1. 2006 Biostatistics Primary MMed (Anaesthesia)

2. Writing a study protocol introduction research question current knowledge research hypothesis research objective

3. Writing a study protocol methodology type of study experiments prospective (randomized, control) observational prospective retrospective audit

4. 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)

5. 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)

6. 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

7. 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 Χ 2 test for discrete data

8. 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

9. 2006 The null hypothesis H0H0

10. 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

11. 2006 Types of research Longitudinal studies Cross-sectional studies

12. 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

13. Types of research - cross-sectional study describes a phenomenon fixed in time description of staging system for cancer laboratory studies of biological processes

14. Comparison longitudinal 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

15. 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

16. 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

17. Designs of a randomised clinical trial: parallel, cross-over Randomisation Test Control Test Control Assessment Control Test Assessment A parallel design clinical trial A two period cross-over design clinical trial Assessment

18. 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

19. 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

20. 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

21. 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

22. 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)

23. 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

24. Progress of a cohort study Population With disease Without disease Exposed Not exposed With disease Without disease With disease Without disease Time

25. 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

26. 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

27. 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

28. 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

29. Case-control studies Subjects with disease (case) Subjects without disease (control) Exposed Not exposed Exposed Not exposed Time (Retrospective)

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. Cross-sectional studies problems bias in the type of responders and non- responders exposures have to be determined by a retrospective history

37. 2006 Calculation of sample size

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

39. Control group response it is first necessary to postulate the response of the control group patients denoted by π 1, to distinguish it from the value that will be obtained from the trial, denoted p 1 experience of other studies may provide π 1

40. 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 p 2 anticipated benefit δ = π2 - π1

41. 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)

42. 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

43. 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

44. One or two-sided Frequency μ = 0 P<2.5% (one-sided)P<5% (two-sided) 1.96σ z

45. 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

46. 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

47. 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

48. 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 α2α2 α2α2 –z α +z α

49. 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

50. 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)

51. 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%

52. 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

53. Statistical Testing Errors Real difference No Yes Significance No effect1-α Type II error tests seen β-error Effect Type I error Power seen α-error 1-β

54. Calculation of sample size calculations depend on a function (z α + z 2β )2, where z α and z 2β 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) z 2β z α (z α +z 2β ) 2 0.3 0.70.524 1.960 6.172 0.2 0.80.842 1.960 7.849 0.1 0.91.2821.96010.507

55. Calculation of sample size comparison of proportions wish to detect a difference in proportions δ = π 2 - π 1 e.g. response rate to placebo and treatment drug for Χ 2 test, the number in each group should be at least m = (z α + z 2β ) 2 {π 1 (1-π 1 ) + π 2 (1-π 2 )} δ 2

56. Calculation of sample size comparison of means (unpaired data) wish to detect a difference in means δ = μ 2 - μ 1 the number in each group should be at least m = 2(z α + z 2β ) 2 σ 2 δ 2 where σ is presumed to be the same with both drugs

57. Calculation of sample size comparison of means (paired data) with cross over trial the number in each group should be at least m = (z α + z 2β ) 2 σ w 2 δ 2 where σ w is the standard deviation of the paired difference between treatment

58. 2006 Randomisation

59. 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 !

60. 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

61. 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

62. 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.

63. 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

64. 2006 Data description

65. 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)

66. 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

67. 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 Vascular surgeon A(1) B(2) C(3) Postoperative bleeding (ml) -------- xx xxx xxx xxx x

68. 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

69. 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

70. 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)

71. Summary of measurements nominal name, birthplace ordinal mild | moderate | severe interval meaningful distance between values ratio allows study of absolute magnitude

72. 2006 Summarising data

73. 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

74. 2006 Measurement of location or central tendency

75. 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 values size of sample ΣxΣx n

76. 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

77. 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

78. 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 21 03 102 0022345578 173 00011223455678899 124 111223344678 55 00124 46

79. 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

80. 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

81. 2006 Measure of dispersion or variability

82. range difference between minimum & maximum values interquartile range (largest value 3rd quartile) - (largest value 1st quartile) standard deviation degrees of freedom

