class: left, middle, inverse, title-slide .title[ # Introduction to Time-to-Event ] .subtitle[ ## Mabel Carabali ] .author[ ### Risk & Hazards ] .institute[ ### EBOH, McGill University ] .date[ ### 01-08-2023 Updated: ( 2024-09-24) ] --- class: middle <img src="images/L6survival_meme.jpg" width="65%" style="display: block; margin: auto;" /> --- class: middle ### What to do, if you want to: - Estimate median survival times, plot survival over time after treatment, or estimate the probability of surviving beyond a prespecified time interval (eg, 5-year survival rate)? - Assess whether survival times are related to covariates and/or adjust for potential confounders. - Account for censoring and avoid .red[lead time bias] -- # Objectives 1. Review the concept of time-to-event a.k.a. "survival" analysis 2. Provide an introduction to epidemiological and statistical methods for the appropriate analysis of time-to-event data .pull-right[.red[More about this on 705]] --- class: middle ### RECALL <img src="images/fig2_2.png" width="60%" style="display: block; margin: auto;" /> .small[(Szklo M, Nieto FJ. Epidemiology : Beyond the Basics. Fourth ed.)] --- class: middle ### Relationship between Incidence Rate and Incidence Proportion - The method of calculating **risks over a time period with changing incidence rates** is known as **.red[survival analysis.]** - "_The cumulative probability of the event during a given interval lasting `\(m\)` units of time and beginning at time `\(x\)`, is the proportion of new events during that period of time in which the denominator is the initial population corrected for losses_". .footnote[(Szklo M, Nieto FJ. Epidemiology : Beyond the Basics. Fourth ed.)] --- class: middle ### .red[Assumptions] in the Estimation of Cumulative Incidence Based on Survival Analysis - **Uniformity of Events and Losses** Within Each Interval (Classic Life Table). - Events and losses are approximately uniform during each defined interval. - If risk changes rapidly within a given interval, then calculating a cumulative risk over the interval is not very informative. - The rationale underlying the method to correct for losses—that is, subtracting one-half of the losses from the denominator also depends on the assumption that losses occur uniformly. - **Independence of censoring AND Survival** - **No secular trends!** --- class:middle ### Incidence Proportion & Survival Proportion Survival proportion: complementary to Incidence Proportion - "Proportion of a closed population at risk that does not become diseased within a given period of time." - `\(S = 1 − R\)`, where `\(R\)` = incidence proportion; `\(S\)` = survival proportion - Equivalently, the proportion of remaining disease free alive individuals by the end of the follow-up period. .small[Lash, T, et al. Modern Epidemiology 4th, Wolters Kluwer Health, 2021] --- class:middle ### Survival probability - **Survival probability** at a certain time, `\(S(t)\)`, is a conditional probability of surviving beyond that time, given that an individual has survived just prior to that time. - Can be estimated as the number of individuals who are alive without loss to follow-up at that time, divided by the number of individuals who were alive just prior to that time. -- - The **Kaplan-Meier** estimate of survival probability is the product of these conditional probabilities up until that time. - At time 0, the survival probability is 1, i.e. `\(S(t_0) = 1\)` --- class:middle ## Product Limit Formula **Kaplan-Meier Formula** - The product limit formula shows us how to calculate the **survival proportion** (and thereby the incidence proportion) over a period of time when the risk changes. - It shows that we need to multiply the survival proportions over all the intervals to calculate the overall survival proportion. <img