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2 Right Censoring and Kaplan-Meier Estimator
- Janel Watson
- 2 years ago
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1 2 Right Censoring and Kaplan-Meier Estimator In biomedical applications, especially in clinical trials, two important issues arise when studying time to event data (we will assume the event to be death. It can be any event of interest): 1. Some individuals are still alive at the end of the study or analysis so the event of interest, namely death, has not occurred. Therefore we have right censored data. 2. Length of follow-up varies due to staggered entry. So we cannot observe the event for those individuals with insufficient follow-up time. Note: It is important to distinguish calendar time and patient time Figure 2.1: Illustration of censored data x x x x x o x x o x o o Study Calendar Study starts time ends 0 Patient time (measured from entry to study) In addition to censoring because of insufficient follow-up (i.e., end of study censoring due to staggered entry), other reasons for censoring includes loss to follow-up: patients stop coming to clinic or move away. deaths from other causes: competing risks. Censoring from these types of causes may be inherently different from censoring due to staggered entry. We will discuss this in more detail later. PAGE 11
2 Censoring and differential follow-up create certain difficulties in the analysis for such data as is illustrated by the following example taken from a clinical trial of 146 patients treated after they had a myocardial infarction (MI). The data have been grouped into one year intervals and all time is measured in terms of patient time. Table 2.1: Data from a clinical trial on myocardial infarction (MI) Number alive and under Year since observation at beginning Number dying Number censored entry into study of interval during interval or withdrawn [0, 1) [1, 2) [2, 3) [3, 4) [4, 5) [5, 6) [6, 7) [7, 8) [8, 9) [9, 10) Question: Estimate the 5 year survival rate, i.e., S(5) = P [T 5]. Two naive and incorrect answers are given by 1. F (5) = P [T <5] = 2. F (5) = P [T <5] = 76 deaths in 5 years 146 individuals =52.1%, Ŝ(5) = 1 F (5) = 47.9%. 76 deaths in 5 years (withdrawn in 5 years) = 65%, Ŝ(5) = 1 F (5) = 35%. Obviously, we can observe the following PAGE 12
3 1. The first estimate would be correct if all censoring occurred after 5 years. Of cause, this was not the case leading to overly optimistic estimate (i.e., overestimates S(5)). 2. The second estimate would be correct if all individuals censored in the 5 years were censored immediately upon entering the study. This was not the case either, leading to overly pessimistic estimate (i.e., underestimates S(5)). Our clinical colleagues have suggested eliminating all individuals who are censored and use the remaining complete data. This would lead to the following estimate F (5) = P [T 5] = 76 deaths in 5 years (censored) =88.4%, Ŝ(5) = 1 F (5) = 11.6%. This is even more pessimistic than the estimate given by (2). Life-table Estimate More appropriate methods use life-table or actuarial method. The problem with the above two estimates is that they both ignore the fact that each one-year interval experienced censoring (or withdrawing). Obviously we need to take this information into account in order to reduce bias. If we can express S(5) as a function of quantities related to each interval and get a very good estimate for each quantity, then intuitively, we will get a very good estimate of S(5). By the definition of S(5), we have: S(5) = P [T 5] = P [(T 5) (T 4)] = P [T 4] P [T 5 T 4] = P [T 4] {1 P [4 T<5 T 4]} = P [T 4] q 5 = P [T 3] P [T 4 T 3] q 5 = P [T 3] {1 P [3 T<4 T 3]} q 5 = P [T 3] q 4 q 5 = = q 1 q 2 q 3 q 4 q 5 where q i =1 P [i 1 T < i T i 1],i =1, 2,..., 5. So if we can estimate q i well, then we will get a very good estimate of S(5). Note that 1 q i is the mortality rate m(x) atyear x = i 1 by our definition. PAGE 13
