# cox proportional hazards model example

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The estimated coefficients in the Cox proportional hazards regression model, b1, for example, represent the change in the expected log of the hazard ratio relative to a one unit change in X1, holding all other predictors constant. trailer n���o�oI��ˀw���U��tz�uv�PG��(~���/-x�C�6���S�Z�v+~��ڦx�$���&⓶ I need help to interpret my cox proportional model? The goal of this page is to illustrate how to test for proportionality in STATA, SAS and SPLUS using an example from Applied Survival Analysis by Hosmer … 3 The Cox Proportional-Hazards Model As mentioned, survival analysis typically examines the relationship of the survival distribution to covariates. As an alternative to the standard Cox model, in this article we present a method that uses inverse probability (IP) weights to estimate the effect of a baseline exposure on a time-to-event outcome. Cox Proportional Hazards Model Model for hazard rate at time t for a patient with covariate values Z Suppose Z=1 if patient in group A, Z=0 if patient in group B ht h t(|) ()exp()ZZβ' The chapter focuses on other advances of the proportional hazard model, such as the hazard model with time‐dependent covariates, the stratified proportional hazard model, and the management of left truncated survival data. Again, the parameter estimates represent the increase in the expected log of the relative hazard for each one unit increase in the predictor, holding other predictors constant. The parameter estimates are generated in SAS using the SAS Cox proportional hazards regression procedure12 and are shown below along with their p-values. An analysis is conducted to investigate differences in all-cause mortality between men and women participating in the Framingham Heart Study adjusting for age. 0000001593 00000 n Forty six percent of the sample are male, the mean age of the sample is 56.8 years (standard deviation = 8.0 years) and the ages range from 45 to 82 years at the start of the study. In contrast, the 95% confidence intervals for the non-significant risk factors (total serum cholesterol and diabetes) include the null value. All of the parameter estimates are estimated taking the other predictors into account. Proportional hazards models are a class of survival models in statistics. The principle of the Cox proportional hazards model is to link the survival time of an individual to covariates. Suppose we consider additional risk factors for all-cause mortality and estimate a Cox proportional hazards regression model relating an expanded set of risk factors to time to death. endstream endobj 409 0 obj<>/W[1 1 1]/Type/XRef/Index[26 366]>>stream The hazard is a function of time. 392 18 Additional tests and graphs for examining a Cox model are described in the textbooks cited. x�bb1cakf�c@ >�+s��Ś)��Le� [�n/�*rߑ$�1��U��'gA�Mݦ���P��GC��E[�hDPP��A��!4� Xė��5 ���ˀ�iɃ����8�0�'Nf�Z/��Ч� ��5Dk5�i@�m�(0 m�(6 Dear partners, Cox proportional hazards regression is a very efficient and elegant method for analyzing survival data. 0(t) is called the baseline hazard. In the study of n=3,937 participants, 543 develop CVD during the study observation period. ��x�缗�A��k)6�ސms������[s�TsJ)��հ��40����]����e�2]���B4�E�=������E�S��y��h��gI1�EB��l=�FbmWE��ؖ�5!N������X�Y*.8 �i&���q�Ŧ%pZ:T�P�8ߟ_�N�M�m%��4r� However, it is often of interest to assess the association between several risk factors, considered simultaneously, and survival time. �d; �n�͠�Z�\@p' �EWg�8u�~�C��Bx.��@T����}��A��� � �ޛ~Az Cox proportional hazard regression is commonly used to model censored survival data. If the hazard ratio for a predictor is close to 1 then that predictor does not affect survival. How are Schoenfeld residuals calculated and how to use them to test the proportional hazards assumption of the Cox model. The incidence of CVD is higher in participants classified as overweight and obese as compared to participants of normal weight. 394 0 obj<>stream This is not to say that these risk factors are not associated with all-cause mortality; their lack of significance is likely due to confounding (interrelationships among the risk factors considered). The purpose of the Cox proportional hazards regression model (CM) is to model the simultaneous effect of multiple factors on the survival. A fully parametric proportional hazards model makes the same assumptions as the Cox regression model but, in addition, also assumes that the baseline hazard function, h0 (t), can be parameterised according to a specific model for the distribution of the survival times. For age, exp(0.11149) = 1.118. What are Cox proportional hazards models. 