2006

Biostatistics

Primary MMed (Anaesthesia)

Writing a study protocol introduction

research question current knowledge research hypothesis research objective

Writing a study protocol methodology

type of study experiments

prospective (randomized, control) observational

prospective retrospective audit

Writing a study protocol methodology

sample size calculation guided by expected difference in either

proportion or numerical data, standard deviation of known data, α and β values, and intended power of study

plan to recruit more subjects in case of drop outs

size of drop outs will affect the basis of assumptions and the initial sample size calculation, and the research hypothesis may show no significant difference (P>0.05)

Writing a study protocol methodology

patient consent inclusion criteria exclusion criteria, restriction process of randomization procedure (how to go about carrying out the

data collection) control group test group(s)

Writing a study protocol methodology

monitoring of the patients during the period of procedure

routine - heart rate, blood pressure, oxygen saturation

data / observations side effects of test protocol

rescue therapy safety of patient

treatment plan criteria for withdrawal from study

Writing a study protocol statistical analysis

the data will be subjected to a test for normality

statement about treatment of normal or nonparametric distribution of data

normal distribution expressed as mean and standard

deviation, with 95% confidence interval t-test for comparison of means obtained

from 2 groups of data analysis of variance test for comparison of

means obtained from more than 2 groups Χ2 test for discrete data

Writing a study protocol statistical analysis

non-normal distribution expressed as median and range, with

limits of 25th and 75th percentiles Mann-Whitney U test for analysis of data

from 2 groups Kruskal-Wallis test for analysis of data

from more than 2 groups state the P value and decide on the

significance level, conventionally it is P<0.05

submit protocol for ethics committee approval

2006

The null hypothesis

H0

Null hypothesis study hypothesis

investigator conducting a study usually has a theory in mind

however, very difficult to prove the hypothesis

simpler to disprove a hypothesis than proving it

null hypothesis differences observed is not due to exposure

to factor, and is by chance always phrased in the negative and that is

why it is termed null

2006

Types of research

Longitudinal studiesCross-sectional studies

Types of research - longitudinal study investigates a process over time

the effect of external factors on human subjects

3 types clinical trial, a cohort study, case-control

study prospective or retrospective studies

in prospective studies, subjects are grouped according to ‘exposure’ to some factor

in retrospective studies, subjects are grouped according to outcome, the ‘exposure’ effect is then determined retrospectively

Types of research - cross-sectional study describes a phenomenon fixed in time

description of staging system for cancer laboratory studies of biological processes

Comparisonlongitudinal studies prospective studies

according to “exposure”

randomised non-randomised

observational studies

retrospective studies according to

outcome determine

“exposure”

cross sectional studies disease description diagnosis and

staging abnormal ranges disease severity

disease processes

Randomised clinical trial randomisation

is a procedure in which the play of chance enters into the assignment of a subject to the alternatives (control and test groups) under investigation, so that the assignment cannot be predicted in advance

tends to produce study groups comparable in unknown as well as known factors likely to influence outcome apart from the actual treatment being given itself

guarantees that the probabilities obtained from statistical tests will be valid

Randomised clinical trial - designs parallel designs

one group receives the test treatment, and one group the control

cross-over designs the subjects receive both the test and the

control treatments in a randomised order each subject acts as own control, allowing

paired or matched analysis, and provides an estimate of the difference between test and control

useful in chronic disease that remain stable over time, such as diabetes and asthma, where the purpose of treatment is palliative, not cure

Designs of a randomised clinical trial: parallel, cross-over

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Test

Control

Test

Control

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A parallel design clinical trial

A two period cross-over design clinical trial

Ass

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Problems of cross-over design cross-over effect

possibility that the effect of the particular treatment used in the first period will carry over to the second period, and may interfere with how the treatment scheduled for the second period will act, thus affecting the final comparison between the two treatments

to allow for this possibility, a washout period, in which no treatment is given, should be included between successive treatment periods

Problems of cross-over design disease may not remain stable over the trial

period subject drop-outs

more will occur in this design trials than in parallel design trials, due to extended treatment period

Non randomised studies historical controls

when randomisation is not possible problem of bias selection already

occurring - those who did not receive transplants may be more ill or may not have satisfied the criteria

survival of patients who received heart transplants and patients who did not

Non randomised studies pre-test–post-test studies

a group of individuals are measured, then subjected to treatment or intervention, and then measured again

purpose of the study is to study the size of the effect of treatment or intervention (e.g. campaign)

major problem is ascribing the change in measurement to the treatment since other factors may also have changed in that interval

Cohort a cohort is a component of a population

identified so that its characteristics can be ascertained as it ages through time designated groups of persons either born in

a certain year or traced over a period of time (who ever worked in a factory)

Cohort studies cohort study

one in which subsets of a defined population can be identified who have been exposed (or will be exposed) to a factor which may influence the probability of occurrence of an outcome (given disease)

usually confined to studies determining and investigating aetiological factors and do not allocate the equivalent of treatments

also for post-marketing surveillance, comparing adverse effects of new drug with alternative treatment

may be referred to as follow-up, longitudinal or prospective study

often termed observational studies, since they observe the progress of individuals over time

Progress of a cohort studyPop

ula

tion With

disease

Without disease

Exposed

Not expose

d

With disease

Without disease

With disease

Without disease

Time

Problems with cohort study exposure to factor may be by natural selection

or up to the individuals’ decision bias may influence the measure of interest other associated factors may also influence

measure of interest e.g. cohort study of cardiovascular risk in

men sterilised by vasectomy e.g. incidence of breast cancer with

consumption of alcohol

Problems with cohort study size of the study

the required size of a cohort study depends not only on the size of the risk being investigated but also the incidence of the particular condition under investigation

cohort studies not suitable for investigating aetiological factors in rare diseases

Problems with cohort study problems with interpretation

bias pool of study subjects when cohort is made up of employed

individuals, the risk of dying in the first few years of follow up is less than in general population, this is known as healthy worker effect; people who are sick are less likely to be employed

incomplete representation e.g. people who respond (to

questionnaires) and people who are lost to follow up

Case-control studies case-control study

also known as a case-reference study or retrospective study

starts with identification of persons with the disease (or other outcome variable) of interest, and a suitable control group of persons without the disease

the relationship of a risk factor to the disease is examined by comparing the two groups with regard to how frequently the risk factor is present

Case-control studies

Subjects with disease (case)

Subjects without disease (control)

Exposed

Not exposed

Exposed

Not exposed

Time(Retrospective

)

