The Kaplan-Meier (KM) survival curves represent the visual description of the survival phenomenon in a given patient population. The creation of these curves requires first of all the collection of survival times and the status of the patients, split in censored (not checked endpoint) and patients with checked endpoint. In Table 2 an example of data collection is shown: the censored patients are shown in column 3 using the number 0, while the checked ones (patients showing the event) using the number 1. The related KM survival curve is shown in Fig. 1. In KM survival curves the proportion of surviving patients in a given population is shown, step by step, and the height of the steps is proportional to the number of patients showing the occurring event (death, local relapse, metastases finding etc.) during follow up time and inversely proportional to the number of subjects at risk at the moment of the event. In column 7 of Table 2 each step contributing to the graph appearance is calculated. A typical aspect of a KM survival curve is given by the height of the steps, which grows moving away from the beginning of the observation time. This is a consequence of the fact that each step represents a single patient for whom the event occurs at a definite time, while the small vertical lines along the horizontal segments of the curve represent the censored patients. The absolute number of remaining patients after a sequence of occurred events is increasingly smaller, so being the fraction of each patient showing the event over the number of remaining patients consistently larger (1/n _i ), and finally being represented by a longer vertical segment in the curve.

When the patient with the highest follow up time shows the event ‘1’ the KM curve falls to 0 in the value of Survival Probability, and generally events in the rightmost part of the curve seem more effective at influencing the outcome than events in the leftmost part. Researchers have to be very careful in selecting adequate follow up times to be displayed and the number of patients in order to prevent unreliable behaviors shown by KM survival curves, which could lead readers to incorrect conclusions. Another feature that can be easily obtained by the KM survival curves is the median survival. In Fig. 1 it has been found by drawing a horizontal line starting from the 50 % of survival probability on the vertical axis and finding the intersection with a vertical segment of the curve, the value of the median survival is given by the intersection of a vertical line starting from this segment with the horizontal axis. Despite the value of the median survival, which is widely reported in literature, it is important to point out that it is not always a good indicator to compare different treatments administered to different patients cohorts. Furthermore the impact of median follow up time and the shape of the KM survival curve should always be considered together. A basic but good tool to achieve a summary is the possibility, provided by many statistical software packages, to add under the x axis of the graph the ‘number at risk’ at regular intervals (also shown in Fig. 1). The ‘number at risk’ is defined as the number of patients who are known not to have experienced some event at that time-point nor having been censored before the time-point: as a rule of thumb the curve can be particularly unreliable when the number of patients remaining at risk is less than 15 (Machin et al. 2006).

Another critical topic in the evaluation of KM survival curves is the evaluation of follow up maturity, and this is usually reported by indicating the median follow up time (MFUP). Different ways for calculating the MFUP have been proposed but the most widely used is the one using the ‘reverse Kaplan-Meier’ estimation of the OS (Schemper and Smith 1996): patients still alive (censored) are patients for whom the follow up time is known, while for patients that experienced death we cannot know if there would have been further follow up after the time of death. So the solution is to create a column in the statistical software that swaps the value of the overall survival endpoint, setting ‘1’ for censored patients (‘0’ in the column of the overall survival status) and ‘0’ for dead patients (‘1’ in the column of the overall survival status). The results should be similar to column 3 (OS status) and 4 (FUP status) of Table 2. Finally a new KM curve can be calculated by using this follow up status column as a series of events, and then the MFUP can be calculated in the same way as for calculating the median survival described before (Fig. 2, MFUP = 15 months). The value of MFUP should always be taken into account (and always mentioned in publications describing survival statistics) when reading statements about median OS. Figure 3 shows how the value of different MFUP compared with identical median OS can impact the reliability of the latter calculated by KM survival curves.

A good evaluation of the reliability of the median overall (and other kinds of) survival can be achieved by comparing it to the value of the MFUP. No exact criteria are known to give a judgment about this comparison; but large differences in the two values are suspicious for unreliable median survival value. Finally much care has to be used in considering the MFUP as the only parameter to evaluate the maturity of the follow up of a patient series because it has been proven misleading in some cases (Shuster 1991).

It is possible to refer the hazard to an instantaneous value, defining it as hazard rate. It shows the risk in a specific time interval related to a given observed population and it can be calculated using Eq. (3):
where d is the number of events observed in the given time interval t, while the denominator is the cumulated time-to event since the beginning of the interval. For example, if during our chosen time interval of 30 days among the 10 patients under observation one exhibits an event at day 5 and another one shows an event at day 15, we have λ _t  = 2/(20 + 240) = 2/260 = 0.00769 because 2, the numerator, is the number of events, 20 is the sum of 5 and 15 (cumulated time-to-event for patients who show the event) and we add the product of 8 patients who did not show the event by 30 days (interval duration), giving 240. This is the same as assigning 30 days as time-to-event for patients with no events. The method is perfectly consistent because the number of affected patients is well represented by the numerator of the fraction. In some clinical conditions it is possible to assume that the hazard rate doesn’t change during a long observation time, thus remaining constant. In this case the survival function can be computed as an exponential function in which survival is given in the form of Eq. (4):
and the only parameter involved in the function is the constant λ, since time t is the independent variable of Eq. (4). In this particular condition it could be very easy to fit the survival function to a single parameter, and use it to predict the outcome in a given patients population. Unfortunately, analyzing the majority of clinical situations, the hazard rate is modified during time, being λ itself a value subject to change over time. For example the rate of local recurrence after the surgery, or the mortality due to postoperative complications, can be higher in the first months after surgery than after some years of follow up. In these conditions it is sometimes feasible to assess the variation of the hazard rate as a function of time t. A model that can describe this behavior is the Weibull distribution described by Eq. (5):

In this equation the parameter κ can assume different values reflecting the trend in the hazard rate variation during time:

The appearance of three Weibull distribution curves according different values of κ is shown in Fig. 5. Figure 6 shows an example of non parametric distribution (Kaplan-Meier) with corresponding fitted exponential and Weibull distributions.

