Accepted Manuscript A guide to survival analysis for manual therapy clinicians and researchers Mark J. Hancock, Christopher G. Maher, Lucíola da Cunha Menezes Costa, Christopher M. Williams PII:
S1356-689X(13)00153-7
DOI:
10.1016/j.math.2013.08.007
Reference:
YMATH 1491
To appear in:
Manual Therapy
Received Date: 25 July 2013 Accepted Date: 25 August 2013
Please cite this article as: Hancock MJ, Maher CG, Costa LdCM, Williams CM, A guide to survival analysis for manual therapy clinicians and researchers, Manual Therapy (2013), doi: 10.1016/ j.math.2013.08.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Title: A guide to survival analysis for manual therapy clinicians and researchers
Authors:
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*Mark J Hancock, Faculty of Human sciences, Macquarie University, Australia Christopher G Maher, The George Institute for Global Health and Sydney Medical School, University of Sydney, Australia
Lucíola da Cunha Menezes Costa, Universidade Cidade São Paulo, São Paulo, Brazil
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Christopher M Williams, The George Institute for Global Health and Sydney Medical School,
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University of Sydney, Australia; Hunter New England Population Health, Australia.
* Corresponding author Mark J Hancock,
75 Talavera Rd North Ryde, Faculty of Human Sciences, Macquarie University, NSW 2109, Australia Ph: +61298506622
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Fax: +61298506630
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Email: [email protected]
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ABSTRACT: Survival analysis is a statistical approach used to study time to an event of interest. The approach is highly applicable to the manual therapy discipline, yet is rarely used. This masterclass aims to
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present a simple overview of survival analysis aimed at both clinicians and researchers in the manual therapy field. Specifically the masterclass aims to 1) describe situations were survival analysis is appropriate to use 2) describe unique characteristics of survival data including censoring of
participants 3) describe analysis methods for survival data including log rank test and cox regression
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4) describe the interpretation of results from survival analyses including survival plots and hazard
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ratios.
Introduction:
Survival analysis is a statistical approach which is relevant to many studies in the manual therapy discipline. However, to date it is used far less commonly than other approaches (Moseley et al.,
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2009). This may be in part due to lack of understanding of the availability of this approach. It is also likely that clinicians may find the approach difficult to understand due to lack of familiarity with the terminology and reporting. Therefore the aim of this masterclass is to present a simple overview of
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survival analysis aimed at both clinicians and researchers in the manual therapy field. Specifically the review will aim to 1) describe situations were survival analysis is appropriate to use 2) describe
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unique characteristics of survival data including censoring of participants 3) describe analysis methods for survival data including log rank test and cox regression 4) describe the interpretation of results from survival analyses including survival plots and hazard ratios
When is survival analysis used?
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The decision of what statistical approach to use is based largely on the nature of the primary outcome (dependent variable). For example in the landmark study by O’Sullivan and colleagues (O'Sullivan et al., 1997) investigating spinal stabilisation exercise for chronic low back pain, the
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primary outcome was pain, measured on a 0-100 scale, at 0, 3, 6 and 30 months. If the participants were injured workers it could be argued that measuring work status (full duties, partial duties, off work) or claim closure (yes/no) at each of those time points would also be informative. The pain, work status and claim closure measures require a different statistical technique (as they are
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continuous, ordinal and dichotomous data respectively). However, they share a common approach in that each of the 3 outcomes (pain, work status and claim closure) have been used to characterise
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health status at key time points. A quite different approach is required if the interest is in how long it takes to reach a key outcome (e.g. days to become pain-free, days to return to work or days to claim closure). These are situations where survival analysis is useful, as we wish to analyse the time until
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an event occurs (Altman & Bland, 1998).
Survival analysis is used to study time to an event. Traditionally it was used to study well people at risk of experiencing a negative event such as disease, disablement or death (hence the terminology);
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however, it can equally be used to study sick people at risk of experiencing a positive event. In the manual therapy field there are many scenarios where time to an event may be an important if not
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optimal outcome. Examples of positive events include time to return to work, time to become painfree, time to discharge; whereas examples of negative events include time to development or onset of a new health condition, time to recurrence, time to fall, etc. Survival analysis can be used in studies of prognosis (Costa Lda et al., 2009; Henschke et al., 2008)and clinical trials testing prevention(Soukup et al., 1999) or treatment (Hancock et al., 2007; Williams et al., 2010). It can be used to study time to adverse events such as device failure or need for revision surgery in orthopaedic trials (Baker et al., 2013). It is worth noting that survival analysis can be used when the
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outcome of interest is time to almost any event. An innovative example from the physical therapy field is time from presentation of a study at a conference to publication in a peer-reviewed journal
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(Smith et al., 2011).
Survival analysis has several advantages and some disadvantages over analyses used in most manual therapy trials, including the O’Sullivan and colleagues (O'Sullivan et al., 1997) example, which focus
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on outcomes at a particular point or points in time (e.g. pain at 3 months, 6 months and 12 months after starting treatment). An advantage of survival analysis is that rather than taking a snap shot at
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one single time point (which may not be the ideal time to assess the effect of the intervention) survival analysis provides data on the impact of intervention, or the natural course of a condition, over the whole time period investigated (Bradburn et al., 2003). Survival analysis avoids the problem of multiplicity (increased type 2 error rate) associated with multiple analyses at different time points and so there is no need to select the key (primary) time point to measure. Survival analysis is able to
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cope with different follow-up periods for participants (discussed in more detail below) and accommodates loss to follow-up more simply than data imputation. Survival analysis also copes with skewed (non-normally distributed) data. A practical disadvantage with survival analysis is the need
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to collect data regularly to ensure precise estimates of time to event. The use of daily patient diaries and electronic data capture methods are options used to overcome this (Axen et al., 2012; Hancock
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et al., 2007).
