Adobe Acrobat 4.0 User Guide - School of Mathematics and Statistics
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1. In the second stage these summary measures are analysed by simple statistical techniques as though they were raw data The method is statistically valid and likely to be more relevant to the study questions If this method is borne in mind when the experiment is being planned it should promote studies with enough subjects and sufficient observations at critical times to enable useful conclusions to be drawn Use of summary measures to analyse serial measurements though not new is potentially a useful and simple tool in medical research Introduction A common study design in medical research is to give patients some intervention and then observe what happens to them over time For example blood glucose concentrations may be measured several times after a glucose drink In many cases there may be more than one group of patients possibly randomised to different treatments Despite its apparent simplicity the analysis of this form of study presents statistical problems which judged from published work are not widely appreciated The purpose of this paper is to propose a general simple method for a clinically useful and statistically valid analysis We consider only studies in which each patient receives a single treatment or intervention so excluding escalating dose studies or crossover trials which require more complicated analysis Though we also restrict attention to outcome variables that are quantitative because these occur most2. different between groups Growth Is the eventual valueof Final value of outcome the outcome variable the measure or difference same betwe n groups between last and first values or percentage change between first and last Growth Is the response in one Time to reach a particular group delayed relative to value for example a fixed the other percentage of baseline Appendix II Calculation of area under the curve The area under the curve is calculated by adding the areas under the graph between each pair of consecutive observations If we have measurements y and y at times t and tz then the area under the curve between those two times is the product of the time difference and the average of the two measurements Thus we get t2 t1 yj y2 2 This is known as the trapezium rule because of the shape of each segment of the area under the curve If we have n 1 measurements y at times t i 0 n then the area under the curve AUC is calculated as n l AUC E Geir Wit vier is 0 Consider the data for one ill patient in the aspirin absorption study At times zero 5 10 15 20 30 40 60 75 90 and 120 minutes the aspirin concentrations are 0 8 3 21 6 33 9 35 5 47 2 38 3 20 5 13 3 0 and Oumol I respectively Thus we have AUC 5 x22 4 10 5 x 8 3 21 6 2 2 15 10 a ae 90 75 x gt 120 90 x 2191 pmol min I If we standardise by the length of the study 120 mi
3. and cube root transformed data BMJ 14 16 18 Original data Group 2 N a ODN DO O O m Day particularly in group 3 Figure 6 also shows the data after a cube root transformation which converts a volume mm into a linear measure which might be termed tumour size in mm The growth rates of the transformed data seem to be more linear than before the one disparate animal in group 3 with apparent tumour regression stands out with clarity The trans formed plots suggest that the tumour growth rate for each animal determined for example by linear regression of tumour size on day summarises the interesting information in the data This illustrates the general point that it may be necessary to transform the data on to a different scale before a simple summary analysis is possible Table II shows the transformation of the data on two rats together with the slope obtained from linear regression A one way analysis of variance excluding the outlier shows that there is no significant TABLE 11 Full data on two rats in tumour growth study with transformed values and corresponding summary measures Tumour volume Rat l Rat 2 Day mm Cube root mm mm Cube root mm 7 55 0 3 803 70 0 4121 11 95 0 4 563 129 7 5 062 12 205 9 5 905 196 0 5 809 13 205 9 5 905 205 8 5 904 14 270 0 6 463 375 7 7 216 15 307 3 6 748 419 1 7 484 17 405 1 7 39Q 421 2 7 496 18 726 0 8 988 573 4 8 308 19 950 4 9 832 7