src="images/Rothman-ch004-image005.jpeg" width="40%" style="display: block; margin: auto;" /> --- class: middle ### Product Limit Formula Example: | |Start (0) | 2| 4| 8 | 14 |19 (End)| |-------|----------|----|-------|----|------|--------| |Index ( _k_) | |1 |2 |3 |4 |5| |Nb Outcomes ( `\({A_k}\)` )| 0 |1| 2|1|1|0| |Nb at Risk ( `\({N_k}\)` )| | | | | | | |% Surviving ( `\({S_k}\)` ) | | | | | | | | .footnote[Nine people over 20 years] --- class: middle ### Product Limit Formula (Kaplan-Meier Formula) $$ Incidence\ Proportion = 1-S$$ $$S =\prod_{k = 1}^{v} (\frac{N_k - A_k} {N_k}) $$ where _v_ are sub-intervals PLF = We need to multiply the survival proportions over all the intervals to calculate the overall survival proportion. --- class: middle ### Product Limit Formula (Kaplan-Meier Formula) | |Start (0) | 2| 4| 8 | 14 |19 (End)| |-------|----------|----|-------|----|------|--------| |Index ( _k_) | |1 |2 |3 |4 |5| |Nb Outcomes ( `\({A_k}\)` )| 0 |1| 2|1|1|0| |Nb at Risk ( `\({N_k}\)` )| 9|9 |8 | 6| 5|4| |% Surviving ( `\({S_k}\)` )| | 8/9| 6/8 | 5/6 |4/5 | 4/4 |Interval length ( `\({N_k}\Delta{A_k}\)` )| | | | | | | |Person-time ( `\(\Delta{A_k}\)`) | | | | | | | |Incidence Rate ( `\({IR_k}\)` )| | | | | | | --- class: middle ### Product Limit Formula (Kaplan-Meier Formula) Hand calculations for 9 people followed for 19 months. | |Start (0) | 2| 4| 8 | 14 |19 (End)| |-------|----------|----|-------|----|------|--------| |Index ( _k_) | |1 |2 |3 |4 |5| |Nb Outcomes ( `\({A_k}\)` )| 0 |1| 2|1|1|0| |Nb at Risk ( `\({N_k}\)` )| 9|9 |8 | 6| 5|4| |**% Surviving ( `\({S_k}\)` )**| | 8/9| 6/8 | 5/6 |4/5 | 4/4| |Interval length ( `\({N_k}\Delta{A_k}\)` )| | 2| 2| 4|6 |5 | |Person-time ( `\(\Delta{A_k}\)`) | | 18|16 |24 |30 |20 | |Incidence Rate ( `\({IR_k}\)` )| | 1/18| 2/16| 1/24 | 1/30| 0/20 | `\(S = (8/9) {\times}(6/8){\times} (5/6){\times} (4/5) {\times} (4/4)\)` `\(S =\)` 0.444 ** `\(Incidence\ Proportion = 1 - S\)`** = 0.556 --- class: middle ### Kaplan-Meier Formula - The Kaplan–Meier approach involves the calculation of the probability of each event at the time it occurs. - The denominator for this calculation is the population at risk at the time of each event’s occurrence - The probability of each event is a “conditional probability” `\(\to\)` conditioned on being at risk (alive and not censored) at the event time. - The advantage of the Kaplan-Meier Product-Limit method over the life table method is that the resulting estimates do not depend on the grouping of the data (into a certain number of time intervals). - However, the Product-Limit method and the life table method are identical if the intervals of the life table contain at most one observation. - Regardless of the method used in the calculation (actuarial or Kaplan–Meier), the cumulative incidence is a proportion, unitless, and its value range from 0 to 1 (or 100%) --- class: middle ### Kaplan-Meier Formula `\(Survival = {S_i}\)` , Cumulative probability of the event `\((1 – {S_i})\)` `\(\to\)` `\(1–{S_{24}} = 1 – 0.18 = 0.82\)` <img src="images/table2_3.png" width="70%" style="display: block; margin: auto;" /> .small[ _*Obtained by multiplying the conditional probabilities in column (5)—see text. † Examples of how to determine how many individuals were at risk at three of the event times (1, 3, and 17 months) are shown with vertical arrows in Figure 2-3_ (Szklo M, Nieto FJ. Epidemiology : Beyond the Basics. Fourth ed.)] --- class: middle ### Simulated Examples .pull-left[ **Recall from the previous scenarios** <img src="L6_Hazards_Intro1_files/figure-html/unnamed-chunk-5-1.svg" width="130%" style="display: block; margin: auto;" /> ] .pull-right[ **Some coding** ```r surv_object<-Surv(time = dat$t, event = dat$O2) eventKM<- survfit(surv_object ~ 1, data = dat, type="kaplan-meier") ``` ] -- **Output** ``` ## Call: survfit(formula = surv_object ~ 