83. Range or interquartile range range is given as the smallest and the largest observations vulnerable to outliers interquartile range the distance between the 25 th and 75 th percentile not vulnerable to outliers displayed as box-whisker plots Postoperative bleeding (ml) ------------ A B C Surgeon

84. 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-1 s = 

85. 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-1 s 2 =

86. 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)

87. 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

88. 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

89. 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 

90. 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

91. 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  +ve +ve   -ve-ve 

92. Normal probability plot normality of observation can be confirmed from the Normal probability plot Normal ordinates, z Observation x x x x x x x x x x x

93. 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

94. 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

95. 2006 Generating data from sample to population

96. 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

97. 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

98. 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

99. 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

100. 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

101. 2006 Normal distribution

102. 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

103. Standard Normal Distribution Frequency μ = 0 2.5% 1.96 σ σ = 1 0 95% 2.5%

104. +1 SD -1 SD -1.5 SD -2 SD Scoring of fine motor skills Frequency above averageaverage borderline impaired severely impaired +2 SD 68.26%

105. IQ Curve

106. 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 1.96σ

107. 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

108. 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

109. 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

110. 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

111. Comparing s and SE standard 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

112. 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

113. 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.

114. 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

115. 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 H 0 is rejected or not) since they provide a range of plausible values for the unknown parameter

116. 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

117. 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 1.96σ

118. 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

119. 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

120. 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) - -

121. 2006 t distribution

122. 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

123. Normal distribution, t-distribution α/2 –z α +z α α/2 –t α +t α

124. 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

125. t-Test reflecting this increased conservatism, the critical value for the t-test Significancet-Testz-Test P<0.05 2.26 1.96 P<0.01 3.25 2.58

126. 2006 Statistical tests the type of test used depends upon the sample size

127. Non-parametric tests Use Wilcoxon signed rank Test of test difference between paired observations Wilcoxon rank sum test Comparison of 2 Mann-Whitney U test groups Kruskal-Wallis one-way Comparison of analysis of variance several groups Spearman rank Measure of correlation association between 2 variables Χ 2 goodness of fit test Comparison of an observed frequency distribution with a theoretical one Parametric tests Paired t test 2-sample t test One-way analysis of variance Pearson correlation

128. 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 Χ 2 -statistics categorical or qualitative variables

129. 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

130. 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

131. 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

132. 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

133. 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  (t 0.05  s/  n)

134. 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

135. 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

136. Two sample or unpaired t-test the standard error for the difference between the means, SE (x 1 -x 2 ) = s  (1/n 1 + 1/n 2 ) where s is the common standard deviation and is derived from s 1 and s 2 the t value is calculated as t = (x 1 -x 2 ) / s  (1/n 1 + 1/n 2 ), d.f. = n 1 + n 2 - 2 confidence interval is CI = (x 1 -x 2 ) ± (t’  SE (x 1 -x 2 ))

137. 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

138. 2006 Comparison of several means analysis of variance

139. 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

140. 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

141. 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

142. 2006 Non-normal distribution non-parametric tests

143. 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

144. 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

145. 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

146. 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

147. 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

148. Kruskal Wallis one way analysis of variance a non-parametric equivalent of the one way analysis of variance for normal distribution Postoperative nausea and vomiting score 5- 4- 3- 2- 1- A B C Groups

149. 2006 Binomial distribution

150. 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)

151. 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)

152. 2006 The Chi-squared test for contingency tables

153. Contingency table expected frequencies AB C(x C  y A )/n (x C  y B )/n x C D(x D  y A )/n (x D  y B )/n x D y A y B Total=n condition or intervention yes or no outcome

154. Contingency table can be used to represent qualitative variables discrete quantitative variables continuous quantitative variables whose values have been grouped

155. The Χ 2 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

156. The Χ 2 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

157. The Χ 2 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 Χ 2 test is calculated from, d.f.= 1 for a 2  2 table the Χ 2 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 Χ 2 = Σ (O-E) 2 E

158. Yates’ correction for continuity Factor A PresentAbsent 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 Χ 2 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 Χ c 2 = Σ (|O-E|-½) 2 E Χ c 2 = N(|ad-bc|-½N) 2 mnrs

159. 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 Χ 2 test is not a good enough approximation even with a continuity correction and the Fisher’s exact test for a 2  2 table should be used

160. Fisher’s exact test for a 2  2 table Factor A PresentAbsentTotal 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 Χ 2 test is not strictly valid the probability of observing the particular table is N!a!b!c!d! m!n!r!s!

