Mathematics and Statistics Flashcards
1
Q
Select true or false for each of the following statements.
A. A graph charting the blood glucose measured by a blood glucose meter (BM) versus laboratory glucose measurement should display the ‘line of identity’ if the monitor is accurate
B. The third gas law describes the relationship between the pressure and temperature of a fixed quantity of gas at a fixed volume. The graphical representation of this is a square hyperbola
A
A. True. This should be a linear relationship, with the line passing through the origin with a gradient of 1.
B. False. The general gas law is described as PV = nRT. Rearranged this gives:
therefore, this is a linear relationship.
2
Q
Select true or false for each of the following statements.
A. Ohm’s law describes the relationship between resistance (R), potential (V), and current (I) as
R = V/I
Current flow has a linear and an inverse hyperbolic relationship with potential and resistance respectively
B. To maintain a PaCO2 of 4 kPa with a VCO2 of 200 ml/min, VA must be 5 L/min
A
A. True. Current is directly proportional to potential, i.e. a linear relationship, and inversely proportional to resistance, i.e. an inverse hyperbolic relationship.
B. True.
4 kPa = FACO2 4% = 0.04 = 40 ml/L
40 x 5 = 200 ml/L
3
Q
Regarding simple transformations in the plane, the change between Fig 1 and Fig 2 represents:
A. Translation in the y-direction
B. Reflection in the line y = x
C. Scaling in the x-direction
D. Reflection in the line x = 0
E. Translation in both the x- and y-directions simultaneously
A
A. Incorrect.
B. Correct.
C. Incorrect.
D. Incorrect.
E. Incorrect.
4
Q
The red dotted line in Fig 1 represents the function y = x2. What does the blue line represent?
A. Translation by two units to the right
B. Reflection in the y axis
C. Scaling by a factor of two
D. Translation by two units upwards
E. Scaling by a factor of 0.5
A
A. Incorrect.
B. Incorrect.
C. Correct. All output values are doubled and the resultant equation is y = 2x2.
D. Incorrect.
E. Incorrect.
5
Q
Fig 1 represents the function y = x2. If we were to shift the parabola to the right by two units and scale it by two, the resultant equation would be:
A. y = (x - 2)2
B. y = 2(x - 2)2 + 2
C. y = 2 + x2
D. y = 2(x - 2)2
E. y = 2(x + 2)2
A
A. Incorrect. y = (x - 2)2 only shifts the parabola to the right by 2 units.
B. Incorrect. y = 2(x - 2)2 + 2 shifts the parabola to the right by 2 units, scales it by 2 and shifts the parabola 2 units upwards.
C. Incorrect. y = 2 + x2 only shifts the parabola 2 units upwards.
D. Correct.
E. Incorrect. y = 2(x + 2)2 would shift the parabola 2 units to the left.
6
Q
Which of the following statements are correct?
A. If f(x) is the original function, f’(x) is the derivative and f’‘(x) is the second derivative of f(x)
B. Calculus provides a way to describe algebraically the rate of change of one variable with respect to another
C. The second derivative describes the rate of change of the original function
D. Integration is the reverse of differentiation and, in principle, it enables us to retrieve the original function from its derivative
A
A. Correct.
B. Correct.
C. Incorrect. The first derivative is the rate of change of the original function.
D. Correct.
7
Q
Which graph below describes which common function?
A
8
Q
Select true or false for each of the following statements.
A. The derivative of y = x2 is 2x
B. The derivative of x3 is 3x
C. The derivative of y = mx + c is c
D. If f(x) = kx2, then is f’(x) = 2kx
E. If f(x) = kx3, then is f’(x) = 3kx2
A
A. True.
B. False. The derivative of x3 is 3x2.
C. False. The derivative of y = mx + c is m.
D. True.
E. True.
9
Q
Which of the following statements is not correct?
A. The infinite polynomial series that represents ex:
f(x) = 1+ x + x2/2! + x3/3!… when differentiated term-by-term remains unchanged and this means that the gradient of f(x) = ex is f’(x) = ex
B. Any quantity that is changing at a rate determined by its prevailing value must be a straight line
C. Newton’s Law of Cooling is an example of an exponential function; a hot body loses heat (and hence temperature) at a rate that is determined by the difference between its temperature and that of its surroundings
A
A. Correct.
B. Incorrect.The rate of change of an exponential function is proportional to its prevailing value. This is the only function for which this is true.
C. Correct.
10
Q
Given that in Fig 1, the red wave shows f(x) = sin(x), which of the following statements are correct?
A. The gradient of the sin curve is 0 at the origin
B. The blue dotted line represents cos(x)
C. At the sin peak the gradient is 0
D. When cos is 0, sin is at a maximum or minimum value
A
A. Incorrect. The gradient of the sin curve is 1 at the origin.
B. Correct.
C. Correct.
D. Correct.
11
Q
What is the graphical meaning of integration?
A. It is the slope of a curve
B. It is the sum of a series of y coordinates
C. It is the rate of change of the slope
D. It is the area under the curve
E. It is the sum of the x coordinates
A
A. Incorrect.
B. Incorrect.
C. Incorrect.
D. Correct.
E. Incorrect.
12
Q
For a wash out function y = Ae-kt which of the following statements are correct?
A. If A is large, the area under the curve is large
B. If the rate constant increases, the area under the curve reduces
C. The area under the curve is inversely proportional to A
D. The area is directly proportional to time constant τ
E. If the rate constants of two processes (with the same value of A) are in a ratio of 2:1, the areas under their curves are in the ratio 1:2
A
A. Correct. If A is large, the area under the curve is large.
B. Correct. If the rate constant increases, the area under the curve reduces.
C. Incorrect. The area under the curve is directly proportional to A.
D. Correct. The area is directly proportional to time constant τ.
E. Correct. If the rate constants of two processes (with the same value of A) are in a ratio of 2:1, the areas under their curves are in the ratio 1:2 because the rate constant is inversely proportional to A.
13
Q
The blue curve on Fig 1 represents:
A. logex
B. ex
C. Square root of x
D. x2
E. 1/x
A
A. Incorrect.
B. Incorrect.
C. Correct.
D. Incorrect.
E. Incorrect.
14
Q
How can we express a change in 0.1 pH units?
A. Square root of 3
B. Cubed root of 2
C. log10(1/3)
D. 1/3log10(2)
E. 100.3
A
A. Incorrect.
B. Correct.
C. Incorrect.
D. Incorrect.
E. Incorrect.
15
Q
We wish to simulate the likely number of requests for epidural analgesia per day in our maternity unit, and have collected data on requests for the preceding 12 months. 1543 epidurals were requested in total, varying from none to 10 per day, giving a daily average of 4 requests.
A. No further information is required to create a model to simulate the likely numbers of epidurals requested per day
B. It is important we consider that each request for an epidural is uninfluenced by any other requests
C. The daily rate of epidural requests is a continuous variable
D. A chart of the number of epidural requests per day will produce a symmetrical bell-shaped curve
E. Over the following 12 months there are occasional days when up to 12 requests for epidurals are noted. This must represent either an increase in deliveries, or an increase in the preference for epidurals
A
A. True. A random process of this nature (Poisson process) only requires us to specify an average, or expected value to simulate a model.
B. True. The assumption that each event occurs independently of every other is central to the concept of randomness.
C. False. The number of requests per day is a discrete variable, as we are counting whole numbers. If we choose to consider the time of requests, or interval between requests, then these would be continuous variables, since any fractional value could be represented.
D. False. The distribution will not be symmetrical since there will be a tail of higher numbers of requests, but no negative values will appear. To describe a curve is also somewhat misleading, as we will see jumps between whole numbers counted, with no fractional values.
E. False. We expect to see a range of numbers of requests, with values close to the expected value being more frequent than those further away from it. It is entirely plausible that in a different sample these occasional high values might occur without any actual change in the underlying rate. However, should we begin to see these outlying values occurring more frequently, we should question whether the underlying rate was indeed changing.
16
Q
Consider the population of mean arterial pressure (MAP) measurements from all healthy, non-hypertensive, adult patients.
Question: What type of data are we describing?
Question: What statistical model is appropriate to describe such data
Question: What information do we need in order to precisely describe the graph of that distribution?
A
Answer: Continuous, quantitative, numerical data.
Answer: A normal distribution.
Answer: We need two values: the mean and the variance (which is the square of the standard deviation). Once we know these values we can draw a graph of the normal distribution of all possible MAP readings for our defined population. All populations of data that can be described by a normal probability model are defined by these two parameters: the population mean (μ) and the population variance (σ2).
17
Q
Which parameters are necessary to describe the following statistical distributions?
A
18
Q
Regarding statistical models:
A. The Exponential distribution is based on a Poisson model
B. The Binomial distribution is continuous
C. The normal distribution is describe by two parameters
D. The Poisson distribution is discrete
E. Population parameters are always known
A
A. True.
B. False. The Binomial distribution is a discrete distribution.
C. True. The normal distribution is described by its mean and variance.
D. True. The Poisson distribution looks at the number of events in a given time period, i.e. interval data that is numerical but discrete.
E. False. We rarely know the true population parameters since we cannot take observations from every single member of the defined population.
If you are unsure about the terms ‘discrete’ and ‘continuous’ you may need to revise earlier statistical sessions, especially ‘Random variables and distributions’.
19
Q
For data that is modelled by a normal distribution, which of the following statements are correct?
A. The sample mean is a biased estimator of the population mean
B. The population variance is greater than the within-sample variance
C. The larger the sample, the smaller the standard error of the mean
D. The population mean is always smaller than the sample mean
E. The standard error of the mean is the same as the sample standard deviation
A
A. Incorrect. The sample mean is an unbiased estimator of the (unknown) population mean.
B. Correct. The population variance is greater than the within-sample variance.
C. Correct. The standard error of the mean (SEM) is the standard deviation of the sample divided by the square root of the number of observations SEM = s/√n. Thus the SEM gets smaller as the sample size increases; the SEM is always smaller than than the sample standard deviation.
D. Incorrect. The population mean is unknown; it can lie either side of the sample mean.
E. Incorrect. The SEM is always smaller than than the sample standard deviation.
20
Q
Regarding sample statistics and population parameters:
A. The Binomial parameter p, is always known
B. For a Binomial model, the population mean can be estimated by the product of the number of observations and the sample estimate, p
C. In a Poisson distribution, the variance is the inverse of the mean
D. Sample data from a Poisson model can be used to estimate the population parameter, λ, for the exponential distribution of times between events
E. The variance of a normal distribution can be estimated from sample data using the formula: s2=Σ (xi–m)2/n
A
A. False. The probability of a ‘true’ event is unknown, but the number of trials in a particular Binomial model is known.
B. True.
C. False. In a Poisson distribution, the mean and variance are the same.
D. True.
E. False. The sample variance, when used to estimate the population variance, is the sum of the residuals squared and divided by (n-1), not n.
21
Q
Regarding the statistical models and their parameters:
A. A Binomial model has one unknown parameter, p
B. An exponential distribution is discrete
C. The Poisson distribution has a variance that is equal to 1μ
D. A normal distribution is defined by two independent parameters
E. In a Poisson model, 1/μ = tλ
A
A. True. A Binomial model requires two parameters, but one is always known – n, the number of trials.
B. False. The exponential distribution is a continuous probability distribution.
C. False. The mean and the variance are the same and described by the parameter μ.
D. True. Both the mean and the variance must be known.
E. False. μ = λt.
22
Q
I wish to estimate the proportion, p, of hospital doctors who are vegetarians. I email 75 doctors out of 750 doctors in my own Trust and ask about their dietary preferences. Fifty of these reply, of whom 10 say they are vegetarians.
