology referrals from a population of 500 000.Is this GP’s
When authors give the background information to a study they often quote fi gures of the form a ±b. Although it is usual that a represents the value of the sample mean, it is not always clear what b
A useful mnemonic to decide which measure of variability to use is: ‘If the purpose is Descriptive use Standard Deviation, if the purpose is Estimation, use the Standard Error’.
What is the population from which the sample was taken? Are there any possible sources of bias that may affect the estimates of the population parameters?
Have reference ranges been calculated on a random sample of healthy volunteers? If not, how does this affect your interpretation? Is there any good reason why a random sample was not taken?
Have confi dence intervals been presented? Has the confi dence level been specifi ed (e.g. 95%)?
Has a Normal approximation been used to calculate confi dence intervals for a Binomial proportion or Poisson rate? If so, is this justifi ed?
As the size of a random sample increases:(a) The standard deviation decreases.(b) The standard error of the mean decreases.(c) The mean decreases.(d) The range is likely to increase.(e) The accuracy
A 95% confi dence interval for a mean(a) Is wider than a 99% confi dence interval.(b) In repeated samples will include the population mean 95% of the time.(c) Will include the sample mean with a
Assume that the mid-upper arm circumference in a population of rural Indian children aged 12 to 60 months follows a Normal distribution with unknown mean, m. Ten samples each of four children are
Table 6.6 shows ten random samples of size 16 drawn from the same population of Indian children.(a) Display as a dot plot alongside the previous one, the means of the ten random samples of size 16
(a) Estimate the 95% confi dence interval for one selected sample of size 4 (sample number 10 in Table 6.5) and display this on the dot plot for n = 4.(b) Estimate the 95% confi dence interval for
A surgeon in a large hospital is investigating acute appendicitis in people aged 65 and over. As a preliminary study he examines the hospital case notes over the previous 10 years and fi nds that of
Have clinical importance and statistical signifi cance been confused?
Is it reasonable to assume that the continuous variables have a Normal distribution?
Have confi dence intervals of the main results been quoted?
Is the result medically or biologically plausible and has the statistical signifi cance of the result been considered in isolation, or have other studies of the same effect been taken into account?
Gaffney et al (1994) compared 141 babies who developed cerebral palsy to a control group of (control) babies made up from the babies who appeared immediately after each cerebral palsy case in the
A randomised controlled trial was conducted to investigate the cost-effectiveness of community leg ulcer clinics (Morrell et al 1998). A total of 233 patients were randomly allocated to either
A study by Taylor et al (2002) investigated whether the measles, mumps and rubella (MMR) vaccination was associated with bowel problems and developmental regression in children with autism. The
A UK study by Peacock et al (1995) of factors affecting the outcome of pregnancy among 1513 women reported that the overall incidence of preterm births was 7.5%, SE 0.68% and 95% CI 6.1 to 8.8%.(a)
Have clinical importance and statistical signifi cance been confused?
Has the sample size been taken into account when determining the choice of statistical tests; that is, are small-sample tests used when appropriate?
Is it reasonable to assume that the continuous variables have a Normal distribution?
Have paired tests been utilised in the appropriate places?
Have confi dence intervals of the main results been quoted?
Is the result medically or biologically plausible and has the statistical signifi cance of the result been considered in isolation, or have other studies of the same effect been taken into account?
Table 8.12 shows the 24 hour total energy expenditure of groups of lean and obese women. The aim of this study was to compare total energy expenditure between the lean and obese women.(a) Write out a
Table 8.13 shows the results of a randomised, double-blind, placebo-controlled trial examining whether patients with chronic fatigue syndrome (CFS) improved 6 weeks after treatment with intramuscular
A prospective double-blind clinical trial was conducted in 11 young, health, normally menstruating female subjects to detect any differences in energy intake in the pre and post phases of the
When a correlation coeffi cient is calculated, is the relationship likely to be linear?
Are the variables likely to be Normally distributed?
Is a plot of the data in the paper? (This is a common omission.)

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