0000002791 00000 n If a predictor is dichotomous (e.g., X1 is an indicator of prevalent cardiovascular disease or male sex) then exp(b1) is the hazard ratio comparing the risk of event for participants with X1=1 (e.g., prevalent cardiovascular disease or male sex) to participants with X1=0 (e.g., free of cardiovascular disease or female sex). For example, to assess the functional forme of age, type this: ggcoxfunctional(Surv(time, status) ~ age + log(age) + sqrt(age), data = lung) 0 The Cox model assumes that the hazards are proportional. For example, if the hazard is 0.2 at time t and the time units are months, then on average, 0.2 events are expected per person at risk per month. The Cox proportional-hazards regression model is t in R with the coxph() function, located in the survival package: library("survival") args(coxph) function (formula, data, weights, subset, na.action, init, control, ties = c("efron", "breslow", "exact"), singular.ok = TRUE, robust = FALSE, model = FALSE, x = FALSE, y = TRUE, tt, method = ties,...) General Form of proportional hazard model Parameters Estimation. 392 0 obj<> endobj The expected hazard is 1.973 times higher in men as compared to women, holding age constant. The expected hazards are h (t) = h 0 (t)exp (b 1a) and h (t) = h 0 (t)exp (b 1b ), respectively. A p-value is less than 0.05 indicates that the hazards are not proportional. In the models we include the indicators for overweight and obese and consider normal weight the reference group. H��W�n�H}�Wԣ���}'{�p�fZj��Т���$_�UMR7����d]�U�uN�:}z�^�����8�y���p�4�������~9x;�N�Lo��������K���� However, after adjustment, the difference in CVD risk between obese and normal weight participants remains statistically significant, with approximately a 30% increase in risk of CVD among obese participants as compared to participants of normal weight. Medical researchers use survival models to evaluate the significance of prognostic variables in outcomes such as death or cancer recurrence and subsequently inform patients of their treatment options [1–4].One standard survival model is the Cox proportional hazards model (CPH) [].The CPH is a semiparametric model that calculates the effects of observed covariates on the risk of … H�T�?o�0�w>ō�:�Di%Đ�0�� The associations are quantified by the regression coefficients coefficients (b1, b2, ..., bp). Similarly, exp(0.67958) = 1.973. Most commonly, this examination entails the speciﬁcation of a linear-like model for the log hazard. H��W TSW�Y�D6tơz�R��e���9l&!���Ix�h6�[Q��Z-Uĕ�ڊ�ԥ�L�����B��V�q�Z�jE�:��(h��9s�̙3�{����������?� �����x�)Hs �0U6�� z��tZC�w- �g, v�=��;���${f��6�:@ �s ]�� �N@�F���J��䷍� ? Model Building with Cox Proportional Hazards James Long 3/5/2019. Hi There . 0000000016 00000 n Consider a simple model with one predictor, X 1. If the hazard ratio is less than 1, then the predictor is protective (i.e., associated with improved survival) and if the hazard ratio is greater than 1, then the predictor is associated with increased risk (or decreased survival). Tests of Proportionality in SAS, STATA and SPLUS When modeling a Cox proportional hazard model a key assumption is proportional hazards. Survival analysis methods can also be extended to assess several risk factors simultaneously similar to multiple linear and multiple logistic regression analysis as described in the modules discussing Confounding, Effect Modification, Correlation, and Multivariable Methods. Purpose of Model Building. The predictor variables in a Cox proportional hazards regression model, sometimes referred to as explanatory variables, can be any mixture of continuous, binary, or categorical variables. Survival models relate the time that passes, before some event occurs, to one or more covariates that may be associated with that quantity of time. Example 5 will illustrate estimation of a Cox proportional hazards regression model and discuss the interpretation of the regression coefficients. It produces an estimate for the hazard ratio: The model is based on the assumption that the baseline hazard function depends on time, t, but the predictor variables do not. Note that there is a positive association between age and all-cause mortality and between male sex and all-cause mortality (i.e., there is increased risk of death for older participants and for men). startxref The Cox proportional-hazards model (Cox, 1972) is essentially a regression model commonly used statistical in medical research for investigating the association between the survival time of patients and one or more predictor variables.. The exponential regression survival model, for example, assumes that the hazard function is constant. ��ǰo����6�.