Censoring and survival analysis Before we can analyse time until an event occurs we need to calculate this time. To perform this calculation we first have to define when the time starts and when it ends. The starting point for the time interval is taken to be the point at which the patient is first considered to be at risk. In a randomized controlled trial the starting point is the date of randomization. The end point is simply
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the time when the event occurs. A unique feature of survival analysis is called censoring (Bradburn et al., 2003). This feature allows use of data from people who do not have the event for as long as they are followed, by considering the time at which these people were last to have still been at risk
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(last follow-up completed) but not experienced the event of interest. Let’s imagine a study investigating a treatment (manual therapy) for acute low back pain where it was decided that the
outcome of interest was time to recovery from pain. If all participants in the study had recovered by the end of the study then we would have data on the number of days to recovery for all participants.
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However, it is rarely the case that all people have the event (e.g. recover) by the end of the study
and it is usually not feasible to continue a study for an indefinite period of time. At some point there
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is a need to stop the study and analyse the data. At that point participants who have not experienced the event are censored. There are other reasons why patients will be censored. The most common reasons for censoring are listed below and presented in Figure 1 (Hosmer et al., 2008; Leung et al., 1997).
People who do not recover (have the event) during the study follow up period (subjects 1
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•
and 5)
People who cannot be followed up for the full study follow up period (subject 2)
•
People withdrawn from the study (subject 4)
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•
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Survival time and calendar time
In most studies participants are enrolled as they become available; for example in the Hancock et al trial (Hancock et al., 2007) of manual therapy and NSAIDs for acute LBP it took 16 months to recruit 240 patients from general practice. Studies using time to event outcomes can either follow all participants for a set period of time (e.g. 3 or 12 months) or follow all participants until a set date (e.g. 31st Dec 2015), understanding that some will be in the study longer than others and so followed for longer. In cases where participants are followed until a set date (calendar time) this needs to be converted to survival time as demonstrated in Figure 2. Survival time rather than calendar time is
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the data required for survival analysis and represents the time to event since some entry point. The entry point is often obvious but not always. For manual therapy studies it is often treatment onset or
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onset of the condition.
The decision to design the study so people are followed for a set period (e.g. 1 year) or until a set date is a practical decision involving the availability of participants, frequency of the event, time
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period over which the effect of the predictor was likely to have an influence and budget/time to
complete the study. Arguably it is simpler to follow all participants for a set time period, however if
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recruitment takes place over a long period (e.g. 2 years) then it may make sense to follow until a set date rather than for a set period if additional important data is collected. Again it is important to emphasise that survival analysis copes well if participants are followed for different time periods (as would occur if participants are recruited at different time points but followed to a single date), due
The Kaplan-Meier life table
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to the ability to censor individuals.
A common method to estimate statistics with survival data is the Kaplan-Meier (or product-limit).
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method (Kaplan, 1958). This method is used to estimate the proportion of people who ‘survive/recover’ beyond a specific time point (e.g. 1 month or the end point of the study). The
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calculation and presentation of the Kaplan-Meier estimates are usually displayed in a life table as per the example in Table 1. Here you can see that the probability of survival changes only on the day an event occurs. This feature is known as conditional probability meaning that the probability of an event at a certain time point is calculated on the condition of the events that occurred previously. A new probability of survival is calculated at each time point when an event occurs. You can see this in Table 1 and also graphically in Figure 3 as the steps in the curve. The new probability is simply calculated by using the formula below:
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Number of remaining cases (initial number of cases minus cases who have had the event or been censored) / number of cases at risk (initial number minus cases censored) Let’s consider the example in Table 1. This data could represent a simple cohort study with 100
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participants with LBP where patients’ recovery was monitored every day. While in survival analysis traditionally the ‘probability of survival’ is described, in this scenario the event is positive (recovery) so the traditional terminology is counter-intuitive and hence we will talk about the ‘probability of
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non-recovery’. If on day 1 a single patient recovered then the probability of non-recovery at day 1 would be 0.99 ((100-1)/100). On day 2 if 3 participants recovered the probability of non-recovery
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would be 0.96 ((100-4)/100). Two important points need noting here. Firstly the number of cases at risk changes depending on how many cases have been censored. In the example in Table 1 you can see that 2 participants withdrew from the study on day 3. As a result when the next event occurred (day 4) the number at risk was now 98 (100 minus 2 participants who had withdrawn). The 2 patients who had withdrawn (been censored) were not at risk of recovery as no data was available
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at this time point. This nicely demonstrates how patients who are censored are used in the calculations of survival probability (in our case non-recovery probability) for as long as they are being followed even if they drop out of the study before they experience the event. The second important
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point from the above example is that while time is a truly continuous variable survival curves typically represent blocks of time (in this example days). Because of this it is common to have
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multiple events occurring at one time point (e.g. day 2). Shorter periods between follow-ups will increase the precision of the estimates and curve produced. Whereas longer periods between follow ups may result in ‘interval-censoring’ meaning that the outcome is known only to lie within a time interval, instead of being directly observed. Survival curve The Kaplan-Meier estimates can also be plotted in a survival curve as demonstrated in Figure 3. Here the basic concept of survival analysis can be more easily visualised and understood. The y axis
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represents the proportion of people who have had the event, while the x axis represents time since study entry. Typically the curve starts at 1 on the y axis at time 0 on the x axis to represent the fact that 100% of the participants have “survived” or not had the event at entry into the study. Over time
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the curve represents the probability of surviving at each time point on the x axis. The survival curve in figure 3A represents the probability of remaining unrecovered (surviving) at different time points after an episode of low back pain. If the probability of recovery is of more interest this is simply
calculated by 1 minus probability of being unrecovered and the survival curve can start from a y
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value of 0 as represented in Figure 3B.
