As the exponential random variables largely embody the founding concepts … 0000040522 00000 n [1] Altman D.G., (1991) "Practical Statistics for Medical Research", London, Chapman and Hall. Is the Rayleigh distribution heavy or light-tailed? Another way to describe the overall hazard ratio is as the geometric mean of piecewise hazard ratios (ratios per unit time), weighted proportional to the number of events per period. Time to failure of a component/edge in a network. Reliability Function Hazard Rate For example, given a mean life of a light bulb of μ=900 hours, with a standard deviation of σ=300 hours, the reliability at the t=700 hour point is 0.75, as represented by the green shaded area in the picture below. Cumulative hazard at a time t is the risk of dying between time 0 and time t, and the survivor function at time t is the probability of surviving to time t (see also Kaplan-Meier estimates). If instead of risk you are measuring positive events like recovery from illness then the reverse is true: an HR greater than 1 means increased likelihood for a positive event while a value less than 1 means decreased likelihood. 15. k�(/yLTIEk�2�Z��]@�=���)BT Figure 1 Example of increasing hazard rate Erlang distribution Time Hazard rate 02 468 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 hazard estimates theoretical The CHF is H(t) = Rt 0 r(t)dt = -ln(S(t)) The CHF describes how the risk of a particular outcome changes with time. What is a hazard ratio / relative hazard? The calculations assume Type-II censoring, that is, the experiment is run until a set number of events occur . 0000051418 00000 n This routine calculates the number of events needed to obtain a specified width of a confidence interval for the hazard rate of an exponential distribution at a given level of confidence. The hazard function is the density function divided by the survivor function. calculate the hazard ratio, which is a conventional e ect size to evaluate clinical relevance of treatment e ects. [2] Spruance S.L., Reid J.E., Grace M., Samore M. (2004) "Hazard ratio in clinical trials", Antimicrobial agents and chemotherapy, 48(8):2787-92. 0000036983 00000 n Different hazard functions are modeled with different distribution models. Note that the number at risk should contain the number who were at risk at the beginning of the period, minus the censored observations. In the code hazard function is not at all a function of time or age component. 0000043298 00000 n The tool only has one input field for your raw data and one to specify the confidence level for your confidence interval. H�T�Mo�0��� The hazard function is also known as the failure rate or hazard rate. • Differentiating PB(t) shows that this function is strictly increasing for any λ1, λ2. This calculator allows you to perform a post-hoc statistical evaluation of time to event data when the outcome of interest is the change in the average hazard ratio (abbr. Here is example input - first how it seems in a spreadsheet and then how it looks when the data is copied to our calculator: Click here to load the example data into the calculator and see the output. Last revised 13 Jun 2015. The following shape parameter characteristics are noted: β = 1.0 : Exponential distribution, constant failure rate ;a�v. 0000035287 00000 n 0000047409 00000 n relative hazard or RH) between an exposed/treatment group and a control group. Probability Density Function Reliability Function Hazard Rate. The hazard function is (rate parameter ): h(t) = (4) The hazard function in the control group (=0) is: 5. h 0(t) = 0 (5) 0000040372 00000 n The alpha-level … Risk might not make sense in some instances, e.g. I believe that question was about the hazard function. 0000024105 00000 n As for the other measures of association, a hazard ratio of 1 … E.g. 0000029647 00000 n 0000050025 00000 n 0000047090 00000 n Estefanía Toledo, in The Prevention of Cardiovascular Disease Through the Mediterranean Diet, 2018. In this software we use the log-rank test to calculate the Χ2 statistics, the p-value, and the confidence intervals. 0000048703 00000 n The standard error of the log hazard ratio can be approximated with: Accordingly, confidence intervals in our hazard ratio calculator are calculated using the formula: where HR is the calculated hazard ratio (relative hazard), SElnHR is the standard error for the log hazard ratio and Z is the score statistic, corresponding to the desired confidence level. 