A. The best model to choose is a Binomial model
B. My estimate for the population parameter, p, is 0.13
C. If I take a further sample of 30 doctors, I would expect six to be vegetarians
D. I estimate that the variance in the number of vegetarian doctors in my Trust is 8
E. In an adjacent Trust hospital there are 500 doctors. I would estimate that 100 +/- 80 (mean +/- SD) are vegetarians
A
A. Correct.
B. Incorrect. Only 50 replied, so 10 / 50 = 0.2 is my estimate for p.
C. Correct. 30 x 0.2 = 6.
D. Incorrect. The variance in a Binomial model is npq, where
q = 1 - p. The variance is therefore 750 x 0.2 x 0.8 = 120.
E. Incorrect. The estimate for the mean is correct, but the SD is incorrect.The estimated variance is 500 x 0.2 x 0.8 = 80 so the SD. is the square root of this, which is 8.94.
23
Q
I wish to estimate the number of patients who have an anaphylactic reaction during anesthesia in a hospital with 1600 beds. I have a record of the number of these adverse events for the last 20 years: 3, 4, 3, 6, 5, 3, 4, 3, 5, 4, 3, 5, 4, 5, 3, 5, 3, 5, 4, 3. I believe a Poisson model will be appropriate.
A. My estimate for the parameter µ is 3 reactions per year
B. If I had taken just the first 10 years of data, my estimate for the mean number of anaphylactic reactions per year would be 4
C. An estimate for the related variable, λ, of the corresponding exponential distribution of timesbetween anaphylactic reactions is 91.25 days
D. If my model is suitable, my estimated population standard deviation will be equal to the population mean
E. The probability of a patient in my hospital having an anaphylactic reaction is 0.025
A
A. Incorrect. The mean number of reactions, µ , is 4; the modal number of reactions is 3.
B. Correct.
C. Correct. The population parameter for the corresponding exponential distribution is the mean time between events. In 1 year we estimate that there are four events, so this gives an estimated value of 365 / 4 = 91.25 days.
D. Incorrect. In a Poisson model the variance, not the standard deviation, is equal to the mean.
E. Incorrect. We have no information about the number of anaesthetics given per year; it is not related to the number of beds in the hospital.
24
Q
For data that is modelled by a normal distribution, which of the following statements are true?
A. The data must always be quantitative and discrete
B. The population variance is greater than the within-sample variance
C. The larger the sample, the closer the sample mean is likely to be to the true population mean
D. The population mean is always smaller than the sample mean
E. The standard error of the mean is smaller than the sample standard deviation
A
A. Incorrect. The normal distribution is used for continuous, numerical data.
B. Correct. The population variance is greater than the within-sample variance.
C. Correct. In general, the larger the sample the closer the sample mean is likely to be to the true population mean.
D. Incorrect. The population mean is unknown, it can lie either side of the sample mean.
E. Correct. The standard error of the mean is the sample standard deviation divided by the square root of the sample size, so is always smaller than the sample standard deviation.
25
I’m concerned that a die used in a board game isn’t fair. I seem to roll a 2 more often than any other number. I’d like to estimate the mean number of times this particular die shows a 2 when thrown. Which of the following statements are true:
A. A Poisson model for discrete data would be suitable
B. If the die is fair, I would expect the probability of throwing a 2 to be p = 1/6 in a Binomial model
C. I expect the ratio of p to q in my model to be approximately 1:6 if the die is fair
D. If I roll the die 12 times it will give me a good, reliable estimate for p
E. If I roll the die 360 times and a 2 is seen on 90 occasions, then my calculated estimate for q is 0.75
A. Incorrect. The most appropriate model is a Binomial model. The parameter I need to estimate is p, the probability of throwing a 2. The probability of throwing any other number is q, where
q = 1 – p.
B. Correct. There are six sides to a die; if fair each has an equal likelihood of being thrown. Thus I would expect a 2 to be thrown 1/6 of occasions.
C. Incorrect. Any number other than 2 should be thrown on 5/6 of occasions. The ratio of p to q in this case would be 1:5, not 1:6.
D. Incorrect. To get a reliable estimate for p, the die should be rolled a large number of times; 12 is insufficient.
E. Correct. If a 2 is seen on 90 out of 360 occasions, then my estimate for p will be 90/360 = 0.25, so my calculated estimate for q is 1 – 0.25 = 0.75.
26
You believe that the daily number of emergency admissions to your intensive care unit can be modelled according to a Poisson model. Which of the following would best help you find the population parameter, μ, to describe your model?
A. Do a literature search to identify similar units to yours that have already published their data
B. Keep a record of the number of admissions for the next 7 days and use the average value for your estimate
C. Look over the records for the last 2 months and count the number of admissions in order to find an average value
D. Look at data for the same month last year and use this to find an average rate of admissions as this will better reflect seasonal variation
E. Keep a record of the number of admissions over the next 2 months and use this to estimate the average number of admissions per day
A. Incorrect. You are not interested in other units, which are likely to have a different number of beds and case-mix. They represent different populations.
B. Incorrect. The best method is to collect data prospectively for a reasonable number of days; 7 days is insufficient and will not give a reasonable estimate.
C. Incorrect. Retrospective data, even for the recent past, is not reliable.
D. Incorrect. Seasonal variation may be important; however last year may have differed from this year – for example a different number of beds may have been open.
E. Correct. Prospective collection of data over 2 months is the most reliable method of estimating the mean number of admissions in a day.
27
Consider the simple clinical questions below:
A. Fentanyl acts more quicky in women than in men
B. Sevoflurane is associated with more nausea and vomiting in the first hour postoperatively than desflurane
C. A remifentanil infusion is more likely to cause postoperative hypertension when used with sevoflurane than with propofol
D. A pregnant woman in labour with effective regional analgesia is more likely to need an urgent Caesarean section if ropivacaine is used rather than bupivacaine
A. True. Fentanyl acts more quickly in women than in men.
B. True. More patients report nausea and vomiting in the first hour postoperatively after a sevoflurane than a desflurane anaesthetic.
C. False. Postoperative hypertension following an intraoperative remifentanil infusion is more common when it is used with propofol than with sevoflurane.
D. True. An epidural using ropivacaine for analgesia during labour is more likely to result in an urgent Caesarean section than if bupivacaine had been used.
These clinical hypotheses define our questions. Notice that these hypotheses are very general, they deal with the populations we are interested in, not any specific study group. This is important. We now need to translate these into statistical hypotheses.
28
The null hypothesis:
A. Is tested to provide evidence against a clinical hypothesis
B. Must define the populations being studied
C. Is always accompanied by an explicit alternative hypothesis
D. Is determined by the statistical test to be used
E. For no difference between means can be written:
H0: μ1= μ2
A. False. The null hypothesis is tested so that evidence may be provided against a neutral position, not your clinical hypothesis.
B. True. This ensures that those reading the results are clear about the populations being studied.
C. False. The alternate hypothesis is often implicit, not explicit.
D. False. The null hypothesis is just a statement of the neutral position and is not influenced by the statistical test.
E. True.
29
When interpreting the result of testing the null hypothesis with a significance level of 0.05:
A. The risk of a false positive result is 1 in 50
B. The p-value is the probability that the result could have arisen by chance
C. The significance level depends on the statistical test used
D. The significance probability is the probability that the null hypothesis is incorrect
E. If the p-value is 0.04 then the null hypothesis should be rejected
A. Incorrect. With a significance level of 0.05, there is a 1 in 20 chance of a false positive result.
B. Correct. The p-value is the probability that the result could have arisen by chance.
C. Incorrect. The significance level is chosen by the investigator, not according to the test.
D. Incorrect. The significance probability (or p-value) is the probability that the result could have arisen by chance, in other words the probability that the null hypothesis is correct.
E. Correct. Since 0.04 <0.05, we will reject the null hypothesis at our 0.05 level of significance.
30
Which of the following statements about errors in statistical tests are true?
A. A type I error is the same as the probability of a false negative result
B. The power of the study is 1 – β error
C. The β error is the same as the significance level
D. An α error is the probability of a false positive result
E. The sum of the α and β errors is always 1
A. False. Type I error = α error = probability of a false positive result.
B. True. Power = 1 – β error.
C. False. The investigator sets the significance level, which is the equivalent of the α error.
D. True.
E. False. The α and β errors are not related in this way.
31
When interpreting a hypothesis test at a significance level of 0.01:
A. The chance of a false positive is 0.01
B. If the test result has a probability 0.001 then the null hypothesis is correct
C. The power of the study is affected by the sample size
D. If the null hypothesis is rejected but it is actually true, this is a type I error
E. Specificity is the power of the study
A. True. The significance level is the probability of a false positive.
B. False. A probability of 0.001 is the probability the test result arose by chance; 0.001 <0.01 so the null hypothesis should be rejected.
C. True.
D. True. Rejecting the null hypothesis when it is actually true is a false positive – a type I error.
E. False. Sensitivity is the power of the test, not specificity.
The sensitivity of any test is the likelihood that it will correctly identify a true positive, i.e. it will give a positive result when our hypothesis is actually true.
The specificity of any test is the likelihood it will correctly identify a true negative i.e. it will give a negative result when our hypothesis is actually false.
32
We want to know if ropivacaine has a more rapid onset of action than bupivacaine at equipotent dose when used for epidural anaesthesia in orthopaedic joint replacement surgery. Which of the following statements best defines a suitable null hypothesis?
A. There is no difference in speed of onset of epidural block using ropivacaine compared with bupivacaine
B. Bupivacaine has a slower onset of neuraxial block than ropivacaine
C. There is no clinically important difference between speed of onset of block using ropivacaine compared with bupivacaine
D. There is no statistical difference in speed of onset of epidural block in the group receiving ropivacaine compared with the group receiving bupivacaine
E. There is no difference in speed of onset of epidural block using ropivacaine compared with bupivacaine in patients undergoing orthopaedic joint replacement
A. Incorrect. We are particularly interested in patients undergoing orthopaedic joint replacement surgery, so this population must be defined in the null hypothesis. This statement does not specifically refer to the population we are interested in.
B. Incorrect. This statement may be a statement of our clinical hypothesis – but it is not a neutral stance, as expected by the null hypothesis.
C and D. Incorrect. These statements mention 'clinically important difference', study groups (samples) and 'statistically significant difference' none of which should be included in the null hypothesis.
E. Correct. For the reasons above, this statement is the best choice.
32
200 patients, of whom 100 are known to have the disease being studied, were tested using a new diagnostic test. The sensitivity was found to be 80 % and the specificity 90 %.
Sensitivity is the probability of identifying true positives, whereas specificity is the probability of identifying true negatives.
So 80% of 100 would be true positives (80) and 20% false negatives (20).
There would be 90% of 100 true negatives (90) and 10% false positives (10).
33
When carrying out a hypothesis test of the difference between two means using a two-sample t-test at the 0.01 level of significance:
A. The value of the test statistic depends on the number of observations in each sample
B. If the test result has a probability 0.04 then the null hypothesis is rejected
C. The power of the test is always 0.99
D. If the test result gave a probability of 0.01 the null hypothesis must be false
E. The specificity is 0.99
A. True. The test statistic is dependent on sample size.
B. False. A probability of 0.04 is greater than 0.01, our significance level, so there is insufficient evidence to reject the null hypothesis.