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It is common to summarise or convey the findings from a survival curve (or Kaplan Meier life table) by reporting the time at which specific proportions of participants had experienced the event. The most common example is the median survival time which is the time at which 50% of participants had the event. In Figure 3A you can see that the median time to recovery was approximately 18
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days. This can be determined by drawing a horizontal line from 0.5 on the y axis until it intersects with the curve and then transferring this point to the x axis to determine the time. This can be done for any survival probability as long as the survival probability of the curve falls to lower than the level
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of interest.
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Comparing survival curves
In manual therapy studies we will often want to determine the influence of a certain variable (treatment or prognostic factor) on time to event. The log rank test (Bland & Altman, 2004) is commonly used to test if the survival function (time to event) for people with a characteristic (e.g. female) is different from those without this characteristic (e.g. male). Figure 4A presents an example of different survival curves for people receiving baseline care and additional manual therapy compared to those receiving just baseline care. You can see the curves follow very similar paths and
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as such the p value from the log rank test is not significant (p=0.870). Figure 4B presents survival curves for people who do and do not meet a clinical prediction rule for recovery from acute low back pain. Here you can see the curves appear different and those who meet the prediction rule have a
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faster rate of recovery as indicated by the significant p value. A limitation of the log rank test is it does not describe the effect size and is unable to allow for important covariates or confounders.
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Cox regression
Cox regression is widely used to assess the influence of multiple factors on time to event (Cox, 1972).
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Similarly to other regression analyses, Cox regression produces coefficients which provide an estimate of size of effect due to the variable of interest as well as confidence intervals. These estimates are called hazard ratios and represent the chance of the event occurring in those with the variable of interest (e.g. female) as a ratio of the hazard of the event occurring in those without (e.g.
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male). Therefore a hazard ratio of 1 represents no difference while a hazard ratio of 3 implies that at any time point the event will occur 3 times more commonly in those with the variable of interest compared to control participants. In the case of continuous predictor variables the hazard ratio
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represents the ratio of hazards for each unit of change in the predictor variable. The hazard ratio does not provide information about speed of time to event (or recovery) so it is incorrect to
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interpret a hazard ratio of 3 as saying people with the treatment will recover 3 times faster than those without the treatment(Spruance et al., 2004).
Proportional hazard assumption A key assumption when performing cox regression is that the hazard ratio for the explanatory variable is relatively constant across time. There are several methods to assess the proportional hazards assumption and these are largely beyond the scope of this paper. Readers interested in this
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topic can find more detail in the following resources (Cox, 1972; Hosmer et al., 2008; Ng'andu, 1997). However, simple visualisation of a plot of survival curves is one simple method and helps provide an understanding of the issue. In Figure 5A you can see survival curves where the
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proportional hazards assumption appears to be valid, as any differences in slopes (hazard rates) between curves are relatively consistent over time. In this case further testing should be conducted to provide further evidence that the assumption has not been violated. Further checking of the
proportional hazards assumption may occur by fitting a time-dependant covariate to the model
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and/or by graphical examination of Schenfeld residuals (Ng'andu, 1997). In Figure 5B however, the curves cross and the hazard ratio clearly changes over time. In this case there is strong evidence the
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proportional hazards assumption is violated. An example relevant to manual therapists where you might expect the proportional hazards assumption to be violated would be a trial comparing steroid injection to exercise therapy in people with sub acromial impingement as the hazard ratio would likely change over time. The event of interest could be time to 50% reduction in pain. In the short
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term (e.g. day 1) you would expect more people to have the event if they received steroid injection, however, at a later time point (e.g. 6 weeks) it is likely a larger number of people receiving exercise therapy will have the event. This is a clear violation of the proportional hazards assumption and
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modifications.
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would produce a plot similar to Figure 5B. Cox regression cannot be used in this case without
One strategy to deal with situations where the proportional hazards assumption is violated is to divide the time interval (Hosmer et al., 2008). In other words the hazard ratios are generated for smaller intervals of time where the hazard ratio is more constant. Initially this decision can be made by visual assessment of survival curves. Using the data in Figure 5B it may be sensible to divide the time into intervals of 0-2 weeks and > 2 weeks.
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Summary: This master class aims to give clinicians and researchers a basic understanding of survival analysis so they can correctly interpret published papers and consider using survival analysis in future trials if
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appropriate. The key messages include: 1) Survival analysis has some advantages that make it a good choice for many studies of manual therapy. 2) Survival analysis can be used when the outcome of interest is time to event, regardless of what defines the event. 3) Censoring is a key feature of
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survival analysis. 4) Survival curves plot the proportion of cases having the event at different time
points. 5) Cox regression is commonly used to test the influence of independent variables on survival
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rates. 6) Hazard ratios from cox regression describe the ratio of hazards for different values of the independent variable (e.g. treatment or control). 7) Median time to recovery is commonly used to compare speed of recovery for different values of the independent variable (e.g. treatment or
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control). 8) The proportional hazards assumption needs to be tested when using cox regression.