1, data = dat, type = "kaplan-meier") ## ## time n.risk n.event survival std.err lower 95% CI upper 95% CI ## 1 20 1 0.950 0.0487 0.859 1.000 ## 7 18 2 0.844 0.0826 0.697 1.000 ## 8 16 1 0.792 0.0928 0.629 0.996 ## 10 14 2 0.679 0.1087 0.496 0.929 ## 17 6 1 0.565 0.1373 0.351 0.910 ``` --- class: middle **.red[Other Example]** .pull-left[ <img src="L6_Hazards_Intro1_files/figure-html/unnamed-chunk-8-1.svg" width="90%" style="display: block; margin: auto;" /> ] .pull-right[ **Kaplan-Meier Curve - Outcome X** <img src="images/L6_km_outcomex.png" width="90%" style="display: block; margin: auto;" /> ] --- class: middle ### .red[Assumptions] in the Estimation of Cumulative Incidence Based on Survival Analysis - Uniformity of Events and Losses Within Each Interval (Classic Life Table). - Events and losses are approximately uniform during each defined interval. - .red[If risk changes rapidly within a given interval, then calculating a cumulative risk over the interval is not very informative.] - .red[The rationale underlying the method to correct for losses—that is, subtracting one-half of the losses from the denominator also depends on the assumption that losses occur uniformly.] - Independence of censoring AND Survival - No secular trends! --- class: middle ### Exponential Formula The exponential formula relates the incidence rate to the incidence proportion. **Simplified:** $$ Risk = 1 - e^{-Incidence rate \times\ Time}$$ **.purple[Elaborated:]** Deriving the survival proportion as a function of the incidence rates for each interval: - Total person-time at risk in the interval is `\(N_k \Delta_{tk}\)` - Number of Outcomes at time `\(t_k\)` is `\(A_k\)` - Number of people at Risk (alive by the end of follow-up) `\(N_k\)` -- - Incidence Rate in the time following = `\(IR = \left(\frac{A_k}{N_k\Delta_{tk}}\right)\)` - Incidence Proportion over same sub interval = `\(IP_k = {IR_k\Delta_{tk}}\)` - Survival Proportion for the sub interval = `\(S_k = 1 - {IR_k\Delta_{tk}}\)` .small[Lash, Timothy, L. et al. Modern Epidemiology. (4th Edition). 2020] --- class: middle ### Exponential Formula | |0 | 2| 4| 8 | 14 | 19 | |-------|------|----|-------|----|------|-------| |Index ( _k_) | |1 |2 |3 |4 |5| |Nb Outcomes ( `\({A_k}\)` )| 0 |1| 2|1|1|0| |Nb at Risk ( `\({N_k}\)` )| 9|9 |8 | 6| 5|4| |% Surviving ( `\({S_k}\)` )| | 8/9| 6/8 | 5/6 |4/5 | 4/4| |**Interval length ( `\({N_k}\Delta{A_k}\)` )**| | 2| 2| 4|6 |5| |Person-time ( `\((\Delta{A_k}\)` )| | 18|16 |24 |30 |20 | |**Incidence Rate ( `\({IR_k}\)` )**| | 1/18| 2/16| 1/24 | 1/30| 0/20| -- Survival Proportion for the sub interval = `\(S_k = 1 - IR_k\Delta_{tk} {\cong}\ exp(-IR_k\Delta_{tk})\)` `\(exp(-0(5)-(\frac{1}{30})(6)-(\frac{1}{24})(4)-(\frac{2}{16})(2)-(\frac{1}{18})(2))\)` `\(S_k\)` = 0.483 --- class: middle ### Exponential Formula **Recall:** `\(S = (8/9) {\times}(6/8){\times} (5/6){\times} (4/5) {\times} (4/4)\)` `\(S =\)` 0.444 `\(S_k\)` = 0.483 `\(Risk = 1 - e^{-Incidence\ rate{\times}\ Time}\)` `\(Risk = 1 - e^{-IR_k\Delta_{tk}}\)` ** `\({Risk_{ExpForm}}\)` ** = `\(1-S_k\)` = 0.517 `\({Risk_{c}} = 5/9\)` = 0.556 = `\(1- S\)` = **1 - 0.444** `$$R = 1 - S {\cong} 1 - e^{-\sum_{k=1}^{v}{IR_k\Delta_{tk}}}$$` `\(Risk_c =\)` **0.556 ** `\({\cong}\ Risk_{ExpForm}\)` = **0.517** --- class: middle ## Exponential Formula **Assumptions** 1. Closed population 2. Event under study is inevitable (no competing risk) 3. Number of events at each event time is a small proportion of the number at risk at that time (can be forced with fine measurement of time) Assumptions 1 and 2 are also assumed for the product limit formula. -- **Product-limit and exponential formulas:** Translate incidence-rate estimates from open populations into incidence-proportion estimates for a closed population of interest ! .small[Lash, Timothy, L. et al. Modern Epidemiology. (4th Edition). 