The transitions from the non-ambulatory state to the ambulatory state generally occur after a relatively long period of time less than or equal to 43 weeks with standard error (14.5), while the transitions from the ambulatory state back to the non-ambulatory state generally happen after a relatively short period of time less than or equal to 25 weeks with standard error (1.1).

Even though the goodness-of-fit statistic for 8 weeks, ᵡ2 = 29.04 with degree of freedom equal to 12, provides evidence against this time-homogeneous model specification, only observations up to 50 weeks were used in computing the test due to the dearth of data in the remaining weeks. In actuality, the changes from state 2 to state 1 are seen more frequently during the 12-week exams than throughout the subsequent exams. T = -1.189 is the result of the equation (14) that was used to determine whether the rate of change from state 2 to state 1 varies with time. It's possible that this is both a sign of true time dependency and the result of some heterogeneity in the individual transition rates.

The calculated parameter values are shown in Table 3. The transition rates between the two groups of patients show significant differences: those who could walk alone before receiving therapy often spent almost twice as long in state 2 before transitioning to state 1 and in state 1 before transitioning to death. µ10 are 9.998 and 5.734 weeks, while µ21 are 16.989 and 10.012 weeks.

However, the overall equation (8) of goodness-of-fit statistic (ᵡ2 = 29.04, 12 d.f.), which was used to compute the test, provides evidence against this time homogenous specification of the model. This is because only observations up to 50 weeks were utilized because there were insufficient data for the remaining weeks to compute the test.

This example's main characteristic is that there are just a few distinct, unevenly spaced time intervals where data are available. This demonstrates how the method may be helpful if it is necessary to extrapolate structure-related data from such observations. We advise applying other statistical models, such as the Cox regression model, survival analysis, or mortality rate to estimate the new model.

Introduction

    Let Z(t), t>=0, be a stochastic process where M=(0,1,2,3,4,…, i), i>1, is the finite set of states on which Z(t) is defined. Let zero be an absorbing state and let 1,2,3,4,…., i be transient states. State zero could represent retirement in an employment example or death in a clinical example. Suppose that a random sample (L) of realizations z_ⱷ(t) representing (H) independent individuals is selected, (ⱷƐL). Ideally, to infer properties of Z(t), each z_ⱷ(t) should be observed continuously. However, this is likely to be impossible or too expensive. Commonly, z_ⱷ(t) is observed at discrete times t₁<t₂<t₃<……<t_n, not necessarily identical for a time all (ⱷ), so that the transition between observations is missed and the length of time spent in the states occupied at t₁, t_2, t_3,…., t_n are not known precisely. Thus various standard nonparametric and semi-parametric analyses of the data cannot be performed (Daoud et al. (2022), Aalen et al., 1980), to follow a fully parametric approach, it is necessary to specify the conditional probability of a state being occupied at time t_j, given the state occupied at t_j-1, j=2,3,4,….n. these are found by solving the forward Kolmogorov equations associated with the model chosen to represent the underlying processes. For simplicity time homogenous makove model are often selected at least as a first approximation (Emily et al.,2020), Alamu et al., 2022).

Two test stats are presented to examine the adequacy of this class of models. The first assesses the overall quality goodness of fit, while the second tests local departures toward time in- homogeneity. If the time intervals between adjacent observations are the same for all individuals and are constant, the adequacy of these models can be assessed by comparing the transition frequencies observed at each observation time. However, in many applications, follow-up occurs depending on availability, for example, at times of medical examinations or a survey interview. The tests proposed in this paper do not need fixed intervals of time between observations. The second test does not require the same periods for each individual. Hence the presented procedure is useful when the structure of basic continuous temporal processes is inferred from these data (NHS Choices 2011), Othman et al., 2011)

1-Methodology: 

Believe the realizations zⱷ(t), (ⱷϵL) and (t≥ 0) of the stochastic process Z(t). Denote by

                                                                                                                                  (1)

The provisional probability of state (s) being engaged at time(  )given that state (r) was occupied at time tj-1, for j= 2, 3,..., n.