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References
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Altman DG, Bland JM. Time to event (survival) data. BMJ 1998;317: 468-469. Axen I, Bodin L, Kongsted A, Wedderkopp N, Jensen I, et al. Analyzing repeated data collected by mobile phones and frequent text messages. An example of low back pain measured weekly for 18 weeks. BMC Medical Research Methodology 2012;12: 105. Baker P, Jameson S, Critchley R, Reed M, Gregg P, et al. Center and surgeon volume influence the revision rate following unicondylar knee replacement: an analysis of 23,400 medial cemented unicondylar knee replacements. Journal of Bone & Joint Surgery - American Volume 2013;95: 702-709. Bland JM, Altman DG. The logrank test. BMJ 2004;328: 1073. Bradburn MJ, Clark TG, Love SB, Altman DG. Survival analysis part II: multivariate data analysis--an introduction to concepts and methods. British Journal of Cancer 2003;89: 431-436. Costa Lda C, Maher CG, McAuley JH, Hancock MJ, Herbert RD, et al. Prognosis for patients with chronic low back pain: inception cohort study. BMJ 2009;339: b3829. Cox DR. (1972). Regression models and life-tables (pp. 187). Hancock MJ, Maher CG, Latimer J, McLachlan AJ, Cooper CW, et al. Addition of diclofenac and/or manipulation to advice and paracetamol does not speed recovery from acute low back pain: A randomised controlled trial. The Lancet 2007;370: 1638-1643. Henschke N, Maher CG, Refshauge KM, Herbert RD, Cumming RG, et al. Prognosis in patients with recent onset low back pain in Australian primary care: inception cohort study. BMJ 2008;337: a171. Hosmer D, Lemeshow S, May S. ( 2008). Applied survival analysis : regression modeling of time-toevent data (2nd ed.): Wiley. Kaplan EL. Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association 1958;53: 457-481. Leung KM, Elashoff RM, Afifi AA. Censoring issues in survival analysis. Annual Review of Public Health 1997;18: 83-104. Moseley AM, Elkins MR, Herbert RD, Maher CG, Sherrington C. Cochrane reviews used more rigorous methods than non-Cochrane reviews: survey of systematic reviews in physiotherapy. Journal of Clinical Epidemiology 2009;62: 1021-1030. Ng'andu NH. An empirical comparison of statistical tests for assessing the proportional hazards assumption of Cox's model. Statistics in Medicine 1997;16: 611-626. O'Sullivan PB, Phyty GD, Twomey LT, Allison GT. Evaluation of specific stabilizing exercise in the treatment of chronic low back pain with radiologic diagnosis of spondylolysis or spondylolisthesis. Spine 1997;22: 2959-2967. Smith HD, Bogenschutz ED, Bayliss AJ, Altenburger PA, Warden SJ. Full-text publication of abstractpresented work in physical therapy: do therapists publish what they preach? Physical Therapy 2011;91: 234-245. Soukup MG, Glomsrod B, Lonn JH, Bo K, Larsen S. The effect of a Mensendieck exercise program as secondary prophylaxis for recurrent low back pain. A randomized, controlled trial with 12month follow-up. Spine 1999;24: 1585-1591. Spruance SL, Reid JE, Grace M, Samore M. Hazard ratio in clinical trials. Antimicrobial Agents & Chemotherapy 2004;48: 2787-2792. Williams CM, Latimer J, Maher CG, McLachlan AJ, Cooper CW, et al. PACE-The first placebo controlled trial of paracetamol for acute low back pain: design of a randomised controlled trial. BMC Musculoskeletal Disorders 2010;11: 169.
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Table 1: Life table example Number Censored
Number of Number of remaining cases at cases risk
Probability of nonrecovery
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0 0 0 100 100 1.00 1 1 0 99 100 0.99 2 3 0 96 100 0.97 3 0 2 94 98 * 4 1 0 93 98 0.95 // (25) (3) 19 2 0 63 95 0.66 20 1 0 62 95 0.65 // (20) (3) 58 3 0 36 92 0.39 59 0 1 35 91 * 60 1 0 34 91 0.37 * no value calculated as no event (recovery) occurred on this day
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Time Number Interval Recovered (days)
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End of study
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Patients Start of study
•
6
>
5
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4
•
2
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3
1
0
20
40
80
60
100
120
140
160
>
> 180
Time since start of study (days)
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Figure 1: Data from an imagined study illustrating the common reasons for censoring Patients 3 and 6 had the event (recovered) and censoring was not needed (●); Patients 1 and 5 had not recovered during the study follow-up period and were censored (); Patient 2 could not be followed up for the full study follow up period and was censored; Patient 4 withdrew before the end of the study and was censored.