2020. (pg. 68] --- class:middle ### .red[Recall] Survival probability - **Survival probability** at a certain time, `\(S(t)\)`, is a conditional probability of surviving beyond that time, given that an individual has survived just prior to that time. - Can be estimated as the number of patients who are alive without loss to follow-up at that time, divided by the number of patients who were alive just prior to that time. - The **Kaplan-Meier** estimate of survival probability is the product of these conditional probabilities up until that time. - At time 0, the survival probability is 1, i.e. `\(S(t_0) = 1\)` --- class:middle ## Worked example - The `survfit` function creates survival curves based on a formula. - Let's generate the overall survival curve for the entire `lung` cohort from the [`survival` package](https://github.com/therneau/survival). - Create `survfit` object and assign it to `f1`, (details about `f1` available via `names` or `str`). - Often want to know probability of surviving a specific time (e.g. 1 year)use `summary()`. - Produce survival curve including this information --- class:middle **Get a Table 1** .pull-left[ ```r lung<- survival::lung tab1<- lung %>% tbl_summary() ``` Check the package `gtsummary` for more information and details on formatting ] -- .pull-right[ <div id="yhzqhjovci" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;"> <style>#yhzqhjovci table { font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji'; -webkit-font-smoothing: antialiased; -moz-osx-font-smoothing: grayscale; } #yhzqhjovci thead, #yhzqhjovci tbody, #yhzqhjovci tfoot, #yhzqhjovci tr, #yhzqhjovci td, #yhzqhjovci th { border-style: none; } #yhzqhjovci p { margin: 0; padding: 0; } #yhzqhjovci .gt_table { display: table; border-collapse: collapse; line-height: normal; margin-left: auto; 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<tr class="gt_col_headings"> <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="<strong>Characteristic</strong>"><strong>Characteristic</strong></th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<strong>N = 228</strong><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span>"><strong>N = 228</strong><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span></th> </tr> </thead> <tbody class="gt_table_body"> <tr><td headers="label" class="gt_row gt_left">inst</td> <td headers="stat_0" class="gt_row gt_center">11 (3, 16)</td></tr> <tr><td headers="label" class="gt_row gt_left"> Unknown</td> <td headers="stat_0" class="gt_row gt_center">1</td></tr> <tr><td headers="label" class="gt_row gt_left">time</td> <td headers="stat_0" class="gt_row gt_center">256 (167, 399)</td></tr> <tr><td headers="label" class="gt_row gt_left">status</td> <td headers="stat_0" class="gt_row gt_center"><br /></td></tr> <tr><td headers="label" class="gt_row gt_left"> 1</td> <td headers="stat_0" class="gt_row gt_center">63 (28%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 2</td> <td headers="stat_0" class="gt_row gt_center">165 (72%)</td></tr> <tr><td headers="label" class="gt_row gt_left">age</td> <td headers="stat_0" class="gt_row gt_center">63 (56, 69)</td></tr> <tr><td headers="label" class="gt_row gt_left">sex</td> <td headers="stat_0" class="gt_row gt_center"><br /></td></tr> <tr><td headers="label" class="gt_row gt_left"> 1</td> <td headers="stat_0" class="gt_row gt_center">138 (61%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 2</td> <td headers="stat_0" class="gt_row gt_center">90 (39%)</td></tr> <tr><td headers="label" class="gt_row gt_left">ph.ecog</td> <td headers="stat_0" class="gt_row gt_center"><br /></td></tr> <tr><td headers="label" class="gt_row gt_left"> 0</td> <td headers="stat_0" class="gt_row gt_center">63 (28%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 1</td> <td headers="stat_0" class="gt_row