U fiV ^1 + 2n(t; feT ) ß2/ß J i + 2n(t; êT ) fi2/ß fi- (1 + n(t;feT) fi2/ß) 1

where

(<Pi(t) - 2 - k)\k! V 2ß \Jl + 2n(t; P) fi2/ß (*i(t) - 1 + k)\ ( fi k=0

(<Pi(t) - 1 - k)\k!\ 2ß y/i + 2n(t; feT) fi2/ß

The further restriction of information from observation to the knowledge of only failure times makes the problem more complicated. First it is necessary to find the distribution of the first passage time Zh of the degradation process:

Zh = inf j t : YI (Tn < t) ■Xn > h} = inf {t : Z (t) > h - X0|

k k where Xo is a (possible random) state at time To := 0. The explicit calculation is possible only for some special cases. Further, this distribution contains all parameters we considered and it is nearly impossible to find explicit estimates except for very simple assumptions. Nevertheless, the problem can be solved numerically.

Let us consider again the case of observable counting process and unobservable frailty variable Y. Let 0k(W) and Zk(W) be the empirical ordinary and central moments, respectively, of a random variable W:

According to (2) we can express the k-th ordinary and central moments of &(t) as linear combinations of moments of Y multiplied by powers of the deterministic function n(t). Actually, let S(k, u) denote the Stirling numbers of second kind where S(k, u) can be recursively determined

5 (k,u) = S(k-l,u-l)+u-S (k-l,u), 1 < u < k, S(0,0) := 1, S (0,u) = 0. We make use of k nk = Y S(k, u) n (n - l) • ... • (n - u +1)

and we consider the factorial moments of a Poisson distributed random variable with mean yn(t). Some elementary calculations yield

In particular, we find

p2(<P(t)) = Eg[Y] • n(t; eT) + d(Y) • n(t; eT)2

p3(m) = Eg[Y] • n(t; eT) + 32(Y) • n(t; oT)2 + ^(Y) • n(t; oT)3 .

where pk ( •) denotes the k-th central moment of a random variable. Let us further assume that the deterministic function n(t) is known and that we are interested only in estimating the parameters of distribution of Y. The moments at the left hand site are replaced by its empirical moments. Further, the moments of Y can be expressed in dependence of the moments of @(t) and the function n:

^(Y ) = n(t; eT )-3{ £ ($(t)) - 3 &($(t)) + 2Ee [$(t)]} ■ (17)

Now it is possible to find moment estimates for all parameters of the distribution of Y. Let us consider again the three previous examples:

1. If Y is rectangular distributed in [a,b] we get from (15) and (16) and taking into account 0 < a < b a = -V3D , b = + V3D

with

In difference to the maximum likelihood method an unique admissible estimator exists if a > 0 and D2 > 0. The assumption T>2 > 0 is fulfilled for sufficient large values of m because D2 is a consistent estimate of the variance n2(Y) of Y. 2. If Y — yo is gamma distributed than it has the first three moments

From (15), (16) and (17) we get the unique moment estimators ft ^,2(m) — sim))

with j2

3. For an inverse Gaussian distributed Y with Ee[Y] = ^ and ) = rf3/b the equations (15) and (16) gives the unique estimators gi(*(t)) and ß- gi(*(t))3

For this distribution we get the same p from the moment method as from the maximum likelihood method in section 1.

The advantages of moment estimators in comparison to maximum likelihood estimators are its simple form and the fact that they can be found explicitely. But it is well known that in general maximum likelihood estimates have better properties. In the next section we compare the two methods concerning to its bias and variance.

