A counting process is a stochastic process {N(t), t≥0} with values that are positive, integer, and increasing. If s < t, then N(t) − N(s) is the number of events occurred during the interval [s, t]. The counting processes for the ith subject was represented for N_i(t) and Y_i(t) with t > 0 and where, N_i(t) = I{X_i≤t; δ = 1} and Y_i(t) = I{X_i≥t}.

N(t) is referred as the number of event observed up to and including time t and Y(t) is referred to as at risk process, indicating whether the subject is at risk at time t. Examples of counting processes include Poisson processes and Renewal processes. A counting process is increasing and hence, it is associated with a sub martingale. So, it can be written as: N(t) = M(t) + A(t), with a martingale M(t) and a predictable increasing process A(t). M(t) is called the martingale associated with the counting process N(t) and the predictable process A(t) is called the cumulative intensity of the counting process N(t).

Wang & Chang [4] developed a nonparametric estimator of the survival function for recurrent events. These authors propose an estimator for marginal survival function in case where there is correlation between the times of occurrences. The estimator can be defined using two processes, d* and R^∗. So,

(1)

where, S(t) is the survival function, i is the index for an individual or subject, j represents the index for an event, T_ijis the time from the j − 1th to the jth event for subject i, τ_i is the time between the initial event and the end of follow-up for subject i, C_i the censoring time. d^∗(t) is the summation of the weighted average of the total number of observed uncensored recurrent times for a subject that are equal to t. d^∗(t) represents the sum of the proportion of individuals of times that are equal at inter occurrence when at least an event. d^∗(t) is evaluated at time t as,

(2)

K_i*(t) is the number of uncensored recurrent events for unit i, and K_iis the number of recurrent events for subject i. The function {.} is one if the condition on breakers is true and zero on other case. So, let

(3)

R_i^∗ is the summation of the weighted average the total number of observed uncensored recurrent times for a subject that are greater than or equal to t. R_i is the total mass of the risk set at time t and it is calculated as,

(4)

Peña et al. [5] developed a non-parametric estimator of the survival function to estimate the survival function of recurrent events. The nonparametric estimator of the inter-event time survivor functions under the assumption of a renewal. They consider a structure model which gap-times independent and identical distribution (i.i.d.). The author used two counting processes N and Y. they considered two time scales: one related to calendar time (s) and other related to gap time (t). So,

(5)

where, S (t, s) is the estimator of survival analysis, N(s,w) is one counting process and represents the number of observed events in the calendar period [0,s] with t≤w. It is calculated as,

(6)

And,

(7)

The function {.} is one if the condition on breakers is true and zero in other case. Y(s, w) is other counting processes and represents the number of observed events in the period [0; s] with t≥w. So,

(8)

This function permits estimates the counting processes of WC estimator and permits to plot the occurrences of events for a number wished of study units.

The syntaxes of this function is,

The arguments are details as follow, yy <- dataset, dat <- name of dataset, xy <- Initial number of the unit of the dataset that is wish include on the study. For defect is “1”, xf<- Final number of the unit of the dataset that is wish include on the study. For defect is “1”, colevent<- colour wished for represent event line on the plot. For defect is “blue”, colcensor<-colour wished for represent censor line on the plot. For defect is “red”, ltyx<- Refer to the type of line that user wish represent the event. For defect is “1”. lwyx<- Refer to the type of line that user wish represent the censor. For defect is “1”, S<- Calendar time for the study. For defect is “1”and T<- Gap time for the study. For defect is “1”.

This function permits estimates the counting processes of PSH estimator and permits to plot the occurrences of events for a number wished of study units. The syntaxes of this function is,

The arguments are details as follow, yy <- dataset, dat <- name of dataset, xy <- Initial number of the unit of the dataset that is wish include on the study. For defect is “1”, xf<- Final number of the unit of the dataset that is wish include on the study. For defect is “1”, colevent<- Colour wished for represent the event line on the plot. For defect is “blue”, colcensor<- Colour wished for represent the censor line on the plot. For defect is “red”, ltyx<- Refer to the type of line that user wish represent the event. For defect is “1”, lwyx<- Refer to the type of line that user wish represent the censor. For defect is “1”, S<- Calendar time for the study. For defect is “1”and T<- Gap time for the study. For defect is “1”.

