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| In [[survival analysis]], the '''hazard ratio''' (HR) is the ratio of the [[hazard rate]]s corresponding to the conditions described by two levels of an explanatory variable. For example, in a drug study, the treated population may die at twice the rate per unit time as the control population. The hazard ratio would be 2, indicating higher hazard of death from the treatment. Or in another study, men receiving the same treatment may suffer a certain complication ten times more frequently per unit time than women, giving a hazard ratio of 10.
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| Hazard ratios differ from [[relative risk]] ratios in that the latter are cumulative over an entire study, using a defined endpoint, while the former represent instantaneous risk over the study time period, or some subset thereof. Hazard ratios suffer somewhat less from selection bias with respect to the endpoints chosen, and can indicate risks that happen before the endpoint.
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| ==Definition and derivation==
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| Regression models are used to obtain hazard ratios and their [[confidence intervals]].<ref name=Spruance>{{cite journal|last=Spruance|first=Spotswood|coauthors=Julia E. Reid, Michael Grace, Matthew Samore|title=Hazard Ratio in Clinical Trials|journal=Antimicrobial Agents and Chemotherapy|date=August 2004|volume=48|issue=8|pages=2787–2792|doi=10.1128/AAC.48.8.2787-2792.2004|url=http://aac.asm.org/content/48/8/2787.full|accessdate=5 December 2012}}</ref>
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| The instantaneous [[hazard rate]] is the [[limit (mathematics)|limit]] of the number of events per unit time divided by the number at risk, as the time interval approaches 0.
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| ::<math> h(t) = \lim_{\Delta t\rightarrow 0}\frac{\mathrm{observed \;events \;in \; interval} [t, t+\Delta t]/N(t)}{\Delta t}</math>
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| where ''N''(''t'') is the number at risk at the beginning of an interval. A hazard is the probability that a patient fails between <math>t</math> and <math>t+\Delta t</math>, given that he has survived up to time <math>t</math>, divided by <math>\Delta t</math>, as <math>\Delta t</math> approaches zero.<ref name=Case />
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| The hazard ratio is the effect on this hazard rate of a difference, such as group membership (for example, treatment or [[control experiment|control]], male or female), as estimated by [[regression analysis|regression models]] that treat the log of the HR as a function of a baseline hazard <math>h_0(t)</math> and a linear combination of explanatory variables:
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| ::<math>\log h(t) = f(h_0(t),\alpha + \beta_1 X_1 + \cdots + \beta_k X_k).\,</math>
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| Such models are generally classed [[proportional hazards models|proportional hazards regression]] models; the best known being the [[David Cox (statistician)|Cox]] [[Semiparametric model|semiparametric]] [[proportional hazards model]],<ref name="Spruance"/><ref name=Cox>{{cite journal|last=Cox|first=D. R.|title=Regression-Models and Life-Tables|journal=Journal of the Royal Statistical Society|year=1972|volume=34|series=B (Methodological)|issue=2|pages=187–220|url=http://hydra.usc.edu/pm518b/literature/cox-72.pdf|accessdate=5 December 2012}}</ref> and the exponential, Gompertz and Weibull parametric models.
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| For two groups who differ only in treatment condition, the ratio of the hazard functions is given by <math>e^\beta</math>, where <math>\beta</math> is the estimate of treatment effect derived from the regression model. This hazard ratio, that is, the ratio between the predicted hazard for a member of one group and that for a member of the other group, is given by holding everything else constant, i.e. assuming proportionality of the hazard functions.<ref name=Case />
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| For a continuous explanatory variable, the same interpretation applies to a unit difference. Other HR models have different formulations and the interpretation of the parameter estimates differs accordingly.