C. False. The power is 1 - β error, not 1 - α error.
D. False. Just because the test result reaches significance and provides evidence against the null hypothesis, there is always a possibility that the result arose by chance and the null hypothesis is correct – this is a false positive error.
E. True. Specificity is the ability of the test to identify a true negative, this is 1 – α error = (1 – 0.01) = 0.99.
34
For samples of the same size taken from a single population that is normally distributed:
A. The distribution of sample means has the same standard deviation as the distribution of the actual data values
B. The variance of the distribution of sample means is independent of the sample mean
C. If the sample size increases, the variance of the distribution of sample means decreases
D. For sample size 25, mean 9 and variance 25, the standard deviation and variance of the distribution of sample means are equal
E. The population variance is the same as the sample variance
A. False. Sample means are distributed normally around the population mean but the standard deviation is σ/√n not σ.
B. True. For any normally distributed data, the mean and variance are independent parameters.
C. True.
D. True. The variance of the distribution of sample means is σ2/n = 25/25 = 1 so the standard deviation (σ/√n) is also 1.
E. False. The sample variance estimates the population variance, but this varies from sample to sample.
35
We have taken a sample from a population which is normally distributed. Our sample size is 16, mean 30, variance 36 and 95 % of observations lie within 2.13 SEM (standard error of the mean):
A. The SEM is 2.25
B. The SEM is 1.5
C. The lower limit of the 95 % CI is 26.8 (to one decimal place)
D. The upper limit of the 95 % CI is 34.8 (to one decimal place)
E. There are 16 degrees of freedom for the t-distribution required to calculate the CI
A. Incorrect. The SEM is s/√n, so in this example is 6/4 = 1.5.
B. Correct.
C. Correct. The lower limit is 30 - 2.13 x 1.5 = 26.805. This is 26.8 to one decimal place.
D. Incorrect. The upper limit is 30 + 2.13 x 1.5 = 33.195. This is 33.2 to one decimal place.
E. Incorrect. The number of degrees of freedom is n – 1, where n is the sample size, so the number of degrees of freedom in this example is 15, not 16.
36
I would like to know what proportion (p) of patients admitted with subarachnoid haemorrhage (SAH) are female. At my local Neurological Centre I found that out of 144 patients with SAH, 48 are male. I would like to construct confidence intervals to help describe my findings. Which of the following statements are correct?
A. q = p/2
B. The sample variance is 32
C. CI for p = [1/3 – (1.96 x √2)/36, 1/3 – (1.96 x √2)/36]
D. When calculating the CI for the mean number of female patients with SAH we must find the standard error of the mean
E. CI for mean = 144(2/3 ± 1.96 x √[(2/3) x (1/3)/144])
A. Correct. Out of 144 patients 48 were male, so 96 were female so my estimate for p is 96/144 = 2/3 and my estimate for q = (1 – p) = 1/3. Thus q = p/2 for my sample and A is correct.
B. Correct. The sample variance is given by npq, so is 144 x 2/3 x 1/3 = 288/9 = 32, so B is correct.
C. Incorrect. The CI for p is given by p ± [1.96 x √(pq/n)] = 2/3 ± [1.96√(2/1286)] = 2/3 ± [(1.96 x √2)/36]. C is incorrect because it uses q, 1/3, not p to construct the CI.
D. Incorrect. We use the standard error of the mean only for data when the variance is independent of the mean and is unknown. This is not the case for the Binomial distribution, since the mean and variance are calculated from the same parameter, so D is incorrect.
E. Correct. The CI for the mean number of females with SAH is n times the CI for the proportion p. So CI = 144(p ± 1.96√(pq/n)) = 144 (2/3 ± 1.96 x √[(2/3) x (1/3)/144]) so D is correct.
37
Pupils learning statistics are interested in the mean blood pressure of all the teaching staff in their secondary school. The school nurse has a record of all these values, the mean is 81 mm Hg and the variance 16 mm Hg. The pupils have taken a representative sample of 25 teachers and found the mean blood pressure to be 78 mm Hg with a variance of 25. Which of the following most accurately describes the mean blood pressure of staff teaching in this school as estimated by the pupils?
A. 81 ± 5 mm Hg
B. 78 ± 1 mm Hg
C. 81 ± 1 mm Hg
D. 78 ± 5 mm Hg
E. 81 ± 4 mm Hg
In this scenario we are interested in the pupils’ estimate of the mean blood pressure, not the true population value, so we can discount any options that use the population mean. This eliminates A, C & E and leaves B & D.
Since the pupils are interested in the sample mean value as an estimate of our population value, we should quote mean ± standard error of the mean (SEM) and not mean ± standard deviation (s). s = √sample variance = √25 = 5; SEM = s/√n = 5/5 = 1 Thus D is incorrect, since it quotes mean ± s, and B is our correct choice.
38
I am suspicious that a coin used by my fellow gambler is biased. I would like to estimate the probability (p) of it landing heads-up and decide to toss the coin 900 times. On 300 occasions it lands heads-up. I also want to see if my suspicion is correct by calculating confidence intervals (CI).
Which of the following statements are correct?
A. The sample standard deviation is 10√2 heads-up
B. CI for p = [1/3 – (1.96 x √2)/90, 1/3 + (1.96 x √2)/90]
C. A continuity correction must be used
D. CI for mean number of heads = 300(1/3 ± 1.96 x √[(1/3) x (2/3)/900])
E. q = 2p
A. Correct. The sample mean is 300 heads, the sample variance is given by npq so is 900 x 1/3 x 2/3 = 200, this means the sample standard deviation is √(200) = √(2 x 100) = √2 x √100 = 10√2, so A is correct.
B. Correct. The CI for p is given by p ± [1.96 x √(pq/n)] = 1/3 ± [1.96√(2/8100)] = 1/3 ± [(1.96 x √2)/90], so B is correct.
C. Incorrect. The CI for a proportion may or may not require a continuity correction – in this case the number of observations is very large and the correction would be extremely small (1/2n = 1/1800) compared with the width of the interval [1.96√2)/90] calculated above. It will not alter the width of my CI when calculated to three decimal places therefore in this case the continuity correction is not needed. So C is incorrect.
D. Inorrect. The CI for the mean number of heads is n times the CI for the proportion p. So CI = 900(p ± 1.96√(pq/n)) = 900(1/3 ± 1.96√[(1/3) x (2/3)/900) = 300 ± 900(1.96 x √ [(1/3) x (2/3)/900]) so D is incorrect.
E. Correct. My estimate for p is 300/900 = 1/3, so my estimate for q = (1 – p) = 2/3 and E is correct.
39
Match the method of calculating confidence intervals to the conditions given, where m is the sample mean, p the sample proportion, σ2 the population variance, SEM the standard error of the mean and d the 5 % percentage point for the t-distribution with (n – 1) degrees of freedom.
40
When referring to confidence intervals (CI):
A. If the population variance is unknown you cannot calculate a CI for the mean for normally distributed data
B. A 90 % CI is narrower than a 95 % CI
C. The width of the confidence interval is always the same for samples of the same size
D. On 19 out of 20 occasions the 95 % confidence interval around a sample mean will include the population mean
E. A CI for a proportion requires the 5 % percentage point for a t-distribution on n – 1 degrees of freedom
A. False. The population variance is usually unknown; we estimate it using sample variance and use this to construct a CI for the mean.
B. True.
C. False. Each sample will have a different variance and as the width of the CI is dependent on the sample variance, the width of the CI will vary from sample to sample – even for samples of the same size.
D. True.
E. False. The t-distribution is used to construct CI for the mean for normally distributed data when the population variance is unknown and the sample is small.
41
Here we have nine observations, and their values, ranked in ascending order:
10, 12, 14, 15, 17, 19, 20, 22, 26.
Which of the following are true?
A. The data have been ranked, so cannot be normally distributed
B. The median is 17
C. An estimate for the population variance is 5.07
D. The normal score for the 3rd quantile is -1.25
E. A z-plot could be used to test the hypothesis that these data are normally distributed
A. Incorrect. Any set of numerical data can be ordered and ranked from lowest to highest value. This has nothing to do with the data distribution.
B. Correct. The median value is the (n + 1)/2 value: n is 9 here so we want the 5th value, which is 17.
C. Incorrect. The population variance is 25.7; 5.07 is the population standard deviation.
D. Incorrect. The normal score required is the solution to the equation: Φ(z) = 3/10, i.e. the number of standard deviations from the mean, below which there should be 33 % of observations. We know that only 16 % of observations lie below -1 standard deviations from the mean and 50 % lie below the mean. The value given cannot be correct, since it must lie between -1 and 0; it is actually -0.43.
E. Correct. A normal probability plot, or z-plot, could be used to test for normality.
42
Here we have 11 observations, and their values, ranked in ascending order:
5, 6, 8, 9, 11, 13, 14, 15, 15, 17, 19.
Which of the following are true?
A. For these data, the median is the same as the mean
B. The standard deviation is 4.56
C. The standard normal variate for the 4th value is -1
D. The normal score for the 6th quantile is 0.5
E. The normal score for the 9th quantile is 0.675
A. Incorrect. The median is the (n+1)/2 or the 6th value, which is 13. The mean is 12, so they are not the same.
B. Correct. The standard deviation is 4.56 to 2 decimal places.
C. Incorrect. The 4th value is 9, so the corresponding normal score is (9-12)/4.56 = -0.658. If you had calculated the standard deviation correctly, then it is clear that 9 is less than one standard deviation from the mean, which is 12. You can see the answer must be incorrect without doing any calculations.
D. Incorrect. The normal score is the solution to the equation:
Φ(z) = i/(n+1); here i = 6, which represents the median. The normal score for the median is 0, not 0.5.
E. Correct. The normal score for i = 9 is 0.675. The 9th observation corresponds to the upper quartile, and this correlates with a value 0.675 standard deviations above the mean.
43
The following data have been collected and we wish to compare the means of some of these samples using a two-sample t-test. Assume that all samples are approximately normally distributed.
Sample set 1 (n=7): 4, 6, 9, 12, 15, 18, 20
Sample set 2 (n=9): 2, 5, 7, 9, 13, 15, 19, 22, 25
Sample set 3 (n=6): 10, 19, 28, 33, 40, 44
Working to two decimal places, which of the following are correct?
A. None of these pairs can be compared because all the samples are of a different size
B. The value for F, when an F-test is used to compare the variances of samples 2 and 3, is 2.63
C. When comparing samples 1 and 3, an F-test gives a value for F of 0.22
D. An F-test on (6, 8) degrees of freedom should be used to compare variances of samples 1 and 2
E. An F-test is a one-tailed test
A. Incorrect. The two-sample t-test does not require the sample sizes to be equal, so these samples can be compared.
You should first calculate the variances for all three samples: the variance of sample 1 is 36.33, that of sample 2 is 62.75 and that of sample 3 is 164.8.
B. Correct. The value of the test statistic F when comparing samples 2 and 3 is 164.8/62.75 = 2.63, which is less than 3.
C. Incorrect. The value of F, when comparing samples 1 and 3, is 164.8/36.33 = 4.54, not 36.33/164.8 = 0.22. F always takes a value greater than 1.