gt_center">113 (50%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 2</td> <td headers="stat_0" class="gt_row gt_center">50 (22%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 3</td> <td headers="stat_0" class="gt_row gt_center">1 (0.4%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> Unknown</td> <td headers="stat_0" class="gt_row gt_center">1</td></tr> <tr><td headers="label" class="gt_row gt_left">ph.karno</td> <td headers="stat_0" class="gt_row gt_center"><br /></td></tr> <tr><td headers="label" class="gt_row gt_left"> 50</td> <td headers="stat_0" class="gt_row gt_center">6 (2.6%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 60</td> <td headers="stat_0" class="gt_row gt_center">19 (8.4%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 70</td> <td headers="stat_0" class="gt_row gt_center">32 (14%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 80</td> <td headers="stat_0" class="gt_row gt_center">67 (30%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 90</td> <td headers="stat_0" class="gt_row gt_center">74 (33%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 100</td> <td headers="stat_0" class="gt_row gt_center">29 (13%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> Unknown</td> <td headers="stat_0" class="gt_row gt_center">1</td></tr> <tr><td headers="label" class="gt_row gt_left">pat.karno</td> <td headers="stat_0" class="gt_row gt_center"><br /></td></tr> <tr><td headers="label" class="gt_row gt_left"> 30</td> <td headers="stat_0" class="gt_row gt_center">2 (0.9%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 40</td> <td headers="stat_0" class="gt_row gt_center">2 (0.9%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 50</td> <td headers="stat_0" class="gt_row gt_center">4 (1.8%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 60</td> <td headers="stat_0" class="gt_row gt_center">30 (13%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 70</td> <td headers="stat_0" class="gt_row gt_center">41 (18%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 80</td> <td headers="stat_0" class="gt_row gt_center">51 (23%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 90</td> <td headers="stat_0" class="gt_row gt_center">60 (27%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> 100</td> <td headers="stat_0" class="gt_row gt_center">35 (16%)</td></tr> <tr><td headers="label" class="gt_row gt_left"> Unknown</td> <td headers="stat_0" class="gt_row gt_center">3</td></tr> <tr><td headers="label" class="gt_row gt_left">meal.cal</td> <td headers="stat_0" class="gt_row gt_center">975 (635, 1,150)</td></tr> <tr><td headers="label" class="gt_row gt_left"> Unknown</td> <td headers="stat_0" class="gt_row gt_center">47</td></tr> <tr><td headers="label" class="gt_row gt_left">wt.loss</td> <td headers="stat_0" class="gt_row gt_center">7 (0, 16)</td></tr> <tr><td headers="label" class="gt_row gt_left"> Unknown</td> <td headers="stat_0" class="gt_row gt_center">14</td></tr> </tbody> <tfoot class="gt_footnotes"> <tr> <td class="gt_footnote" colspan="2"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span> Median (Q1, Q3); n (%)</td> </tr> </tfoot> </table> </div> ] --- class:middle **Worked example** ```r f1 <- survfit(Surv(time, status) ~ 1, data = lung) ``` <img src="L6_Hazards_Intro1_files/figure-html/unnamed-chunk-12-1.svg" width="55%" style="display: block; margin: auto;" /> -- **Summary Results** ```r summary(survfit(Surv(time, status) ~ 1, data = lung), times = 365.25) ``` ``` ## Call: survfit(formula = Surv(time, status) ~ 1, data = lung) ## ## time n.risk n.event survival std.err lower 95% CI upper 95% CI ## 365 65 121 0.409 0.0358 0.345 0.486 ``` --- class:middle ### What happens if you use a naive estimate ? 