For continuous time processes,

let

                                                                                                                                      (2)

denote the transition rate from state (R) to state s at time (t) (Lawless, J.F 2003), Bedford, T. and Cook, R.2009). A stochastic process defined on the state space (S) is fully specified when all p_rs(t) are defined. For a time-homogeneous Markov process the transition rates do not vary with time, i.e. p_rs(t)=µ_rs, say, and the conditional probabilities  are functions of x_j= t_j-t_j-1, say, and not of t_j-1 or t_j (Cox and Miller (1965). Denote such time-homogeneous conditional probabilities by (x_j). For the special case S = (0, 1, 2),

                                                                                                                                               (3)

                                                                                                                                              (4)

                                                                                                                                              (5)

Where r=1,2 and , for m , I=1,2; I m,

The parameters in the exponential function, , are the roots of the quadratic equation

                                                                                     (6a)

As q_r0 (x_j) tends to 1 as x_j→∞, the process is absorbed in state (0). The extension of results to more than two transition states is straightforward (Cox and Miller Heterogeneity among the zⱷ(t) may be taken into account by defining the transition

rates (1) as functions of some explanatory variables, for example, by writing (Cox et al.,. 1984)

exp( )                                                                                                                                                          (6b)

where  and y_ⱷ is the vector of explanatory variables. If part of this individual heterogeneity is not recognised, and therefore omitted in the specification of equation (6b), the transition rates will appear to be negatively correlated with time even if the process is truly time homogeneous (Tony Lancaster and Stephen Nickell, 1980; Clayton and Cuzick, 1985).

2-Markove Chain Model for tests for Departures from Time Homogeneous

2.1 Goodness of Fit

Let(  )be the number of the realizations zⱷ(t), , in state (r) at time t, and let Q_rs(x_j) be the number of realisations which have had in state r at time t_j and in state s at t_j+1 = t_j+x_j. For(  ) and (x_j) to be observed all realisations zⱷ(t) must be observed at t_j and t_j+1. Assuming that a time homogeneous Markov process is suitable, the conditional predictable value of  is

Denote by (xj) the maximum likelihood estimate of (x_j) which is obtained, as in equations (3,4 and 5), by considering all spaced by( xj) Then (xj) is efficiently estimated by

                                                                                                                                                              (7)

A goodness-of-fit statistic for the time-homogeneous Markov specification of the process has been defined as the sum of the chi-squared goodness-of-fit statistics defined on each of the n-1 disjoint intervals (t1, t2), (t2, t3),..., (tn-1 , ),

                                                                                                                       (8)

Expression (8) would had have to obtain if all possible transitions among the states in (S) were treated as outcomes of a multinomial distribution. The limiting chi-square distribution of( ᵡ2 )is degrees of freedom equal to the number of independent cells in the multinomial distribution, namely K²(n-1), minus the number of parameters in (xj). While, the transition rates were the same for all the realisations zⱷ(t), as in equations (3,4 and 5), the parameters were K² and the number of degrees of freedom is therefore K²(n-2)

2.2. Inhomogeneity of time

Let µ = { }, r, s  S, is the vector of transition rates specified a time-homogeneous Markov model. Departures from this model in the direction of time-inhomogeneity will imply that at least one of the elements in( µ )varies with time, where time has measured from the origin of the process. For simplicity, consider the case of just one transition rate being weakly time dependent. Let (t) be such a rate and  its value at the origin of the process t = 0. Assuming that it was linear sable over the time interval of interest we can write for small( ),

(t)= +                                                                                                                                                                               (9)

A local test for time-inhomogeneity of the transition rate from state 1 to state 2 is then equivalent to a test for  = 0 in equation (9).