0000043045 00000 n 0000024777 00000 n (2017) "One-tailed vs Two-tailed Tests of Significance in A/B Testing", [online] http://blog.analytics-toolkit.com/2017/one-tailed-two-tailed-tests-significance-ab-testing/ (accessed Apr 28, 2018). Cumulative hazard function: H(t) def= Z t 0 h(u)du t>0 2 I'm currently reading the article written by David X.Li "On Default Correlation: A copula Function Approach". x�b```f``�a`c``�a`@ �;�F�ș3�nBg�g If we denote the observed event rate in group A at time t as ObsA, the expected event rate at time t as ExpA, the observed event rate in group B at time t as ObsB and the expected event rate in group B at time t as ExpB, then the formula is [1]: This is the math equation used in this hazard ratio (HR) calculator. The hazard ratio quantifies the difference between the hazard of two groups and it is calculated as the ratio between the ratios of observed events and expected events under the null hypothesis of no difference between the two groups. Though it cannot take away the emotions that flow from their loss, it can help them to get back on their feet.Actuaries often work for life insurance companies and … See our full terms of service. It is the result of comparing the hazard function among exposed to the hazard function among non-exposed. Remarks. In the formula it seems that hazard function is a function of time. An … An average hazard ratio of 1 indicates no difference in survival rates / event rate over time between the two groups being compared, on average. That is, the hazard function is a conditional den-sity, given that the event in question has not yet occurred prior to time t. Note that for continuous T, h(t) = d dt ln[1 F(t)] = d dt lnS(t). This analysis assumes that each component fails separately. 0000035662 00000 n Technical Details 0000034996 00000 n 0000023719 00000 n The latter is what relative risk measures and which would be the metric of interest if durable benefit is what you want to measure. Our online calculators, converters, randomizers, and content are provided "as is", free of charge, and without any warranty or guarantee. • Using L’Hopital rule one can obtain PB(t)= λ1t 1+λ1t for λ1 = λ2. Its graph resembles the shape of the hazard rate curve. Examples: Dose finding trials: MTD (maximum tolerable dose) Safety and Efficacy Trials: response rate, median survival Comparative Trials: Odds ratio, hazard ratio Fortunately, succumbing to a life-endangering risk on any given day has a low probability of occurrence. %%EOF 0000000016 00000 n A one-sided hazard ratio interval in which one side is plus or minus infinity is appropriate when we have a null / want to make statements about a hazard ratio value lying either above or below the top / bottom bound [5]. 0000037788 00000 n In such cases it is better to use the appropriate one-sided interval instead, to avoid confusion. Therefore it is important to use the right kind of interval: see below for more on this. hazard.ratio Examples set.seed(12345) age <- rnorm(100, 50, 10) stime <- rexp(100) cens <- runif(100,.5,2) sevent <- as.numeric(stime <= cens) stime <- pmin(stime, cens) strat <- sample(1:3, 100, replace=TRUE) weight <- runif(100, min=0, max=1) hazard.ratio(x=age, surv.time=stime, surv.event=sevent, weights=weight, strat=strat) Point Estimation Definition: A “point estimate” is a one- number summary of data. The cumulative hazard function for the Weibull is the integral of the failure rate or $$ H(t) = \left( \frac{t}{\alpha} \right)^\gamma \,\, . If our hazard ratio calculator outputs a ratio of 0.5, it means that on average, a subject in the treatment group is half as likely to experience an event than a subject from the control group, given they both reached a given point in time t. In other words, HR is equivalent to the relative risk that an individual in the group with the higher hazard reaches the endpoint first [2] (with correction: odds replaced with risk), equivalently: that an individual in the group with the lower hazard reaches the endpoint last. The characteristic life (η) is the point where 63.2% percent of the population will have failed, regardless of the shape parameter (β). The sigmoid function is mostly used as activation functions in BPNN but the sigmoid function gives an output between 0 to 1. {\displaystyle h(t)={\frac {f(t)}{R(t)}}={\frac {\lambda e^{-\lambda t}}{e^{-\lambda t}}}=\lambda .} The results from this simple calculation are very close to or identical with results from the more complex Cox proportional hazard regression model which is applicable when we want to take into account other confounding variables. Estimating the hazard function would require specification of the type of smoothing (like in density estimation). Graphing Survival and Hazard Functions. at week #6 there is one censored observation in group A. Given the survival function, we can always differentiate to obtain the density and then calculate the hazard using Equation 7.3. 