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Entry of study
•
6
•
Patients who had the event (i.e. recovered)
4
•
3
2
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Figure 2: Converting calendar time to survival time
>
5
Patients who were censored
1
2
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0
Survival time, months
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Legend:
4
> > 6
A
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B
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Median days to recovery =18
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Figure 3: Survival curves for days to recovery from low back pain. A) Proportion not recovered; B) Proportion recovered. Dashed lines in figure A represent the median time to recovery Figures drawn from raw data from the study reported in Hancock et al, 2007
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A
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B
p=
S1356-689X(13)00153-7
DOI:
10.1016/j.math.2013.08.007
Reference:
YMATH 1491
To appear in:
Manual Therapy
Received Date: 25 July 2013 Accepted Date: 25 August 2013
Please cite this article as: Hancock MJ, Maher CG, Costa LdCM, Williams CM, A guide to survival analysis for manual therapy clinicians and researchers, Manual Therapy (2013), doi: 10.1016/ j.math.2013.08.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Title: A guide to survival analysis for manual therapy clinicians and researchers
Authors:
RI PT
*Mark J Hancock, Faculty of Human sciences, Macquarie University, Australia Christopher G Maher, The George Institute for Global Health and Sydney Medical School, University of Sydney, Australia
Lucíola da Cunha Menezes Costa, Universidade Cidade São Paulo, São Paulo, Brazil
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Christopher M Williams, The George Institute for Global Health and Sydney Medical School,
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University of Sydney, Australia; Hunter New England Population Health, Australia.
* Corresponding author Mark J Hancock,
75 Talavera Rd North Ryde, Faculty of Human Sciences, Macquarie University, NSW 2109, Australia Ph: +61298506622
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Fax: +61298506630
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Email: [email protected]
ACCEPTED MANUSCRIPT
ABSTRACT: Survival analysis is a statistical approach used to study time to an event of interest. The approach is highly applicable to the manual therapy discipline, yet is rarely used. This masterclass aims to
RI PT
present a simple overview of survival analysis aimed at both clinicians and researchers in the manual therapy field. Specifically the masterclass aims to 1) describe situations were survival analysis is appropriate to use 2) describe unique characteristics of survival data including censoring of
participants 3) describe analysis methods for survival data including log rank test and cox regression
SC
4) describe the interpretation of results from survival analyses including survival plots and hazard
M AN U
ratios.
Introduction:
Survival analysis is a statistical approach which is relevant to many studies in the manual therapy discipline. However, to date it is used far less commonly than other approaches (Moseley et al.,
TE D
2009). This may be in part due to lack of understanding of the availability of this approach. It is also likely that clinicians may find the approach difficult to understand due to lack of familiarity with the terminology and reporting. Therefore the aim of this masterclass is to present a simple overview of
EP
survival analysis aimed at both clinicians and researchers in the manual therapy field. Specifically the review will aim to 1) describe situations were survival analysis is appropriate to use 2) describe
AC C
unique characteristics of survival data including censoring of participants 3) describe analysis methods for survival data including log rank test and cox regression 4) describe the interpretation of results from survival analyses including survival plots and hazard ratios
When is survival analysis used?
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The decision of what statistical approach to use is based largely on the nature of the primary outcome (dependent variable). For example in the landmark study by O’Sullivan and colleagues (O'Sullivan et al., 1997) investigating spinal stabilisation exercise for chronic low back pain, the
RI PT
primary outcome was pain, measured on a 0-100 scale, at 0, 3, 6 and 30 months. If the participants were injured workers it could be argued that measuring work status (full duties, partial duties, off work) or claim closure (yes/no) at each of those time points would also be informative. The pain, work status and claim closure measures require a different statistical technique (as they are
SC
continuous, ordinal and dichotomous data respectively). However, they share a common approach in that each of the 3 outcomes (pain, work status and claim closure) have been used to characterise
M AN U
health status at key time points. A quite different approach is required if the interest is in how long it takes to reach a key outcome (e.g. days to become pain-free, days to return to work or days to claim closure). These are situations where survival analysis is useful, as we wish to analyse the time until
TE D
an event occurs (Altman & Bland, 1998).
Survival analysis is used to study time to an event. Traditionally it was used to study well people at risk of experiencing a negative event such as disease, disablement or death (hence the terminology);
EP
however, it can equally be used to study sick people at risk of experiencing a positive event. In the manual therapy field there are many scenarios where time to an event may be an important if not
AC C
optimal outcome. Examples of positive events include time to return to work, time to become painfree, time to discharge; whereas examples of negative events include time to development or onset of a new health condition, time to recurrence, time to fall, etc. Survival analysis can be used in studies of prognosis (Costa Lda et al., 2009; Henschke et al., 2008)and clinical trials testing prevention(Soukup et al., 1999) or treatment (Hancock et al., 2007; Williams et al., 2010). It can be used to study time to adverse events such as device failure or need for revision surgery in orthopaedic trials (Baker et al., 2013). It is worth noting that survival analysis can be used when the
ACCEPTED MANUSCRIPT
outcome of interest is time to almost any event. An innovative example from the physical therapy field is time from presentation of a study at a conference to publication in a peer-reviewed journal
RI PT
(Smith et al., 2011).
Survival analysis has several advantages and some disadvantages over analyses used in most manual therapy trials, including the O’Sullivan and colleagues (O'Sullivan et al., 1997) example, which focus
SC
on outcomes at a particular point or points in time (e.g. pain at 3 months, 6 months and 12 months after starting treatment). An advantage of survival analysis is that rather than taking a snap shot at
M AN U
one single time point (which may not be the ideal time to assess the effect of the intervention) survival analysis provides data on the impact of intervention, or the natural course of a condition, over the whole time period investigated (Bradburn et al., 2003). Survival analysis avoids the problem of multiplicity (increased type 2 error rate) associated with multiple analyses at different time points and so there is no need to select the key (primary) time point to measure. Survival analysis is able to
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cope with different follow-up periods for participants (discussed in more detail below) and accommodates loss to follow-up more simply than data imputation. Survival analysis also copes with skewed (non-normally distributed) data. A practical disadvantage with survival analysis is the need
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to collect data regularly to ensure precise estimates of time to event. The use of daily patient diaries and electronic data capture methods are options used to overcome this (Axen et al., 2012; Hancock
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et al., 2007).