As before, the conditional probabilities associated with this new specification of the Markov model can be found by solving the forward Kolmogorov equations. An approximate solution is, for r, s  M (see Appendix A for the explicit solutions when i=2),

                                                                                                                       (10)

where  is the first derivative of  with respect to & evaluated at  = 0. The contribution to the likelihood function of the n observations on y (t) is

                                                                                                (11)

where  are the states occupied by  at the times . Denoting by , the contribution associated a time-homogeneous specification of the model, we find, under regularity conditions (Crowder, M. 2012), Crowder.and Sweeting,  1991),

                                                                                                                                                 (12)

The score function associated with the right-hand side of equation (12) is

                                                                                                                                           (13)

Where ⱷ is included in the notation to identify the contribution of each realization zⱷ(t). This is valid only under the assumption that equation (9) is a proper approximation for P12(t) and e is close to 0. A local test statistic for =0 in equation (9) can be defined as

                                                                                                                                  (14)

where ( ) is the element of the inverse information matrix corresponding to  which is computed at the maximum likelihood values (10) when =0. Thus (  is the variance of the score function statistic (13) when =0. Assuming asymptotic normality of equation (13), the test statistic has asymptotic standard normal distribution. The sign in equation (14) indicates whether  decreases or increases with time (Barlow, R. and Proschan, F 1975).

3. Kidney Disease Data

Table 1 in appendix B, provides information on 40 individuals with kidney illness who received from Rizgari at the Erbil Hospital. Their capacity or inability to walk unassisted was noted before treatment started as well as at 3 months, 6 months, one year, and two years. Assume that states 0, 1, and 2 correspond to being dead, incapable of walking, and ambulant, respectively. Consequently, the information can be seen as discrete time observations of realizations produced by a time continuous process specified on three states. Also utilized for data analysis are the Statistics Package Social Sciences (spss) and easy fit. We fitted a time homogeneous Markov model with the identical transition rates (µrs) for all patients as a first approximation to this process.

Table 2 shows the maximum likelihood estimates of these parameters. As µ20 has have small compared with its standard error-as well as in absolute value-the transitions from the ambulatory state to death were be likely to occur indirectly, via unobserved state in the non-ambulatory state. A model with µ20 constrained to be equal to 0 is fitted to test this hypothesis, showing that there is no significant gain by estimating µrs ; the likelihood ratio test statistic is equal to 0.027. Noting that the reciprocal of σrs =1/ µrs , represents the expected time spent in state r before a transition to state s occurs, the estimate was values of the other parameters show the following. On average, transitions out of the non-ambulatory state towards the ambulatory state occur after a fairly long spell ( = 43 weeks; standard error, 14.5), while transitions out of the ambulatory state back to the non-ambulatory state occur after a fairly short spent in the non-ambulatory state before death ( =25 weeks; standard error, 1.1).

Table2 :Using Markov Model by (MLE) of the time homogeneous: full and restricted specification

Standard errors are given in parentheses: the standard error of times the standard error of

The overall equation of goodness-of-fit statistic 8, however, shows evidence against this time homogeneous specification of the model, X2  = 29.04, 12 d.f.; only the observations up to 50 weeks were used in computing the test because of sparseness of data in the following weeks. Indeed, the transitions from state 2 to state 1 are observed more frequently at the 12 weeks examinations than at the following examinations. The equation (14) computed to test whether the transition rate from state 2 to state 1 changes as time increases takes a negative, although not significant, value; T = -1.189. This could be an indication of truly time dependence as well as a consequence of some heterogeneity in the individual transition rates.

To explore whether this is feasible we use information about the pretreatment status of each patient, treating it as a proxy for individual frailty (James et al., 1979). Let (z) take value 1 if the patient was in a non-ambulatory status, value 0 otherwise, and

 Table3: MLE of the time homogeneous Markov model with transition rates depending on y

Table 4:Observed and fitted frequencies of transition between states: time homogenous Markov specification.

Specify a time homogeneous Markov model with transition rates depending on y as in equation (6). Table 3 reports the values of the estimated parameters. Significant differences in the transition rates of the two groups of patients surface: those who were able to walk unaided before treatment began have, on average, state spells in state 2 before a transition to state 1 and in state 1 before a transition to death, almost twice as long as the others  is 16.989 weeks and 10.012 weeks and 10 is 9.998 weeks and 5.734 weeks. The overall goodness of fit of this specification is satisfactory (Table 4). The test statistic 8, soft a value of 36.32, however, should not be formally compared with a chi-squared statistic distribution with 24 degrees of freedom since the data in the table are very sparse.