�-��z-����Be�;gp}~1�R� ��n��QPE We now use Cox proportional hazards regression analysis to make maximum use of the data on all participants in the study. Cox proportional hazards regression model The Cox PH model • is a semiparametric model • makes no assumptions about the form of h(t) (non-parametric part of model) • assumes parametric form for the eﬀect of the predictors on the hazard In most situations, we are more interested in the parameter estimates than the shape of the hazard. The numbers of CVD events in each of the 3 groups are shown below. A total of 5,180 participants aged 45 years and older are followed until time of death or up to 10 years, whichever comes first. The latter two models are multivariable models and are performed to assess the association between weight and incident CVD adjusting for confounders. Tests of hypothesis are used to assess whether there are statistically significant associations between predictors and time to event. To facilitate interpretation, suppose we create 3 categories of weight defined by participant's BMI. In most situations, we are interested in comparing groups with respect to their hazards, and we use a hazard ratio, which is analogous to an odds ratio in the setting of multiple logistic regression analysis. There is an 11.8% increase in the expected hazard relative to a one year increase in age (or the expected hazard is 1.12 times higher in a person who is one year older than another), holding sex constant. Date last modified: June 3, 2016. The function cox.zph() [in the survival package] provides a convenient solution to test the proportional hazards assumption for each covariate included in a Cox refression model fit. �����e����mP����移w��ԍ�ȧ�d�zm��yZ�D�� 0JK�S��(,�n���ؘ~����Wg�Vx��}��:m�P)��O����7�x�PU�����"�H2>��jxb����VHt�%������Q��츹���. 0000000656 00000 n After accounting for age, sex, blood pressure and smoking status, there are no statistically significant associations between total serum cholesterol and all-cause mortality or between diabetes and all-cause mortality. The same is true in the model adjusting for age, sex and the clinical risk factors. There are however, other assumptions as noted above (i.e., independence, changes in predictors produce proportional changes in the hazard regardless of time, and a linear association between the natural logarithm of the relative hazard and the predictors). One of the most popular regression techniques for survival outcomes is Cox proportional hazards regression analysis. The proportional hazard assumption may be tested using the R function cox.zph(). There is a 0.11149 unit increase in the expected log of the relative hazard for each one year increase in age, holding sex constant, and a 0.67958 unit increase in expected log of the relative hazard for men as compared to women, holding age constant. In the previous chapter (survival analysis basics), we described the basic concepts of survival analyses and methods for analyzing and summarizing … K�D4�������;� l�@�1��1��?�ײ���[x��c�'�?���uz��9�Mh1���Ɠ�n��Jvt����j�ǘ�K�6�T��t����� �h �ImʶB�#�� �i�Y3��1_Z����W �6G����7̣��W����ޕ�d0@c�w�d��ϡ�"y*�-8�U�+�氈�g�����۫��o�Bx�}����햪�'���� ˘�w����M���\����q�כ?��o���mZ���J���q1r:b7�WFy�w���K#?���֭��^�T}.&%#�Yt�M��cl�Sn��n�j_ѕ~T�eӡ������X#���Dm�i[�_� w����jY�s�@�� �o�ٳZXX�=��m���_w��G:'z���$��ʼ*�d����! Participants are followed for up to 10 years for the development of CVD. In the unadjusted model, there is an increased risk of CVD in overweight participants as compared to normal weight and in obese as compared to normal weight participants (hazard ratios of 1.215 and 1.310, respectively). The hazard ratio is the ratio of these two expected hazards: h0(t)exp (b1a)/ h0(t)exp (b1b) = exp(b1(a-b)) which does not depend on time, t. Thus the hazard is proportional over time. In other studies, it is not. In practice, interest lies in the associations between each of the risk factors or predictors (X1, X2, ..., Xp) and the outcome. �?