4. differ is too vague to indicate the correct statistical analysis There are other methods of analysis of serial data such as repeated measures analysis of variance multivariate analysis of variance Kenward s method and hierarchical models Methods such as the split plot analysis of variance which ignore the correlations within subjects are generally not valid 234 None of these methods provides results that are as easy to understand as the method of summary measures nor are they as easy to use If there are missing values as is common with this type of study the method of summary measures will usually still allow the summary to be calculated but other methods may be difficult to apply If the times of observations differ among subjects then all the methods other than summary measures will fail There are problems with all methods of analysis if the data collection period varies greatly among subjects or if data collection is stopped in some subjects for reasons which may relate to the outcome variable For example if the time dependence was really peaked but some subjects were withdrawn from the study early while their response was still rising it would seem that the time dependence was one of growth If a large proportion of subjects do not complete the study then in common with other types of study this will make sensible analysis of the results very difficult if not impossible We have described just two common ty
5. measure related to the maximum effect of the agent given The time taken to reach this maximum may also be a clinically important variable There may be instances where it is the minimum not the maximum level that is relevant For data in the growth category the rate at which the variable is changing over either the whole or part of the experiment is an important feature A good measure of this rate will often be the slope of a line fitted to the data which is most easily measured by the regression coefficient estimated by least squares In some circum stances for example after giving different drugs to different groups of subjects the final outcome possibly expressed as a difference from baseline may usefully represent some achievable value This may be more appropriate for data that tend to level off over the last few values for example diastolic blood pressure in the treatment of hypertension Once the appropriate summary measure has been calculated for each subject its values can be treated as raw data for an appropriate statistical analysis For example if there are two groups then means or medians of the summary measures could be compared More than one summary measure can be constructed so that different aspects of the response may be investigated It is not easy to give any general rule about how many measures can be constructed though clearly each measure should aim at summarising a different aspect of
6. of dialysis The changes in the variables were shown to be roughly linear with time Other examples of this type of time dependency might be irreversible changes in lung function in people exposed to toxic fumes or growth of children in different social settings In this type of study the important feature is the rate at which the variable changes Peaked curve Growth curve 000 0 O 2 0 0 0 8 o o 0 Oo Ea ke e Ses 0 O Baseline 6 g 3 O Time Time FIG 1 Examples of peaked curve and growth curve 120 5 e 80 3 Healthy patients E Ill patients i lt 60 p lt 005 40 20 0 30 60 90 120 Time minutes FIG 2 Mean and standard deviation of aspirin concentrations in nine healthy and nine ill patients over time Usual method of display 16 K 140 D 120 10 80 Wo 140 12 100 80 160 140 120 Blood glucose mg 100 ml FIG 3 Individual plots of data given by Fikri and Ghalioungui of blood glucose concentrations against time in 18 patients with ancylostoma anaemia 0 1 2 3 0 1 2 Time hours Time hours Time hours 231 Usual method of analysis When two groups are being compared a common but inappropriate analysis is to apply separate two sample tests at each time point for example the test or Mann Whitney U test To illustrate this approach we compare aspirin absorption in patie
7. should be made around the time when the feature is likely to occur For the analysis of such data we recommend the method of summary measures for extracting the useful infor mation from the data This method is in common use in clinical pharmacology and should become m widespread in all branches of clinical research We are grateful to John Williams for producing the figures and Lindsey Izzard for typing the revised drafts We thank David Appleton Michael Healy Niels Keiding Judy Simpson Lene Skovgaard and particularly Martin Gardner for constructive criticism of previous drafts of this paper Illingworth PJ Jung RF Howie PW Isles TE Reduction in postprandial energy expenditure during pregnancy Br Med F 1987 294 1573 6 Bradley JR Evans DB Gore SM Cowley AJ Is dialysis hypotension caused by an abnormality of venous tone Br Med F 1988 296 1634 7 Fikri MM Ghalioungui P Ancylostoma anaemia Lancet 1937 1 800 2 Wishart J Growth rate determination in nutrition studies with the bacon pig and their analysis Biometrika 1938 30 16 28 Oldham PD A note on the analysis of repeated measurements of the same subjects J Chronic Dis 1962 15 969 77 Rowell JG Walters DE Analysing data with repeated observations on each experimental unit Journal of Agricultural Science Camb 1976 87 423 32 Healy MJR Some problems of repeated measurements In Bithell JF Coppi R eds Perspectives in medical statistics London Academic Press 19