4 Table 2.2: Life-table estimate of S(5) assuming censoring occurred at the end of interval duration [t i 1,t i ) n(x) d(x) w(x) m(x) = d(x) n(x) 1 m(x) ŜR (t i )= (1 m(x)) [0, 1) [1, 2) [2, 3) [3, 4) [4, 5) Case 1: Let us first assume that anyone censored in an interval of time is censored at the end of that interval. Then we can estimate each q i =1 m(i 1) in the following way: d(0) Bin(n(0),m(0)) = m(0) = d(0) n(0) = =0.185, q 1 =1 m(0) = d(1) H Bin(n(1),m(1)) = m(1) = d(1) n(1) = =0.155, q 2 =1 m(1) = where H means data history (i.e, data before the second interval). The life table estimate would be computed as shown in Table 2.2. So the 5 year survival probability estimate ŜR (5) = (If the assumption that anyone censored in an interval of time is censored at the end of that interval is true, then the estimator ŜR (5) is approximately unbiased to S(5).) Of course, this estimate ŜR (5) will have variation since it was calculated from a sample. We need to estimate its variation in order to make inference on S(5) (for example, construct a 95% CI for S(5)). However, ŜR (5) is a product of 5 estimates ( q 1 q 5 ), whose variance is not easy to find. But we have log(ŝr (5)) = log( q 1 )+log( q 2 )+log( q 3 )+log( q 4 )+log( q 5 ). So if we can find out the variance of each log( q i ), we might be able to find out the variance PAGE 14
5 of log(ŝr (5)) and hence the variance of ŜR (5). For this purpose, let us first introduce a very popular method in statistics: delta method: Delta Method: If θ a N(θ, σ 2 ) then f( θ) a N(f(θ), [f (θ)] 2 σ 2 ) Proof of delta method: If σ 2 is small, θ will be close to θ with high probability. We hence can expand f( θ) aboutθ using Taylor expansion: f( θ) f(θ)+f (θ)( θ θ). We immediately get the (asymptotic) distribution of f( θ) from this expansion. Returning to our problem. Let φ i = log( q i ). Using the delta method, the variance of φ i is approximately equal to ( ) 2 var( φ 1 i )= var( q i ). q i Therefore we need to find out and estimate var( q i ). Of course, we also need to find out the covariances among φ i and φ j (i j). For this purpose, we need the following theorem: Double expectation theorem (Law of iterated conditional expectation and variance): If X and Y are any two random variables (or vectors), then E(X) = E[E(X Y )] Var(X) =Var[E(X Y )] + E[Var(X Y )] Since q i =1 m(i 1), we have var( q i ) = var( m(i 1)) PAGE 15
6 = E[var( m(i 1) H)] + var[e( m(i 1) H)] [ ] m(i 1)[1 m(i 1)] = E +var[m(i 1)] n(i 1) [ ] 1 = m(i 1)[1 m(i 1)]E, n(i 1) which can be estimated by m(i 1)[1 m(i 1)]. n(i 1) Hence the variance of φ i = log( q i ) can be approximately estimated by ( ) 2 1 m(i 1)[1 m(i 1)] q i n(i 1) = m(i 1) [1 m(i 1)]n(i 1) = d (n d)n. Now let us look at the covariances among φ i and φ j (i i). It is very amazing that they are all approximately equal to zero! For example, let us consider the covariance between φ 1 and φ 2. Since φ 1 = log( q 1 )and φ 2 = log( q 2 ), using the same argument for the delta method, we know that we only need to find out the covariance between q 1 and q 2, or equivalently, the covariance between m(0) and m(1). This can be seen from the following: E[ m(0) m(1)] = E[E[ m(0) m(1) n(0),d(0),w(0)]] = E[ m(0)e[ m(1) n(0),d(0),w(0)]] = E[ m(0)m(1)] = m(1)e[ m(0)] = m(1)m(0) = E[ m(0)]e[ m(1)]. Therefore, the covariance between m(0) and m(1) is zero. Similarly, we can show other covariances are zero. Hence, var(log(ŝr (5))) = var( φ 1 )+var( φ 2 )+var( φ 3 )+var( φ 4 )+var( φ 5 ). Let θ = log(ŝr (5)). Then ŜR (5) = e θ.so var(ŝr (5)) = (e θ ) 2 var(log(ŝr (5))) = (S(5)) 2 [var( φ 1 )+var( φ 2 )+var( φ 3 )+var( φ 4 )+var( φ 5 )], PAGE 16