161. Larger tables chi-squared test can also be applied to larger tables, generally called r  c 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 2  2 chi-squared test should not be applied to tables showing only proportions or percentages Χ 2 = Σ (O-E) 2 E

162. Goodness of fit apart from using contingency tables, the Χ 2 - 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

163. 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

164. 2006 Correlation and linear regression

165. 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

166. 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

167. 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

168. Different correlations yy yy xx xx r = 1r = 0.2 r = 0r = -0.4 strong positive weak positive no correlation weak negative

169. 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 yy xx r > 0r = 0y x r < 0

170. Variance explained, r 2 squaring of the correlation coefficient and multiplying by 100 gives r 2 or variance explained, the proportion of the variance of one variable explained by the other e.g. r of 0.9, r 2 = 0.81 x100; approximately 80% of the variance of one variable can be accounted/explained/predicted by the other

171. 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)

172. 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

173. Test of significance Spearman rank correlation (correlation, r s ) 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 n 3 -n 6Σd26Σd2 r s = 1-

174. 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 r s with the critical values of the Spearman rank correlation coefficient table, significant if r s is > critical value

175. 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

176. 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

177. Regression line regression equation describes the relationship between y and x y = a + bx y = α + βx for population parameters a 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

178. Multiple regression model of multiple regression y = a + b 1 x 1 + b 2 x 2 + ……b k x k applications to look for relationships between continuous variables, allowing for a third (possibly confounding) variable e.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

179. Multiple regression 95% confidence intervals b  t 0.05 SE(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

180. 2006 Degrees of freedom

181. 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

182. Degrees of freedom for Χ 2 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 = n 1 + n 2 minus 2, that is n –1 for each group

183. 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

184. 2006 Diagnostic tests and implications Predictions

185. 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

186. 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

187. Prediction model Occurrence of event/disease Prediction by testpresent (D+) absent (D-) Total positive test (T+) a b a+b negative test (T-)c d c+d Total e f g as 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)

188. 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

189. Predictive value of a test Occurrence of event/disease Prediction by testpresent (D+) absent (D-) Total positive test (T+)a b a+b negative test (T-)c d c+d Total e f g predictive 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

190. 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?

191. 2006 Risk and Odds and ratios

192. Measure of Abbrev n 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

193. 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 cases exposed a b not exposed c d total e f with the disease, odds of event are a/c; without the disease, odds of event are b/d

194. 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 b not exposed c d total e f odds ratio, OR is (a/c)/(b/d) = ad/bc

195. 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

196. Risk +ve cases –ve cases total exposed a b a+b not exposed c d c+d total 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

197. Relative risk +ve cases –ve cases total exposed a b a+b not exposed c d c+d total 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)

198. 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 exposed cohort) 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% significance level

199. 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) reduction0.2-0.15 = 0.05 relative risk of having pain 0.15/0.20 = 0.75 relative risk reduction (1-0.75) x 100% = 25%

200. 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/CIcalculato r.xls.

201. -50 -25 0 25 50 Relative risk reduction (%) -3894159 n=100/gp n=1000/gp Confidence interval around relative risk reduction RRR 0f 25%

202. 2006 Are the likely treatment benefits worth the potential harm and cost? Number needed to treat (NNT) Number needed to harm (NNH)

203. 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

204. 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) = 500 70 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 reduction NNT = number needed to treat

205. 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

206. Number to treat ibuprofen control number of patients 50 50 at least 50% pain relief over 6 hours 27 10 expressed 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

207. Number to treat another way of calculating NNT for the analgesic trial NNT = 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

208. 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

209. 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