5 Comparison of Maximum Likelihood and Moment Estimates

Let Y — y0 Gamma distributed and let n be Weibull, it is n(t; a) = ta+1 with a > —1. We have considered sample sizes of m = 50, m = 100, m = 250 and m = 500. For each realization a path of the degradation process was simulated with true parameter 0Y = (c, y0, b) = (2.4, .5,1.2). The observation is assumed to continue up to time t = 10. We have considered three different values of the parameter a in the deterministic part n(t; a). For a = —.3 the derivative of n(t; a) is a decreasing function. The expected number of jumps up to time t = 10 is 5.01. If a = 0 then we get a linear cumulative intensity or a constant hazard and expected number of jumps up to time t = 10 is 10. Last, for a = .3 the derivative of n(t; a) is increasing and the expected number of jumps up to time t = 10 is 19.95. Such a simulation was repeated 750 times and from these 750 parameter estimates the mean and the variance of the estimator where calculated. The results are shown in table 1.

If the parameter a in the deterministic part is unknown, too, then it can be estimated by

a is a maximum likelihood estimator which does not contain other parameters [WeK04]. In table 2 the results of the same simulation are shown with the difference that now a is unknown and has to be estimated. >From the simulation we get the following results:

1. Influence of a:

In both cases we can see that for a = 0.3 the variances of the estimator y0 (in both cases, MLE and ME) are smaller than for a = 0 or a = —0.3. The variances of the moment estimators for b and c are also smaller for a = 0.3 than for a = 0 or a = —0.3, while a does not influence the variances of the maximum likelihood estimators.

Table 1. Empirical moments of maximum likelihood (MLE) and moment (ME) estimators (9% = (2.4, 0.5, 1.2))

Mean Variance Mean Variance Mean Variance Mean Variance a = -0.3

MLE c 2.614 1.798 2.435 0.762 2.386 0.302 2.401 0.140 yo 0.599 0.018 0.561 0.016 0.520 0.008 0.508 0.004 b 0.861 0.184 0.975 0.190 1.119 0.110 1.171 0.062

ME c 2.910 3.902 2.824 2.208 2.826 1.475 2.758 0.893 yo 0.480 0.047 0.458 0.041 0.436 0.031 0.444 0.023 b 1.680 2.552 1.708 2.026 1.754 1.507 1.650 1.052

MLE c 2.197 0.694 2.205 0.435 2.351 0.262 2.339 0.149

ME c 3.189 2.533 2.846 1.325 2.771 0.876 2.613 0.427 yo 0.431 0.037 0.457 0.027 0.466 0.019 0.489 0.010 b 1.982 2.030 1.716 1.257 1.621 0.855 1.431 0.375

MLE c 2.497 0.706 2.484 0.573 2.392 0.268 2.446 0.143

ME c 3.361 2.066 3.133 1.399 2.758 0.672 2.602 0.347 yo 0.388 0.030 0.418 0.024 0.464 0.013 0.470 0.007 b 2.276 2.110 2.004 1.457 1.596 0.625 1.417 0.279

2. The variances of the moment estimators are 2-4 times larger than the variances of the maximum likelihood estimators .

3. Both, bias and variance are particularly visible smaller if the parameter

is known (with the exception of the independent of n moment estimate b). The ratio of the variances of the maximum likelihood estimators and moment estimators, however, is the same for known and for unknown 0T.

In the paper we have shown the advantages and disadvantages of maximum likelihood and moment estimators. The moment estimates are easy to calculate, where in many cases it is difficult to find maximum likelihood estimates. Moreover, there are problems, in which the maximum likelihood estimate of the parameters does not exist.

On the other hand the maximum likelihood estimators have a noticeable smaller variance as moment estimators. The ratio of the variances of the

Table 2. Empirical moments of maximum likelihood (MLE) and moment (ME) estimators (9% = (2.4, 0.5, 1.2))