7 which can be estimated by [ var(ŝr (5)) = (ŜR (5)) 2 d(0) (n(0) d(0))n(0) + + = (ŜR (5)) 2 4 d(3) (n(3) d(3))n(3) + i=0 d(1) (n(1) d(1))n(1) + ] d(4) (n(4) d(4))n(4) d(2) (n(2) d(2))n(2) d(i) [n(i) d(i)]n(i). (2.1) Case 2: Let us assume that anyone censored in an interval of time is censored right at the beginning of that interval. Then the life table estimate would be computed as shown in Table 2.3. So the 5 year survival probability estimate = (In this case, the estimator Ŝ L (5) is approximately unbiased to S(5).) The variance estimate of ŜL (5) is similar to that of ŜR (5)exceptthatweneedtochange the sample size for each mortality estimate to n w in equation (2.1). Table 2.3: Life-table estimate of S(5) assuming censoring occurred at the beginning of interval duration [t i 1,t i ) n(x) d(x) w(x) m(x) = d(x) n(x) w(x) 1 m(x) ŜL (t i )= (1 m(x)) [0, 1) [1, 2) [2, 3) [3, 4) [4, 5) The naive estimates range from 35% to 47.9% for the five year survival probability with the complete case (i.e., eliminating anyone censored) estimator giving an estimate of 11.6%. The life-table estimate ranged from 40% to 43.2% depending on whether we assume censoring occurred at the left (i.e., beginning) or right (i.e., end) of each interval. More than likely censoring occurs during the interval. Thus ŜL and ŜR are not correct. A compromise is to use the following modification: PAGE 17
8 Table 2.4: Life-table estimate of S(5) assuming censoring occurred during the interval duration [t i 1,t i ) n(x) d(x) w(x) m(x) = d(x) n(x) w(x)/2 1 m(x) ŜLT (t i )= (1 m(x)) [0, 1) [1, 2) [2, 3) [3, 4) [4, 5) That is, when calculating the mortality estimate in each interval, we use (n(x) w(x)/2) as the sample size. This number is often referred to as the effective sample size. So the 5 year survival probability estimate ŜLT (5) = 0.417, which is between ŜL =0.400 and Ŝ R = Figure 2.2: Life-table estimate of the survival probability for MI data Survival probability Time (years) Figure 2.2 shows the life-table estimate of the survival probability assuming censoring occurred during interval. Here the estimates were connected using straight lines. No special significance should be given to this. From this figure, the median survival time is estimated to PAGE 18
9 be about 3 years. The variance estimate of the life-tabble estimate ŜLT (5) is similar to equation (2.1) except that the sample size n(i) is changed to n(i) w(i)/2. That is var(ŝlt (5)) = (ŜLT (5)) 2 4 i=0 d(i) [n(i) w(i)/2 d(i)][n(i) w(i)/2]. (2.2) Of course, we can also use the above formula to calculate the variance of ŜLT (t) atother time points. For example: { } var(ŝlt (1)) = (ŜLT (1)) 2 d(0) [n(0) w(0)/2 d(0)][n(0) w(0)/2] = (146 3/2 27)(146 3/2) = = Therefore SE(ŜLT (1)) = = The calculation presented in Table 2.4 can be implemented using Proc Lifetest in SAS: options ls=72 ps=60; Data mi; input survtime number status; cards; ; proc lifetest method=life intervals=(0 to 10 by 1); time survtime*status(0); freq number; run; PAGE 19
10 Note that the number of observed events and withdrawals in [t i 1,t i ) were entered after t i 1 instead of t i. Part of the output of the above SAS program is The LIFETEST Procedure Life Table Survival Estimates Effective Conditional Interval Number Number Sample Probability [Lower, Upper) Failed Censored Size of Failure Interval Conditional Probability Standard Survival Standard Median Residual [Lower, Upper) Error Survival Failure Error Lifetime Here the numbers in the column under Conditional Probability of Failure are the estimated mortality m(x) = d(x)/(n(x) w(x)/2). The above lifetable estimation can also be implemented using R. Here is the R code: > tis <- 0:10 > ninit <- 146 > nlost <- c(3,10,10,3,3,11,5,8,1,6) > nevent <- c(27,18,21,9,1,2,3,1,2,2) > lifetab(tis, ninit, nlost, nevent) PAGE 20
11 The output from the above R function is nsubs nlost nrisk nevent surv pdf hazard se.surv NA NA se.pdf se.hazard NA NA Note: Here the numbers in the column of hazard are the estimated hazard rates at the midpoint of each interval by assuming the true survival function S(t) is a straight line in each interval. You can find an explicit expression for this estimator using the relation λ(t) = f(t) S(t), and the assumption that the true survival function S(t) is a straight line in [t i 1,t i ): S(t) =S(t i 1 )+ S(t i) S(t i 1 ) t i t i 1 (t t i 1 ), for t [t i 1,t i ). These estimates are very close to the mortality estimates we obtained before (the column under Conditional Probability of Failure in the SAS output.) Kaplan-Meier Estimator The Kaplan-Meier or product limit estimator is the limit of the life-table estimator when intervals are taken so small that only at most one distinct observation occurs within an interval. Kaplan and Meier demonstrated in a paper in JASA (1958) that this estimator is maximum likelihood estimate. PAGE 21