����kxw����f"�5D� ���� rNbt For example, 1/0.2 = 5, which is the expected event-free time (5 months) per person at risk. Using the data in Example 3, the hazard ratio is estimated as: Thus, the risk of death is 4.870 times higher in the chemotherapy before surgery group as compared to the chemotherapy after surgery group. Cox The Cox proportional hazards regression model The Cox model assumes that the hazard for the ith individual (i = 1;:::;n) is i(t) = 0(t)exp( 1X i1 + 2X i2 + + pX ip) = 0(t)exp( 0X i) where 1; 2;:::; p are regression parameters, X i1 is the covariate value for covariate 1 for individual i, etc. It is a regression model for survival data. There are a total of 402 deaths observed among 5,180 participants. Explore how to fit a Cox proportional hazards model using Stata. In a Cox proportional hazards regression model, the measure of effect is the hazard rate, which is the risk of failure (i.e., the risk or probability of suffering the event of interest), given that the participant has survived up to a specific time. 0000013656 00000 n There are several important assumptions for appropriate use of the Cox proportional hazards regression model, including. <<68e2afd157af024880fcce8d8b28999f>]>> Consequently, it does not matter which appears in the numerator of the hazard ratio. Descriptive statistics are shown below on the age and sex of participants at the start of the study classified by whether they die or do not die during the follow up period. The Cox proportional hazards regression model can be written as follows: where h(t) is the expected hazard at time t, h0(t) is the baseline hazard and represents the hazard when all of the predictors (or independent variables) X1, X2 , Xp are equal to zero. endstream endobj 405 0 obj<>stream x�bbdbŃ3� ���ţ� u�4 For each covariate, the function cox.zph () correlates the corresponding set of scaled Schoenfeld residuals with time, to test for independence between residuals and time. independence of survival times between distinct individuals in the sample, a multiplicative relationship between the predictors and the hazard (as opposed to a linear one as was the case with multiple linear regression analysis, discussed in more detail below), and, Overweight as BMI between 25.0 and 29.9, and. For example, the hazard rate when time $$t$$ when $$x = x_1$$ would then be $$h(t|x_1) = h_0(t)exp(x_1\beta_x)$$, and at time $$t$$ when $$x = x_2$$ would be $$h(t|x_2) = h_0(t)exp(x_2\beta_x)$$. �$*��������6��TR�Xs��oD@.�o��bMg��K:#}F�����:�D��'�b46����P�ye���ݞJ�d�%���5��,�T�i-3¹���ד��i[;nI\�g��:�k5}��ZS��m�Lg�������%�hH?���K-^�Uί�q�;97 ��ݔ���1�p ���Y�΄E�ULw�gض��* 2���v؈�ˈR���_��X J�P�3��[_�O0O���۳��qw��� �:�1Ә�G2��s��o8�'#1���� �8^4��+S�e[�Wքp�Z��^�.Y�Fww5w��O���9"����������(��%W ���b��#P*���(>اt�:���3|�4�H4�jB��i�Pk�DC��7�٠�"�O��VC�����Ӹ�ٖ�y�3xT�1kC��~�3 For interpretability, we compute hazard ratios by exponentiating the parameter estimates. In the example above, the explanatory variables were all categorical or binary and included secondary drug prevention, sex, socioeconomic deprivation index, and categorised age. The Cox PH model uses the hazard function to estimate the RR of failure. These men were followed for one year after their release, and the dates of any arrests were recorded. endstream endobj 402 0 obj<> endobj 403 0 obj<> endobj 404 0 obj<>stream %%EOF 0000005602 00000 n Example 3 examined the association of a single independent variable (chemotherapy before or after surgery) on survival. Notice that for the statistically significant risk factors (i.e., age, sex, systolic blood pressure and current smoking status), that the 95% confidence intervals for the hazard ratios do not include 1 (the null value). endstream endobj 393 0 obj<>>>/LastModified(D:20060402191530)/MarkInfo<>>> endobj 395 0 obj<>/ProcSet[/PDF/Text]/ExtGState<>>>/StructParents 0>> endobj 396 0 obj<> endobj 397 0 obj<> endobj 398 0 obj<> endobj 399 0 obj<> endobj 400 0 obj<> endobj 401 0 obj<>stream In a Cox proportional hazards regression analysis, we find the association between BMI and time to CVD statistically significant with a parameter estimate of 0.02312 (p=0.0175) relative to a one unit change in BMI. Cox proposed a partial likelihood for β without involving baseline hazard function λo(t) .The parameters of the Cox model can still be estimated by the method of partial likelihood without specifying the baseline hazard.The likelihood of the event to be observed occurring for subject j at time Xj can be written as In an observational study with a time-to-event outcome, the standard analytical approach is the Cox proportional hazards regression model. A prospective cohort study is run to assess the association between body mass index and time to incident cardiovascular disease (CVD). 