Mean Variance Mean Variance Mean Variance Mean Variance

MLE |
c |
2.187 |
3.217 |
2.000 |
0.913 |
2.125 |
0.481 |
2.197 |
0.407 |

yo |
0.683 |
0.040 |
0.666 |
0.040 |
0.623 |
0.027 |
0.607 |
0.024 | |

b |
0.818 |
0.157 |
0.891 |
0.141 |
1.026 |
0.084 |
1.072 |
0.054 | |

a |
-0.328 |
0.011 |
-0.337 |
0.010 |
-0.337 |
0.009 |
-0.333 |
0.009 | |

ME |
c |
2.801 |
3.942 |
2.729 |
3.067 |
2.641 |
1.622 |
2.607 |
1.144 |

yo |
0.548 |
0.066 |
0.533 |
0.066 |
0.523 |
0.049 |
0.527 |
0.041 | |

b |
1.621 |
2.179 |
1.739 |
2.357 |
1.669 |
1.410 |
1.572 |
0.916 | |

a= |
0.0 | ||||||||

MLE |
c |
2.194 |
1.156 |
2.177 |
0.695 |
2.209 |
0.625 |
2.299 |
0.403 |

yo |
0.594 |
0.029 |
0.594 |
0.028 |
0.602 |
0.028 |
0.592 |
0.019 | |

b |
1.072 |
0.304 |
1.157 |
0.207 |
1.194 |
0.147 |
1.137 |
0.084 | |

a |
-0.031 |
0.014 |
-0.040 |
0.014 |
-0.048 |
0.012 |
-0.036 |
0.010 | |

ME |
c |
3.128 |
3.691 |
2.800 |
1.645 |
2.660 |
1.400 |
2.503 |
0.678 |

yo |
0.438 |
0.051 |
0.471 |
0.041 |
0.516 |
0.038 |
0.532 |
0.025 | |

b |
2.082 |
2.271 |
1.885 |
1.264 |
1.721 |
0.826 |
1.485 |
0.394 | |

a= |
0.3 | ||||||||

MLE |
c |
2.571 |
0.924 |
2.652 |
1.001 |
2.394 |
0.495 |
2.363 |
0.286 |

yo |
0.510 |
0.014 |
0.520 |
0.014 |
0.546 |
0.011 |
0.552 |
0.010 | |

b |
1.424 |
0.516 |
1.455 |
0.378 |
1.269 |
0.148 |
1.160 |
0.059 | |

a |
0.286 |
0.010 |
0.280 |
0.009 |
0.277 |
0.007 |
0.278 |
0.006 | |

ME |
c |
3.324 |
2.478 |
3.234 |
1.748 |
2.687 |
0.900 |
2.494 |
0.497 |

yo |
0.394 |
0.038 |
0.421 |
0.032 |
0.489 |
0.020 |
0.517 |
0.015 | |

b |
2.297 |
1.926 |
2.171 |
1.535 |
1.639 |
0.703 |
1.409 |
0.250 |

maximum likelihood estimators and moment estimators becomes smaller with increasing sample size.

References

[ABG93] Anderson, P., Borgan, 0., Gill, R., Keiding, N.: Statistical Models Based on Counting Processes. Springer, New-York (1993)

[BaN01] Bagdonavicius, V., Nikulin, M.: Estimation in Degradation Models with Explanatory Variables. Lifetime Data Analysis, 7, 85-103, (2001)

[BBK02] Bagdonavicius, V., Bikelis, A.,Kazakevicius, A., Nikulin, M.: Non-parametric Estimation from Simultaneous Degradation and Failure

Time Data. Comptes Rendus, Academie des Sciences de Paris, 335, 183-188, (2002)

[BBK03] Bagdonavicius, V., Bikelis, A.,Kazakevicius, A., Nikulin, M.: Estimation from Simultaneous degradation and Failure Data, In: Lindquist, B.H., Doksum, K.A. (eds) Mathematical and Statistical Methods in Reliability. World Scientific Publishing Co., New Jersey London Singapore HongKong (2003) [Bre81] Bremaud, P.: Point Processes and Queues. Springer, New York

Berlin Heidelberg (1981) [Cra66] Cramer, H.: The Elements of Probability Theory and Some of Its