8. the difference between the groups is expressed as the ratio of the geometric means together with the 95 confidence interval There is strong evidence that there is both a greater peak value and a higher overall level for blood aspirin concentrations in healthy subjects than in ill subjects Figure 5 gives the plot of peak aspirin concentrations in the healthy and ill patients by the time the maximum occurs as derived from fig 4 This shows clearly that peak values tend to be lower and occur later in the ill patients which can also be deduced less easily from fig 4 Nevertheless there also seems to be a negative relation between the size of the peak and its time of occurrence that is the higher peaks occur earlier particularly in healthy patients which is not easily seen in fig 4 GROWTH DATA A second example describes changes in tumour volume in three groups of 10 rats at 11 time points after different injections The injections were tissue culture medium group 1 tissue culture medium and normal spleen cells group 2 and normal spleen cells immune RNA and tumour antigen group 3 Figure 6 plots the results original data which clearly belong to the growth category Plainly the tumours are growing in almost all animals but at a variable rate BMJ Tumour volume mm Tumour size mm FIG 6 Individual plots of tumour volume against time in three groups of 10 rats expressed as original data
9. 01 8 8 887 20 661 5 8 713 21 798 6 9 278 Summary measures slope of cube root of tumour volume on day rat 1 0 444 mm day rat 2 0 404 mm day 20 22 6 8 10 12 14 16 18 20 22 6 8 10 12 14 16 18 20 22 Day difference in the rate of growth of tumours in the three groups mean tumour growth rates 0 438 0 438 and 0 435 mm day for groups 1 2 and 3 respectively Fz 26 0 0 p 1 0 Discussion It is common in medical research for methods of statistical analysis to become standard for particular types of data Their use becomes widely accepted and little thought is given to whether they are truly appropriate to the clinical question being posed We have shown that there are serious problems associated with the common use of comparisons at each time point when analysing serial measurements on patients In particular the method is inappropriate because it does not provide clear answers to clinically relevant ques tions In addition there are important statistical deficiencies We have described the method of summary measures which avoids all of the problems identified with the more usual analysis Its main disadvantage is that it may be difficult to specify in advance an appropriate summary measure The use of this type of analysis should encourage researchers to think about the features of the data that will be of most interest to them when designing the study Posing the question simply as How do the groups
10. 81 155 7 Yates F Regression models for repeated measurements Biometrics 982 38 850 3 De Klerk NH Repeated warnings re repeated measures Aust N Z J Med 1986 16 637 8 wu b w w a Si lt a 10 Armitage P Berry G Statistical methods in medical research Oxford Blackwell Scientific 1987 355 360 411 11 Gardner MJ Altman DG Confidence intervals rather than P values estimation rather than hypothesis testing Br Med F 1986 292 746 50 12 Tsai K T Koziol JA VARCOV I a computer program for the multivariate analysis of growth and response curves Comput Methods Programs Biomed 1988 27 69 74 13 Morrison DF Multivariate statistical methods New York McGraw Hill 1976 14 Kenward MG A method for comparing profiles of repeated measurements Applied Statistics 1987 36 296 308 15 Goldstein H Multilevel models in educational and social research London Griffin 1987 5 1 60 Appendix I Some summary measures Type of data Question to be answered Summary measure Peaked Is the overall value of the Overall mean equal time outcome variable the intervals Area under same in different groups curve unequal time intervals Peaked Is the maximum Maximum minimum minimum response value different between groups Peaked Isthetimetomaximum Time to maximum minimum response minimum response different between groups Growth Istherateofchangeof Regression coefficient the outcome variable
11. Division of Medical Statistics University of Newcastle upon Tyne The Medical School Newcastle upon Tyne NE2 4HH J N S Matthews PHD lecturer Medical Statistics Laboratory ICRF Lincoln s Inn Fields London WC2A 3PX Douglas G Altman Bsc director Medical Statistics and Computing University of Southampton Southampton General Hospital Southampton SO9 4XY MJ Campbell PHD senior lecturer Department of Medical Physics Royal Postgraduate Medical School London W12 ONN Patrick Royston MSC senior lecturer Correspondence to Dr Matthews Br Med J 1990 300 230 5 BMJ 33 90 COPYRIGHT 1990 Reprinted from the BRITISH MEDICAL JOURNAL 27th January 1990 300 230 235 Analysis of serial measurements in medical research J N S Matthews Douglas G Altman M J Campbell Patrick Royston Abstract In medical research data are often collected serially on subjects The statistical analysis of such data is often inadequate in two ways it may fail to settle clinically relevant questions and it may be statistically invalid A commonly used method which compares groups at a series of time points possibly with 1 tests is flawed on both counts There may however be a remedy which takes the form of a two stage method that uses summary measures In the first stage a suitable summary of the response in an individual such as a rate of change or an area under a curve is identified and calculated for each subject
