{"id":996,"date":"2020-06-02T12:36:02","date_gmt":"2020-06-02T16:36:02","guid":{"rendered":"http:\/\/pressbooks.library.upei.ca\/montelpare\/?post_type=chapter&p=996"},"modified":"2020-08-24T14:19:48","modified_gmt":"2020-08-24T18:19:48","slug":"survival-analysis","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.upei.ca\/montelpare\/chapter\/survival-analysis\/","title":{"raw":"Survival Analysis","rendered":"Survival Analysis"},"content":{"raw":"

Essential Background in Survival Analysis<\/span><\/h2>\r\n

Survival analysis can be considered in its simplest form as a method to analyze longitudinal data for a cohort, or for a comparison of cohorts with a specific interest in the proportion of individuals that reached or exceeded a definite point on a time scale. <\/span><\/p>\r\nIn survival analysis, the demarcation point for the event of interest on a time scale is referred to in a variety of ways but is dependent upon the perspective of the researcher.\u00a0 For example, if the researcher is interested in the application of survival analysis to estimate mortality as a result of a given treatment regimen then the demarcation point may be used to count the number of individuals that died within the interval up to a specific time, versus the number of individuals that lived beyond the selected time (i.e. survived).\u00a0 However, given the intention of the research, the mathematics of survival analysis need not be limited to only counting deaths (or survival), rather, the approaches of survival analyses may be thought of as a set of mathematical functions that enable statistical techniques which can be applied to the evaluation of any selected event at a specific period of time. Hence, there are several methods that can be used to perform survival analysis, however, in this chapter, the focus will be on the application of SAS for survival analysis using life tables, the calculation of the log-rank test, and the application of the Cox Proportional Hazard Model.\r\n

Important Functions Used in Survival Analysis<\/h2>\r\nThe progression of information about functions used in the computation of survival analyses is presented in Figure 19.1. In the following section, we will review the important concepts of the probability density function for a random discrete variable and a random continuous variable, the cumulative distribution function, the survival function, and the hazard function.\r\n