Case-control studies designs

matched design control subjects can be chosen to match

individual cases for certain important variables such as age, gender and weight

unmatched design controls can be a sample from a suitable

non-diseased population

Case-control studies selection of controls

not required that the control group are alike in every aspect to the cases, usually 2 or 3 variables which presumably will influence outcome are matched, such as age, gender, social class

main purpose is to control for confounding variables that might influence the case-control comparison

confounding arises when the effects of two processes are not separated, e.g. disease related to 2 exposure factors

Case-control studies selection of controls

matching can be wasteful if matching criteria leads to many available controls being discarded because they fail the matching criteria

if controls are too closely matched to their respective cases, the relative risk may be underestimated

Limitations of case-control studies ascertainment of exposure relies on previously

recorded data or on memory, and it is difficult to ensure lack of bias between the cases and controls the cases may be more motivated to recall

possible risk factors difficulty with selection of suitable control

group a major source of criticism

Cross-sectional studies subjects are included without reference to

either their exposure or their disease usually deals with exposures that do not

change, such as blood type, or chronic smoking habit

resembles a case-control study except that the number of cases are not known in advance, but are simply the prevalent cases at the time of survey

Cross-sectional studies sampling methods

quota sample to ensure that the sample is

representative of general population in say, age, gender and social class structure

not recommended in medical research grab or convenience sample

only subjects who are available to the interviewer can be questioned

Cross-sectional studies problems

bias in the type of responders and non-responders

exposures have to be determined by a retrospective history

2006

Calculation of sample size

Calculation of sample size consider

control group response the anticipated benefit (of the treatment) significance level power

Control group response it is first necessary to postulate the response

of the control group patients denoted by π1, to distinguish it from the value

that will be obtained from the trial, denoted p1

experience of other studies may provide π1

The anticipated benefit it is also necessary to postulate the size of the

anticipated response in treatment group patients, denoted by π2, to distinguish it from the value that will be obtained from the trial, denoted p2

anticipated benefit δ = π2 - π1

Type I error the error of incorrectly rejecting the null

hypothesis when it (the hypothesis) is true the error of concluding that the differences

seen in the result is significant when in fact it is not

wrongly accepting that differences in the results as significant when there is no difference

designated α equivalent to the false positive rate (1-

specificity)

One or two-sided null hypothesis (H°) is that there is no

significant difference, and chance has occurred no assumption about the direction of change

or variation alternate hypothesis states that the difference

is real, further that it is due to some specific factor, where no direction of change is specified

both ends of the distribution curve are important, and the test of significance is two-sided, or two-tailed

where the direction is specified, then only one tail of the curve is relevant, and the test of significance is one-sided, or one-tailed

One or two-sided the critical value is the value of a test statistic

at which we decide to accept or reject H° critical value for a one-sided test at

significance p, will be equivalent to that for a two-sided test at 2p

one-sided p = 0.025 / two-sided p = 0.05 thus, it is tempting to use one-sided tests as

the significance is greater, but the decision should be made before the data is collected, not after the direction of change is observed and should be clearly stated when presenting results

One or two-sided

Frequency

μ = 0

P<2.5% (one-sided)

P<5% (two-sided)

1.96σ z

Significance level the significance level, α, is the probability of

making a Type I error and is set before the test is carried out

in most cases, it will be two-sided

Significance level denoted by the letter P, represents the

probability of the observed value being due solely to chance variation

can be interpreted as the probability of obtaining the observed difference, or one more extreme, if the null hypothesis is true

the smaller the value of P, the less likely the variation is to be due to chance and the stronger the evidence for rejecting the null hypothesis

Significance level most scientific work, by accepted convention,

rejects the null hypothesis at P < 0.05 probability of the observed value being due

solely to chance is < 0.05 (or < 1 in 20) this means that we shall reject the null

hypothesis on 5% of occasions, when it is in fact true, i.e. there was simply a chance variation and that the 2 treatment are equally effective

Significance levels α

the probability that a random variable, (Normally distributed with mean = 0 and standard deviation = 1) will be greater than z or less than -z

z the value on the horizontal

axis of a Normal distribution corresponding to the probability α

α2 α2

–zα +zα

Significance level if α is 0.05, then the corresponding z is 1.96

to link z with the corresponding α, we write z0.05 = 1.96

zα is the value along the axis of a Normal distribution

thus 0.05/2 = 0.025 is to the left of z = –1.96 0.025 is to the right of z = +1.96

Type II error error in incorrectly accepting the null

hypothesis of no difference between treatments, when it (the hypothesis) is in fact false (and should be rejected) accepting the null hypothesis when there

should be significant difference in the results accepting that the differences seen in the

result is not statistically significant and making the conclusion P>0.05

the probability of making a type II error is designated β

equivalent to the false negative rate (1-sensitivity)

Type II error 2 main factors produce β error

chance alone unusual data sample which does not

support a difference statistical methods can produce incorrect

conclusions too small a sample size

the smaller n, the greater must be the real difference before statistical difference may be shown

usually set at a value of β = 0.2 or 20%

Power the probability that a study can predict a

difference, when a real difference actually exists, is termed the statistical power of the study

it is the probability of rejecting the null hypothesis when it (the hypothesis) is false

first decide how much false negative or type II error (β) rate is reasonable power equals 1-β

the higher the power of the study, the smaller the difference which may be detected

Statistical Testing Errors

Real difference No Yes

Significance No effect1-α Type II errortests seen β-error

Effect Type I error Power seen α-error 1-β

Calculation of sample size calculations depend on a function (zα + z2β)2,

where zα and z2β are the ordinates for the normal distribution the value of z for the corresponding α and 2β

are read off from the Table of Normal distribution

Table to assist in sample size calculations, α = 0.05 β Power (1-β) z2β zα (zα +z2β)2

0.3 0.7 0.524 1.960 6.1720.2 0.8 0.842 1.960 7.8490.1 0.9 1.282 1.960

10.507

Calculation of sample size comparison of proportions

wish to detect a difference in proportions δ = π2 - π1

e.g. response rate to placebo and treatment drug

for Χ2 test, the number in each group should be at least m = (zα + z2β)2{π1(1-π1) + π2(1-π2)}

δ2

Calculation of sample size comparison of means (unpaired data)

wish to detect a difference in means δ = μ2 - μ1

the number in each group should be at leastm = 2(zα + z2β)2 σ2

δ2 where σ is presumed to be the same with

both drugs

Calculation of sample size comparison of means (paired data)

with cross over trial the number in each group should be at least

m = (zα + z2β)2 σw2

δ2 where σw is the standard deviation of the paired difference between treatment

2006

Randomisation

Methods of randomisation simple randomisation

either manually by tossing coin or throwing a six-sided die or from table of random numbers

a good method in large trials but does not guarantee equal numbers of

patients in each of the two groups in smaller trials there is a high chance of

getting notable imbalance between the groups

groups of different sizes presence of allocation bias

!