The value of the hazard rate can be a function of time and can be expressed as λ(t). This variability in most real clinical situations cannot be expressed correctly using parametric survival functions. In order to assess this trend it is possible to find the relation between hazard rate and time by using the specific plot shown in Fig. 7, where two survival curves are plotted against the complementary log transformation (CLT), that is against . This graph points out the correlation between survival rate and time, allowing to find if there is a constant rate rather than a variable rate of λ. When the CLT shows a good fit of the data points around a regression line (Fig. 7a2) it means that the hazard rate is roughly constant, and the survival function can be expressed in the exponential form shown in Eq. (4). In other cases (Fig. 7b2) there could not be a good fit of the data along the regression line function. In order to enhance these findings it is also possible to display the difference between the CLT points and the data points predicted by the regression function using the plot of residuals (Fig. 7a3, b3). This graph shows the distance between each point from the predicted one on the y axis, and gives a direct measure of the agreement of the survival predicted function with the observed data. In the ideal condition of a parametric fitting with exponential form the observed points should align on the horizontal line traced from the value of ‘0’ in the y axis as in Fig. 7a3, but in all other cases the results are much more similar to the plot shown in Fig. 7b3, where there is no alignment in the plot of residuals. In this case λ has to be expressed as function of time in the form λ(t). In Fig. 8 the plot of two survival curves is compared to the corresponding CLTs. The two series show a different survival rate and the CLT graph shows that the hazard rate is not constant, not having a linear-like appearance. But the two CLT lines show similar trends, seemingly ‘parallel’ because there is an approximately constant vertical distance between them at any given time. This typical appearance means that there is a proportion between the survival rates in the two populations. Starting from this kind of findings in the survival curves among populations differentiated by specific factors D. R. Cox published his fundamental paper where he described the proportional hazards regression model (PHRM, Cox 1972). In the main assumption of the model the hazard for a particular patient has to be related to the average underlying hazard of the whole examined population. We denote this average hazard as λ₀(t), and the hazard for the specific patient can be specified as:


where h(t) is a function of time t (Machin et al. 2006). In this condition it does not matter which is the function of distribution of the hazards allowing to model any survival data collection characterized by a time-dependent hazard rate. The only assumption to be satisfied in the PHRM is that the hazards in the groups must remain proportional to each other over time. In the PHRM usually the factors that can affect the outcome are named covariates. The covariates are variables that can be numerical and continuous (e.g. age, hemoglobin level, a serum-marker level etc.) or can be categorical (e.g. sex, tumor clinical stage, therapy protocol etc.). The relation that exists among survival and covariates can be calculated by using the following assumptions and equations:


where H ₀ (t) is the baseline hazard at time t and S ₀(t) is the baseline survival at time t. They are the value of the hazard function and survival function achieved just with all the covariates values set to ‘0’ at the time t. The statistical software packages usually provide either a table with the values of H ₀ (t) or the values of S ₀(t) after the computation of the PHRM. Note that the survival at mean of covariate (further output usually calculated) cannot be used in place of S ₀(t) that is the survival with all covariates values set to ‘0’, as previously stated. An example of application of PHRM is given in the following section, the dataset is shown in Appendix 1.

A population of 144 patients has been followed after the diagnosis of a kind of cancer for a maximum follow-up time of 70 months (MFUP = 29 months). The first problem to solve when defining a PHRM is the sample size. As a rule of thumb, the maximum number of covariates that can be analyzed in a binary outcome model (such as a PHRM) is given by the minimum value of the frequencies of the two response levels divided by 10 (Harrel 2001). So in this example we have 39 cases with censor status = 1 and 105 cases with censor status = 0. In this case being the minimum value of the frequencies = 39 we cannot analyze more than 4 (4×10 = 40) covariates in the model. So a set of 4 factors has been investigated: two factors (Factor 1 and Factor 2) are divided in two categories each; the last two factors are numerical and continuous: the age of the patients and a serum marker level. A series of Kaplan-Meier survival curves has been drawn: in Fig. 9 the appearance is shown of the subpopulations of patients categorized according to Factor 1 (categories ‘1’ and ‘0’), Factor 2 (categories ‘1’ and ‘0’), the two age classes (using a cut-off value of 37 years) and the two classes of serum marker level (generically defined ‘low’ and ‘high’ level). When analyzing continuous variables in order to detect relationships between those values and survival it can be useful to distinguish classes by dividing the population according to a specific cut-off. The cut-off value can be identified by the researcher looking at the data for ‘anomalous’ correlations or by using simple statistical methods, such as dividing the population in classes found by using the mean or, better, the median value of the continuous variable. This is an effective method because the median value of a continuous variable bisects the overall population into two subgroups with (nearly) the same number of cases (if the population is even the two subpopulations are equivalent, if it is odd the number of cases for each subpopulation is different for one case). Looking at the example for all the analyses the distinction between the curves is statistically significant according to the Logrank test (Fig. 9). But when one wants to analyze the impact of combined factors

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