The flow of function processing in survival analysis<\/strong><\/h3>\r\n\"\"\r\n\r\nThere are several ways to demonstrate survival analysis, but we will begin here by reviewing the basic terminology and the elements of the different functions used in the calculation of survival analysis so that we can measure the risk of an event happening at a specific period of time.\r\n\r\nThe probability density function represents a value that describes the probability of an outcome or a combination of outcomes occurring within a known outcome space \u2013 such as an interval.\r\n\r\nThe probability density function (pdf) can refer to either the associated probability value from a discrete random variable or from a continuous random variable.\u00a0 When the pdf refers to a discrete random variable then it is also referred to as the probability mass function (pmf) for a positive discrete random variable. In this case, we define a positive discrete random variable as a variable that holds numbers from the whole number line, meaning that the scores are whole numbers (ranging from 0 to + \u221e) and may resemble (0,1,2,3, \u2026, \u221e) without decimal values.\r\n
Probability Density Function (pdf) Related to Tossing a Single die<\/h6>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Possible outcome expressed as [latex]P(X = x)[\/latex]<\/td>\r\nThe probability associated with the outcome<\/td>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]P(X = 1)[\/latex]<\/td>\r\n1\/6<\/td>\r\n<\/tr>\r\n
[latex]P(X = 2)[\/latex]<\/td>\r\n1\/6<\/td>\r\n<\/tr>\r\n
[latex]P(X = 3)[\/latex]<\/td>\r\n1\/6<\/td>\r\n<\/tr>\r\n
[latex]P(X = 4)[\/latex]<\/td>\r\n1\/6<\/td>\r\n<\/tr>\r\n
[latex]P(X = 5)[\/latex]<\/td>\r\n1\/6<\/td>\r\n<\/tr>\r\n
[latex]P(X = 6)[\/latex]<\/td>\r\n1\/6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nA graph of the frequency distribution for these data would produce a platykurtic (flat) distribution profile since each outcome value has a frequency of 1.\r\n\r\nHowever, we could create a graph to demonstrate the cumulative outcomes for the probabilities of the random discrete variable (X) ranging from 1 to 6; which would be to consider the discrete outcome ranging as follows: \u03a1(X=1) \u2264 \u03a1(X=6).\r\n\r\nThe Cumulative Distribution Function commonly referred to as the c.d.f. and written as F(x)=P(X\u2264x)\u00a0 represents the set of values associated with the probabilities of the random variable (X) occurring equal to or less than a given value (x) in an outcome space.\r\n\r\nIn the example of the toss of a fair six-sided die, the outcome space is based only on the discrete numbers 1 through 6, as shown in the following outcome chart.\r\n
Cumulative Distribution Function (c.d.f) Related to Tossing a Single die<\/h6>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Possible outcome expressed as [latex]P(X \\le x)[\/latex]<\/td>\r\nProbability associated with the outcome<\/td>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]P(X \\le 1)[\/latex]<\/td>\r\n1\/6 = 0.17<\/td>\r\n<\/tr>\r\n
[latex]P(X \\le 2)[\/latex]<\/td>\r\n2\/6 = 0.33<\/td>\r\n<\/tr>\r\n
[latex]P(X \\le 3)[\/latex]<\/td>\r\n3\/6 = 0.50<\/td>\r\n<\/tr>\r\n
[latex]P(X \\le 4)[\/latex]<\/td>\r\n4\/6 = 0.67<\/td>\r\n<\/tr>\r\n
[latex]P(X \\le 5)[\/latex]<\/td>\r\n5\/6 = 0.83<\/td>\r\n<\/tr>\r\n
[latex]P(X \\le 6)[\/latex]<\/td>\r\n6\/6 = 1.00<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhile the example presented here describes the c.d.f. for discrete random variable outcomes (and their associated probabilities based on the probability mass function (pmf) or probability density function (pdf)), the c.d.f. is also relevant for continuous variable values and the pdf is based on the outcomes (X<\/em>) in an interval (a<\/em>, b<\/em>) represented by P<\/em>(a<\/em>\u00a0<\u00a0X<\/em>\u00a0<\u00a0b<\/em>), where all numbers from the real number line are eligible within the interval of the distribution, typically ranging from 0 to 1.\r\n\r\nIf the data for the c.d.f. were attributed to a continuous random variable such as time, then the graph of the set of probabilities for all possible outcomes of the c.d.f. is presented as a positive S-shaped<\/em> curve ranging from 0 to 1, as shown in the figure below.\r\n
\"\"<\/h6>\r\n
Schematic of a c.d.f. for a Continuous Random Variable<\/h6>\r\n
\r\n

The SAS code to generate this image was written by Wicklin (2011)[1]<\/a> and was processed unedited in SAS Studio shown here.<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\ndata cdf;\r\n\r\ndo x = -3 to 3 by 0.1;\r\n\r\ny = cdf(\"Normal\", x);\r\n\r\noutput; end;\r\n\r\nx0 = 0;\r\n\r\ncdf0 = cdf(\"Normal\", x0);\r\n\r\noutput;\r\n\r\nx0 = 1.645; cdf0 = cdf(\"Normal\", x0); output;\r\n\r\nrun;\r\n\r\nods graphics \/ height=500;\r\n\r\nproc sgplot data=cdf noautolegend;\r\n\r\ntitle \"Normal Cumulative Probability\";\r\n\r\nseries x=x y=y;\r\n\r\nscatter x=x0 y=cdf0;\r\n\r\nvector x=x0 y=cdf0 \/xorigin=x0 yorigin=0 noarrowheads lineattrs=(color=gray);\r\n\r\nvector x=x0 y=cdf0 \/xorigin=-3 yorigin=cdf0 noarrowheads lineattrs=(color=gray);\r\n\r\nxaxis grid label=\"x\";\r\n\r\nyaxis grid label=\"Normal CDF\" values=(0 to 1 by 0.05);\r\n\r\nrefline 0 1\/ axis=y;\r\n\r\nrun;\r\n\r\n<\/div>\r\n<\/div>\r\nThe c.d.f. is an important step in the computation of the survival analysis because it is part of the computation of the survival function. In a time relevant model as is typical in a biostatistics application, the cumulative distribution function can be represented as [latex]F(t)=P(T \\le{t}) \\textit{where t}[\/latex]\u00a0 is the value of the random variable representing a measured time and [latex]{t}[\/latex] is the value of the intended time at the event.\r\n\r\nThe survival function [latex]S{(t)}[\/latex] provides the estimate of the duration of time to an event, be it a failure, death, or a specified incident. The survival function begins at 1, the point where an individual enters the dataset and ends at 0 the point where data monitoring stops, usually because the event of interest has occurred.\r\n\r\nIn simple terms, the Survival Function is the complement of the c.d.f. and is computed as [latex]S{(t)}= 1- F(t)\\textit{, where t >0}[\/latex]. More important, the survival function is the denominator in the computation of the Hazard Function, which is a main element in one approach to the computation of the survival analysis. The survival function can show the probability of surviving up to a designated event, based on units of time.\r\n
\r\n