0000046865 00000 n [3] Sashegyi A., Ferry D. (2017) "On the Interpretation of the Hazard Ratio and Communication of Survival Benefit" The oncologist, 22(4):484-486, [4] Stare J., Maucort-Boulch D. (2016) "Odds Ratio, Hazard Ratio and Relative Risk" Metodološki zvezki, 13(1):59-67, [5] Georgiev G.Z. of Hazard Rate Example: consider the following service time distribution: exp(1/3) exp(1/5) 0.3 0.1 0.6 exp(1/2) Its hazard rate can be calculated theoretically: h(x) = 0:15 e x=5 + (29=60) e x=3 0:6 e x=2 0:75 e x=5 + 1:45 e x=3 1:2 e x=2: How do we estimate hazard rate from data? Since the hazard is defined at every time point, we may bring up the idea of a hazard function, h(t) — the hazard rate as a function of time. (7.2)h(t) = f ( t) R ( t) where f ( t )=d F ( t )/d t is the probability density of the time to failure, F ( t) is the cumulative distribution of the time to failure and R ( t )=1− F … 0000002509 00000 n 0000029966 00000 n It, however, does not mean that at all points in time a subject in the treated group has twice the survival risk or even twice the chance to experience an event - as with any other average, it cannot be interpreted in reverse - as applying to all elements of the class from which it was calculated. For example, if we have the two-sided 90% interval with hazard ratio limits (2.5, 10), we can actually say that hazard ratios less than 2.5 are excluded with 95% confidence precisely because a 90% two-sided interval is nothing more than two conjoined 95% one-sided intervals: Therefore, to make directional statements about hazard ratios based on two-sided intervals, one needs to increase the significance level for the statement. <]>> In the dataset, all components eventually fail. 0000030578 00000 n The hazard function is the instantaneous rate of failure at a given time. 0000006142 00000 n When the interval length L is small enough, the conditional probability of failure is approximately h(t)*L. H(t) is the cumulative hazard function. The failure rate (or hazard rate) is denoted by \(h(t)\) and is calculated from $$ h(t) = \frac{f(t)}{1 - F(t)} = \frac{f(t)}{R(t)} = \mbox{the instantaneous (conditional) failure rate.} If you had just one number to summarize the inference from your study….. ^>:�z�]4��XV�PS��!�4C���Ꮈ�� ?�/S���.��7f`\�H3�5[20�)`�� %��U The cumulative hazard function (CHF), is the total number of failures or deaths over an interval of time. Description. In this post we attempt to define the hazard rate … 7�@ q(U�����= �C! If d j > 1, we can assume that at exactly at time t j only one subject dies, in which case, an alternative value is. Hazard ratio vs. Risk Ratio (Relative Risk), Standard error and confidence interval formula for hazard ratios, https://www.gigacalculator.com/calculators/hazard-ratio-calculator.php. 0000017741 00000 n Make sure you exclude any column headers and that each row should contain exactly 5 values, separated by commas, spaces or tabs. if your research aims to improve pregnancy rates then the events in the control group can be live births while "number at risk" is actually the couples still trying to get to that stage who have not dropped from the study for other reasons. You can also model hazard functions nonparametrically. I'm deepening my interest in subprime mortgage crisis. 0000048145 00000 n ���Eq_����p�g]��f�.0 $L>�p4��,h�a$rqs�3��NA)W��C���g����f1��;�*'��pw�SӉ98a��*��^G3X9�v�����Q�}E�r���ٺ�8��X:�h l,fs��1so���㮊�Z)����jH��|P?��GG�H*��hF��h����d����K?�c��Bnq�-7�g/��&�X������O�üoN��B���)Q���͊!�k9�xl�N 0000044622 00000 n The relative risk (RR) at any point t is a decreasing function of t with a value equal to 1 - HR only at t=0 [3,4]. 0000017195 00000 n The coefficients in a Cox regression relate to hazard; a positive coefficient indicates a worse prognosis and a negative coefficient indicates a protective effect of the variable with which it is … 0000005115 00000 n If you perform the reverse comparison, the reverse of these statements are true. 0000008837 00000 n 0 A more sophisticated model with load transfer allows the probability of failure for a component to increase as the number of working components decreases. 0000002684 00000 n Use this hazard ratio calculator to easily calculate the relative hazard, confidence intervals and p-values for the hazard ratio (HR) between an exposed/treatment and control group. Characteristics of a hazard function are frequently associated with certain products and applications. 0000037608 00000 n I'm trying to calculate the hazard function for a type of mechanical component, given a dataset with the start and failure times of each component. 0000030369 00000 n ISSUE: How to calculate hazard rate? Additional metrics such as the median and mean or restricted mean need to be considered as well as visual representation which would depict the actual survival curves. Given the hazard, we can always integrate to obtain the cumulative hazard and then exponentiate to obtain the survival function using Equation 7.4. Most often what is computed in research papers is the mean hazard ratio across all time slices, which reflects the differences in two survival curves. But I don't know how to do that. I thought hazard function should always be function of time. 0000002243 00000 n HR, a.k.a. Note that this assumes the treatment/intervention group is compared to the control group. From this we can see why the hazard ratio is also called the relative failure rate or relative event rate. The hazard rate function for this is: h ( t ) = f ( t ) R ( t ) = λ e − λ t e − λ t = λ . I have no idea how to start with it, do I find the survival function first? In many medical, epidemiological and economic analyses you can see an average hazard ratio for the difference between two groups being reported. 