D. Incorrect. For F values, the larger variance is the numerator and the smaller the denominator, and the degrees of freedom are quoted in this order too. So when comparing samples 1 and 2, the degrees of freedom will be [(9 - 1), (7 - 1)], i.e. (8, 6) not (6, 8).
E. Correct. The F-test is a one-tailed test: we are looking only at one end of the distribution by always placing the larger over the smaller variance.
44
You are asked to compare two samples of data, both normally distributed, using a t-test. There are 12 observations in the first group and 16 in the second. The mean and standard deviation of the first group are 26 and 8, respectively, and 22 and 6 of the second group.
Regarding these samples:
A. The test statistic T is distributed according to a t-distribution with 27 degrees of freedom
B. A two-sample t-test cannot be used because the difference in variance between the two samples is too great
C. A z-plot could be used to confirm that both samples are approximately normally distributed
D. The value of the pooled variance is 47.85
E. The two-sample t-test cannot be used because the samples are of different sizes
A. False. A two-sample t-test is needed; the appropriate number of degrees of freedom is n1 + n2 - 2, which is 26.
B. False. The variances of the two groups are 64 and 36, respectively. This is close enough - the larger is no more than 2.5 times the smaller.
C. True.
D. True. The pooled variance is
[(n1 - 1)s12 + (n2 - 1)s22]/(n1 + n2 - 2)
= [(11 x 64) + (15 x 36)]/26
= 1244/26
= 47.85.
E. False. The samples do not have to be the same size for a two-sample t-test.
45
When comparing two data sets:
A. The F-test is a two-tailed test of the ratio of variances of two samples
B. A normal probability plot takes the same form as a cumulative frequency plot
C. If the result of an F-test is p >0.05, then the null hypothesis is rejected
D. A Mann-Whitney U-test can be used to compare the samples if the z-plot is non-linear
E. If the samples are of equal size, a single-sample t-test must be used
A. False. The F-test is a one-tailed test.
B. False. A normal probability plot takes the same form as a z-plot, not a cumulative frequency plot.
C. False. We accept the null hypothesis if p>0.05 but reject it if p<0.05.
D. True. The Mann-Whitney U-test is the non-parametric equivalent of a two-sample t-test.
E. False. Just because the samples are of equal size, it does not automatically follow that a single-sample t-test should be used. If the samples are paired, then we may use it, but not otherwise.
46
When comparing two data sets:
A. The two-sample t-test can only be used if both samples are approximately normally distributed and have similar variances
B. The null hypothesis for the two-sample t-test is that the variances of the two populations are the same
C. The test statistic for the ratio of variances test is T
D. The single-sample t-test is a test of the mean of the differences between matched observations being zero
E. The degrees of freedom associated with a two-sample t-test with sample sizes 15 and 17 is 32
A. True.
B. False. The null hypothesis is that the means are the same.
C. False. The test statistic is F for the ratio of variances test (an F-test).
D. True.
E. False. The number of degrees of freedom is 15 + 17 - 2 = 30.
47
Match the pairs of samples to the appropriate test. The samples are as follows:
Sample set 1: Heights of 9 year old girls, n = 12, m = 140 cm, s = 10 cm.
Sample set 2: Heights of 9 year old boys, n = 15, m = 130 cm, s = 9 cm.
Sample set 3: Heights of 15 year old girls, n = 12, m = 160 cm, s = 8 cm.
Sample set 4: Heights of 15 year old boys, n = 15, m = 175 cm, s = 15 cm.
The number of degrees of freedom (DF) for a two-sample t-test is the total number of observations from both groups minus two.
For 28 DF We must be comparing the two larger groups i.e. samples 2 and 4. For 22 DF We must be comparing the two smaller groups i.e. samples 1 and 3.
We now need to match the remaining pairs, samples 1 and 2 and samples 3 and 4, with their correct F-test.
The DF for the F-test are given in the same order as the size of the variances of the two groups, i.e. largest variance first.
For (11, 14) DF, the variance of the small group must be greater than that of the large group, so we must be comparing samples 1 and 2.
For (14, 11) DF, the variance of the large group must be greater than that of the small group, so we are comparing samples 3 and 4.
48
We have collected data on the number of RBCs in urine samples from two groups of patients; group 1 have proven carcinoma of the kidney, group 2 have infections of the renal tract.
Group 1: 29, 30, 35, 37, 45, 50, 54, 56, 63, 70, 71
Group 2: 14, 17, 19, 24, 27, 30, 33, 36, 41, 43, 46
We wish to compare the two groups to see if there is a difference in mean RBC count between the two groups of patients.
Which of the following is our best approach?
A. There are equal numbers in each group, so we can use a paired t-test
B. An F-test will show that the variances are very different, so we should use a Mann-Whitney U-test
C. A normal probability plot could be done to check that the differences between pairs of observations are normally distributed, followed by a paired t-test
D. A z-plot could be done to check both groups are normally distributed, followed by a two-sample t-test
E. An F-test will show that the variances are similar, so we can use a two-sample t-test
A. Incorrect.
B. Incorrect.
C. Incorrect.
D. Correct.
E. Incorrect.
Although there are equal numbers in each group, they are not matched, so a paired t-test is inappropriate, but a two-sample t-test might be used if the pre-conditions are met.
We can calculate a value for F, since the variance of group 1 is 231.3 and that of group 2 is 118.2, thus F<2, and we know the variances are similar.
The other pre-condition to meet is normality for the two groups. A z-plot or a normal probability plot could be used for this.
A two-sample t-test is a parametric test that determines whether or not the difference between the means of two samples is different from zero
This differs from the single-sample t-test , which is used to determine whether the mean difference between paired observations differs from zero
Before a two-sample t-test is used, it is essential to confirm that both samples come from normal distributions and that the two samples have similar variances
49
Fig 1 shows bivariate datum pairs. What is the correlation coefficient?
A. -1
B. -0.8
C. 0
D. +0.8
E. +1
F. None of the above
The data are positively correlated because y increases with x. As a perfect straight line with a positive slope would have a correlation coefficient of +1, the approximate value must be +0.8.
The equation is calculated using linear regression analysis.
A. Incorrect.
B. Incorrect.
C. Incorrect.
D. Correct.
E. Incorrect.
F. Incorrect.
50
Fig 1 shows bivariate datum pairs. What is the correlation coefficient?
A. -1
B. -0.5
C. 0
D. +0.5
E. +1
F. None of the above
The data are negatively correlated. As a perfect straight line with a negative slope would have a correlation coefficient of -1, the approximate value must be -0.5.
A. Incorrect.
B. Correct.
C. Incorrect.
D. Incorrect.
E. Incorrect.
F. Incorrect.
51
Fig 1 shows bivariate datum pairs. What is the correlation coefficient?
A. -1
B. -0.5
C. 0
D. +0.5
E. +1
F. None of the above
The datum pairs are randomly scattered and there is no correlation (coefficient = zero).
A. Incorrect.
B. Incorrect.
C. Correct.
D. Incorrect.
E. Incorrect.
F. Incorrect.
52
Fig 1 shows bivariate datum pairs. What is the correlation coefficient?
A. -1
B. -0.5
C. 0
D. +0.5
E. +1
F. None of the above
There is clearly a non-linear relationship between the two variables so determination of a (linear) correlation coefficient is inappropriate.
A. Incorrect.
B. Incorrect.
C. Incorrect.
D. Incorrect.
E. Incorrect.
F. Correct.
+
53
Regarding a Bland-Altman plot:
A. It is only valid for normally distributed bivariate data
B. It shows the mean of each individual datum pair on the y-axis
C. It is the preferred method of showing the relationship between height and weight of patients
D. The limits of agreement are equal to +/- 1 SD from the mean of the individual differences between datum pairs
E. Zero bias means that the two methods of measurement are equivalent
A. False. The data do not need to be normally distributed.
B. False. The difference between datum pairs is shown on the y-axis.
C. False. Correlation and linear regression analysis are more appropriate.
D. False. +/- 2 SD.
E. False. Even though the bias (average discrepancy) of the two methods may be zero, if the limits of agreement are very wide, the two methods of measurement cannot be viewed as equivalent.
54
Regarding the Pearson correlation coefficient:
A. It may be used for non-parametric bivariate data
B. A value of 2.4 corresponds to a good correlation
C. A value of 0.2 is not statistically significant
D. A value of 0 indicates the analysis was inappropriate
E. It corresponds to the slope of the regression line
F. It may take a negative value
A. False. It is used for normally distributed bivariate data.
B. False. It must lie within the range –1 to +1.
C. False. Statistical significance is not directly related to the value of r.
D. False. This indicates no correlation between the two variables.
E. False. The slope of the regression line is determined by linear regression analysis.
F. True. For negatively correlated data (y decreases with x).
55
The linear regression line (line of best fit):
A. May only be validly determined if r>0.5 or <-0.5
B. Passes through the point corresponding to the mean of all x values and the mean of all y values
C. Is drawn so as to minimize the sum of the residuals
D. Takes the form y = ax + b where a and b are constants
E. Is indicated on a Bland-Altman plot
A. False. Validity is independent of r although the p-value for r must be statistically significant.
B. True.
C. False. The line of best fit is drawn so as to minimize the sum of the squared residuals.
D. True. This is the equation of a straight line.
E. False.
56
The Bland-Altman analysis:
A. Is appropriate when examining the possible relationship between age and time to recovery after a standardized anaesthetic
B. Is appropriate when comparing time to 80 % recovery of train-of-four ratio using both electromyography and acceleromyography
C. Gives narrow limits of agreement provided r is highly statistically significant (p<0.001)
D. Calculates the bias which corresponds to the mean discrepancy between two methods of measurement
E. Is a form of scatterplot
A. False. Correlation and linear regression analysis are appropriate.
B. True. These are two different methods of measuring the same thing.
C. False. The limits of agreement or not directly related to the significance of the correlation coefficient.
D. True.
E. True. With the mean of each datum pair on the x-axis and the difference between each datum pair on the y-axis.
57
Which data may be analyzed most appropriately with which statistical test?
The Pearson correlation coefficient is used if the data are normally distributed.
Either the Spearman or Kendall coefficients may be used when the data are not normally distributed.
Once a statistically significant correlation has been found, linear regression analysis may be undertaken to define mathematically the relationship between the two variables.
Bland-Altman analysis is more appropriate and useful than determination of a correlation coefficient if the two variables are in fact the same variable measured by two different methods.
If the correlation coefficient is not statistically significant (p>0.05), no further analysis is valid.
58
Raw categorical data is difficult to interpret, particularly when there is a lot of it. It may be summarized and presented in one of two ways; either as tables or graphs.
Which do you think are the specific advantages of using graphs over tables to present data?
A. They are precise
B. They have more visual impact
C. They provide all the data
D. They are useful to quickly identify patterns
A, C. Incorrect. Tabular representations of data are precise and provide the reader with all the data so that they are able to perform any statistical calculations for themselves.
B, D. Correct. Graphs are less precise but have more visual impact in presentations. They are useful in quickly identifying patterns in the data.
59
Which of these variables are nominal categorical, ordinal categorical or continuous data? Note that VAS is the visual analogue scale.
Height, weight and visual analogue scale pain scores are interval scale data as they are continuous.
Gender and survivor status are nominal categorical variables - there is no hierarchy between the categories.
The nausea verbal rating score is an example of ordinal categorical data – there is an implied rank order between the categories although they cannot be treated as continuous data.