0000014116 00000 n Get started. All Rights Reserved. ۾.x�n����5[�f��bWT����X�4��퇧���mA#����^sAqǯ+!c~EsBye�]���$�.��9��i��)��;��(�8�����'\���"V��Jk~�ҋQ�)]���ذ�2��b��t�E^���+mi}^�q�9�5 ��~�mP���[�6Fԋݮܷ��w�4�#K-nդ��ŵ�ۖ��|oI[Guc�?�e>�j'C���7��Q��[26��'�X_��������a2X��Ж]�����E�}��x8iG��Q���q�i��rd�!�>�|uO���u�� �Mp��ya��eͬ�n�Z�w)�&~vrr�ڇ~-�މ�������߷[�7a�Y, �yzLx��Λ�r{�X4s�mFó��d0�^�Y��y;��2[k������(�u�p�o�P8&�x�ˌP�,�a0��Pmˢ�u����&�H8H�����Xl�4�^�$-���"e�S1'���8 i�g���/�C�$2�l�Pԁm���9 The Cox proportional hazards model is called a semi-parametric model, because there are no assumptions about the shape of the baseline hazard function. Cox proportional-hazards model is developed by Cox … Because we have three weight groups, we need two dummy variables or indicator variables to represent the three groups. Notice that the right hand side of the equation looks like the more familiar linear combination of the predictors or risk factors (as seen in the multiple linear regression model). This might help to properly choose the functional form of continuous variable in the Cox model. However, the hazard represents the expected number of events per one unit of time. A one unit increase in BMI is associated with a 2.3% increase in the expected hazard. Cox Models If we exponentiate the parameter estimate, we have a hazard ratio of 1.023 with a confidence interval of (1.004-1.043). 0000004990 00000 n h(t) = P{ t < T < (t + ∆) | T >t} = f(t) / (1 - F(t)) = f(t) / S(t) The hazard function describes the concept of the risk of an outcome (e.g., death, failure, hospitalization) in an interval after time t, conditional on the subject having survived to time t. The technique for estimating the regression coefficients in a Cox proportional hazards regression model is beyond the scope of this text and is described in Cox and Oakes.9 Here we focus on interpretation. The immune-related risk score and clinical information, such as age, sex, World Health Organization (WHO) grade, and isocitrate dehydrogenase 1 (IDH1) mutation, were used to independently validate and develop a prognostic … One of the most popular regression techniques for survival analysis is Cox proportional hazards regression, which is used to relate several risk factors or exposures, considered simultaneously, to survival time. {��*��8g�I��D�!R~ڴBl�':�F�[��Ձ�{�8K��=�\���2S�K���N6&�'� .^{� Wayne W. LaMorte, MD, PhD, MPH, Boston University School of Public Health. Sometimes the model is expressed differently, relating the relative hazard, which is the ratio of the hazard at time t to the baseline hazard, to the risk factors: We can take the natural logarithm (ln) of each side of the Cox proportional hazards regression model, to produce the following which relates the log of the relative hazard to a linear function of the predictors. 26 The CM aims to estimate hazard ratios over time. In Example 3 there are two active treatments being compared (chemotherapy before surgery versus chemotherapy after surgery). The examples that follow illustrate these tests and their interpretation. The Cox proportional hazards model is: Suppose we wish to compare two participants in terms of their expected hazards, and the first has X 1 = a and the second has X 1 = b. In a proportional hazards model, the unique effect of a unit increase in a covariate is multiplicative with respect to the hazard rate. A�%x�9��n�e��3\��N�5���d0�� �k����./��3�c�v� Notice that the predicted hazard (i.e., h(t)), or the rate of suffering the event of interest in the next instant, is the product of the baseline hazard (h0(t)) and the exponential function of the linear combination of the predictors. 