Censoring and survival analysis Before we can analyse time until an event occurs we need to calculate this time. To perform this calculation we first have to define when the time starts and when it ends. The starting point for the time interval is taken to be the point at which the patient is first considered to be at risk. In a randomized controlled trial the starting point is the date of randomization. The end point is simply
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the time when the event occurs. A unique feature of survival analysis is called censoring (Bradburn et al., 2003). This feature allows use of data from people who do not have the event for as long as they are followed, by considering the time at which these people were last to have still been at risk
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(last follow-up completed) but not experienced the event of interest. Let’s imagine a study investigating a treatment (manual therapy) for acute low back pain where it was decided that the
outcome of interest was time to recovery from pain. If all participants in the study had recovered by the end of the study then we would have data on the number of days to recovery for all participants.
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However, it is rarely the case that all people have the event (e.g. recover) by the end of the study
and it is usually not feasible to continue a study for an indefinite period of time. At some point there
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is a need to stop the study and analyse the data. At that point participants who have not experienced the event are censored. There are other reasons why patients will be censored. The most common reasons for censoring are listed below and presented in Figure 1 (Hosmer et al., 2008; Leung et al., 1997).
People who do not recover (have the event) during the study follow up period (subjects 1
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People who cannot be followed up for the full study follow up period (subject 2)
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People withdrawn from the study (subject 4)
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Survival time and calendar time
In most studies participants are enrolled as they become available; for example in the Hancock et al trial (Hancock et al., 2007) of manual therapy and NSAIDs for acute LBP it took 16 months to recruit 240 patients from general practice. Studies using time to event outcomes can either follow all participants for a set period of time (e.g. 3 or 12 months) or follow all participants until a set date (e.g. 31st Dec 2015), understanding that some will be in the study longer than others and so followed for longer. In cases where participants are followed until a set date (calendar time) this needs to be converted to survival time as demonstrated in Figure 2. Survival time rather than calendar time is
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the data required for survival analysis and represents the time to event since some entry point. The entry point is often obvious but not always. For manual therapy studies it is often treatment onset or
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onset of the condition.
The decision to design the study so people are followed for a set period (e.g. 1 year) or until a set date is a practical decision involving the availability of participants, frequency of the event, time
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period over which the effect of the predictor was likely to have an influence and budget/time to
complete the study. Arguably it is simpler to follow all participants for a set time period, however if
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recruitment takes place over a long period (e.g. 2 years) then it may make sense to follow until a set date rather than for a set period if additional important data is collected. Again it is important to emphasise that survival analysis copes well if participants are followed for different time periods (as would occur if participants are recruited at different time points but followed to a single date), due
The Kaplan-Meier life table
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to the ability to censor individuals.
A common method to estimate statistics with survival data is the Kaplan-Meier (or product-limit).
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method (Kaplan, 1958). This method is used to estimate the proportion of people who ‘survive/recover’ beyond a specific time point (e.g. 1 month or the end point of the study). The
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calculation and presentation of the Kaplan-Meier estimates are usually displayed in a life table as per the example in Table 1. Here you can see that the probability of survival changes only on the day an event occurs. This feature is known as conditional probability meaning that the probability of an event at a certain time point is calculated on the condition of the events that occurred previously. A new probability of survival is calculated at each time point when an event occurs. You can see this in Table 1 and also graphically in Figure 3 as the steps in the curve. The new probability is simply calculated by using the formula below:
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Number of remaining cases (initial number of cases minus cases who have had the event or been censored) / number of cases at risk (initial number minus cases censored) Let’s consider the example in Table 1. This data could represent a simple cohort study with 100
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participants with LBP where patients’ recovery was monitored every day. While in survival analysis traditionally the ‘probability of survival’ is described, in this scenario the event is positive (recovery) so the traditional terminology is counter-intuitive and hence we will talk about the ‘probability of
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non-recovery’. If on day 1 a single patient recovered then the probability of non-recovery at day 1 would be 0.99 ((100-1)/100). On day 2 if 3 participants recovered the probability of non-recovery
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would be 0.96 ((100-4)/100). Two important points need noting here. Firstly the number of cases at risk changes depending on how many cases have been censored. In the example in Table 1 you can see that 2 participants withdrew from the study on day 3. As a result when the next event occurred (day 4) the number at risk was now 98 (100 minus 2 participants who had withdrawn). The 2 patients who had withdrawn (been censored) were not at risk of recovery as no data was available
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at this time point. This nicely demonstrates how patients who are censored are used in the calculations of survival probability (in our case non-recovery probability) for as long as they are being followed even if they drop out of the study before they experience the event. The second important
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point from the above example is that while time is a truly continuous variable survival curves typically represent blocks of time (in this example days). Because of this it is common to have
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multiple events occurring at one time point (e.g. day 2). Shorter periods between follow-ups will increase the precision of the estimates and curve produced. Whereas longer periods between follow ups may result in ‘interval-censoring’ meaning that the outcome is known only to lie within a time interval, instead of being directly observed. Survival curve The Kaplan-Meier estimates can also be plotted in a survival curve as demonstrated in Figure 3. Here the basic concept of survival analysis can be more easily visualised and understood. The y axis
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represents the proportion of people who have had the event, while the x axis represents time since study entry. Typically the curve starts at 1 on the y axis at time 0 on the x axis to represent the fact that 100% of the participants have “survived” or not had the event at entry into the study. Over time
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the curve represents the probability of surviving at each time point on the x axis. The survival curve in figure 3A represents the probability of remaining unrecovered (surviving) at different time points after an episode of low back pain. If the probability of recovery is of more interest this is simply
calculated by 1 minus probability of being unrecovered and the survival curve can start from a y
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value of 0 as represented in Figure 3B.