60
Categorical data obtained during the course of an investigation may be subjected to statistical analysis. The first step is to construct a contingency table as previously described.
Which are true of a valid contingency table?
A. The table must be exhaustive
B. Only 2x2 tables are valid for statistical analysis
C. Subjects may be included in more than one cell
D. The table must be mutually exclusive
E. Each cell contains the data expressed as a percentage
A. True. The table must be exhaustive – every subject must be included in the table somewhere.
B. False. Any size of contingency table is valid for statistical analysis.
C. False. The table must be mutually exclusive – each subject may only be included in one of the cells.
D. True. As option C.
E. False. Each cell in the table must contain the actual frequency of the data and not be expressed as a percentage.
61
In a 2 × 2 contingency table:
A. The cells may contain frequency or relative frequency (%)
B. The cells must be either exhaustive or mutually exclusive
C. Fisher's Exact test is appropriate for paired data
D. Chi-squared test may be used with Yates' correction
E. Interval scale (continuous) data may be analyzed
A. False. The cells must contain frequency data.
B. False. They must be both exhaustive and mutually exclusive.
C. False. The paired test for a 2 × 2 table is McNemars test.
D. True. Although it will be less accurate than Fisher's Exact test.
E. False. A 2 × 2 contingency table is used for categorical data only.
61
Which of the following are examples of categorical data?
A. Height
B. Weight
C. Eye colour
D. Nausea scores on a verbal rating scale
E. Mean arterial blood pressure
F. Hypertensive/normotensive
Submit
C, D, F. Correct. Eye colour, nausea verbal rating scores and hypertensive/normotensive are examples of categorical data. Subjects are assigned to one, and one only, of the categories which are mutually exclusive.
Eye colour and hypertensive/normotensive are examples of nominal categorical data – there is no implied rank order between the categories.
Nausea verbal rating scores are ordinal categorical data – there is an implied rank order. For example, a score of 2 implies more severe nausea than a score of 1 and a score of zero implies the absence of nausea. Although it is sometimes tempting to treat ordinal data as if it were continuous, this approach is not valid – a score of 2 does not imply twice as much nausea as a score of 1. Similarly, a mean score of 1.6 is invalid as it is not defined.
A, B, E. Incorrect. Height, weight and mean arterial blood pressure are continuous data. The data are measured on an interval scale and there is a precise mathematical relationship between subjects with different values on the scale. Thus a mean arterial blood pressure of 100 mm Hg is twice as big as a mean arterial blood pressure of 50 mm Hg. Individuals can assume any value on the scale, limited only by the precision in its measurement
62
When interpreting a bar chart:
A. The x-axis is dimensionless
B. The size of the data set must always be explicitly quoted
C. There should be gaps between the bars
D. The bars should be arranged in descending size
E. The bars may represent either nominal or ordinal data
A. True. In contrast with a histogram.
B. False. If the y-axis denotes frequency, then the size of the dataset can be calculated (implicit quotation of the size of the dataset).
C. True. In contrast with a histogram.
D. False. They may be arranged in any order though the chart may be clearer when arranged in ascending or descending order of frequency count.
E. True.
63
Regarding the relative likelihood of two outcomes occurring in two different treatment groups:
A. Relative risk may be used for prospective data
B. Odds ratio may be used for prospective data
C. Relative risk may be used for retrospective data
D. Odds ratio may be used for retrospective data
E. The two measures give identical results if the data is prospective
A. True.
B. True.
C. False. As the number at risk is only known when data are prospective.
D. True.
E. False. Although both are statistically valid results.
64
A new drug for hypertension (drug A) is compared with an established treatment (drug B) in 200 patients (100 per group) and the occurrence of an adverse side-effect recorded over a one year period.
If the occurrence is 20/100 in group A and 10/100 in group B, then regarding the occurrence of this adverse side-effect in group A compared with group B, which of the following statements is true?
A. Relative risk = 0.5 and Odds ratio = 0.44
B. Relative risk = 2.25 and Odds ratio = 2.0
C. Relative risk = 2.0 and Odds ratio = 2.25
D. Relative risk = 2.0 and Odds ratio = 2.0
E. Relative risk = 2.0 and Odds ratio is invalid
Relative risk of occurrence of the adverse side-effect in group A compared with group B is the ratio of the proportion of all the subjects from group A with side-effects to the proportion of subjects from group B with side-effects.
RR = 20/(20+80) / 10/(10+90) = 0.2/0.1 = 2
Odds ratio is defined as the ratio of the odds of subjects from group 1 having outcome 1 to the odds of subjects from group 2 having outcome 1
OR = 20/80 / 10/90 = 0.25 / 0.11 = 2.25
These measures are equally valid statistically as the data are prospective and hence the size of the ‘at risk’ population is known. Relative risk usually accords more closely with our ‘commonsense’ notion of relative risk in a colloquial sense and is therefore preferred when valid.
In the example given here, RR and OR are similar. The lower the incidence of the outcome of interest, the closer is the approximation of RR and OR. The higher the incidence, the more they diverge. You can work out for yourself that if the same study was repeated but noted occurrence of another side-effect in 80/100 patients in group A compared with 40/100 patients in group B, RR would remain at 2.0 but OR would be 6.0.
65
Which categorical data in the contingency table described may be analyzed appropriately with which statistical test?
66
Parametric statistical tests have non-parametric equivalents. Check your knowledge of the appropriate choice of parametric statistical test before continuing.
67
Which parametric test is the Mann Whitney U test analogous to?
A. Repeated measures analysis of variance (ANOVA)
B. One sample Student's t-test
C. Two sample paired Student's t-test
D. One-way analysis of variance (ANOVA)
E. Two sample unpaired Student's t-test
The Mann Whitney U test is equivalent to the parametric two sample unpaired Student's t-test.
A. Incorrect.
B. Incorrect.
C. Incorrect.
D. Incorrect.
E. Correct.
68
Which parametric test is the Mann Whitney U test analogous to?
A. Repeated measures analysis of variance (ANOVA)
B. One sample Student's t-test
C. Two sample paired Student's t-test
D. One-way analysis of variance (ANOVA)
E. Two sample unpaired Student's t-test
The Mann Whitney U test is equivalent to the parametric two sample unpaired Student's t-test.
A. Incorrect.
B. Incorrect.
C. Incorrect.
D. Incorrect.
E. Correct.
68
Which parametric test is the Wilcoxon Matched pairs test analogous to?
A. Repeated measures analysis of variance (ANOVA)
B. One sample Student's t-test
C. Two sample paired Student's t-test
D. One-way analysis of variance (ANOVA)
E. Two sample unpaired Student's t-test
The Wilcoxon Matched pairs test is equivalent to the parametric two sample paired Student's t-test.
A. Incorrect.
B. Incorrect.
C. Correct.
D. Incorrect.
E. Incorrect.
69
Which parametric test is the Kruskal-Wallis test analogous to?
A. Repeated measures analysis of variance (ANOVA)
B. One sample Student's t-test
C. Two sample paired Student's t-test
D. One-way analysis of variance (ANOVA)
E. Two sample paired Student's t-test
The Kruskal-Wallis test is equivalent to the parametric one-way analysis of variance (ANOVA).
A. Incorrect.
B. Incorrect.
C. Incorrect.
D. Correct.
E. Incorrect.
70
Which parametric test is the Friedman test analogous to?
A. Repeated measures analysis of variance (ANOVA)
B. One sample Student's t-test
C. Two sample paired Student's t-test
D. One-way analysis of variance (ANOVA)
E. Two sample paired Student's t-test
The Friedman test is equivalent to the repeated measures analysis of variance (ANOVA).
A. Correct.
B. Incorrect.
C. Incorrect.
D. Incorrect.
E. Incorrect.
71
Non-parametric data:
A. Are normally distributed
B. May be measured on an interval scale
C. Are discrete rather than continuous
D. May be negatively skewed
E. In two sample groups are compared using Student's t-test
F. Follow a frequency distribution curve defined by the mean and standard deviation of the data
A. False. By definition, non-parametric data are not normally distributed.
B. True.
C. False. Non-parametric data usually refers to distribution-free continuous data. However, ordinal categorical data are also distribution-free and so are analyzed using non-parametric statistical tests.
D. True.
E. False. This test is used with normally distributed data.
F. False. This is true of normally distributed data.
72
Is this distribution skewed or not?
A. Not skewed
B. Positively skewed
C. Negatively skewed
The data is positively skewed as there is more data in the upper than lower tail.
A. Incorrect.
B. Correct.
C. Incorrect.
73
Assume the distributions of the following samples of interval data are unknown. Which non-parametric test is most appropriate to compare them?
74
The Sign test:
A. May be used to compare the median of a single sample dataset with a known population median
B. May be used as an alternative to the Mann-Whitney U test
C. May be used as an alternative to the Wilcoxon
Matched Pairs test
D. Is only valid for sample sizes >100
E. Depends on the binomial distribution for its calculation
F. Is only valid for continuous interval data
A. True. It is also often used to compare paired non-parametric data when it is an alternative to the Wilcoxon Matched Pairs test.
B. False.
C. True.
D. False. The Sign test is valid for small datasets.
E. True.
F. False. The Sign test is equally valid when the data is ordinal categorical.
75
The shaded area in one of these graphs shows the PPV numerator (number of subjects with +ve test result and the disease). Which graph is it?
76
Some subjects are normal, but have test results that exceed those with the disease. Where is the range of results in which a subject could be either normal or have the disease?
Select 1, 2 or 3.
76
Using the definition of sensitivity below, which graph shows the position of line AB if the sensitivity of the test was 90%?
77
As defined below, specificity can be thought of as the inverse of sensitivity. Which graph shows the position of line AB if the specificity of the test was 90%?
78
The shaded area in one of these graphs shows the first term of the PPV denominator (the number of normal subjects with a +ve test result). Which graph is it?
79
Which range of values shows Specificity=1 and Positive predictive value=1?
If the diagnostic threshold is chosen within the range indicated by line 3, the specificity=1 and PPV=1.
Note that the sensitivity will be low, as a large number of patients with the disease will have measured variables below the diagnostic threshold, limiting the usefulness of the test.
80
Where is the ROC curve when the diagnostic threshold is in range 3 of the x-axis?
81
Where is the ROC curve when the diagnostic threshold is in range 2 of the x-axis?
82
Which graph shows these values: Sensitivity=1, Specificity=1, Positive predictive value=1?
83
Match each term with its equation.
83
Label the axes of this ROC curve.
84
Which region of the ROC curve would probably be optimal to select the diagnostic threshold?
85
There are generally two types of table: a reference table and a demonstration table. Reference tables show data to a high degree of accuracy. There is a logical order to the entries, but there isn't a message that is intended to be gleaned from looking at the table. On the other hand, demonstration tables are intended to convey a message, and the figures are often rounded.
Below is a list of both demonstration and reference tables. Which are the demonstration tables?
A. A price list
B. A balance sheet
C. A list of the top 20 bestselling books
D. A train timetable
E. A Chi-squared table
F. A list of MRSA rates in local hospitals
A. False. Reference. The user's intention is to look up the exact price.
B. True. Demonstration. All balance sheets are rounded to the nearest 1 000 to allow comparisons between various financial aspects of a company's status. Think of the way a hospital's budget is described as £200 million, not £204 234 178.
C. True. Demonstration. When we're told that the latest Harry Potter sold 11 million copies in a single day, that doesn’t mean it sold exactly that figure.