12 Figure 2.3: An illustrative example of Kaplan-Meier estimator x x o x o x x o x o Patient time (years) m(x) : Ŝ(t) : We will illustrate through a simple example shown in Figure 2.3 how the Kaplan-Meier estimator is constructed. By convention, the Kaplan-Meier estimate is a right continuous step function which takes jumps only at the death time. The calculation of the above KM estimate can be implemented using Proc Lifetest in SAS as follows: Data example; input survtime censcode; cards; ; Proc lifetest; PAGE 22
13 time survtime*censcode(0); run; And part of the output from the above program is The LIFETEST Procedure Product-Limit Survival Estimates Survival Standard Number Number SURVTIME Survival Failure Error Failed Left * * * * * Censored Observation The above Kaplan-Meier estimate can also be obtained using R function survfit(). The code is given in the following: > survtime <- c(4.5, 7.5, 8.5, 11.5, 13.5, 15.5, 16.5, 17.5, 19.5, 21.5) > status <- c(1, 1, 0, 1, 0, 1, 1, 0, 1, 0) > fit <- survfit(surv(survtime, status), conf.type=c("plain")) Then we can use R function summary() to see the output: > summary(fit) Call: survfit(formula = Surv(survtime, status), conf.type = c("plain")) time n.risk n.event survival std.err lower 95% CI upper 95% CI Let d(x) denote the number of deaths at time x. Generally d(x) is either zero or one, but we allow the possibility of tied survival times in which case d(x) may be greater than one.let n(x) PAGE 23
14 denote the number of individuals at risk just prior to time x; i.e., number of individuals in the sample who neither died nor were censored prior to time x. Then Kaplan-Meier estimate can be expressed as KM(t) = ( 1 d(x) ). x t n(x) Note: In the notation above, the product changes only at times x where d(x) 1;, i.e., only at times where we observed deaths. Non-informative Censoring In order that the life-table estimates give unbiased results there is an important assumption that individuals who are censored are at the same risk of subsequent failure as those who are still alive and uncensored. The risk set at any time point (the individuals still alive and uncensored) should be representative of the entire population alive at the same time. If this is the case, the censoring process is called non-informative. Statistically, if the censoring process is independent of the survival time, then we will automatically have non-informative censoring. Actually, we almost always mean independent censoring by non-informative censoring. If censoring only occurs because of staggered entry, then the assumption of non-informative censoring seems plausible. However, when censoring results from loss to follow-up or death from a competing risk, then this assumption is more suspect. If at all possible censoring from these later situations should be kept to a minimum. Greenwood s Formula for the Variance of the Life-table Estimator The derivation given below is heuristic in nature but will try to capture some of the salient feature of the more rigorous treatments given in the theoretical literature on survival analysis. For this reason, we will use some of the notation that is associated with the counting process approach to survival analysis. In fact we have seen it when we discussed the life-table estimator. PAGE 24