12. commonly the methods can also be applied to ordered data such as pain scores Types of time dependency It is helpful to distinguish two main ways in which the outcome variable may change with time Peaked In many studies the outcome variable starts from a baseline sometimes zero rises to a peak and then returns to baseline This is displayed as a peaked curve fig 1 For example in a study of post prandial energy expenditure during pregnancy the metabolic rate was measured in women after a 12 hour fast and then at 30 minute intervals for two hours in response to a test meal This was done during pregnancy and again once lactation had stopped The metabolic rate rose toa peak after about 60 minutes and then fell steadily Women were found to have a reduced energy expenditure during pregnancy In that study the interest lay in both the total response and the time to reach the maximum value Growth Sometimes the outcome variable steadily increases or decreases with time and does not start to return to its initial value over the period of study This is displayed as a growth curve fig 1 A recent study investigated the role of peripheral vascular tone in All rights of reproduction of this reprint are reserved in all countries of the world hypotension induced by dialysis Each patient had sessions of dialysis with acetate fluid and with bicarbonate fluid Blood pressure was measured every 15 minutes during the four hours
13. ealthy and ill patients which clearly belong to the peaked category It is clear that the mean curves in fig 2 hide considerable variability The basic question posed by the researcher was Do ill patients have reduced absorption of aspirin We could answer this by using two of the summary measures given in appendix I We could calculate the area under the curve for each patient and we could also look at the maximum value Both these measures are meaningful and are familiar to pharmacologists Table I gives statistics of these two summary measures for comparing the two groups The distributions of both summary measures not shown are skewed indicating that the data should be trans formed or analysed by a non parametric method TABLE I Analysis of data from aspirin study Maximum Area under curve concentration Healthy patients n 9 Arithmetic mean SD umol l 26 5 8 8 86 0 41 5 Geometric mean mol l 25 4 77 8 Ill patients n 9 Arithmetic mean SD pmol l 17 5 5 0 46 7 26 3 Geometric mean pmol l 16 8 41 2 Ratio of geometric means 1 52 1 89 95 Confidence interval 1 11 to 2 08 1 14 to 3 13 t Test 2 83 df 16 2 66 df 16 p Value 0 01 0 02 Also the standard deviations increase with the mean concentration This suggests that the data should be transformed by a logarithmic transformation Table I gives the results of an analysis where the summary measures have been log transformed and where
14. imple to use and widely applicable is what we refer to as the method of summary measures Though not new it has been described several times since 1938 it does not seem to be widely known among medical research workers The method considers the individual as the basic unit and uses the responses for each individual subject to construct a single number which summarises some aspect of that subject s response curve This approach avoids all the difficulties outlined above Appendix I lists examples of some summary measures When the outcome measure is the concentration of a substance that has been given for example a drug or glucose the response will often be peaked The total uptake of the substance may be of interest and can be measured by the area under the response curve in an individual subject Appendix II gives a method for calculating the area under the curve and shows that it may be interpreted as a type of weighted average of the responses Thus the area under the response curve may be useful even in cases when a direct interpretation such as the amount of substance absorbed is not possible An alternative measure of the overall value is the simple mean of the observations If the intervals between successive observations are the same this will be closely related to the area under the curve Another feature of peaked data that is frequently of interest is the maximum value often denoted by Cmax which may be interpreted as a