0000002020 00000 n The parameter estimates are again generated in SAS using the SAS Cox proportional hazards regression procedure and are shown below along with their p-values.12 Also included below are the hazard ratios along with their 95% confidence intervals. Open in app. ... For example the book discusses “adjusting for potential confounders” but a satisfactory definition of confounding requires a formal causal inference framework. As a result, the hazard in a group can exceed 1. The expected hazards are h(t) = h0(t)exp (b1a) and h(t) = h0(t)exp (b1b), respectively. The Cox proportional hazards model relates the hazard rate for individuals or items at the value Xi, to the hazard rate for individuals or items at the baseline value. Because we model BMI as a continuous predictor, the interpretation of the hazard ratio for CVD is relative to a one unit change in BMI (recall BMI is measured as the ratio of weight in kilograms to height in meters squared). There are other regression models used in survival analysis that assume specific distributions for the survival times such as the exponential, Weibull, Gompertz and log-normal distributions1,8. The way I understand cox regression is that it works on the assumption that the hazard curves for... Cox proportional hazards - how to interpret summary output Hi all, I've been using a cox proportional hazard model to do survival analysis in R. For example, in the medical domain, we are seeking to find out which covariate has the most important impact on the survival time of a patient. There are a number of basic concepts for testing proportionality but the implementation of these concepts differ across statistical packages. Cox proportional-hazards model is developed by Cox in 1972. We now estimate a Cox proportional hazards regression model and relate an indicator of male sex and age, in years, to time to death. %PDF-1.4 %���� This video provides a demonstration of the use of the Cox proportional hazards model using SPSS. The Cox proportional hazards model is: Suppose we wish to compare two participants in terms of their expected hazards, and the first has X1= a and the second has X1= b. At baseline, participants' body mass index is measured along with other known clinical risk factors for cardiovascular disease (e.g., age, sex, blood pressure). * Adjusted for age, sex, systolic blood pressure, treatment for hypertension, current smoking status, total serum cholesterol. Immune-related differentially expressed genes (DEGs) were screened to establish a prognostic signature by a multivariate Cox proportional hazards regression model. For the melanoma data, p=0.222, indicating that the hazards are, at least approximately, proportional. The Cox proportional-hazards model (Cox, 1972) is essentially a regression model commonly used statistical in medical research for investigating the association between the survival time of patients and one or more predictor variables. 0000013887 00000 n The antilog of an estimated regression coefficient, exp(bi), produces a hazard ratio. 0000005140 00000 n However, after adjustment for age and sex, there is no statistically significant difference between overweight and normal weight participants in terms of CVD risk (hazard ratio = 1.067, p=0.5038). 0000003114 00000 n 0000001346 00000 n Fitted lines with lowess function should be linear to satisfy the Cox proportional hazards model assumptions. Cox Proportional Hazards Model An hazard rate is the probability estimate of the time it takes for an event to take place. A probability must lie in the range 0 to 1. Cox regression provides a better estimate of these functions than the Kaplan-Meier method when the assumptions of the Cox model are met and the fit of the model is strong. A����C1�x� � n5H�",�o"���d!���Y�&k6T��4�2^G�"��W���u��ɤa�]S�) ��C���@���g�� V�t���RpZ�g�ĒR�]��u�}��/��s;���>�����nV>s����L�}�w��w�4-�5'Ë�Y�u�Oϑ�GA�ӫ�=b�wF�osN�Y��г'���_VmkL�x�n��v�\�u沦��+g6n�8��&>����BI�ZJ��(�u8�N�,�*:��s{a7�{ElN�֐QW>�DM]�o��+��w@��v�,W�����.Ȋ9��bv�n�ې�BŽ�C��uY��˜�'[�\�� It is nota probability, but an event rate (e.g., death): the number of events per unit of population and per unit of time. The variable in the data set are: • WEEK: The week of the first arrest after release, ;b]P!orr.aN����%31���WO6����b�����sK����zR�c\�� k�|�����f#�*��h&l�9{(&��/��/�!,�K @�F��!���1 ãM�� �QT2�HQv2*4�_�'k��ը1�Hh �)�d�Q4��lIW���"��@��Κ M sI#!I9L�\�#s�3 )�RF��K�t��9t�n��5Q���Rpb�C4I �h�. The event can be anything ranging from death of an organism or failure of a machine or any other time to event setting. For example, taking a drug may halve one's hazard rate for a stroke occurring, or, changing the material from which a manufactured component i 0000002551 00000 n 0000000924 00000 n Cijz ��.��B�"�$":�%I�x:�#�㰘>?�xS��B��Pa֑$�b�l28l\$���C�3�2u��f�yaR>�F�s�t�T�yӂ��7YGQ�Q6����; Another interpretation is based on the reciprocal of the hazard. For example, a parametric model based on the exponential distribution may be written as Thus, the predictors have a multiplicative or proportional effect on the predicted hazard. 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