Probability survival data

Kaplan and Meier (1958) introduced a methodology for estimating, from censored survival data, the probability of being event-free as a function of time. If the event is death then we are estimating the probability of surviving and the resultant plots of the estimated probability of surviving as a function of time are called either Kaplan-Meier (KM) curves or survival curves. 

Whatever data we have generally available in most publications are mean concentrations of litter or leaves of all species in an area. For example, it is not all species which contribute to fixation of atmospheric nitrogen and contribute to a general improvement of soil fertility. Calcicole species can probably survive well only on soils with neutral or alkaline pH. The ecological significance of the differences in relative importance of functional groups such as Al-accumulators or Leguminosae in native communities is rarely discussed in the literature (Haridasan and Araujo 1988). 

Kaplan-Meier procedure. A metliod of estimating survival probabilities from data on survival times for individuals in a cohort which can handle the case where a number of the individuals are still alive, so that their eventual survival times are censored. 

Within each visit window, the number of deaths, survival probability, and associated confidence intervals are obtained whenever a death occurs. The values are retained and are output to the data set once per visit at the last record, where the number of subjects remaining at risk is captured in the left variable from the ProductLimitEstimates data set. 

Evaluation of the data surrounding the death by physicians who are unassociated with the clinical trial lends additional credibility to the report and conclusions. Physician biases probably will strongly influence their decision regarding the association of a patient s death with the clinical trial, and this factor must be considered in interpreting their report. This is particularly true for developing survival curves in cancer or other often fatal diseases, when deaths unrelated to the disease or to the treatment are excluded from the analysis. 

Figure 10.5. Viabilities of D. vulgaris wild-type and Arbo strains following expo-snre to either air or air plus 10 pM paraquat. A, the surviving colony-forming units (CPUs) vs the times of either air or air plus paraquat exposure. B, The same data as percent survival of the air plus paraquat-exposed cells normahzed to the survivals of the air-exposed cells. The greater absolute survivals for the air-only exposed Arbo strain relative to wild type is an artifact due probably to the greater initial cell density used for the Arbo than for the wild-type strain in these experiments. Reprinted with permission from Lumppio et al. 2001, copyright 2001 American Society for Microbiology.

This measure is likely to be a reasonable proxy for disease-specific health outcomes for two reasons. First, the proportion of deaths occurring above a certain age can be interpreted as the probability of survival until that age, for example, age 65 (Lichtenberg 2005b). Second, there is a statistically positive relationship between life expectancy at birth and the proportion of deaths occurring above a specific age, based on comparisons of time series data within a country or cross-sectional data across countries. For example, with life expectancy at birth on the vertical axis and the proportion of deaths occurring above age 65 for the whole population at the horizontal axis using time series data from Taiwan for 1971-2002, there is a significantly positive relationship, for both males and females (Fig. 13.4). Life expectancy at birth increases as the age at death increases. 

Using data shown in Figure 13.4, we used ordinary least squares to estimate the effect of the probability of survival to age 65 on life expectancy at birth and found a significantly positive association between these two measures a 10% increase in probability of survival to age 65 was associated with a 1.3% increase in life expectancy. This result, combined with the estimates in Tables 13.2 and 13.3, implies that a 10% increase in the stock of pharmaceutical innovation would lead to an increase in life expectancy at birth by 0.10% (i.e., 0.8% X 1.3%) to 0.18% (1.4% x 1.3%). 

Probably the second most common data type is binary. Examples of binary data include cured/not cured, responder/non-responder, died/survived. Here the measure is based on a dichotomy. 

It is this specific feature that has led to the development of special methods to deal with data of this kind. If censoring were not present then we would probably just takes logs of the patient survival times and undertake the unpaired t-test or its extension ANCOVA to compare our treatments. Note that the survival times, by definition, are always positive and frequently the distribution is positively skewed so taking logs would often be successful in recovering normality. 

Traces of the complex carbohydrates contained in the juice survive the processing and appear in the molasses analytical data indicate that they probably consist largely of pectins and pentosans.10 Quantitative estimations based on the ash-free solids of Louisiana molasses revealed the presence of 2% uronic acids and 0.5% methoxyl.16 Heat-modified... 

In Figure 10.30 the survival rate of the total sedimentary mass for the different Phanerozoic systems is plotted and compared with survival rates for the total carbonate and dolomite mass distribution. The difference between the two latter survival rates for each system is the mass of limestone surviving per unit of time. Equation 10.1 is the log linear relationship for the total sedimentary mass, and implies a 130 million year half-life for the post-Devonian mass, and for a constant sediment mass with a constant probability of destruction, a mean sedimentation rate since post-Devonian time of about 100 x 1014 g y 1. The modem global erosional flux is 200 x 1014 g y-1, of which about 15% is particulate and dissolved carbonate. Although the data are less reliable for the survival rate of Phanerozoic carbonate sediments than for the total sedimentary mass, a best log linear fit to the post-Permian preserved mass of carbonate rocks is... 

---