121 of the 228 patients died by `\(1\)` year so: `$$\Big(1 - \frac{121}{228}\Big) \times 100 = 47\%$$` - You get an **incorrect** estimate of the `\(1\)`-year probability of survival when you ignore the fact that 42 patients were censored before `\(1\)` year. - Recall the **correct** estimate of the `\(1\)`-year probability of survival was 41%. - Ignoring censoring leads to an **overestimate** of the overall survival probability, because the censored subjects only contribute information for part of the follow-up time, and then fall out of the risk set, thus pulling down the cumulative probability of survival --- class:middle ## Comparing survival times between groups - We can conduct between-group significance tests using a log-rank test. - The log-rank test equally weights observations over the entire follow-up time and is the most common way to compare survival times between groups. - Other methods weight according to early or late follow-ups (see `?survdiff` for different test options). -- The `survdiff` function provides the log-rank p-value. For example, we can test whether there was a difference in survival time according to sex (SAB) in the `lung` data ```r survdiff(Surv(time, status) ~ sex, data = lung) ``` ``` ## Call: ## survdiff(formula = Surv(time, status) ~ sex, data = lung) ## ## N Observed Expected (O-E)^2/E (O-E)^2/V ## sex=1 138 112 91.6 4.55 10.3 ## sex=2 90 53 73.4 5.68 10.3 ## ## Chisq= 10.3 on 1 degrees of freedom, p= 0.001 ``` --- class:middle ## The Cox regression model We may want to quantify an effect size for a single variable, or include more than one variable into a regression model to account for the effects of multiple variables. The Cox regression model is a semi-parametric model that can be used to fit univariable and multivariable regression models that have survival outcomes. `$$h(t|X_i) = h_0(t) \exp(\beta_1 X_{i1} + \cdots + \beta_p X_{ip})$$` `\(h(t)\)`: hazard, or the instantaneous rate at which events occur `\(h_0(t)\)`: underlying baseline hazard -- Some key assumptions of the model: - Non-informative censoring - Proportional hazards - Semi-parametric --- class:middle ## The Cox regression model We can fit regression models for survival data using the `coxph` function, which takes a `Surv` object on the left hand side and has standard syntax for regression formulas in `R` on the right hand side. ```r coxph(Surv(time, status) ~ sex, data = lung) ``` ``` ## Call: ## coxph(formula = Surv(time, status) ~ sex, data = lung) ## ## coef exp(coef) se(coef) z p ## sex -0.5310 0.5880 0.1672 -3.176 0.00149 ## ## Likelihood ratio test=10.63 on 1 df, p=0.001111 ## n= 228, number of events= 165 ``` --- class:middle ## Formatting Cox regression results We can see a tidy version of the output using the `tidy` function from the `broom` package: ```r broom::tidy(coxph(Surv(time, status) ~ sex, data = lung), exp = TRUE) %>% kable() ``` <table> <thead> <tr> <th style="text-align:left;"> term </th> <th style="text-align:right;"> estimate </th> <th style="text-align:right;"> std.error </th> <th style="text-align:right;"> statistic </th> <th style="text-align:right;"> p.value </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> sex </td> <td style="text-align:right;"> 0.5880028 </td> <td style="text-align:right;"> 0.1671786 </td> <td style="text-align:right;"> -3.176385 </td> <td style="text-align:right;"> 0.0014912 </td> </tr> </tbody> </table> Or use `tbl_regression` from the `gtsummary` package ```r coxph(Surv(time, status) ~ sex, data = lung) %>% gtsummary::tbl_regression(exp = TRUE) ``` <div id="lhpadwsitq" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;"> <style>#lhpadwsitq table { font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji'; 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} </style> <table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false"> <thead> <tr