Methods of randomisation random permuted block (RPB) randomisation

a method for ensuring that group sizes never get too far out of balance and avoids assigning different numbers to each study group

combinations (e.g. AABB, ABBA) obtained from blocked randomisation can be assigned numbers (say 1-6), the sequence of the combinations can then be dictated by numbers from table of random numbers

a potential problem with the method is that if the block length becomes know, the method is predictable and selection bias can arise

randomly varying block length can help

Methods of randomisation stratified randomisation

if there are important prognostic factors which, if they were distributed unequally between the treatment groups, would give rise to a serious bias, then it may be prudent to intervene in the randomization process to ensure balance between these factors to ensure balanced treatment allocation for patients within each group or centre

in stratification, RPBs are used within each stratum defined by the prognostic factors

stratification can be cumbersome if there are too many prognostic factors and minimization is a method which is can provide balance in a less cumbersome way

Methods of randomisation unequal randomization (1:2, 2:1 for sample

sizes of test : control groups) although maximum power can be obtained

when the allocations to the groups are in the ratio 1:1, the loss in power is slight if the ratio departs only slightly from 1

there can be practical advantages to unequal allocation, which might be worth considering in some applications.

Methods of randomisation carrying out randomisation

randomisation list should be prepared and held by a person not involved in the investigation and not the investigator determining patient eligibility

this person serves as a check for the trial when a patient is confirmed as eligible for

the trial, randomisation is then revealed over the telephone by opening sequentially numbered

envelopes

2006

Data description

Types of data qualitative data (varying by description)

nominal data ordered categorical or ranked data or

ordinal data numerical or quantitative data (varying by

number) numerical discrete data (differ only by fixed

amount) numerical continuous data (differ by any

amount)

Qualitative data nominal data

data that one can name or describe not measured but simply counted expressed in number (or frequency) or

percentage (or relative frequency) 2 or more groups of observation

2 groups - gender of patients, did or did not get exposure to factor

> 2 groups - blood groups, racial groups, anaesthetists, surgeons

Qualitative data ordered categorical or ranked data

if there are more than two categories of classification, it may be possible to order them in some way or assign ranks to categories to facilitate statistical analysis

e.g. ordinal data : mild, moderate, severe

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Quantitative data numerical discrete

data consists of counts number of general anaesthetics and

regional anaesthetics performed in this hospital in a year

number of anaesthetic trainees passing the Final MMed in the past 5 years

Quantitative data - numerical continuous such data are measurements that can take

any value in a given range age, estimated blood volume, hourly blood

loss interval scale,

position of zero is arbitrary, a difference between two measurements has meaning, but not their ratio

meaning may change if a ratio or percentage is applied to a different scale, e.g. 10% increase in body temperature in Celsius and Fahrenheit scales

Quantitative data - numerical continuous ratio scale,

value of zero has real meaning, negatives are invalid

meaning does not change when ratio or percentage is applied to different units of measure, e.g. 10% increase in body weight in kilograms or pounds

continuous data is often dichotomised to make nominal data and then ordered or ranked for statistical analysis e.g. diastolic blood pressure which is

continuous can be converted to hypertension (>90 mmHg) or normotension (90 mmHg)

Summary of measurements nominal

name, birthplace ordinal

mild | moderate | severe interval

meaningful distance between values ratio

allows study of absolute magnitude

2006

Summarising data

Summarising data for the study measurement of location or central tendency

mean or arithmetic average interval or ratio of a quantitative variable

median and quartiles interval or ratio, (± ordinal)

mode nominal, ordinal, interval or ratio

measurement of dispersion or variability range or interquartile range standard deviation

measures of symmetry

2006

Measurement of location or central tendency

the mean (x, pronounced xbar) or average of n observations is the sum of the observations, Σx divided by their number, n (arithmetic average)

advantage mean uses all the data values, is statistically

efficient disadvantage

vulnerable to outliers, single observations (not erroneous

measurements) which if excluded from the calculations, have noticeable influences on the results

Mean or average

=x =sum of all sample valuessize of sample

Σxn

Median and quartiles lower, median and upper quartiles divide the

data into 4 equal parts approximately equal numbers of

observations in the 4 sections (equal only when n is divisible by 4)

estimation of quartiles the data is first ordered from smallest to

largest, and then counting upwards the number of observations

Median median or middle quartile

value above and below which half the measurements fall

for odd number of observations, is the observation at the centre of the ordering

for even number of observations, is the average of the ‘middle’ two observations

advantage not affected by outliers

disadvantage not statistically efficient as it does not make

use of all the individual data values

Lower and upper quartiles practical method of calculating lower or upper

quartiles are by the stem-and-leaf plot observations

10,13,20,20,22,22,23,24,25,25,27,28,30,30,30,31,31,32,32,33,34,35,35,36,37,38,38,39,39,41,41,41,42,42,43,43,44,44,46,47,48,50,50,51,52,54

2 1 0310 2 002234557817 3 0001122345567889912 4 111223344678 5 5 0012446

Percentile 25th percentile

the value above which 75 percent of the observed cases fall and below which 25 percent of the observed cases fall

50th percentile the median, the value above and below

which half of the observed values of a variable fall

75th percentile the value above which 25 percent of the

observed values of a variable fall and below which 75 percent of the observed values of a variable fall

Mode this is the value that occurs most frequently,

or if the data is grouped, the grouping with the higher frequency

not much use in statistical analysis as its value depends on the accuracy with which the data are measured

bimodal distribution describes a distribution with two peaks in it

2006

Measure of dispersion or variability

Measure of dispersion or variability range

difference between minimum & maximum values

interquartile range (largest value 3rd quartile) - (largest value

1st quartile) standard deviation degrees of freedom

Range or interquartile range range

is given as the smallest and the largest observations

vulnerable to outliers interquartile range

the distance between the 25th and 75th percentile

not vulnerable to outliers displayed as box-whisker plots

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Standard deviation, s a measure of dispersion, i.e. how far variables

are away from their mean, often abbreviated as SD expressed in the same units of measurement

as the observations

the value Σ(x-x)2 is interpreted as from each x value subtract the mean x, square this difference, then add each of the n squared differences

n-1 (or the degree of freedom) compensates for small sample sizes (n < 30)

and higher probability of falling outside the SD

Σ(x-x)2

n-1s =

Standard deviation, s s reflects the variability in the data

if x’s are widely scattered about x, then s would be large

variance a measure of the dispersion of values about

the mean the square of the standard deviation

coefficient of variation expresses the SD as a percentage of the

sample mean c.v. = s 100%

x

Σ(x-x)2

n-1s2 =

Standard deviation, s for data with a normal distribution, there is

a 68.26% chance that the actual value will be within 1 standard deviation above or one standard deviation below the mean value

a 95.45% chance that the actual value will be within 2 standard deviations

a 99.7% chance that the actual value will be within 3 standard deviations

68%x-

34.13% (1SD)