A key feature of this example is that data are available only at a few separate unequally spaced time points. It illustrates how the procedure may be useful whenever it is required to recover information about the structure of a continuous time process from such observations.

Appendix A:

The expressions in equation 10 are derived as follows, for M =(0,1,2), Suppose that at time t state r is occupied, Thus the Kolmogorov forward equations is for  (Crowder  2012), Lemke. 2016).

  (t, )= -              a1

  (t, )=               a2

Combining equations 9 and (a1 and a2), we find

(t, )  -          a3

(t, )           a4

A first – order Taylor expansion of   around  yield

                      a5

r,s=1,2, where x=  and (t, ) is the first derivative of  with respect to  evaluated at  . Combining equations (a5) and (a3, a4) and equating the coefficients of  we find

        a6

           a7

This yields solutions

                                a8

                                 a9

Where  are the roots of equation 6a and , for I, m =1,2, I

                   a10

               a11

          a12

             a13

Combining equations (a8, a9 and a3) we have

                              a14

                              a15

Where  and m =1,2, were defined in equation (3,4 and 5). Since

                                           a16

The limiting event of the process is, once again, absorbing in state 0. Better approximation to the solution of equation a1 and a2 can be obtained by expanding equation a5 to higher orders and equating, in sequence, the coefficient of ɛ, ɛ2, ɛ3,….

Appendix B:

Table 1: Kidney Disease data from Rizgari hospital Erbil Between 2020-22

State 0 ≡ death; state 1≡ non-ambulant status; state 2 ≡ ambulant status; Un.k ≡ alive but unknown status.

Conclusion and Recommendation:

Typically, the Markov model is used to examine the homogeneity of time series data. Due to this, we used two statistical tests to determine whether this model was adequate. The first test evaluates the model's quality of fit, whereas the second evaluates the homogeneity of the model using maximum likelihood. Utilizing weekly patient data on patients with kidney disorders gathered from Rizgari Hospital for the years 2020 to 2022, by using social science software packages. The time continuous process must be specified in terms of three states while representing the study's data as discrete time observations. As previously mentioned, states 0 and 1 denote death and non-ambulant condition; state 2 respectively. We fitted a time homogeneous Markov model with the identical transition rates µrs for all patients as a first approximation to this process.

The transitions from the non-ambulatory state to the ambulatory state generally occur after a relatively long period of time less than or equal to 43 weeks with standard error (14.5), while the transitions from the ambulatory state back to the non-ambulatory state generally happen after a relatively short period of time less than or equal to 25 weeks with standard error (1.1).

Even though the goodness-of-fit statistic for 8 weeks, ᵡ2 = 29.04 with degree of freedom equal to 12, provides evidence against this time-homogeneous model specification, only observations up to 50 weeks were used in computing the test due to the dearth of data in the remaining weeks. In actuality, the changes from state 2 to state 1 are seen more frequently during the 12-week exams than throughout the subsequent exams. T = -1.189 is the result of the equation (14) that was used to determine whether the rate of change from state 2 to state 1 varies with time. It's possible that this is both a sign of true time dependency and the result of some heterogeneity in the individual transition rates.

The calculated parameter values are shown in Table 3. The transition rates between the two groups of patients show significant differences: those who could walk alone before receiving therapy often spent almost twice as long in state 2 before transitioning to state 1 and in state 1 before transitioning to death. µ10 are 9.998 and 5.734 weeks, while µ21 are 16.989 and 10.012 weeks.

However, the overall equation (8) of goodness-of-fit statistic (ᵡ2 = 29.04, 12 d.f.), which was used to compute the test, provides evidence against this time homogenous specification of the model. This is because only observations up to 50 weeks were utilized because there were insufficient data for the remaining weeks to compute the test.

This example's main characteristic is that there are just a few distinct, unevenly spaced time intervals where data are available. This demonstrates how the method may be helpful if it is necessary to extrapolate structure-related data from such observations. We advise applying other statistical models, such as the Cox regression model, survival analysis, or mortality rate to estimate the new model.