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It is common to summarise or convey the findings from a survival curve (or Kaplan Meier life table) by reporting the time at which specific proportions of participants had experienced the event. The most common example is the median survival time which is the time at which 50% of participants had the event. In Figure 3A you can see that the median time to recovery was approximately 18
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days. This can be determined by drawing a horizontal line from 0.5 on the y axis until it intersects with the curve and then transferring this point to the x axis to determine the time. This can be done for any survival probability as long as the survival probability of the curve falls to lower than the level
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of interest.
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Comparing survival curves
In manual therapy studies we will often want to determine the influence of a certain variable (treatment or prognostic factor) on time to event. The log rank test (Bland & Altman, 2004) is commonly used to test if the survival function (time to event) for people with a characteristic (e.g. female) is different from those without this characteristic (e.g. male). Figure 4A presents an example of different survival curves for people receiving baseline care and additional manual therapy compared to those receiving just baseline care. You can see the curves follow very similar paths and
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as such the p value from the log rank test is not significant (p=0.870). Figure 4B presents survival curves for people who do and do not meet a clinical prediction rule for recovery from acute low back pain. Here you can see the curves appear different and those who meet the prediction rule have a
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faster rate of recovery as indicated by the significant p value. A limitation of the log rank test is it does not describe the effect size and is unable to allow for important covariates or confounders.
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Cox regression
Cox regression is widely used to assess the influence of multiple factors on time to event (Cox, 1972).
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Similarly to other regression analyses, Cox regression produces coefficients which provide an estimate of size of effect due to the variable of interest as well as confidence intervals. These estimates are called hazard ratios and represent the chance of the event occurring in those with the variable of interest (e.g. female) as a ratio of the hazard of the event occurring in those without (e.g.
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male). Therefore a hazard ratio of 1 represents no difference while a hazard ratio of 3 implies that at any time point the event will occur 3 times more commonly in those with the variable of interest compared to control participants. In the case of continuous predictor variables the hazard ratio
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represents the ratio of hazards for each unit of change in the predictor variable. The hazard ratio does not provide information about speed of time to event (or recovery) so it is incorrect to
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interpret a hazard ratio of 3 as saying people with the treatment will recover 3 times faster than those without the treatment(Spruance et al., 2004).
Proportional hazard assumption A key assumption when performing cox regression is that the hazard ratio for the explanatory variable is relatively constant across time. There are several methods to assess the proportional hazards assumption and these are largely beyond the scope of this paper. Readers interested in this
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topic can find more detail in the following resources (Cox, 1972; Hosmer et al., 2008; Ng'andu, 1997). However, simple visualisation of a plot of survival curves is one simple method and helps provide an understanding of the issue. In Figure 5A you can see survival curves where the
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proportional hazards assumption appears to be valid, as any differences in slopes (hazard rates) between curves are relatively consistent over time. In this case further testing should be conducted to provide further evidence that the assumption has not been violated. Further checking of the
proportional hazards assumption may occur by fitting a time-dependant covariate to the model
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and/or by graphical examination of Schenfeld residuals (Ng'andu, 1997). In Figure 5B however, the curves cross and the hazard ratio clearly changes over time. In this case there is strong evidence the
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proportional hazards assumption is violated. An example relevant to manual therapists where you might expect the proportional hazards assumption to be violated would be a trial comparing steroid injection to exercise therapy in people with sub acromial impingement as the hazard ratio would likely change over time. The event of interest could be time to 50% reduction in pain. In the short
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term (e.g. day 1) you would expect more people to have the event if they received steroid injection, however, at a later time point (e.g. 6 weeks) it is likely a larger number of people receiving exercise therapy will have the event. This is a clear violation of the proportional hazards assumption and
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modifications.
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would produce a plot similar to Figure 5B. Cox regression cannot be used in this case without
One strategy to deal with situations where the proportional hazards assumption is violated is to divide the time interval (Hosmer et al., 2008). In other words the hazard ratios are generated for smaller intervals of time where the hazard ratio is more constant. Initially this decision can be made by visual assessment of survival curves. Using the data in Figure 5B it may be sensible to divide the time into intervals of 0-2 weeks and > 2 weeks.
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Summary: This master class aims to give clinicians and researchers a basic understanding of survival analysis so they can correctly interpret published papers and consider using survival analysis in future trials if
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appropriate. The key messages include: 1) Survival analysis has some advantages that make it a good choice for many studies of manual therapy. 2) Survival analysis can be used when the outcome of interest is time to event, regardless of what defines the event. 3) Censoring is a key feature of
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survival analysis. 4) Survival curves plot the proportion of cases having the event at different time
points. 5) Cox regression is commonly used to test the influence of independent variables on survival
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rates. 6) Hazard ratios from cox regression describe the ratio of hazards for different values of the independent variable (e.g. treatment or control). 7) Median time to recovery is commonly used to compare speed of recovery for different values of the independent variable (e.g. treatment or
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control). 8) The proportional hazards assumption needs to be tested when using cox regression.