D. False. Reference.
E. False. Reference.
F. True. Demonstration. Although there is value in knowing the exact rate, the figure you will remember is a rounded one. If you want people to take away a message, a rounded figure is easier to digest.
86
Of the 2 208 passengers aboard the RMS Titanic when she sank, we know the ages of just over a thousand of them. Obviously, it would be impossible to see any pattern in data like these if they were presented as a table. A histogram is a useful way of summarizing this.
Which of the following age ranges had the largest numbers of passengers?
A. 55 - 60
B. 20 - 25
C. 25 - 30
A. False.
B. True.
C. False.
87
This is from an American study into the risk factors for heart disease. One of the variables measured was HbA1c levels, which varied between 2 and 17. A value of more than seven is usually taken as evidence of diabetes. There were approximately 400 participants.
Regarding this data, which of the statements below is correct?
A. A histogram is another word for a bar chart
B. HbA1c levels are continuous data
C. HbA1c levels could be displayed in a histogram
D. A histogram normally has 10 - 20 bins. In this case, however, we’ll need more because there are almost 500 participants in the study
E. Histograms show the distribution of continuous data but at the expense of some of the detail
A. False.
B. True.
C. True.
D. False.
E. True.
88
Two diagnostic tests are available for a disease, the respective ROC curves are shown below. Which would be a better choice?
ROC 1 is the better choice, as this has a larger area under the curve. There could be other considerations, e.g. costs, side-effects, that may influence the decision.
89
Due to better treatment, a disease has become much less prevalent in the population. The sensitivity and specificity of the diagnostic test have not changed, but the positive predictive value has decreased. Which terms have changed as disease prevalence has decreased?
A. Number of subjects with +ve test result and the disease
B. Number of normal subjects with a +ve test result
C. Number of normal subjects with a -ve test result
A. Correct.
B. Incorrect.
C. Incorrect.
90
Which line represents a test with a likelihood ratio of 1 and a baseline prevalence of the disease of 10%?
C
91
It is decided that 10 bins are enough. Looking at the graph, this seems reasonable.
A. These data are clustered around a single central point
B. There are more patients with HbA1c less than six than greater than eight
C. The area of the bar in a histogram represents the frequency
D. This distribution has a ‘tail’ to the right
A. True.
B. False.
C. True
D. True. These data are ‘right skewed’.
+
91
A scatterplot normally shows that one variable controls another. To be more scientific, we say that a change in the ‘independent’ variable causes a change in the ‘dependent’ one.
Look at the graph on the right. Which is the independent variable: SBP or Age.
A. SBP
B. Age
On this scatterplot, age is the independent variable and blood pressure the dependent. (No one would suggest for a moment that the reason you get older is that your blood pressure goes up!)
92
Traditionally, the independent variable is plotted across the bottom of the graph – the x-axis (or 'abscissa'), while the dependent variable is indicated on the y-axis (or 'ordinate'). Sometimes, however, cause and effect can be difficult to distinguish.
Below are some pairs of variables that a researcher wants to plot as a scatterplot. She has put the independent variables first. Is she correct?
In which pairs does the independent variable come first?
A. Date vs Rainfall
B. Height vs Age
C. Height vs Total Lung Capacity
D. Blood Cholesterol Level vs Blood Pressure
A. True. The rainfall varies according to the date. When the independent variable is time, the graph is referred to as a 'time series'.
B. False. Height varies according to age. This, too, is a time series.
C. True. Lung capacity varies according to height.
D. True. This one is not so straightforward, but it wouldn’t be unreasonable to suggest that blood pressure level varies according to blood cholesterol.
93
Which data type matches which type of graph?
94
The data you want to display is:
High oxygen group: 13 patients with infection, 237 patients without infection.
Low oxygen group: 28 patients with infection, 222 patients without infection.
Which of the following methods would be most suitable for displaying these data?
A. Bar chart
B. Pie chart
C. Table
D. Histogram
A. False. A bar chart would work, but the graphic would not add to your understanding.
B. False. Pie charts are rarely useful. They could work here, but you would need two, and the graphic would not add to your understanding.
C. True.
D. False. A histogram displays a distribution; this is two proportions.
95
A table is a very suitable way to display this data. What statistical test would you need to use?
A. ANOVA
B. t-test
C. Chi-squared test
A. False.
B. False.
C. True. A chi-squared test is an ideal choice for data in a table.
96
It is generally felt that obesity causes diabetes. In our study of risk factors, our researcher decides to plot HbA1c against weight in a boxplot, and feels that Hba1c is the dependent variable.
Looking at these scatterplots, would you say:
A. Scatterplot two is more correct
B. These charts demonstrate a correlation between weight and HbA1c
C. Scatterplots compare continuous to categorical data
D. For any given datapoint, it is possible to read weight and HbA1c level
E. The patient with the highest HbA1c is among the heaviest patients
A. True.
B. False.
C. False. Scatterplots compare continuous with continuous.
D. True
E. False. He’s actually among the lightest.
97
Do you remember Jenny, the specialty trainee who is conducting a postal survey into consultant anaesthetic practice? She is keen to learn what agents are commonly used to anaesthetise emergency cases. Below are some of the questions she asked.
Which will give her categorical data?
A. How often do you anaesthetize emergencies? Daily, weekly, monthly, or less than once a month?
B. How long have you been a consultant?
C. What induction agent do you use? Thiopentone, propofol, etomidate, other?
D. If you use Thiopentone, what dose do you give?
E. Would you say that most of your emergencies are (a) in office hours or (b) out of hours?
A. True. Categorical.
B. False. Continuous. If she had asked them to tick a box saying 'less than five years, 5 to 10 years' etc., that would be categorical.
C. True. Categorical.
D. False. Continuous.
E. True. Categorical. (Binary, to be exact!)
98
One way of assessing the clinical significance is to look at risk. In this study, 250 patients were on normal postoperative levels of oxygen. 28 developed infections. Thus, your risk of infection on normal O2 is 28/250 or 0.112. You might explain this to a patient as 'On normal oxygen, you have roughly one chance in nine of developing an infection.'
Now look at your risk of infection on high oxygen. What is it?
A. 13/222
B. 13/250
C. 13/41
D. 13/500
A. False.
B. True. The risk is 0.052, or 1 in 19.
C. False.
D. False.
99
RR = (a × n2) / (c × n1)
What is the relative risk of contracting the disease if you've been exposed to the risk factor in this case?
A. 0.5
B. 2
C. 7/8
D. 0.8
A. False.
B. True. The sum is 8 × 2/1 × 8 = 16/8 = 2.
C. False.
D. False.
+
100
An odds ratio of 1 means no difference between the two treatments.
Below are a number of 95 % confidence intervals for a number of odds ratios. Which two are statistically significant?
A. 1.82 to 2.68
B. 0.56 to 1.48
C. 0.99 to 1.99
D. 0.76 to 0.97
E. 0.85 to 5.05
A. True.
B. False.
C. False.
D. True
E. False.
Options A and D are the only ones that are statistically significant because all the other ranges include 1, and thus include the possibility that there is no difference.
To summarize:
An odds ratio greater than 1 is a greater risk, less than 1 is a smaller risk.
Odds ratios don't give you an easily understandable figure for how much risk.
Odds ratios are not affected by the manner in which the 'table' is constructed.
101
142 patients were chosen, randomized into two groups, and given either xenon anaesthesia or a total intravenous anaesthetic (TIVA) using propofol. The incidences of post-operative nausea and vomiting were recorded.
A. This is a cohort study
B. In this study, TIVA is being compared to a control group
C. In this study, TIVA is being compared to xenon
D. Both odds ratios and risk ratios would be easy to calculate for this study
E. Odds ratios would be more useful than risk ratios for explaining what risk a patient would have of feeling sick after a particular type of anaesthetic
A. True.
B. False.
C. True.
D. True.
E. False. Odds ratios are more difficult to understand.
102
These are the results obtained:
Xenon group: 71 patients, of whom 47 experienced nausea.
TIVA group: 71 patients, of whom 19 experienced nausea.
103
Regarding measures of association:
A. A measure of association is an indication of the amount by which one factor depends on another
B. The chi-squared test is a measure of association
C. Confidence intervals cannot be calculated for a measure of association
D. A measure of association can be a ratio
E. A measure of association cannot be used to judge statistical significance
A. True.
B. False. The chi-squared test will tell us that a table is unlikely to have the figures it has through chance alone; it doesn't tell us how various factors depend on each other.
C. False.
D. True. For example, risk ratios or odds ratios.
E. False. If the 95 % confidence interval for an RR or OR does not cross 1, we can assume statistical significance.
103
Carol is scheduled for day-case surgery, but is worried about nausea afterwards. Her anaesthetist is planning to use TIVA, but Carol wonders whether xenon would be a better choice. Look again at the figures opposite, then chose the best statement from the following list to explain Carol’s risk to her.
A. Xenon causes less nausea
B. Xenon is more than five times more likely to make you feel ill
C. The risk of nausea is about two and a half times greater if you use xenon
A. False.
B. False.
C. True. RR for nausea using xenon: 2.47. OR for nausea using xenon: 5.36.
104
The 95 % confidence intervals for the two measures of association are:
RR 1.62 to 3.76
OR 2.61 to 11.01
Therefore, only one of the following statements is correct.
A. Both are statistically significant
B. Neither is statistically significant
C. The OR is statistically significant but the RR isn’t
D. The RR is statistically significant but the OR isn’t
E. The RR is not a valid test
A. True. In both cases, the confidence intervals do not include 1.
B. False.
C. False.
D. False.
E. False.
104
Use the results to calculate the risk ratio.
Select one option from the answers below.
Possible answers:
A. 2.47
B. 24.7
C. 0.4
D. 5.36
E. 47
A. True. (47 × 71) / (19 × 71) = 47 / 19 = 2.47
B. False.
C. False.
D. False.
E. False.
104
Now calculate the odds ratio.
A. 2.47
B. 24.7
C. 0.4
D. 5.36
E. 47
A. False.
B. False.
C. False.
D. True. (47 × 52) / (19 × 24) = 2444/ 456 = 5.36
E. False.
105
Regarding cohort studies:
A. Cohort study compares at least two groups
B. A control group is a group which receives either a placebo or the current standard treatment
C. In a study to determine the incidence of nausea after two types of anti-emetic, the control group is the group with no nausea
D. In a study to investigate an antihypertensive, the null hypothesis is that there is no link between the drug and blood pressure
E. A cohort study can prove a link between an agent and a disease
A. True.
B. True.
C. False. Nausea is an outcome.
D. True. For example, risk ratios or odds ratios.
E. False. Like any statistical method, a cohort study can provide evidence to support a particular view, but cannot be said to 'prove' something.
106
Continuing with measures of association:
A. Odds ratios and risk ratios are both valid measures of association for cohort studies
B. A risk ratio can be defined as the risk of a certain outcome given certain conditions, divided by the risk of that outcome given other conditions
C. An odds ratio is the probability of a given outcome, divided by the probability of the alternative outcome
D. Risk ratios are more intuitive than odds ratios
E. Risk ratios can be affected by the incidence of the disease we are studying
A. True.
B. True.
C. False. This is simply the odds.
D. True. For example, risk ratios or odds ratios.
E. True. Common diseases generate smaller risk ratios.
107
The following studies are either case-control or cohort. Think about whether they look to the future (cohort) or to the past (case-control). Which of them are case-control studies?