15. lly respond over time and give no information about variation among subjects in their response over time The error bars often shown fig 2 relate only to between subject variation at each time point From point 3 it is evident that in addition there are convincing statistical arguments against multiple tests Separate significance tests at different time points are often interpreted as if they gave independent information about the relative location of the groups That this is untrue may be illustrated by considering what would happen if the blood aspirin concentrations had been measured every minute The number of significant p values would increase enormously even though the increase in the amount of information about the separation of the groups would be minimal The tests are clearly not independent and so their interpretation is much more difficult Further dividing the results into significant and not significant introduces an artificial dichotomy into serial data Most biological variables change over time in a smooth and continuous manner and so the idea that at one point in time the difference between two variables is not significant whereas at the next point in time it is significant is artificial Thus separate significance tests are not a sensible way to assess the difference between sets of repeated measurements Recommended method of analysis use of semmary measures One method that is clinically relevant s
16. nts who are either healthy or ill Each patient was given the same dose of aspirin per kg body weight and had his or her blood aspirin concentration measured at time zero and after 5 10 15 20 30 40 60 75 90 and 120 minutes Analyses of mean concentrations at each time point would result in the graph shown in fig 2 which is typical of graphs in many published papers The important features of the analysis shown in fig 2 are a the lines joining the means at each time point are drawn for each group b error bars often undefined are attached at each time point c an indicator of statistical significance is placed by each time point to summarise the results of the separate significance tests What is wrong with this approach There are several criticisms that can be made 1 The curve joining the means may not be a good descriptor of a typical curve for an individual Important variation in the shapes and locations of curves for different subjects may be hidden We illustrate this by some data published over 50 years ago Figure 3 shows individual glucose tolerance curves for 18 subjects with ancylostoma anaemia From the mean curve also shown we might deduce that for a typical patient the venous blood sugar concentration rises to a maximum at about one hour after drinking a glucose solution and then falls back to normal around 90 mg 100 ml 5 0mmol l after some three hours From the individual curves however it i
17. nutes we get 2191 120 18 3 pmol l which is close to the mean value of the observations of 19 9 umol l Accepted 27 November 1989 235 Printed in Great Britain by Bourne Offset Limited Iver Bucks BMJ
18. pes of serial data but other forms of response over time are sometimes of interest For example in studies of basal body temperature around the time of ovulation the feature of interest is whether there has been a change in the temperature level and when this occurred In studies assessing the effectiveness of antihypertensive treatment the outcome of interest may be the level at which the blood pressure stabilises rather than the change from baseline or the rate of change One consequence of replacing the measurements on a subject by a single summary measure is that what seemed to be a lot of data suddenly seems rather small This will be the case when trying to make up for lack of subjects by measuring each subject many times The strong dependency between measurements close in time on the same patient means that the original wealth of data was to some extent illusory the summary measures will give a more honest indication of the amount of information that has been collected When a researcher is planning a study in which serial measurements on each subject will be collected the intended method of analysis should be considered It is valuable to have a clear idea of the features of the data that will be of prime interest The sample size of the study should be taken as the number of subjects not the total number of observations If it is known that the timing of a particular feature such as a peak is of interest than extra observations