15 It is useful when considering the product limit estimator to partition time into many small intervals, say, with interval length equal to x where x is small. Figure 2.4: Partition of time axis Patient time x Let x denote some arbitrary time point on the grid above and define = number of individuals at risk (i.e., alive and uncensored) at time point x. = number of observed deaths occurring in [x, x + x). Recall: Previously, was denoted by n(x) and was denoted by d(x). It should be straightforward to see that w(x), the number of censored individuals in [x, x+ x), is equal to {[ Y (x + x)] }. Note: In theory, we should be able to choose x small enough so that { > 0and w(x) > 0} should never occur. In practice, however, data may not be collected in that fashion, in which case, approximations such as those given with life-table estimators may be necessary. With these definitions, the Kaplan-Meier estimator can be written as { KM(t) = 1 }, as x 0, all grid points x such that x + x t which can be modified if x is not chosen small enough to be { } LT (t) = 1, w(x)/2 all grid points x such that x + x t where LT (t) means life-table estimator. If the sample size is large and x is small, then is a small number (i.e., close to zero) and as long as x is not close to the right hand tail of the survival distribution (where may PAGE 25
16 be very small). If this is the case, then exp { } { 1 Here we used the approximation e x 1+x when x is close to zero. This approximation is exact when =0. }. Therefore, the Kaplan-Meier estimator can be approximated by { KM(t) exp } { =exp all grid points x such that x + x t here and thereafter, {x <t} means {all grid points x such that x + x t}. }, If x is taken to be small enough so that all distinct times (either death times or withdrawal times) are represented at most once in any time interval, then the estimator will be uniquely defined and will not be altered by choosing a finer partition for the grid of time points. In such a case the quantity is sometimes represented as t Basically, this estimator take the sum over all the distinct death times before time t of the number of deaths divided by the number at risk at each of those distinct death times. 2. The estimator hazard function Λ(t) = t 0 λ(x)dx. Thatis Recall that S(t) = exp( Λ(t)). is referred to as the Nelson-Aalen estimator for the cumulative Λ(t) =. By the definition of an integral, Λ(t) = t 0 λ(x)dx λ(x) x. grid points x such that x + x t PAGE 26
17 By the definition of a hazard function, λ(x) x P [x T<x+ x T x]. With independent censoring, it would seem reasonable to estimate λ(x) x, i.e., the conditional probability of dying in [x, x + x) given being alive at time x by. Therefore we obtain the Nelson-Aalen estimator Λ(t) =. We will now show how to estimate the variance of the Nelson-Aalen estimator and then show how this will be used to estimate the variance of the Kaplan-Meier estimator. time x. For a grid point x, leth(x) denote the history of all deaths and censoring occurring up to H(x) ={dn(u),w(u); for all values u on our grid of points for u<x}. Note the following 1. Conditional on H(x), we would know the value of (i.e., the number of risk at time x) and that would follow a binomial distribution denoted as H(x) Bin(,π(x)), where π(x) is the Conditional probability of an individual dying in [x, x + x) giventhat the individual was at risk at time x (i.e., π(x) =P [x T<x+ x T x]). Recall that this probability can be approximated by π(x) λ(x) x. 2. The following are standard results for a binomially distributed random variable. (a) E[ H(x)] = π(x), PAGE 27
18 (b) Var[ H(x)] = π(x)[1 π(x)], [ ] (c) E H(x) = π(x), {[ ][ ][ ] } (d) E 1 H(x) = π(x)[1 π(x)]. Consider the Nelson-Aalen estimator Λ(t) =.Wehave [ ] E[ Λ(t)] = E = [ ] E = [ [ ]] E E H(x) = π(x) λ(x) x t 0 λ(x)dx =Λ(t). Hence E[ Λ(t)] = π(x). If we take x smaller and smaller, then in the limit π(x) goestoλ(t). Namely Λ(t) is nearly unbiased to Λ(t). How to Estimate the Variance of Λ(t) The definition of variance is given by Var( Λ(t)) = E[ Λ(t) E( Λ(t))] 2 [ = E ] 2 π(x) [ { }] 2 = E π(x). Note: The square of a sum of terms is equal to the sum of the squares plus the sum of all cross product terms. So the above expectation is equal to E { } 2 π(x) + { }{ dn(x } π(x) ) Y (x x x <t ) π(x ) = [ ] 2 E π(x) + [{ }{ dn(x }] π(x) ) Y (x ) π(x ) E x x <t PAGE 28