class="gt_col_headings"> <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="<strong>Characteristic</strong>"><strong>Characteristic</strong></th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<strong>HR</strong><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span>"><strong>HR</strong><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span></th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<strong>95% CI</strong><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span>"><strong>95% CI</strong><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span></th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<strong>p-value</strong>"><strong>p-value</strong></th> </tr> </thead> <tbody class="gt_table_body"> <tr><td headers="label" class="gt_row gt_left">sex</td> <td headers="estimate" class="gt_row gt_center">0.59</td> <td headers="conf.low" class="gt_row gt_center">0.42, 0.82</td> <td headers="p.value" class="gt_row gt_center">0.001</td></tr> </tbody> <tfoot class="gt_footnotes"> <tr> <td class="gt_footnote" colspan="4"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span> HR = Hazard Ratio, CI = Confidence Interval</td> </tr> </tfoot> </table> </div> --- class:middle ## Hazard ratios - The quantity of interest from a Cox regression model is a **hazard ratio (HR)**. The HR represents the ratio of hazards between two groups at any particular point in time. - The HR is interpreted as the instantaneous rate of occurrence of the event of interest in those who are still at risk for the event. It is **not** a risk, though it is commonly interpreted as such. - If you have a regression parameter `\(\beta\)` (from column `estimate` in our `coxph`) then HR = `\(\exp(\beta)\)`. - A HR < 1 indicates reduced hazard of death whereas a HR > 1 indicates an increased hazard of death. --- class:middle ### Hazard ratios So our HR = 0.59 implies that around 0.6 times as many females are dying as males, at any given time. ```r fit4 <- survfit(Surv(time, status) ~ sex, data = lung) ggsurvplot(data = lung, fit = fit4, xlab = "Months", xscale = 30.4, break.x.by = 182.4, fun = "cumhaz", legend.title = "", legend.labs = c("Male", "Female"), risk.table = TRUE, risk.table.y.text = FALSE) ``` <img src="L6_Hazards_Intro1_files/figure-html/unnamed-chunk-18-1.svg" width="70%" style="display: block; margin: auto;" /> --- ### Hazard ratios <br> .red[Are there potential issues with the hazard ratio?] -- <br> <br> Yes, the average HR ignores the distribution of events during the follow-up. <br> <br> But a possible solutions of time specific HRs poses another problem - selection bias due to depletion of susceptibles. <br> <br> Possible solutions i) report survival curves ii) accelerated survival models. <br> .small[[Hernán,The Hazards of Hazard Ratios](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3653612/)] --- class:middle ### What are competing risks? When subjects have multiple possible events in a time-to-event setting (e.g recurrence, death from disease, death from other causes). So what’s the problem? Why not just use KM approach and treat competing events as censored events? .purple[Remember basic KM assumption - censored patients have same risk as those remaining under observation]. .red[Unobserved dependence among event times is the fundamental problem that leads to the need for special consideration.] Two approaches to analysis in the presence of competing risks. 1. **Cause-specific hazards** instantaneous rate of occurrence of the given type of event in subjects who are currently event‐free estimated using Cox regression (coxph function) 2. **Subdistribution hazards** instantaneous rate of occurrence of the given type of event in subjects who have not yet experienced an event of that type estimated using Fine-Gray regression (`crr()` in `cmprsk` package) --- class:middle ### Hazard of the Hazards --- class: middle ### QUESTIONS? ## COMMENTS? # RECOMMENDATIONS?