47.73% (2SD)

49.85% (3SD)

34.13% (1SD)

47.73% (2SD)

49.85% (3SD)

Measures of symmetry symmetric distribution

if the distribution is symmetric then the median and mean will be close

data expressed as mean and standard deviation

skewed distribution a distribution is skewed to the right (left) if

the longer tail is to the right (left) data expressed as median and interquartile

range mean and standard deviation are sensitive

to the skewness

Skewness an index of the degree to which a distribution

is not symmetric, or to which the tail of the distribution is skewed or extends to the left or right

calculation of skewness, sk

normality can be confirmed if mean and median are close

skewness is used, along with the kurtosis statistic, to assess if a variable is normally distributed

sk = 3(mean-median) SD

Skewness the normal distribution

is symmetric, and has a skewness value of zero

a distribution with a significant positive skewness has a long right tail

a distribution with a significant negative skewness has a long left tail

+1

-1

Kurtosis a measure of the extent to which observations

are clustered in the tails kurtosis can be used, along with the skewness

statistic, to assess whether a variable is normally distributed

for samples from a normal distribution, the values of kurtosis will fluctuate around 0

Kurtosis for a normal distribution,

the value of the kurtosis statistic is 0

if a variable has a negative kurtosis, its distribution has lighter tails than a normal distribution

if a variable has a positive kurtosis, a larger proportion of cases fall into the tails of the distribution than into those of a normal distribution

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Normal probability plot normality of

observation can be confirmed from the Normal probability plot

Normal ordinates, z

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Normality bell-shaped distribution of a continuous

variable symmetrical about its mean median and mean will be close skewness value is zero kurtosis value is zero normal probability plot

sk = 3(mean-median) SD

Mean or median? mean and median

convey different impressions of the location of the data

both give useful information if the distribution is symmetric, mean is a

better summary statistic if the distribution is skewed, the median is less

influenced by the tails for nominal or ordered categorical data, mean

is the proportion of each group

2006

Generating data from sample to population

Populations and samples a population is a theoretical concept used to

describe an entire group (any collection of people, objects, events, or observations) this is usually too large and cumbersome to

study so investigation is usually restricted to one or more samples drawn from the study population

Populations and samples samples are taken from populations to provide

estimates of the population parameters the purpose of summarising the behaviour

of a particular group is usually to draw some inference about a wider population of which the group is a sample, such as determining the reference normal range

statistics describes the sample parameters describe characteristics of the

population

Populations and samples to allow true inferences about the study

population from a sample there are a number of conditions, the study population must be clearly defined every individual in the population must have

an equal chance of being included in the sample, i.e. a random sample

random does not refer to the sample, but the manner in which it was selected

the opposite of random sampling is purposive sampling, i.e. every 2nd patient

Sampling errors sampling errors

the smaller the sample size, the greater the error

the greater the variability of the observations, the greater the error

non-sampling errors these do not necessarily decrease as the

sample size increases result in bias or systematic distortion of the

results

Central Limit Theorem definition: even when the variable is not

normally distributed the sample mean will tend to be normally distributed

if random samples of n measurements are repeatedly drawn from a population with a finite mean μ, and a standard deviation σ, then when n is large, the relative frequency histogram for the (repeated) sample means will tend to be distributed normally

2006

Normal distribution

Normal distribution bell-shaped distribution of a continuous

variable symmetrical about its mean described by

population mean, μ population standard deviation, σ bell is tall and narrow for small standard

deviation, and short and wide for large standard deviation

a skewed distribution can be transformed into Normal distribution shape by taking the logarithm of the measurements or working with the square root of the observations

σ

μ

Standard Normal Distribution

Frequency

μ = 0

2.5%

1.96σ

σ = 1

0

95%

2.5%

+1

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Frequency

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68.26%

IQ Curve

Normal distribution mathematical property of the Normal

distribution 68.26% of the distribution lies between μ ±

1σ 95.45% of the distribution lies between μ ±

2σ μ – 1.96 σ and μ + 1.96 σ (for exactly

95%) 99% of the distribution lies between μ ± 3σ

μ – 2.58 σ and μ + 2.58 σ

1-α

α/2 α/21.96σ

1.96σ

Normal distribution in practice, the parameters μ and σ must be

estimated from the sample data for this purpose, a random sample from the

population is first taken if the sample is taken from a Normal

distribution, and provided that the sample is not too small, similarly, approximately 95% of the collected data will be within x – 1.96 s to x + 1.96 s

1.96 is the 5% percentage point of the normal distribution

2.58 is the 1% percentage point of the normal distribution

Standard error SD(x)/n the standard deviation of the sampling

distribution for a statistic can apply to: mean, difference between

means, skewness, kurtosis, Pearson correlation, regression coefficient, proportion, difference between proportion

a measure of how much the value of a test statistic may vary from sample to sample

Standard error of the mean standard deviation of the mean, SD(x) or

standard error of mean SE(x) or SE defines the precision with which a mean is

estimated SE(x) or SD(x) = SD(x)/n or s/n

Exercise: Calculation of SE Mean (x) alanine aminopeptidase value for 25

subjects is 1.0 U; SD is 0.3 U SE = SD/√n

= 0.3/√25= 0.3/5= 0.06

Comparing s and SEstandard deviation, s is a measure of the

variability between individuals with respect to the measurement under consideration

describes the sample

standard error of mean is a measure of the

uncertainty in the sample statistic,

always refers to an estimate of a population parameter

the larger the sample size, the smaller the standard error of the mean

Standard error of skewness a measure of the variability of the skewness

statistic examine how far it is from zero by dividing the

measure of skewness by its standard error the larger the absolute value of this

quotient, the less reasonable it is to assume that the variable comes from a distribution with zero skewness, such as the normal distribution

Standard error of kurtosis a measure of the variability of the kurtosis

statistic examine how far it is from zero by dividing the

measure of kurtosis by its standard error (SE Kurt) the larger the absolute value of this

quotient, the less reasonable it is to assume that the variable comes from a distribution with zero kurtosis, such as the normal distribution.