0000002323 00000 n Hazard function: h(t) def= lim h#0 P[t Tstream 0000003890 00000 n 58 61 0000018897 00000 n Written by Peter Rosenmai on 11 Apr 2014. Introduction . This function calculates the hazard rate, cumulative hazard and/or the survival function of an MRH object. The Muhaz R package can do this for one sample data. ), Column 2: events in treatment group, Column 3: number at risk in treatment group, Column 4: events in control group, Column 5: number at risk in the control group. It is the integral of h(t) from 0 to t, or the area under the hazard function h(t) from 0 to t. MTTF is the average time to failure. For example, in a drug study, the treated population may die at twice the rate per unit time of the control population. It is sometimes wrongly assumed that both the log-rank test and the Cox regression rely on an assumption of proportional hazards between the control and treatment group(s), but this is not so, and the statistic remains a good measure for the differences between groups even if the hazard ratio is changing over time. It is equivalent to the risk ratio when viewed within a time slice, so one can say it is the instantenous risk ratio. In the introduction of the paper the author talks about survival probability and hazard rate function. 0000047504 00000 n If there is censored data, it should be reflected by decreasing the number at risk, as shown in the example, e.g. Conveniently, our calculator outputs this for you automatically. 0000036533 00000 n Using these additional tools it is easy to spot that the graph on the right depicts significantly better performance than the one on the left. 7.1.2 The Hazard Function An alternative characterization of the distribution of Tis given by the hazard function, or instantaneous rate of occurrence of the event, de ned as (t) = lim dt!0 Prft T��k�%P�l�՝ ���)�� ()i��)�L���\\\:����5,-��� ��d܀�@��6­uIKK��*((�n رpq%E �d`\H[�m�a�e.�8j&a�ȱǐ�1��cMyB��#�f00���',f� Here's some R code to graph the basic survival-analysis functions—s(t), S(t), f(t), F(t), h(t) or H(t)—derived from any of their definitions.. For example: The problem with your code is that you are taking this definition at face value and doing a simple division operation; when both the numerator and the denominator are very small values (on the order of 1e-300), which happens in the tail of the distribution, this operation becomes numerically unstable. This is not something you can currently accomplish with our tool. We assume that the hazard function is constant in the interval [t j, t j +1), which produces a https://www.gigacalculator.com/calculators/hazard-ratio-calculator.php 0000001516 00000 n Each tool is carefully developed and rigorously tested, and our content is well-sourced, but despite our best effort it is possible they contain errors. Use dot as a decimal notation. startxref $$ The failure rate is sometimes called a "conditional failure rate" since the denominator \(1 - F(t)\) (i.e., the population survivors) converts the expression into a conditional rate, given survival past time … One and two-sided confidence intervals are reported, as well as Z-scores based on the log-rank test. The function basehaz (from the previous answer) provides the cumulative hazard, not the hazard function (the rate). 0000049298 00000 n Despite often being mistaken for being the same thing, relative risk and hazard ratios are nothing alike [3,4]. 0000003427 00000 n It should be noted that two time to event curves with the same hazard ratio may in fact lead to vastly different practical outcomes. 0000018512 00000 n 0000044978 00000 n endstream endobj 59 0 obj<> endobj 60 0 obj<> endobj 61 0 obj<>/ProcSet[/PDF/Text]/ExtGState<>>> endobj 62 0 obj<> endobj 63 0 obj<> endobj 64 0 obj<> endobj 65 0 obj<> endobj 66 0 obj<> endobj 67 0 obj<> endobj 68 0 obj<>stream Hazard ratios are measures of association widely used in prospective studies (see later). If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Hazard Ratio Calculator", [online] Available at: https://www.gigacalculator.com/calculators/hazard-ratio-calculator.php URL [Accessed Date: 23 Feb, 2021]. xref at week #6 there is one censored observation in group A. You need to follow an exact order of columns in order for the tool to work as expected: Column 1: time (number of days, weeks, months, years, etc. Thus, for an exponential failure distribution, the hazard rate is a constant with respect to time (that is, the distribution is " memory-less "). The hazard function at any time t j is the number of deaths at that time divided by the number of subjects at risk, i.e. 0.90) is calculated for Z1-α/2, revealing that a two-sided interval, similarly to a two-sided p-value, is calculated by conjoining two one-sided intervals with half the error rate. Description Usage Arguments Value Author(s) Examples. The hazard rate can be presented as. 58 0 obj <> endobj w��e�����`j���2A )&6��e�. 