Applications. Krieger Publishing Co., Melbourne, Florida (1966) [Cox55] Cox, D.R.: Some statistical methods connected with series of events. J. R. Statist. Soc B, 17, 129-164 (1955) [DoN96] Doksum, K.A., Normand, S.T.: Models for degradation processes and event times based on gaussian processes. In: Jewell, N.P. et al. (eds) Lifetime data: Models in reliability and survival analysis. Kluwer academic publishers, Dordrecht (1996) [Gra91] Grandell, J.: Aspects of risk theory. Springer, New York (1991) [Gra97] Grandell, J.: Mixed Poisson Processes. Chapman & Hall, London

[KaL98] Kahle, W., Lehmann, A.: Parameter Estimation in Damage Processes: Dependent Observations of Damage Increments and First Passage Time. In: Kahle, W. et al. (eds) Advances in Stochastic Models for Reliability, Quality and Safety. Birkhauser, Boston

[KaW04] Kahle, W., Wendt, H.: On a Cumulative Damage Process and Resulting First Passage Times. Applied Stochastic Models in Business and Industry, 20, 17-26 (2004) [LaB95] Last, G., Brandt, A.: Marked Point Processes on the Real Line.

Springer, New York Berlin Heidelberg (1995) [Wen99] Wendt, H.: Parameterschätzungen für eine Klasse doppelt-sto-chastischer Poisson Prozesse bei unterschiedlichen Beobachtungsinformationen. PhD-Thesis, Otto-von-Guericke-University, Magdeburg (1999)

[WeK04] Wendt, H., Kahle, W.: On Parameter Estimation for a Position-Dependent Marking of a Doubly Stochastic Poisson Process. In: Nikulin, N. et al. (eds) Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life. Birkhauser, Boston Basel Berlin (2004) [Whi95] Whitmore, G.A.: Estimating degradation by a Wiener diffusion process subject to measurement error. Lifetime data analysis, 1, 307-319 (1995)

[WhS97] Whitmore, G.A., Schenkelberg, F.: Modeling accelerated degradation data using Wiener diffusion with a time scale transformation, Lifetime data analysis, 3, 27-45, (1997)

Use of statistical modelling methods in clinical practice

Klyuzhev V.M., Ardashev V.N., Mamchich N.G., Barsov M.I.,Glukhova S.I. Burdenko Main Military Clinical Hospital, Moscow, Russia [email protected]

The necessity to generalize the great amount of information concerning the investigated physiological systems, the possibility to predict the body functional reserves have lead to the wide use of statistical modelling methods in the medical practice. The present paper based on the experience of collaborative work of medical specialists and statisticians is devoted to the review of some methods of multivariate statistics used in medicine. The statistical analysis of correlation matrices allowing to carry out the systemic approach to the phenomena under discussion underlies these methods. The data processing using factor and cluster analysis allows to gather the signs into groups identical to the concept of disease syndrome, to obtain the patient grouping, to reveal the connections between the signs, and, according to it, to form the new hypotheses about the revealed causes of dependence. At the stage of diagnostic decision the regression and discriminant analysis can be used.

The main statistical methods used in diagnosis and prediction according to the problems and their clinical significance are shown in Table 1. Not dwelling upon the well-known Student t-test, Walsh t-test and Hotelling T2-test as they are discussed in the available literature, we shortly describe the multivariate statistical methods. More about these and other statistical methods one can see, for example, in [BS83], [GN96],[VN93],[VN96],[BR04],[KK02],[Zac71], etc...

Problem

Comparison of statistically significant difference between the separate signs

Comparison of statistically significant difference by sign population

Unification of signs into groups identical to the concept of disease nsyndromez

Hierarchical patient classification Evaluation of signs for diagnosis and prediction

Differential evaluation of sign population for diagnosis and prediction Scheme of medicament choice

Method

Student t-test Walsh t-test

Hotelling T2-test

Factor and cluster analyses

Cluster and factor analyses Regression analysis

Discriminant analysis

Logical programming

Purpose and Model

Testing of statistical significance of multi-variate measurements Syndrome approach model Analysis of connections between the signs. Hypothesis forming

Classification and diagnosis model Diagnosis and prediction model

Differential diagnosis and prediction model

Appropriate therapy choice model

Table 1. The main methods

Factor analysis is based on the general idea according to which the values of all analyzed symptoms are under the influence of a rather small set of factors. These factors cannot be measured directly and that is why they are called latent. To a certain extent, the factors play the part of causes and the observed symptoms act as the consequences. As the number of latent factors is significantly lower than the number of analyzed signs the aim of factor analysis is to reduce the dimensionality of sign space.