Confidence interval gives an estimated range of values which is

likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data

if independent samples are taken repeatedly from the same population, and a confidence interval calculated for each sample, then a certain percentage (confidence level) of the intervals will include the unknown population parameter

confidence intervals are usually calculated so that this percentage is 95%, but 90%, 99%, 99.9% confidence intervals for the unknown parameter can be produced

Confidence interval the width of the confidence interval gives us

some idea about how uncertain we are about the unknown parameter

a very wide interval may indicate that more data should be collected before anything very definite can be said about the parameter

confidence intervals are more informative than the simple results of hypothesis tests (where H0 is rejected or not) since they provide a range of plausible values for the unknown parameter

Confidence limits (xx, yy) the lower and upper boundaries or values of a

confidence interval the values which define the range of a

confidence interval

Confidence interval for a mean define a range of values within which the

population mean μ is likely to lie that is, a range of values that is likely to

cover the true but unknown population mean value (say, 95% of time)

95% confidence interval for a large sample (>60) 95% CI = x 1.96 s/n where s/n is

SE(x)

95% confidence interval

α/2 α/21.96σ 1.96σ

Confidence interval for a mean the upper and lower values are the 95%

confidence limits a reported CI from a particular study may or

may not include the actual population mean but if the study were to be repeated 100

times, of the 100 resulting 95% CI, we would expect 95 of these to include the population mean

Confidence interval for a mean small samples

less precise statements about population parameters can be made than with large samples

x and s will not always be necessarily close to μ and σ, respectively

sample size is already taken into account in the calculation of the standard deviation of the mean, SD(x), using n, i.e. s/n

of practical importance only when the sample size is very small (less than 15) and when the distribution in the population is extremely non-normal

Exercise: Calculation of CI Mean (x) alanine aminopeptidase value for 25

subjects is 1.0 U; SD is 0.3 U Calculate 95% CI 95% CI = x + (1.96 x SE) where SE

= SD/√n= 1.0 + (1.96 x 0.3/√25)= 1.0 + (1.96 x 0.06)= 1.0 + (0.1176)

95% CI is from 0.8824 to 1.1176 lower and upper boundaries of 95% CI =

(0.8824, 1.1176)

-

-

2006

t distribution

t distribution t distribution with (n–1) degrees of freedom

introduced by WS Gossett, who used the pen-name ‘Student’, and is often called Student’s t distribution

like the normal distribution, the t distribution is a symmetrical bell-

shaped distribution with a mean of zero, but is more spread out, having longer tails

the exact shape of the t distribution depends on the degree of freedom (d.f.), n–

1, of the standard deviation s degrees of freedom refers to the number of

observations completely free to vary, the fewer the degrees of freedom, the more the t distribution is spread out

Normal distribution, t-distribution

α/2 α/2

–zα +zα

α/2 α/2

–tα +tα

Confidence interval using t distribution confidence interval

is calculated using t’, the appropriate percentage point of the t distribution with (n–1) degrees of freedom

small sample CI = x (t’ s/n) for small degrees of freedom,

the percentage points of the t distribution are larger in value than the corresponding percentage points of the normal distribution

because sample standard deviation s may be a poor estimate of the population σ, and when this uncertainty is taken into account, the resultant CI is wider

t-Test reflecting this increased conservatism, the

critical value for the t-test

Significance t-Test z-TestP<0.05 2.26 1.96P<0.01 3.25 2.58

2006

Statistical tests

the type of test used depends upon the sample size

Non-parametric tests UseWilcoxon signed rank Test oftest difference

between paired observations

Wilcoxon rank sum test Comparison of 2 Mann-Whitney U test groupsKruskal-Wallis one-way Comparison of analysis of variance several groupsSpearman rank Measure of correlation association

between 2 variablesΧ2 goodness of fit test Comparison of

an observed frequency distribution with

a theoretical one

Parametric testsPaired t test

2-sample t test

One-way analysis of variance

Pearson correlation

Tests statistics a statistic derived from sample data, used to

measure the difference between the observed data and what would be expected under the null hypothesis: z-statistics

principal of the standard normal deviate t-statistics

small samples, with limited degrees of freedom

Χ2-statistics categorical or qualitative variables

Normal test, z test the Normal test or z test requires that,

the sample size is large (n > 30) the population standard deviation, σ is

known the variable is assumed to be normally

distributed commonly the population σ is unknown,

however it is possible to use the sample standard deviation, s as an estimate of σ

Large sample if the sample size is large, n > 30

then the sample standard deviation, s, is considered to be an adequate estimate of the population σ

thus, the standard error of the sample mean becomes,

SE(x) = (s/√n) under these circumstances the z-test can

again be used to test the significance of the difference between the population mean μ and the sample mean x'

assuming the population fits a normal distribution

Small samples if n < 30

the sample standard deviation, s, is not an adequate estimate of the population σ

Student's t-test is employed (Gosset in 1908)

the t-distribution describes a series of curves,

dependent upon the number of degrees of freedom

as for the normal distribution, these are symmetrical with a mean μ = 0

Paired samples t-test attributes and demographic data, disease

condition are matched to make two groups of subjects as similar as possible

the two groups can be two groups of subjects in a matched

case-control study can be of the same subjects observed before

and after a treatment as in a cross-over trial this test is a statistical test of the null

hypothesis that two population means are equal

any observed differences between the groups, if statistically significant, can be attributed to the variable of interest

Paired t-test also known as the related test, or matched

test paired t = x /(s/n), d.f. = n-1

the corresponding P value or significance level, is obtained from the Table of percentage point for Student’s t distribution

95% CI = x (t0.05 s/n)

One sample t-test tests whether a sample mean is different from

some specified value, μ, which need not be zero

t = (x-μ) / (s/n), d.f. = n–1

Two sample or unpaired t-test also known as independent sample, or

unrelated test, the t test is used for small samples (n<30)

for analysing data in 2 groups of subjects in a parallel group clinical trial or the unmatched case-control study

requires that the population distributions are normal

when comparing 2 means, the validity of the t test also depends on the equality of the 2 population standard deviations

Two sample or unpaired t-test the standard error for the difference between

the means, SE (x1-x2) = s(1/n1 + 1/n2) where s is the common standard deviation

and is derived from s1 and s2

the t value is calculated as t = (x1-x2) / s(1/n1 + 1/n2), d.f. = n1 +

n2 - 2 confidence interval is

CI = (x1-x2) ± (t’ SE (x1-x2))

Small samples, unequal SD first approach is to seek a suitable change of

scale to remedy this, so that the t test can be used taking logarithms of the individual values

alternatives are to use a non-parametric test or to use either the Fisher-Behrens or the Welch tests