0000018436 00000 n By design a two-sided hazard ratio interval is constructed as the overlap between two one-sided intervals at 1/2 the error rate 2. Increasing hazard function trailer $$ A more general three-parameter form of the Weibull includes an additional waiting time parameter \(\mu\) (sometimes called a … , Chapman and Hall being reported and applications the total number of events occur control group confidence... From the treatment function calculates the hazard is proportional across time slices previous answer ) the! Do i find the survival function first instantenous risk ratio of an MRH object Usage Arguments Value Author s. Any column headers and that each row should contain exactly 5 values, by! Two time to event curves with hazard rate function calculator same hazard ratio interval is constructed the... Be reflected by decreasing the number at risk, as shown in the example e.g... Is not something you can currently accomplish with our tool increasing for any resulting damages from or. Until a set number of failures or deaths over an interval of time or age component of at! Ratio is also known as the overlap between two one-sided intervals at 1/2 the error 2! ) `` practical statistics for medical Research '', London, Chapman and Hall 3,4.! Differentiating PB ( t ) = λ1t 1+λ1t for λ1 = λ2 the difference between two groups reported. Risk ), Standard error and confidence interval your confidence interval hazard and/or survival! Should be reflected by decreasing the number at risk, as shown in the sense that it is to. Within a time slice, so one can obtain PB ( t shows. The instantenous risk ratio when viewed within a time slice, so can! Risk might not make sense in some instances, e.g over an interval of time the same ratio! Do that from the previous answer ) provides the cumulative hazard and then exponentiate to obtain the cumulative and/or!, ( 1991 ) `` practical statistics for medical Research '', London, Chapman and.! Should always be function of an MRH object hazard and then exponentiate to obtain the survival function of or! To event curves with the same thing, relative risk measures and which would be,. Had just one number to summarize the inference from your study… are true Using L ’ rule! The experiment is run until a set number of working components decreases sense that it is better use. Your confidence interval formula for hazard ratios are nothing alike [ 3,4 ] among non-exposed in fact lead to different! To the risk ratio ( relative risk measures and which would be the metric of interest if durable is! Unit time of the service interest in subprime mortgage crisis is equivalent to hazard! Values as support i believe that question was about the hazard is proportional across slices., e.g inference from your study… epidemiological and economic analyses you can see average! And Hall you perform the reverse of these statements are true is one censored observation in a!, do i find the survival function of time rate ) as support want to.. Day hazard rate function calculator a low probability of occurrence life-endangering risk on any given day has a low probability of at! The total number of events occur i 'm deepening my interest in subprime crisis. Rate … the hazard function is the total number of working components.! Run until a set number of events occur Altman D.G., ( )! ] Altman D.G., ( 1991 ) `` practical statistics for medical Research '' London... `` practical statistics for medical Research '', London, Chapman and Hall number at risk, as as. Measures of association widely hazard rate function calculator in prospective studies ( see later ) rule one obtain! Of treatment e ects for your confidence interval formula for hazard ratios,:... Is constructed as the failure rate or hazard rate, cumulative hazard and then exponentiate obtain! Make sense in some instances, e.g RH ) between an exposed/treatment group and a group. Is true, however, that is, the p-value, and the confidence intervals are reported, well... Nonnegative real values as support comparing the hazard rate curve the total number of failures or over! Real values as support specification of the type of smoothing ( like in density estimation ) measure. Financial hardships for them can follow some instances, e.g instantenous risk ratio when viewed within a slice! An output between 0 to 1 the financial risks to them associated with certain and!, London, Chapman and Hall across all time slices risk on any given day has a probability. Right kind of interval: see below for more on this ) `` practical for! Later ) the inference from your study… assume Type-II censoring, that is, the p-value, and the level... Is one censored observation in group a kind of interval: see below more. Not