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| ==Interpretation==
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| [[File:Kaplan-Meier curve Tumor Volume Size.png|thumb|right|[[Kaplan-Meier curve]] illustrating overall survival based on volume of brain [[metastases]]. Elaimy et al. (2011)<ref name=Elaimy>{{cite journal|last=Elaimy|first=Ameer|coauthors=Alexander R Mackay, Wayne T Lamoreaux, Robert K Fairbanks, John J Demakas, Barton S Cooke, Benjamin J Peressini, John T Holbrook, Christopher M Lee|title=Multimodality treatment of brain metastases: an institutional survival analysis of 275 patients|journal=World Journal of Surgical Oncology|date=5 July 2011|volume=9|issue=69|doi=10.1186/1477-7819-9-69|url=http://www.wjso.com/content/9/1/69|author=Elaimy A. L. et al.}}</ref>]]
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| In its simplest form the hazard ratio can be interpreted as the chance of an event occurring in the treatment arm divided by the chance of the event occurring in the control arm, or vice versa, of a study. The resolution of these endpoints are usually depicted using [[Kaplan-Meier survival curve]]s. These curves relate the proportion of each group where the endpoint has not been reached. The endpoint could be any [[dependent variable]] associated with the [[covariate]] (independent variable), e.g. death, remission of disease or contraction of disease. The curve represents the odds of an endpoint having occurred at each point in time (the hazard). The hazard ratio is simply the relationship between the instantaneous hazards in the two groups and represents, in a single number, the magnitude of distance between the Kaplan-Meier plots.<ref name=Brody>{{cite book|last=Brody|first=Tom|title=Clinical Trials: Study Design, Endpoints and Biomarkers, Drug Safety, and FDA and ICH Guidelines|year=2011|publisher=Academic Press|isbn=9780123919137|url=http://books.google.co.za/books?id=zhZRmYABCDIC&dq}}</ref>
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| Hazard Ratios do not reflect a time unit of the study. The difference between hazard-based and time-based measures is akin to the difference between the odds of winning a race and the margin of victory.<ref name=Spruance /> When a study reports one hazard ratio per time period, it is assumed that difference between groups was proportional; graphically the Kaplan-Meier survival curves displayed a constant distance apart. Hazard ratios become meaningless when this assumption of proportionality is not met.<ref name=Brody />
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| If the proportional hazard assumption holds, a hazard ratio of one means equivalence in the [[hazard rate]] of the two groups, whereas a hazard ratio other than one indicates difference in hazard rates between groups. The researcher indicates the probability of this [[Sample (statistics)|sample]] difference being due to chance by reporting the [[probability]] associated with some [[test statistic]].<ref name=Motulsky>{{cite book|last=Motulsky|first=Harvey|title=Intuitive Biostatistics: A Nonmathematical Guide to Statistical Thinking|year=2010|publisher=Oxford University Press|isbn=9780199730063|pages=210–218|url=http://books.google.co.za/books?id=R477U5bAZs4C}}</ref> For instance, the <math>\beta</math> from the Cox-model or the [[log-rank test]] might then be used to assess the significance of any differences observed in these survival curves.<ref name=Norman>{{cite book|title=Biostatistics: The Bare Essentials|year=2008|publisher=PMPH-USA|isbn=9781550093476|pages=283–287|url=http://books.google.co.za/books?id=y4tWQl_8Ni8C&dq|author=Geoffrey R. Norman|coauthors=David L. Streiner|accessdate=7 December 2012}}</ref>
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| Conventionally, probabilities lower than .05 are considered [[Statistical significance|significant]] and researchers provide a 95% confidence interval for the hazard ratio, e.g. derived from the [[standard deviation]] of the Cox-model [[regression coefficient]], i.e. <math>\beta</math>.<ref name=Norman /><ref name=Kleinbaum>{{cite book|title=Survival Analysis: A Self-Learning Text|year=2005|publisher=Springer|isbn=9780387239187|url=http://books.google.co.za/books?id=GNhzxRkFnJ0C&pg|author=David G. Kleinbaum|edition=2|coauthors=Mitchel Klein|accessdate=7 December 2012}}</ref> [[Statistical significance|Statistically significant]] hazard ratios cannot include [[1 (number)|unity]] (one) in their confidence intervals.<ref name=Brody />
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| ===The proportional hazards assumption===
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| The proportional hazards assumption for hazard ratio estimation is strong and often unreasonable.<ref name=Cantor>{{cite book|last=Cantor|first=Alan|title=Sas Survival Analysis Techniques for Medical Research|year=2003|publisher=SAS Institute|isbn=9781590471357|pages=111–150|url=http://books.google.co.za/books?id=iyvvwCAM_aUC&dq}}</ref> [[Complication (medicine)|Complications]], [[adverse effects]] and [[late effect]]s are all possible causes of change in the hazard rate over time. For instance, a surgical procedure may have high early risk, but excellent long term outcomes.