2006

Comparison of several means

analysis of variance

Analysis of variance the t-test is generalised to more than 2 groups

by means of a technique termed analysis of variance

for this method, there are both between- and within-groups degrees of freedom the between-groups and within-groups

degrees of freedom are quoted in this order for every case

depending on the number of factor(s) included for analysis

one-way analysis of variance two-way analysis of variance

One-way analysis of variance used to compare the means of several groups one-way analysis of variance is used when the

subgroups to be compared are defined by just one factor e.g. comparison of means between different

socioeconomic classes, or different ethnic groups, or by a disease process

this method assesses how much of the overall variation in the data is attributed to differences between the group means, and comparing this with the amount attributable to differences within group

Two-way analysis of variance used when the data are classified in 2 ways

e.g. by age-group and gender balanced and unbalance study designs

a balanced design if there are equal numbers of observations in each group

an unbalanced design if there are not equal numbers of observations in each group

multiple regression test can also be applied

2006

Non-normal distribution

non-parametric tests

Non-parametric tests non-parametric statistical tests are for

analysing numerical data that make no assumption about the underlying normality of distribution

particularly useful when there is obvious non-normality in a small data set which cannot be corrected with a suitable transformation

Wilcoxon signed rank test a non-parametric statistical test equivalent of

the paired t test, analysing differences between paired observations

it makes no assumptions about the shapes of the distribution of the two variables

the absolute values of the differences between the two variables are calculated as (+/-) for each case and ranked from smallest to largest

the test statistic is based on the sums of ranks for negative and positive differences

Wilcoxon signed (+/-) rank test procedure

the absolute values of the differences between the paired observation are calculated (+/-) for each case and ranked from smallest to largest

exclude any differences which are zero, then rank in order, ignoring signs (i.e. + or –)

2 pairs having the same difference are given the mean of what would have been their successive ranks (i.e. 2nd & 3rd would have been ranked as 2.5 & 2.5)

add up the ranks of positive differences and negative differences separately

each of the (+) & (–) ranks is totaled, and the smaller referred to the Table of critical values for Wilcoxon matched pairs signed rank test for P value

Wilcoxon rank sum test a non-parametric equivalent of the unpaired t

test or two-sample test procedure

rank the observations from both groups together in ascending order of magnitude

if any of the values are equal, average their ranks

add up the ranks in the group with the smaller sample size

compare this sum with the critical ranges in the Table of critical ranges for the Wilcoxon rank sum test

Mann-Whitney U a non-parametric equivalent of the unpaired t

test or two-sample test that two independent samples come from the same population

similar approach as Wilcoxon rank sum test with entirely comparable results

Kruskal Wallis one way analysis of variance a non-parametric equivalent of the one way

analysis of variance for normal distribution

Post

op

era

tive

nause

a a

nd

vom

itin

g s

core 5-4-3-2-1- A B C

Groups

2006

Binomial distribution

Binomial distribution data (discrete variables) which can take only 0

or 1 response, such as treatment failure or treatment success, follow the binomial distribution providing the underlying population response rate does not change

proportion (p) of respondents (to treatment) in sample p = R/n , R is the number of respondents

and n is the total number in the sample potential variation about this expectation is

expressed by SD(R)

Binomial distribution p can be used to estimate the population

response rate, π number of successful response in the

population can be estimated to be nπ standard error of p is SE(p) = (pq/n) where

q = 1 - p, is the proportion of unsuccessful responders

95% CI for π is p + 1.96 SE(p)

2006

The Chi-squared test for contingency tables

Contingency table expected frequencies

A B

C (xC yA)/n (xC yB)/n xC

D (xD yA)/n (xD yB)/n xD

yA yB Total=n

condition or interventio

n

yes or no outcome

Contingency table can be used to represent

qualitative variables discrete quantitative variables continuous quantitative variables whose

values have been grouped

The Χ2 test is used to

examine the association between discrete (categorical or qualitative) variables

test whether an observed frequency distribution differs significantly from a postulated theoretical one

test whether there is an association or dependence between the row variable and the column variable, or

test whether the distribution of individuals among the categories of one variable is independent of their distribution among the categories of the other

The Χ2 test advantage

it allows comparison of many more categories, drawn-up into a contingency table

the null hypothesis is that any number of categories have equal chance of any other factor

when the table has only 2 rows or 2 columns, this is equivalent to the comparison of proportions

applicable for a parallel group clinical trial, an unmatched

case-control study, or a cross-sectional survey

The Χ2 test in 2 2 tables first calculate the values expected in the 4 cells

of the table assuming the null hypothesis is true estimated cell value = row total x column

total / total N the Χ2 test is calculated from

, d.f.= 1 for a 2 2 table

the Χ2 test is valid (Cochran 1954) when n is > 40, regardless of the expected values, and if less than 20% of the expected values are less than 5 and none are less than 1

when n is between 20 and 40, the test is only valid if all the expected values are at least 5

Χ2 = Σ(O-E)2

E

Yates’ correction for continuity Factor A

Present Absent Total

Factor B Present a c m Absent b d n

Total r s N Yates’ correction for continuity

for converting the discrete data, as Χ2 distribution used to calculate the P value is continuous, like the Normal distribution

, where || means take the value as positive

for ease of calculation, this is equivalent to

Χc2 = Σ

(|O-E|-½)2

E

Χc2 = N(|ad-bc|-½N)2

mnrs

Yates’ correction for continuity the use of the continuity correction is always

advisable although it has most effect when the expected numbers are small

when the numbers are very small, the Χ2 test is not a good enough approximation even with a continuity correction and the Fisher’s exact test for a 22 table should be used

Fisher’s exact test for a 2 2 table Factor APresent Absent Total

Factor B Present a c m Absent b d n

Total r s N Fisher’s exact test to determine P value

to be used if any expected counts in a 2 2 table is less than 5, as the P value given by the Χ2 test is not strictly valid

the probability of observing the particular table is

N!a!b!c!d!m!n!r!s!