Example Nº 1. For this example, we will use TBCplapy dataset of the TestSurvRec package [8]. On the procedure, we estimate the counting processes of WC estimator for the patient with id 72. Figure 1 shows occurrence times of tumours (weeks) on the unit number 72. The figure shows the calendar times of apparitions of the tumours on the patient with id 72: S_i=72,j=1= 8, S_{i=72, j=2}= 15, S_{i=72; j=3}= 18, S_{i=72; j=4}= 20, S_{i=2, j=5}= 22, S_{i=72; j=6}= 25, S_{i=72; j=7}= 38, S_{i=72; j=8}= 40 and S_i=72= 48. The last observation is the time of study of the patient. Its deduced that, t_{i=72; j=1}= 8, t_{i=72; j=2}= 7, t_{i=72; j=3}= 3, t_{i=72; j=4}= 2, t_{i=72; j=5}= 2, t_{i=72; j=6}= 3, t_{i=72; j=7}= 13, t_{i=72; j=8}= 2 and C_i=72= 8. We shows as calculates the counting processes for the WC estimator,

Estimation of d*_i=72 (t = 3). Calculation of d*_i=72(t =3) with S = 42,

Numbers of apparitions of the tumour on the unit with id = 72. K^∗_i=72= 8, (see Figure 1).

As, {.} is one if the condition is true and zero in other case, as K_i≥0 is true, then {K_i=72^∗ > 0} = 1, so

Figure 1
Graphics representation of the occurrence times of tumours on unit number 72.

See that, there is only one time of occurrence with T = 3 on the unit with id = 72, then,

d*_i=72(T = 3) = 1/8 * 2

d*_i=72 (T = 3) = 0,25

For the estimation of R*_i=72 (t = 3) is used the equation (4). So, for i =72,

Then,

R*_I=72(t = 3) = 1/8*[5+ 0x0],

R*_i=72(t = 3) = 0,625

It is certificates with the R code,

The output is,

Example Nº 2. On this example, we were used TBCplapy dataset of TestSurvRec package. On the procedure, were estimates the counting processes and survival function of the estimator WC for all patients of the dataset, and was used the Plot.Counting.Processes.WC function created on R language.

To continuation, we show an abstract of the output of the function “Plot.Counting.Processes.WC”. On the abstract, we present the results of the outputs of the first and the last calculations of the counting processes for the units of study, as it is appreciate below,

The Table 1 presents the estimations of the survival curves S (t) using WC model. For these estimation are used the equations (1), (2), (3) and (4).

Table 1
Estimation of S(t), WC Model.

Example Nº 3. For this example, also we use TBCplapy dataset of TestSurvRec package. On the procedure, we explicate as to estimate the counting processes of the estimator PHS for the patient with id equal to 72. To continuation, we shows the calculates, For the calculate of Ni(s; t) is used the equation (6), i = 72, s = 64 and t = 3,

On the figure Nº 1 is shows that, K_i=72 (s = 64) = 8

and

For, t=2 and s=64,

So, N_i=72 (s = 64; t = 2) = 3

See that, for, t=2 and s=64,

So, it is obtained the same result that before method and,

For the calculate of Y_i(s, t) is used the equation (8). So with i = 72, t = 3 and s = 64,

Of the figure 1, we have that K_i=72 (s = 64) = 9 and

so,

Y_i=72 (s = 64; t = 3) = 6

It is certificates with the R code,

Output of the R code,

Example Nº 4. For this example, also were use the TBCplapy dataset of the TestSurvRec package. On the procedure, were estimated the counting processes of the estimator PSH for all patients and was used the function created on R language. Code on R-language for the estimation of PSH estimator.

To continuation is shows the output of the function “Plot.Counting.Processes.PSH”. On the abstract, its present the results of the outputs of the first and the last calculations of the counting processes, its show as follow,

The Table 2 present the estimations of S(s,t) using PSH model. For these calculates are used the equations (5), (6), (7) and (8).

Table 2
Estimation of S(s,t), PSH Model.