For example, consider the following data set in which a measure of time to an event is recorded.<\/p>\r\n\r\n<\/header>\r\n

The cutoff time is set at 48\u00a0 (totally arbitrary units)<\/em> <\/strong>so that any value above 48 is assigned a censor score of 1 and any value less than 48 is a value of 0.<\/div>\r\n<\/div>\r\nTable depicting number of individuals that exceeded the time to event\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Patient ID<\/td>\r\nTime to Event: The measure of the length of time to the event happening<\/td>\r\nEvent Counter variable (0=event has not happened, 1=event has happened)<\/td>\r\n<\/tr>\r\n<\/thead>\r\n
01<\/td>\r\n40<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
02<\/td>\r\n38<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
03<\/td>\r\n54<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
04<\/td>\r\n56<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
05<\/td>\r\n28<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
06<\/td>\r\n36<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
07<\/td>\r\n42<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
08<\/td>\r\n51<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
09<\/td>\r\n45<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
10<\/td>\r\n49<\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n

The data are processed with the following SAS code[2]<\/a> to produce a graph of the survival curve shown below.<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\ntitle 'program to show a survival curve';\r\n\r\ndata survcurv;\r\n\r\ninput id t_event censor;\r\n\r\ndatalines;\r\n\r\n01 40 0\r\n\r\n02 38 0\r\n\r\n03 54 1\r\n\r\n04 56 1\r\n\r\n05 28 0\r\n\r\n06 36 0\r\n\r\n07 42 0\r\n\r\n08 51 1\r\n\r\n09 45 0\r\n\r\n10 49 1\r\n\r\n;\r\n\r\nproc lifetest data=survcurv(where=(censor=1)) method=lt\r\n\r\nintervals=(45 to 60 by 1) plots=survival; time t_event*censor(0); run;\r\n\r\n<\/div>\r\n<\/div>\r\nThe results of this analysis include the table of the survival estimates and the survival curve below \u2013 note that the failure point was set at 48. The curve shows the probability of surviving to 48 and then beyond 48.\r\n\r\nNotice that the entire group begins at probability = 1 and ends at probability = 0.\r\n\r\nFigure of SAS representation of the survival function for n=10 with censoring at x=48.\r\n\r\n\"\"\r\n\r\nThe Hazard Function is determined by the ratio of the probability density function (pdf) to the survival function [latex]S{(t)}[\/latex] and can be written as:[latex]\\lambda = {(p.d.f.) \\over{S(t)}}[\/latex]\r\n\r\nThe following explanation may help to describe the elements of the hazard function<\/strong> in greater detail. In this annotated formula the hazard function is shown to represent the likelihood of an event such as death or survival occurring within an interval at time [latex]{(t)}[\/latex].\r\n

[latex]\\lambda(t) = {\\lim\\limits_{\\Delta{t}\\to {0}}} {P(t \\le{T} \\lt{t} + \\Delta{t} \\mid{T} \\ge{t})\\over{\\Delta(t)}}[\/latex]<\/h3>\r\n\"\"\r\n

Annotated image of the Hazard Function Equation<\/h6>\r\n