Larger tables chi-squared test can also be applied to larger

tables, generally called rc tables r denotes rows and c denotes columns

, d.f.= (r-1)(c-1)

there is no continuity correction or exact test for contingency tables larger than 22

chi-squared test should not be applied to tables showing only proportions or percentages

Χ2 = Σ(O-E)2

E

Goodness of fit apart from using contingency tables, the Χ2-

statistic can be used to see if any observed set of data follows a particular distribution e.g. by calculating the expected frequencies

from say a Poisson distribution, and then comparing these with the observed data

the degrees of freedom will be the number of observations, n - 1

Paired comparison in contingency tables McNemar’s test is a test on a 2x2 classification

table when the difference between paired proportions is tested in cross-over trials and matched-pair case-

control studies

2006

Correlation and linear regression

Correlation and linear regression techniques for dealing with the relationship

between 2 or more continuous variables correlation

examines linear association between 2 variables not whether one variable predicts another

variable strength of the association summarised by the

correlation coefficient regression

examines dependence of one variable, the dependent variable, on the other, the independent variable

relationship summarised by a regression equation consisting of a slope and an intercept

Correlation coefficient, r summarises the strength of the association

between 2 variables allows testing of the hypothesis that the

population correlation coefficient r is zero i.e. whether an apparent association between the variables would have arisen by chance

Pearson correlation coefficient when the correlation coefficient is based on

the original observations Spearman rank correlation coefficient

when it is calculated from the ranks of the data

Correlation coefficient, r dimensionless quantity ranging from -1 to +1

a positive correlation is one in which both variables increase together

a negative correlation is one in which one variable increases as the other decreases

when variables are exactly linearly related, then the correlation either equals +1 or -1

unaffected by units of measurement

Different correlations

y y

y y

x x

x x

••••

••••

••

•• •

••

••••

••

••

•••••

• ••

•

•••

•••

••

•

r = 1 r = 0.2

r = 0 r = -0.4

strong positive

weak positive

no correlation weak negative

Correlation coefficient, r should not be used

if the relationship is non-linear in situations where one of the variables is

determined in advance should be used with caution

in the presence of outliers when the variables are measured over more

than one distinct group e.g. disease and healthy groups

y y

x x

r > 0

r = 0 y

x

r < 0

• ••••

• ••

••

••

••

• • ••••

•

Variance explained, r2

squaring of the correlation coefficient and multiplying by 100 gives r2 or variance explained, the proportion of the variance of one variable explained by the other e.g. r of 0.9, r2 = 0.81 x100; approximately

80% of the variance of one variable can be accounted/explained/predicted by the other

Test of significance Pearson correlation (coefficient, r)

this is a parametric measure of the degree of association between 2 numerical variables

to test whether this is significantly different from zero, calculate

SE(r) = {(1-r2)/(n-2)} and t = r/SE(r) and compare this with the Table of t-

distribution with n-2 degrees of freedom for P value

r ={Σ(x-x)2 Σ(y-y)2}

Σ(x-x)(y-y)

Test of significance assumes both variables are random samples

and at least one has a normal distribution outlying points away from the main body of

the data suggest the variable may not have a normal distribution in this case it may be better to replace the

observation by their ranks and use the Spearman rank correlation coefficient

Test of significance Spearman rank correlation (correlation, rs)

this is a non-parametric measure of the degree of association between 2 numerical variables

the values of each variable are independently ranked and the measure is based on the differences between the pairs of rank of the 2 variables

where d is the difference between each pair

of ranks

n3-n6Σd2rs= 1-

Test of significance Spearman rank correlation

this correlation will have a value between -1 and 1, and its interpretation is similar to that of the Pearson correlation coefficient

the significance of the association is tested by comparing rs with the critical values of the Spearman rank correlation coefficient table, significant if rs is > critical value

Regression looking for dependence of one variable, the

dependent variable, on the other, the independent variable

relationship summarised by a regression equation consisting of a slope and an intercept the slope represents the amount the dependent

variable increases with unit increase of the independent variable

the intercept represents the value of the dependent variable when the independent variable is zero

multiple (multivariate) regression examines simultaneous relationship between one dependent variable and a number of independent variable

Linear regression assumption

a change in one variable, x, will lead directly to a change in another variable, y

e.g. haemoglobin increase with age, not the other way around

y variable (resultant change) is termed dependent variable

x variable is termed the independent variable

Regression line regression equation describes the relationship

between y and x y = a + bx

y = α + βx for population parametersa is the intercept and b is the regression

coefficient calculation of b calculation of intercept, a = y – bx unwise to use equation to predict an outcome

based on extrapolation of observed parameters

b =Σ(x-x)(y-y)

Σ(x-x)2

Multiple regression model of multiple regression

y = a + b1x1 + b2x2 + ……bkxk

applications to look for relationships between continuous

variables, allowing for a third (possibly confounding) variablee.g. predicted haemoglobin = 5.24 + 0.11(age) + 0.097(PCV)

corresponding t-value is b/SE(b) with d.f. of n minus the number of estimated parameters

from these, P value is derived from Table of t distribution

Multiple regression 95% confidence intervals

b t0.05SE(b) if the interval includes zero, the conclusion

should be that that relationship between y and x remains the same whether or not x changes

2006

Degrees of freedom

Degrees of freedom the number of degrees of freedom depends on

2 factors the number of groups we wish to compare the number of parameters we need to

estimate to calculate the standard deviation of the contrast of interest

Degrees of freedom for Χ2 test for comparison of 2 proportions, 1 df for t test

2 sets of degrees of freedom one degree of freedom between-groups one for within-groups

for comparing 2 means, 1 df for between groups, there are also dfs for estimating σ

for paired data, df = number of subjects minus 1

for unpaired data, df = n1 + n2 minus 2, that is n –1 for each group

Degrees of freedom for linear regression

given n independent pairs of observations, 2 degrees of freedom are removed for the 2 parameters that have been estimated, thus d.f. = n-2

for multiple regression d.f. are n minus the number of estimated

parameters

2006

Diagnostic tests and implications

Predictions

Sensitivity what is the probability that the test result will

be positive when a disease is present? that is when the test result is positive, how accurate is it in relation to overall presence of the disease? presence of acute myocardial infarction and

positive T-Troponin test restriction of atlanto-occipital joint extension

and difficult laryngoscopy

Specificity what is the probability that the test result will

be negative when a disease is absent? that is when the test result is negative, how accurate is it in relation to overall absent of the disease? absence of acute myocardial infarction and

negative T-Troponin test Mallampati class I and II and no difficulty in

laryngoscopy

Prediction model Occurrence of event/disease

Prediction by test present (D+) absent (D-) Totalpositive test (T+) a b a+bnegative test (T-) c d c+d

Total e f gas fraction or percentage:prevalence of disease = e/g or P(D+)sensitivity of a test = a/e or P(T+ given D+)specificity of a test = d/f or P(T- given D-)false negative rate = 1- sensitivity = c/e (Type II error)false positive rate = 1- specificity = b/f (Type I error)

Sensitivity and specificity both are useful statistics because they will

yield consistent results for the diagnostic test in a variety of patient groups with different disease prevalences

important point sensitivity and specificity are characteristics

of the test, not the population to which the test is applied

Predictive value of a test Occurrence of event/disease

Prediction by test present (D+) absent (D-) Totalpositive test (T+) a b a+bnegative test (T-) c d c+d

Total e f gpredictive value of a positive test attempts to calculate the

probability of the patient having the disease (D+) when the test is positive T+, or P(D+ given T+) = a/(a + b)

predictive value of a negative test attempts to calculate the probability of the patient not having the disease (D-) when the test is negative T-, or P(D- given T-) = d/(c + d)

discrimination - overall correct classification rate, how well the model separates those in D+ and D- = (a+d)/g

false classification rate = (c+b)/g or = 1 - discrimination

Predictive value or sensitivity? predictive value is what the clinician wants

positive test, is the disease present? negative test, is the disease absent? correct classification rate

sensitivity is supplied with the statistical test disease present, was the test positive? no disease, was the test negative?