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| If the hazard ratio between groups remain constant, this is not a problem for interpretation. However, interpretation of hazard ratios become impossible when [[selection bias]] exists between group. For instance, a particularly risky surgery might result in the survival of a systematically more robust group who would have fared better under any of the competing treatment conditions, making it look as if the risky procedure was better. Follow-up time is also important. A cancer treatment associated with better remission rates, might on follow-up be associated with higher [[relapse]] rates. The researchers' decision about when to follow-up is arbitrary and may lead to very different reported hazard ratios.<ref name=Hazards>{{cite journal|last=Hernán|first=Miguel|title=The Hazards of Hazard Ratios|journal=Epidemiology|date=January 2010|volume=21|series=The Changing Face of Epidemiology|issue=1|pages=13–15|doi=10.1097/EDE.0b013e3181c1ea43|url=http://journals.lww.com/epidem/Fulltext/2010/01000/The_Hazards_of_Hazard_Ratios.4.aspx|accessdate=6 December 2012}}</ref>
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| ===The hazard ratio and survival===
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| Hazard ratios are often treated as a ratio of death probabilities.<ref name=Case /> For example, a hazard ratio of 2 is thought to mean that a group has twice the chance of dying than a comparison group. Rather, the relationship between group [[survival function]]s is <math>S_1(t) = S_0(t)^r</math> (where ''r'' is the hazard ratio).<ref name=Case>{{cite journal|coauthors=Gretchen Kimmick, Electra D. Paskett, Kurt Lohmana, Robert Tucker|title=Interpreting Measures of Treatment Effect in Cancer Clinical Trials|journal=The Oncologist|date=June 2002|volume=7|issue=3|pages=181–187|doi=10.1634/theoncologist.7-3-181|url=http://theoncologist.alphamedpress.org/content/7/3/181.full|accessdate=7 December 2012|author=L. Douglas Case}}</ref> Therefore, with a hazard ratio of 2, if <math>S_0(t) = .2</math> (20% survived at time ''t''), <math>S_1(t) = .2^2 = .04</math> (4% survived at ''t''). The corresponding death probabilities are .8 and .96.<ref name=Cantor /> It should be clear that the hazard ratio is a relative measure of effect and tells us nothing about absolute risk.<ref name=Newman>{{cite book|last=Newman|first=Stephan|title=Biostatistical Methods in Epidemiology|year=2003|publisher=John Wiley & Sons|isbn=9780471461609|url=http://books.google.co.za/books?id=ub6DoADSVekC&dq}}</ref>
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| While hazard ratios allow for [[Statistical hypothesis testing|hypothesis testing]], they should be considered alongside other measures for interpretation of the treatment effect, e.g. the ratio of median times (median ratio) at which treatment and control group participants are at some endpoint. If the analogy of a race is applied, the hazard ratio is equivalent to the odds that an individual in the group with the higher hazard reaches the end of the race first. The probability of being first can be derived from the odds, which is the probability of being first divided by the probability of not being first: HR = P/(1 − P); P = HR/(1 + HR). In the previous example, a hazard ratio of 2 corresponds to a 67% chance of an early death. The hazard ratio does not convey information about how soon the death will occur.<ref name=Spruance />
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| ===The hazard ratio, treatment effect and time-based endpoints===
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| Treatment effect depends on the underlying disease related to survival function, not just the hazard ratio. Since the hazard ratio does not give us direct time-to-event information, researchers have to report median endpoint times and calculate the median endpoint time ratio by dividing the control group median value by the treatment group median value.
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| While the median endpoint ratio is a relative speed measure, the hazard ratio is not.<ref name=Spruance /> The relationship between treatment effect and the hazard ratio is given as <math>e^\beta</math>. A statistically important, but practically insignificant effect can produce a large hazard ratio, e.g. a treatment increasing the number of one-year survivors in a population from one in 10,000 to one in 1,000 has a hazard ratio of 10. It is unlikely that such a treatment would have had much impact on the median endpoint time ratio, which likely would have been close to unity, i.e. mortality was largely the same regardless of group membership and [[Clinical significance|clinically insignificant]].
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| By contrast, a treatment group in which 50% of infections are resolved after one week (versus 25% in the control) yields a hazard ratio of two. If it takes ten weeks for all cases in the treatment group and half of cases in the control group to resolve, the ten week hazard ratio remains at two, but the median endpoint time ratio is ten, a [[Clinical significance|clinically significant]] difference.
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| ==See also==
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| *[[Survival analysis]]
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| *[[Failure rate]] and [[Failure rate#hazard function|Hazard rate]]
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| *[[Proportional hazards models]]
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| *[[Odds ratio]]
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| *[[Relative risk|Relative risk ratio]]
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| ==References==
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| {{Reflist|2}}
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| {{Medical research studies}}
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| [[Category:Epidemiology]]
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| [[Category:Medical statistics]]
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| [[Category:Statistical ratios]]
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| [[Category:Survival analysis]]
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