19. s evident that this summary hides a wide variety of curves including multiple peaks cases 4 and 10 and a steady rise case 18 The maximum rise above the fasting value time zero occurs at times ranging from 30 minutes to two hours after the glucose drink The summary of the results was that 12 out of 18 patients with ancylostoma anaemia showed abnormal glucose tolerance informa tion which could not be gleaned from the mean curve 2 No account in the analysis is taken of the fact that measurements at different time points are from the same subjects It is inherent in the design that the main 160 N G7 Ph aa 4 a TR 160 Mean 140 120 100 s0 Po ae Time at BMJ BMJ interest lies in the way individual subjects respond over time yet this is ignored when each time point is analysed separately 3 Successive observations on a given subject are likely to be correlated The value at one time point is likely to influence successive time points so that the significance tests will not be independent If a test at one point in time gives a significant result then it is likely that tests performed at points close in time will also give significant results The usual approach to the analysis of serial data may thus be criticised on both clinical and statistical grounds Points 1 and 2 together indicate that the usual analysis may give a misleading impression of the way that individual subjects typica
20. summary measures by hand or using an ad hoc computer program is quite feasible Graphic display Graphs are an effective way to display data and illustrate conclusions Nevertheless it is more difficult to create informative and truthful graphs of repeated measures than might be supposed The most informa tive approach and one which is strongly recommended when the data are first analysed is to produce separate graphs of the responses against time for each subject In order to aid visual comparisons the same axis scaling should be used on all graphs The plots may be arranged into a panel or grid with separate panels for each group It may help to order the plots in some way such as by increasing mean or maximum value Depending on the sample size it may be feasible to include plots of the raw data in a paper for example 232 160 140 120 Aspirin umol l 60 40 20 80 Aspirin umol l a O D N oO as was done in the blood glucose example and we recommend this If raw data cannot be shown it may be possible to classify the curves and plot representative examples How the representative examples were chosen for example a one in five random sample should be described in any publication Fikri and Ghalioungui considered that there were seven classes of curve in their data fig 3 The question whether the mean of a given set of responses represents a typical subject s cur
21. the response There are seldom more than two or three interesting aspects of a response curve and so two or at most three different measures should be enough for a complete analysis of the data As the principle is to reduce a large number of dependent observations to a smaller number of summaries the method would be vitiated if too many summaries were used The summary measures should have some clear clinical or biological relevance and ideally they should be chosen before the data are collected This avoids any temptation to choose a particular summary measure because it shows a maximal difference between groups Indeed an advantage of the method is that by thinking in terms of relevant summary measures the researcher is compelled to decide on specific questions that the data are required to answer By thinking in these terms in advance it may be possible to improve the design of a study For cxamplc if the important summary measure is the time to maximum response then to get a precise estimate frequent measurements would be needed around the time that the maximum response is expected It would be dogmatic however to insist that a summary measure could not be chosen after the data have been examined especially when the shape of the time response curve is uncertain There are few computer packages that allow computation of summary measures in a routine manner Nevertheless studies of this nature are often fairly small and calculating
22. ve is not simple to answer The setting in which the mean curve is most likely to be useful occurs when all the peak responses occur at the same time for example when subjects all respond quickly to a stimulus Summary measures have considerable advantages for plotting because the usual graphical methods such as histograms scatter plots and so on may be applied to them Insight may be gained from a scatter plot of any two summary measures In particular a useful exercise is to plot the maximum or minimum value for each subject against the time that the maximum or Healthy patients III patients 60 90 120 Time minutes FIG 4 Individual plots of aspirin concentrations against time in healthy patients and ill patients Maximum aspirin concentration umol l Healthy patients 10 20 Il patients 30 40 0 10 20 30 40 Time of maximum minutes FIG 5 Scauer plot of maximum aspirin concentrations by time of maximum in heulthy patients and ill patients 233 minimum occurred This enables large quantities of data to be plotted succinctly and may disclose a relation between the two variables that could not be discerned in the plot of the raw data Examples We illustrate the points discussed above with two examples relating respectively to peaked data and to growth data PEAKED DATA We first reconsider the study of aspirin absorption Figure 4 shows the individual responses in the h
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