A. A study of mouthwash on reducing dental plaque. One group is given mouthwash, another a placebo. Levels of plaque are measured after 3 months
B. A study on whether seatbelts reduce the risk of head injury. One group is all the patients presenting with serious head injuries (fitting defined criteria) from car accidents; a group of similar size is that of patients from car accidents who did not have head injuries. Did they use seat belts or not?
C. A study to investigate the link between smoking and cancer. Two groups: one suffering from lung cancer; the other people (of similar sex and age) admitted to hospital on the same day who did not have cancer. Smoking histories are then taken
A. False. Note that both groups had dental plaque. Also, this study looks forward. A case-control study should look back into the history, not ahead to future outcomes.
B. True. Unlike the study in ‘A’, only one group suffered from the ‘disease’, in this case, head injury.
C. True. Once again, only one group had the disease.
108
In a case-control study, it is not appropriate to use a risk ratio as your measure of association. This is because the experimenter sets the risk when choosing the groups. Odds ratios do work, however.
The table opposite is from a study on the link between smoking and lung cancer.
What is the odds ratio? (Remember that odds ratio = (a × d)/(c × b) for a table whose rows are a-b and c-d.)
A. 14
B. 0.14
C. 1.08
D. 0.0045
E. 28
A. True. It is rounded off.
(axd) – 647 x 27 = 17 469
(cxb) – 622 x 2 = 1 244
(a × d)/(c × b) – 17 469/1 244 = 14.0426045
B. False.
C. False.
D. False.
E. False.
109
The cases below are examples of the three types of bias: selection, information and confounding. Which of them is/are confounding?
A. Incidence of nausea is estimated in patients who were discharged on the day of their surgery, compared to those who were admitted overnight
B. A study chooses to look at the incidence of nausea in day case patients compared to those sent home. In this case the groups are chosen at random; one group sent home, one kept in. The group kept in are asked about nausea every hour. The group sent home are phoned the following morning and asked whether they were nauseous at any stage the previous night
C. A RCoA survey asks two questions: did you use the e-learning, and did you pass the primary FRCA exam? On the basis of this, they decide that using the e-learning leads to more exam passes
A. False. An example of selection bias: the patients may have been admitted due to nausea.
B. False. An example of information bias: constant questioning might make the inpatient group slightly more likely to report any minor sensations of nausea, which the home group might not have noticed so readily.
C. True. This is the case of confounding. Students who study more efficiently may be more likely to use the ELA in addition to books and other modes of learning.
110
Which one of the following is most likely to be a phase III study?
A. A study involving 10 healthy volunteers
B. A study involving 50 laboratory mice
C. A randomized controlled trial involving 200 patients
D. A randomized controlled trial involving 20 000 patients
E. A study of the 20 000 patients 5 years later
A. False. Healthy volunteers are tested in phase I.
B. False. Clinical trials involve humans.
C. False. Phase III studies usually provide evidence to license a drug; a drug is unlikely to be licensed on testing in just 200 patients.
D. True. Phase III studies are large multi-centre trials, often involving tens of thousands of patients.
E. False. This is phase IV.
110
A new antacid is given to 100 people for a month. At the end of the month:
40 people have stopped taking the drug, because of side effects
30 people have reported an improvement in their symptoms
Using intention-to-treat analysis, what is the percentage of patients reporting an improvement?
A. 60 %
B. 50 %
C. 40 %
D. 30 %
E. 17 %
A. False.
B. False.
C. False.
D. True. 30/100=30 %. With intention-to-treat analysis, we include those who dropped out.
E. False.
111
A new antacid is given to 100 people for a month. At the end of the month:
40 people have stopped taking the drug, because of side effects
30 people have reported an improvement in their symptoms
Using per protocol analysis, what is the percentage of patients reporting an improvement?
A. 60 %
B. 50 %
C. 40 %
D. 30 %
E. 17 %
A. False.
B. True. 30/60=50 %. With per protocol analysis, we do not include those who dropped out.
C. False.
D. False.
E. False.
112
Which of the following are true of cohort studies and which of case-control studies?
113
Regarding planning a clinical trial:
A. A study flowchart needs to be able to account for everyone who took part in a study
B. A per protocol analysis only requires that you know how many people were originally allocated to a group
C. Intention-to-treat analysis takes side effects into account
D. Per protocol analysis can be used to judge the acceptability of a treatment
E. The numbers shown in a study flowchart are ranges
A. True.
B. False. You need to know exactly how many people actually took part in the test.
C. True. Because intention-to-treat analysis includes those who dropped out, it will include information on acceptability, side-effects, etc.
D. False. Because per protocol analysis excludes those who dropped out, it won't include information on acceptability, side-effects, etc.
E. False. They are precise.
113
Match the criterion with its definition.
114
Regarding bias, which of the following are true?
A. Information bias is always due to faulty or misused equipment
B. Selection bias is minimized by asking for volunteers
C. Confounding implies the existence of a factor which hasn't been recognized in the study design
D. Information bias is less likely in a double-blinded trial
E. If a study is shown to have minimized bias, its results must be true
A. False. It may well be due to the people involved, whether they be testees or researchers.
B. False. Even with volunteers, rigorous checking is required to make sure that the two test groups differ only in the presence, or otherwise, of the disease.
C. True.
D. True.
E. False. Its results are more likely to be true. However, we can only minimize bias, not eliminate it altogether.
115
Which of the following are good questions according to the criteria we have outlined on the previous pages?
A. What kind of pain relief is suitable for the first two post-op days in patients who have had hip replacements?
B. Do McCoy laryngoscopes give shorter intubation time than McIntosh Laryngoscopes in patients graded Mallampatti 3 or 4?
C. In all patients undergoing general anaesthesia, does TIVA or sevoflurane (without nitrous oxide) lead to lower post-op nausea scores?
D. Are fluid warmers better than warm-air blankets?
E. In patients with tension headaches, is aspirin better than placebo?
A. Incorrect. It is not specific enough - there is no mention of intervention, comparison or outcome.
B. Correct.
C. Correct.
D. Incorrect. There is no outcome; what do we define as ‘better’? And in what patient group?
E. Incorrect. It has the same problems as the question above.
116
He decides to search for a systematic review, having been told that one already exists. Where should he look? Order these sources according to their reliability.
The Cochrane Database – As a collection of well written systematic reviews, this is respected as a very good source. The reviews here are normally of excellent quality.
Medline - This is a fairly good place to start, but again will return much that is irrelevant.
Google – It may find the review, but among thousands of other webpages which may be of limited value.
Textbooks – Sadly, by the time most textbooks are published, a lot of medicine has moved on.
117
Once we have decided that it is appropriate to combine the results, meta-analysis can be performed. The results of this can be illustrated mathematically or graphically.
It is tempting to take our different odds ratios and just take an average, but there are problems with this. Can you think why?
A. It doesn’t take into account how often these studies have been cited
B. Odds ratios can’t be averaged
C. It doesn’t take into account the size of the individual studies
D. It doesn’t take into account who conducted the study or which journal it appeared in
A. Incorrect.
B. Incorrect.
C. Correct.
D. Incorrect.
A straight average doesn’t take into account the size of the study – smaller studies are more likely to have less reliable results, but as a straight average these are all taken as equal. If you have four really good studies indicating an odds ratio (OR) of 2, and one very small study indicating an OR of 12, our average is going to be brought up to 4 because of the least convincing study!
118
Which name goes with which stage of the process?
119
Paul anaesthetizes for an ENT list every week, a list which normally has at least three children for tonsillectomies. He is fairly happy with the pain relief he provides. While reading an article on tonsillectomy, he learns that the author regularly gives intravenous steroids to his tonsillectomy patients. He starts to think about whether he should be doing the same, so decides to formulate a clinical question using the PICO template.
Which of the elements of a PICO question are already at least partially present?
A. Patient
B. Intervention
C. Comparison
D. Outcome
A. Correct. He is looking at children undergoing tonsillectomy
B. Correct. The intervention is steroids, but he needs to settle on a dose.
C. Correct. Although not explicit, he wants to compare his current regime to a regime plus steroids.
D. Incorrect. There are a number of outcomes he might like to look at. Post-op pain scores, nausea scores, time to first meal, time to discharge, etc. If Paul is just going to search for systemic reviews, this isn’t too much of an issue. If on the other hand he intends to write a systemic review, he needs to think very carefully about what he uses as his outcome.
120
The review reported no side effects, the intervention is inexpensive, the NNT is good and the outcomes are desirable. Paul decides to incorporate this into his anaesthetic. A colleague suggests that the steroid might be given orally pre-op. He argues that it could be timed so that blood levels are similar intra-operatively to when given IV.
Is this a valid argument?
A. Yes
B. No
A. Incorrect.
B. Correct. Although it seems to be a very common sense view, this wasn’t what the review looked at. It may very well work, and might even warrant researching, but the evidence we have been looking at specifies iv intra-op steroids. It would be unwise to extrapolate beyond that.
121
The NNT to prevent nausea was calculated to be 4. Paul correctly decides that this is good. However, before he decides to change his practice to include intra-operative steroid, which other factors would he be interested in?
A. The PPV of the test for nausea
B. The cost of the steroid
C. Any side effects
A. Incorrect.
B. Correct.
C. Correct. The review looked specifically at the outcomes listed, but would report any side effects. If these were common, the drug may not be as desirable.
122
When finding evidence:
A. It is important to find unpublished work in order to avoid selection bias
B. If we only considered level I evidence in our meta-analysis, we would exclude non-randomized experiments
C. Google is useful to find studies, but should not be used on its own
D. Case-control studies are considered more valuable than randomized controlled trials
E. Expert opinion should always be highly regarded
A. True.
B. True.
C. True.
D. False. Randomized controlled trials are considered the gold standard. In a case-control study, there is always difficulty in finding a set of controls with a similar background to the cases.
E. False. Without the evidence to support it, an opinion is just an opinion. Experts can be wrong.
123
When combining evidence:
A. Tarone’s test assesses the value of individual studies
B. Failing the Breslow-Day test indicates that the studies are too different, and there might be another factor confounding the results
C. Averaging the odds ratios of all the studies is a good method of combining the results
D. A Mantel-Haentzel odds ratio gives a weighting according to the size of the studies
E. In a forest plot, the longer the line the bigger the study
A. Incorrect. It tests the homogeneity of different studies.
B. Correct.
C. Incorrect. Doing so would not allow for the size of the different studies.
D. Correct.
E. Incorrect. They represent the 95% confidence interval for the OR.
124
Regarding test results:
A. Certain medical tests can predict whether or not a patient has or has not got a condition with 100% accuracy
B. The positive predictive value of a test is the chance that you have got the condition, given that the test says you do
C. A good test will always have a high PPV
D. A test for a rare condition will always have a low PPV
E. PPV can be used to decide whether a patient is likely to have a particular condition
A. False. No test is 100% accurate.
B. True.
C. False. Rare conditions always have low PPVs, no matter how good the test is.
D. True.
E. True.
125
When deciding whether a treatment is appropriate:
A. Treatment decisions need to consider efficacy, cost and acceptability
B. The NNT is the number of different drugs a patient needs to take to relieve his or her symptoms
C. A drug with an NNT above 15 is ineffective and should not be used
D. NNTs are easily calculated
E. NNTs can be used during discussions with patients to help them decide whether a treatment will benefit them
A. Correct.
B. Incorrect. It is the number of patients that you need to give a particular treatment to in order to achieve the outcome you are looking for in one of them.