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References
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Altman DG, Bland JM. Time to event (survival) data. BMJ 1998;317: 468-469. Axen I, Bodin L, Kongsted A, Wedderkopp N, Jensen I, et al. Analyzing repeated data collected by mobile phones and frequent text messages. An example of low back pain measured weekly for 18 weeks. BMC Medical Research Methodology 2012;12: 105. Baker P, Jameson S, Critchley R, Reed M, Gregg P, et al. Center and surgeon volume influence the revision rate following unicondylar knee replacement: an analysis of 23,400 medial cemented unicondylar knee replacements. Journal of Bone & Joint Surgery - American Volume 2013;95: 702-709. Bland JM, Altman DG. The logrank test. BMJ 2004;328: 1073. Bradburn MJ, Clark TG, Love SB, Altman DG. Survival analysis part II: multivariate data analysis--an introduction to concepts and methods. British Journal of Cancer 2003;89: 431-436. Costa Lda C, Maher CG, McAuley JH, Hancock MJ, Herbert RD, et al. Prognosis for patients with chronic low back pain: inception cohort study. BMJ 2009;339: b3829. Cox DR. (1972). Regression models and life-tables (pp. 187). Hancock MJ, Maher CG, Latimer J, McLachlan AJ, Cooper CW, et al. Addition of diclofenac and/or manipulation to advice and paracetamol does not speed recovery from acute low back pain: A randomised controlled trial. The Lancet 2007;370: 1638-1643. Henschke N, Maher CG, Refshauge KM, Herbert RD, Cumming RG, et al. Prognosis in patients with recent onset low back pain in Australian primary care: inception cohort study. BMJ 2008;337: a171. Hosmer D, Lemeshow S, May S. ( 2008). Applied survival analysis : regression modeling of time-toevent data (2nd ed.): Wiley. Kaplan EL. Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association 1958;53: 457-481. Leung KM, Elashoff RM, Afifi AA. Censoring issues in survival analysis. Annual Review of Public Health 1997;18: 83-104. Moseley AM, Elkins MR, Herbert RD, Maher CG, Sherrington C. Cochrane reviews used more rigorous methods than non-Cochrane reviews: survey of systematic reviews in physiotherapy. Journal of Clinical Epidemiology 2009;62: 1021-1030. Ng'andu NH. An empirical comparison of statistical tests for assessing the proportional hazards assumption of Cox's model. Statistics in Medicine 1997;16: 611-626. O'Sullivan PB, Phyty GD, Twomey LT, Allison GT. Evaluation of specific stabilizing exercise in the treatment of chronic low back pain with radiologic diagnosis of spondylolysis or spondylolisthesis. Spine 1997;22: 2959-2967. Smith HD, Bogenschutz ED, Bayliss AJ, Altenburger PA, Warden SJ. Full-text publication of abstractpresented work in physical therapy: do therapists publish what they preach? Physical Therapy 2011;91: 234-245. Soukup MG, Glomsrod B, Lonn JH, Bo K, Larsen S. The effect of a Mensendieck exercise program as secondary prophylaxis for recurrent low back pain. A randomized, controlled trial with 12month follow-up. Spine 1999;24: 1585-1591. Spruance SL, Reid JE, Grace M, Samore M. Hazard ratio in clinical trials. Antimicrobial Agents & Chemotherapy 2004;48: 2787-2792. Williams CM, Latimer J, Maher CG, McLachlan AJ, Cooper CW, et al. PACE-The first placebo controlled trial of paracetamol for acute low back pain: design of a randomised controlled trial. BMC Musculoskeletal Disorders 2010;11: 169.
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Table 1: Life table example Number Censored
Number of Number of remaining cases at cases risk
Probability of nonrecovery
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0 0 0 100 100 1.00 1 1 0 99 100 0.99 2 3 0 96 100 0.97 3 0 2 94 98 * 4 1 0 93 98 0.95 // (25) (3) 19 2 0 63 95 0.66 20 1 0 62 95 0.65 // (20) (3) 58 3 0 36 92 0.39 59 0 1 35 91 * 60 1 0 34 91 0.37 * no value calculated as no event (recovery) occurred on this day
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Time Number Interval Recovered (days)
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End of study
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Patients Start of study
•
6
>
5
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4
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2
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3
1
0
20
40
80
60
100
120
140
160
>
> 180
Time since start of study (days)
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Figure 1: Data from an imagined study illustrating the common reasons for censoring Patients 3 and 6 had the event (recovered) and censoring was not needed (●); Patients 1 and 5 had not recovered during the study follow-up period and were censored (); Patient 2 could not be followed up for the full study follow up period and was censored; Patient 4 withdrew before the end of the study and was censored.
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Entry of study
•
6
•
Patients who had the event (i.e. recovered)
4
•
3
2
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Figure 2: Converting calendar time to survival time
>
5
Patients who were censored
1
2
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0
Survival time, months
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Legend:
4
> > 6
A
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B
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Median days to recovery =18
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Figure 3: Survival curves for days to recovery from low back pain. A) Proportion not recovered; B) Proportion recovered. Dashed lines in figure A represent the median time to recovery Figures drawn from raw data from the study reported in Hancock et al, 2007
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A
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B
p=