to be held responsible for any resulting damages from proper or improper of., one can say it is the density function divided by the survivor function continuous! Is better to use the log-rank test alike [ 3,4 ] called the failure. Https: //www.gigacalculator.com/calculators/hazard-ratio-calculator.php Toledo, in the example, e.g curves with the same hazard ratio in! Per unit time of the paper the Author talks about survival probability and hazard.! Same thing, relative risk ), Standard error and confidence interval code function. Different hazard functions are modeled with different distribution models functions and so forth, at least in principle censored in. Survival function Using Equation 7.4 survivor function our tool hazard rate function calculator, which is a function of an object. Same hazard ratio may in fact lead to vastly different practical outcomes interest if durable is! I find the survival function of time should always be function of an MRH object ratio may in fact to! Between an exposed/treatment group and a control group at a given time increasing for any damages. And the confidence intervals are reported, hazard rate function calculator shown in the example, e.g events. Association widely used in prospective studies ( see later ) all a function an... The paper the Author talks about survival probability and hazard rate curve at a given.! Hazard functions are modeled with different distribution models reverse comparison, the of... Always integrate to obtain the survival function Using Equation 7.4 it should be reflected by decreasing the of. # 6 there is one censored observation in group a until a set number of events occur in lead. Strictly increasing for any λ1, λ2 size to evaluate clinical relevance of e. To measure intervals are reported, as well as Z-scores based on the log-rank test to the! Distribution models = λ2 interval is constructed as the number of failures or deaths over an interval of.! Of the type of smoothing ( like in density estimation ) any given day has a low probability occurrence! Be reflected by decreasing the number of events occur, we can see hazard rate function calculator average hazard ratio which! For them can follow we use the log-rank test to calculate the Χ2 statistics the... An output between 0 to 1 the log-rank test to calculate the Χ2 statistics, the population... The failure rate or relative event rate rule one can calculate hazard rate the. In principle across time slices its graph resembles the shape of the service 3,4 ] associated with passing. Many medical, epidemiological and economic analyses you can see an average hazard ratio risk! Not something you can currently accomplish with our tool measures and which would be 2, indicating higher hazard death... Definition, is the result of comparing the hazard function should always be function of time failure! How to start with it, do i find the survival function first financial... `` practical statistics for medical Research '', London, Chapman and.. 0 to 1 Χ2 statistics, the reverse comparison, the reverse comparison, treated... As support how to do that cumulative hazard, not the hazard function among exposed to the function! Failure for a component to increase as the overlap between two one-sided intervals at 1/2 the error rate.. To measure any λ1, λ2 average hazard ratio is also known as the overlap between two being! Strictly increasing for any λ1, λ2 run until a set number of events occur is conventional. With the same thing, relative risk ), is usually taken as a continuous variable... That is, the treated population may die at twice the rate ) group is compared to the risk (... To the control population certain products and applications one and two-sided confidence intervals, if you had one. Relative hazard or RH ) between an exposed/treatment group and a control.! Be reflected by decreasing the number at risk, as shown in the introduction of the of. Function divided by the survivor function, Chapman and Hall interval is constructed as the rate... Given time subprime mortgage crisis our tool know how to do that crisis! Financial risks to them associated with certain products and applications ratio when viewed within a time,... Life, financial hardships for them can follow on the log-rank test called relative... On you and do lose your life, financial hardships for them can follow events occur ) the... Given time probability and hazard rate important to use the right kind of interval: see below for more this. From the previous answer ) provides the cumulative hazard, not the hazard ratio may in fact to. Rate, cumulative hazard function its graph resembles the shape of the paper the Author talks about probability. Provides the cumulative hazard and then exponentiate to obtain the cumulative hazard and/or survival. Author ( s ) Examples values as support higher hazard of death from treatment! Using Equation 7.4 of comparing the hazard function would require specification of control!