C. Incorrect. If a treatment is inexpensive and well tolerated, it may be useful to use it in spite of a low NNT.
D. Correct.
E. Correct.
126
From your existing knowledge of statistics, which of the following are examples of categorical data?
A. Colour of eyes: blue, brown, green
B. Number of patients with and without nausea after a drug
C. American Society of Anesthesiologists (ASA) score
D. Smokers/non-smokers
E. ABO blood group
Categorical data are discrete and qualitative. Each patient belongs to only one category and there is no intrinsic ranking, nor order, to the categories.
A. True.
B. True.
C. False. ASA scores have an intrinsic ranking, and are ordinal data.
D. True.
E. True.
127
Which of the following are examples of ordinal data?
A. Glasgow Coma Scores (GCS)
B. Grades of oedema: mild, moderate and severe
C. Number of children in family
D. Age in years
When categorical data are stratified into groups with an implied rank order, the data are termed ordinal. A common example is the use of verbal rating pain scores.
A. True.
B. True. As there is some intrinsic order to the categories mild, moderate and severe, these are commonly given numbers like one, two and three, and analyzed using tests for ordinal data.
C. False. The number of children in a family is numerical data and is an actual observation describing a property of the subject. However, the tests for ordinal data are commonly used for this type of numerical data.
D. False. The age in years can take any value and is continuously variable numerical data.
128
Given that numerical data are either discrete or continuously variable, which of the following are examples of continuously variable data?
A. Heart rate in bpm
B. Weight in kilos
C. Pain scores, using a visual analogue scale
D. Duration of stay in hospital in days
A. True. Many of these observations are usually expressed as integers, but theoretically heart rate, weight, etc. can take any value and therefore are continuously variable data.
B. True.
C. False. Pain scores are properly ordinal data, as the score represents a rank; a pain score of 8 is worse than that of 4, but it is not twice as bad in the same way 8 kg is twice the weight of 4 kg. In practice, we will often see this data treated as continuously variable.
D. True.
129
Sometimes the data points in one group have a unique corresponding data point in the other group; this is known as paired data - even if there are more than two groups! Decide which of the following are paired data.
A. Weights of patients before and after a diet
B. The resting heart rate of a set of twins
C. The GCS of patients with head injuries in Glasgow and Edinburgh
A. True.
B. True.
C. False.
The next page shows the tests for paired data.
130
If we treat skewed data as ordinal data, which of these tests would be used?
A. ANOVA
B. Contingency table
C. Kruskal-Wallis
D. Mann-Whitney
E. t-test
Two groups: Mann-Whitney
More than two groups: Kruskal-Wallis
A. False.
B. False.
C. True.
D. True.
E. False.
131
Another way to deal with skewed data is to transform the data so that it approximates sufficiently to a normal distribution. What transformation can be used to change positively skewed data to a normal distribution?
A. Exponential transformation
B. Linear transformation
C. Logarithmic transformation
D. Square root transformation
A. False.
B. False.
C. True.
D. False.
132
After transformation, parametric tests can be used on the logarithmic data that are now normally distributed. Which of these tests would be used for this?
A. ANOVA
B. Friedman
C. Kruskal-Wallis
D. Mann-Whitney
E. t-test
Two groups: t-test
More than two groups: ANOVA
A. True.
B. False.
C. False.
D. False.
E. True.
133
Identify the criteria that are used to select the appropriate test for a research project.
A. Type of data
B. Number of subjects in each group
C. Number of groups
D. Each subject has a unique correlate in the other group
A. True.
B. False. There are small corrections that may be required in contingency tables with small numbers of subjects.
C. True.
D. True. This would require a paired test.
133
Choose the most appropriate test for each of the research projects below.
Twins study: there are 3 possible tests that could be done here; between twins before the diuretic, between twins after the diuretic, and a 'before and after' the diuretic comparison within each individual. All these tests are paired.
Length of stay: the data are continuously variable, but almost certainly positively skewed. Without investigating the data further, non-parametric test should be used, i.e. the Mann-Whitney.
134
Regarding data:
A. A parameter is a measurable characteristic of a sample
B. A variable is a measurable characteristic of a population
C. A parameter has a fixed value
D. Interval data may be converted into categorical data
E. Ordinal data form a subset of categorical data
False.
False.
True.
True.
True.
135
In a 2 x 2 contingency table:
A. The rows usually represent outcome
B. The upper row usually represents the control group
C. In order for it to be valid for statistical analysis, the observations must be independent
D. The cells may contain either frequency or relative frequency (%)
E. The cells must be either exhaustive or mutually exclusive
False.
False.
True.
False.
False.
136
Considering two sets of paired interval data
A. When the data relate to two different methods of measuring the same variable, the correlation coefficient is not a good indicator of the comparative agreement between the two methods
B. The range of the Pearson correlation coefficient is between 0 and 1
C. A p-value < 0.001 for the correlation coefficient indicates a high correlation
D. The square of the Pearson correlation coefficient (r2) equals the amount of shared variance between them
E. The Pearson correlation coefficient is used for normally distributed data
True.
False.
False.
True.
True.
137
Regarding a histogram:
A. The x-axis is dimensionless
B. The rectangles may be of different width
C. The frequency within each class interval is always proportional to the height of the rectangle associated with that class interval
D. Class intervals of zero frequency must be included
E. Class intervals may be of unequal size
False.
True.
False.
True.
True.
138
When determining the central tendency of a sample dataset:
A. The mode is always used for nominal categorical data.
B. The median is preferred to the mean for asymmetric distributions.
C. The mean is less susceptible to outliers than the median.
D. The mean is always preferred when the data are normally distributed.
E. The median of an even number of values equals the mean of the two central values.
False.
False.
True.
True.
True.
139
Regarding measures of the spread of data about a central tendency:
A. The range is useful in statistical analysis.
B. There are three quartiles.
C. The interquartile range of a normal distribution equals ± 1 standard deviation.
D. The standard deviation of a sample equals (i = 1i = n(xi - )2/n).
E. A box and whisker plot may only be used to present normally distributed data.
False.
True.
False.
False.
False.
140
A normal distribution:
A. Is fully described by its mean and standard deviation.
B. Is a probability curve with an area under curve (AUC) = 1.
C. May be positively skewed.
D. The mean and median are identical.
E. Has approximately 95% of the data lying within one standard deviation of the mean.
True.
True.
False.
True.
False.
140
The standard normal distribution:
A. Has a mean of one.
B. Has z-values which indicate the number of standard deviations above or below the mean a given datum value lies.
C. Is a useful way of comparing data from two different normal distributions.
D. Has a kurtosis of one.
E. Has a skewness of zero.
False.
True.
True.
False.
True.
141
Regarding Student's two sample t-tests:
A. There are two versions for use with normally distributed data and data that do not follow a normal distribution.
B. There is a series of t-distributions.
C. Should only be used if the sample standard deviations or variances are equal.
D. The unpaired version gives the same result as ANOVA for two groups of normally distributed data.
E. The paired version is more powerful.
False.
True.
True.
True.
True.
142
The following statistical tests are appropriate for the data described:
A. Student's paired t-test - comparison of two groups of normally distributed data.
B. ANOVA - comparison of four groups of normally distributed data.
C. Student's unpaired t-test - comparison of two groups of data that are not normally distributed.
D. Mann-Witney U-test - comparison of three groups of data that are not normally distributed.
E. Fisher Exact test - analysis of a 2 x 3 contingency table.
True.
True.
False.
False.
False.
143
A binomial distribution:
A. Approximates to a normal distribution as n increases.
B. May be used to describe a situation where there are n independent trials with two mutually exclusive outcomes.
C. Has a mean = n/p.
D. Has a standard deviation = np(1-p).
E. Can be used to predict the probability of obtaining 70 or more heads when a coin is tossed 100 times.
True.
True.
False.
False.
True.
144
If two events A and B are considered:
A. P(A AND B) = 0 if they are mutually exclusive.
B. P(A OR B) = P(A) + P(B) if they are mutually exclusive.
C. P(A OR B) = 1 if A and B are exhaustive and mutually exclusive.
D. P(A AND B) = P(A) x P(B) if they are independent and not mutually exclusive.
E. P(A OR B) = P(A) + P(B) - P(A AND B) if they are independent and not mutually exclusive.
True.
True.
True.
True.
True.
145
Considering p-values:
A. If p < 0.05 and the null hypothesis is true, a type II error occurs.
B. If p > 0.05 and the null hypothesis is false, a type I error occurs.
C. The occurrence of type II errors may be reduced by performing a power calculation.
D. If p = 0.09 then the null hypothesis is rejected.
E. If p > 0.95, the 95% confidence interval for the difference between the means of two groups excludes zero.
A. False. This is a type I error
B. False. This is a type II error
C. True.
D. False. The null hypothesis is conventionally rejected only if p<0.05.
E. False. The 95% confidence interval for the difference between the means of two groups excludes zero when p<0.05.
146
The power of a statistical test:
A. Is greater when two groups of normally distributed data are compared than when two groups of data that do not follow a normal distribution are compared.
B. Is defined as (1 - alpha 1 ) x 100% where alpha 1 is the probability of a type I error.
C. Increases with sample size.
D. Increases when a lower p-value for statistical significance is required.
E. When applied to a clinical study should be at least 50%.
A. True. Parametric tests are more powerful than non-parametric.
B. False. The correct formula is (1-beta) x 100% where beta is the probability of a type II.
C. True.
D. False.
E. False. At least 80% power is usually required by reviewers/editors.
147
Regarding statistical tests:
A. The null hypothesis proposes that there are no differences between study groups with respect to confounding variables.
B. A one-tailed test generates a smaller p-value than a two-tailed test for the same data.
C. Two tailed-tests are generally preferred.
D. A p value < 0.001 is highly clinically significant.
E. When comparing 10 study groups versus placebo, multiple t-tests are appropriate if the data is normally distributed.
A. False. Although a good study design attempts to ensure that there are no differences between confounding variables (e.g. by randomisation), the null hypothesis refers only to the variable(s) of interest.
B. True.
C. True.
D. False. This is a highly statistically significant result but it may not be clinically significant
E. False. If t-tests are used, Bonferroni's correct should be applied. A better approach is to employ ANOVA.
148
Systematic reviews:
A. Are always quantitative.
B. Use Funnel plots to detect heterogeneity.
C. The ideal funnel plot is inverted and symmetrical.
D. Heterogeneity refers to differences in the trials that are combined.
E. Replication bias occurs if two similar studies give the same result.
A. False. Systematic reviews may be qualitative (narrative).
B. False. Funnel plots are used to detect publication bias.
C. True.
D. True.
E. False. Replication bias occurs if data from the same study is published in more than one of the source references.
149
Concerning clinical trial design:
A. Retrospective studies enable groups to be reliably matched and randomized.
B. The prevalence of a disease refers to the overall proportion of the population afflicted with a disease.
C. Misallocation bias is a feature of cross-sectional surveys.
D. Selecting every other patient in a clinic is a good method of randomization.
E. Prospective observational cohort studies may be randomized.
A. False. By definition, these studies are observational and cannot incorporate random allocation.
B. True.
C. False. Misallocation is a feature of case-control studies.
D. False. This is an ineffective method of randomisation as there is no allocation concealment and is therefore open to researcher bias.
E. False. By definition, these studies are observational and cannot incorporate random allocation.