The first factor accumulates maximal information about symptom interconnections and reveals the most distinct signs of the phenomenon under investigation. The second and the subsequent factors comprise the signs that supplement this information by giving some additional significant and indi vidual disease features. The factors are not correlated and ordered according to the variance decrease (the highest is in the first one). It allows to gather the different clinical signs into groups similar to the concept of disease syndrome and to rank them according to significance degree.

The main information about the investigated phenomenon can be presented graphically as vectors in space, the axes of which are the values of first, second and subsequent factors. The use of factor analysis method allows to establish the connection between the diseases, to reveal the signs having no direct connection (or having slight connection) with the given disease.

We give the examples of statistical processing of the material in patient group with different variants of a coronary disease (CHD).

The clinical signs included in the factor which we named cardiorrhexis during myocardial infarction are presented in Table 2.

Signs |
Factor load |

The first myocardial infarction |
0,8 |

Female sex |
0,7 |

History of hypertensive disease |
0,7 |

Severe, sometimes intolerable cardiac pain |
0,5 |

Pain syndrome lasts more than 4 h, pain relapse |
0,6 |

Arterial hypertension |
0,7 |

Trinomial cardiac rhythm |
0,6 |

Systolic murmur above the whole cardiac surface |
0,5 |

Leukocytosis |
0,5 |

Increase in sialic acid level |
0,4 |

High activity of creatine phosphokinase |
0,4 |

Table 2. The results of factor analysis in patients with acute myocardial infarction complicated by cardiorrhexis

Table 2. The results of factor analysis in patients with acute myocardial infarction complicated by cardiorrhexis

One can see that these signs have factor loads ranging from 0,4 to 0,8. They show the acuity of disease manifestation and allow us to formulate the hypothesis of mechanical incompetence of myocardial connective stroma. Car-diorrhexis occurs during the first myocardial infarction with a background of pre-existent hypertensive disease. Pain severity, high fermentative activity, increased level of sialic acids, high leukocytosis reflect the severity of pathological process and probable myocardial stroma destruction.

So, with the help of factor analysis, it is possible to order the system volumes according to the levels and to create the hierarchical classification of the phenomenon under investigation.

The aim of cluster analysis is to partition a set of objects into preset or unknown number of classes based on a certain mathematical criterion of classification quality (a cluster is a group of elements characterized by any general feature). It can be used for disease class detection, patient attribution to appropriate groups and classification of disease symptoms. Based on the measurement of similarity and differences between the patterns (clinical profile) the clusters or groups of subjects investigated are selected. The selected clusters are compared with disease actual outcomes. Depending on their concurrence or difference the problem whether the clinical profile of the disease corresponds to the actual outcomes is solved. Such grouping based on the simple diagnostic principle, i.e. the similarity of one patient with another is the mathematical model of classification. The use of clinical signs allowed us to divide the investigated subjects into 4 groups. Each group corresponds to a disease functional class. The accuracy of classification obtained is 95%.

The most successful model of differential diagnosis is discriminant analysis [OW61]. Its aim is to include the subject (according to a certain rule) in one of the classes (k) depending on the parameters observed (p). This problem is solved with the help of step discriminant analysis. At every step the variable exercising the most significant influence on group division is entered into discriminant equation. As a result the following evaluation of discriminant function for i population is obtained aii X1 + ai2 + ... + aip xp + ci, where i = 1,..., k.

When k equals 2 (two populations) the investigated subject belongs to group 1 if the following condition is carried out:

p y^aj Xj > C2 — ci, aj = aij + a2j, j = 1,...,p.

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