2006

Risk and Odds and ratios

Measure of Abbrevn Description No effect Total effect success

Absolute risk ARR Absolute change in risk: the risk of an ARR=0% ARR=initial reduction event in the control group minus the risk

risk of an event in the treated group; usually expressed as a percentage

Relative risk RRR Proportion of the risk removed by RRR=0% RRR=100%reduction treatment: the absolute risk reduction

divided by the initial risk in the control group; usually expressed as a percentage

Relative risk RR The risk of an event in the treated group RR=1 or RR=0

divided by the risk of an event in the RR=100% control group; usually expressed as a

decimal proportion, sometimes as a percentage

Odds ratio OR Odds of an event in the treated group OR=1 OR=0

divided by the odds of an event in the control group; usually expressed as a decimal proportion

Number NNT Number of patients who need to be NNT= NNT= 1/needed to treat treated to prevent one event; this is the initial reciprocal of the absolute risk reduction

risk (when expressed as a decimal fraction); it is usually rounded to a whole number

Odds given that a subject has or does not have a

disease, odds always measures the incidence of

event (exposure) to non event (non-exposure)

+ve cases -ve casesexposed a bnot exposed c dtotal e f

with the disease, odds of event are a/c; without the disease, odds of event are b/d

Odds ratio odds ratio compares the incidence of event (of

exposure) to non-event (non-exposure) among 2 groups of subjects with 2 opposing outcomes

odds ratio is the ratio of two odds +ve cases -ve cases

exposed a bnot exposed c dtotal e f

odds ratio, OR is (a/c)/(b/d)= ad/bc

Odds Ratio 95% CI 1 includes 1 > 1 does not

include 1

< 1 does not

include 1

includes 1

Interpretation No association Positive association between exposure and outcome at the 5% significance level (the odds of exposure is greater in cases Than in controls) Negative association between exposure and outcome at the 5%significance level (the odds of exposure is smaller in cases than controls)Association of exposure and outcome is not proven by the study at the 5% significance level

Risk +ve cases –ve cases

totalexposed a b a+bnot exposed c d c+dtotal e f g

risk of an event is the probability that an event will occur within a stated period of time

the risk of developing the disease within the follow-up time is a/(a+b) for the exposed population c/(c+d) for the unexposed population

Relative risk +ve cases –ve cases

totalexposed a b a+bnot exposed c d c+dtotal e f g

the relative risk is a summary of the outcome of a cohort study

RR = (a/(a+b))/(c/(c+d))or a(c+d)/c(a+b)

Relative risk 95% CI 1 includes 1 > 1 does not

include 1

< 1 does not

include 1

includes 1

Interpretation No association Positive association between exposure and outcome at the 5% significance level (outcome is more likely in the exposedcohort)Negative association between exposure and outcome at the 5% significance level (outcome is less likely in the exposed cohort)Association of exposure and outcome is not proven by the study at the 5% significancelevel

How large was the treatment effect?

PrOpofol - LignOcaine Trial

+ lignocaine control

# of patients randomized 100 100# (%) patients with pain 15(15%)

20(20%)absolute risk (pain) reduction 0.2-0.15 = 0.05relative risk of having pain 0.15/0.20 =

0.75relative risk reduction (1-0.75) x 100% = 25%

Depends on the sample size sample size of 100 each arm

95% CI for RRR of 25% is -28.15 to 77.76 lignocaine + propofol probably no benefit

sample size of 1000 each arm 95% CI for RRR of 25% is (8.35 to 41.61) confident that true RRR is close to 25%

http://ptwww.cchs.usyd.edu.au/Pedro/CIcalculator.xls.

-50 -25 0 25 50

Relative risk reduction (%)

-38 9 41 59n=100/gp

n=1000/gp

Confidence interval around relative risk reduction

RRR 0f 25%

2006

Are the likely treatment benefits worth the potential harm and cost?

Number needed to treat (NNT)Number needed to harm (NNH)

Number needed to treat …the number of patients who must receive an intervention of therapy

during a specific period of time to prevent

one adverse outcome or produce one positive outcome

Number needed to treat … Risk of perioperative AMI NNT without β-blocker with β-blocker* (1/ARR) (ARR)

40 year old man 2% 1.8% 1/0.002

(0.2% or 0.002) = 50070 year old man 40% 36%

1/0.04 (4% or 0.04) = 25

*assuming 10% relative risk reduction (RRR) with preoperative administration of β-blocker

ARR = absolute risk reductionNNT = number needed to treat

Number needed to treat … if there is a higher probability that a patient

will experience an adverse outcome if we do not treat, the more likely the patient will benefit from

treatment and the fewer such patients we need to treat to

prevent one adverse outcome

Number to treat ibuprofen

controlnumber of patients 50 50at least 50% pain relief over 6 hours 27 10expressed in percentage 54%

20% absolute risk reduction = 54%-20% (or 0.54-

0.20) = 34% (or 0.34)

number needed to treat = 1/0.34 = 2.94 or 3

Number to treat another way of calculating NNT for the

analgesic trialNNT = 1/(the proportion of patients with at

least 50% pain relief with analgesic minus the proportion of patients with at least 50% pain relief with placebo)

= 1/((27/50) - (10/50)) = 1/(0.54 - 0.20)

= 1/0.34 = 2.9

or 3

Number to treat the best NNT would be 1,

every patient with treatment benefited no patient given control benefited

generally NNTs between 2 and 5 are indicative of effective treatments, but NNTs of 20, 50 or 100 may be useful for prophylactic treatments, like interventions to reduce death after heart attack

relevance of which depends on the intervention and the consequences

Number to harm for adverse effects, the number needed to

harm (NNH) can be calculated in exactly the same way as an NNT

for an NNH, large numbers are obviously better than small numbers, because that means that the adverse effect occurs with less frequency