19 We will first demonstrate that the cross product terms have expectation equal to zero. Let us take one such term and let us say, without loss of generality, that x<x. [{ }{ dn(x }] E π(x) ) Y (x ) π(x ) [ [{ }{ dn(x } ]] = E E π(x) ) Y (x ) π(x ) H(x ) Note: Conditional on H(x ),,Y(x) andπ(x) are constant since x<x. Therefore the above expectation is equal to [{ } [{ dn(x } ]] E π(x) ) E Y (x ) π(x ) H(x ) The inner conditional expectation is zero since { dn(x } ) E Y (x ) H(x ) = π(x ) by (2.c). Therefore we show that [{ }{ dn(x }] E π(x) ) Y (x ) π(x ) =0. Since the cross product terms have expectation equal to zero, this implies that Var( Λ(t)) = [ ] 2 E π(x) Using the double expectation again, we get that [ ] 2 E π(x) } 2 = E E π(x) H(x) [ [ ]] = E Var H(x) [ ] π(x)[1 π(x)] = E. Therefore, we have that Var( Λ(t)) = [ ] π(x)[1 π(x)] E. PAGE 29
20 If we wanted to estimate π(x)[1 π(x)], then using (2.d) we might think that [ ] 1 may be reasonable. In fact, we would then use as an estimate for Var( Λ(t)) the following estimator; summing the above estimator over all grid points x such that x + x t. Var( Λ(t)) = [ 1 ]. In fact, the above variance estimator is unbiased for Var( Λ(t)), which can be seen using the following argument: [ ] E 1 [ ] = E 1 { } = E E 1 H(x) = [ ] π(x)[1 π(x)] E (by (2.d)) = Var[ Λ(t)]. (double expectation again) What this last argument shows is that an unbiased estimator for Var[ Λ(t)] is given by [ 1 ]. Note: If the survival data are continuous (i.e., noties)and x is taken small enough, then would take on the values 0 or 1 only. In this case and [ ] 1 Var( Λ(t)) = = Y 2 (x), Y 2 (x), PAGE 30
21 which is also written as t 0 Y 2 (x). Remark: We proved that the Nelson-Aalen estimator π(x). We argued before that in the limit as x goes to zero, is an unbiased estimator for becomes t 0. We also argued that π(x) λ(x) x, hence as x goes to zero, then π(x) goes to t 0 λ(x)dx. These two arguments taken together imply that t 0 is an unbiased estimator of the cumulative hazard function Λ(t) = t 0 λ(x)dx, namely, E [ t 0 ] =Λ(t). Since Λ(t) = is made up of a sum of random variables that are conditionally uncorrelated, they have a martingale structure for which there exists a body of theory that enables us to show that PAGE 31
22 Λ(t) is asymptotically normal with mean Λ(t) and variance Var[ Λ(t)], which can be estimated unbiasedly by Var( Λ(t)) = [ 1 ] ; and in the case of no ties, by Var( Λ(t)) = Y 2 (x). Let us refer to the estimated standard error of Λ(t) by se[ Λ(t)] = [ 1 ] 1/2. The unbiasedness and asymptotic normality of Λ(t) aboutλ(t) allow us to form confidence intervals for Λ(t) (at time t). Specifically, the (1 α)th confidence interval for Λ(t) is given by Λ(t) ± z α/2 se( Λ(t)), where z α/2 is the (1 α/2)th quantile of a standard normal distribution. That is, the random interval [ Λ(t) z α/2 se( Λ(t)), Λ(t)+z α/2 se( Λ(t))] covers the true value Λ(t) with probability 1 α. This result could also be used to construct confidence intervals for the survival function S(t). This is seen by realizing that S(t) =e Λ(t), in which case the confidence interval is given by [e Λ(t) zα/2 se( Λ(t)), e Λ(t)+zα/2 se( Λ(t)) ], PAGE 32
23 meaning that this random interval will cover the true value S(t) with probability 1 α. An example: We will use the hypothetical data shown in Figure 2.3 to illustrate the calculation of Λ(t), Var Λ(t), and confidence intervals for Λ(t) ands(t). For illustration, let us take t = 17. Note that there are no ties in this example. So Λ(t) = t = 0 = =0.804, Var[ Λ(t)] = t Y 2 (x) = 0 Y 2 (x) = =0.145, 2 ŝe[ Λ(t)] = = So the 95% confidence interval for Λ(t) is ± = [0.0572, 1.551]. and the Nelson-Aalen estimate of S(t) is Ŝ(t) =e Λ(t) =e = The 95% confidence interval for S(t) is [e 1.551, e ]=[0.212, 0.944]. Note The above Nelson-Aalen estimate Ŝ(t) =0.448 is different from (but close to) the Kaplan-Meier estimate KM(t) = It should also be noted that above confidence interval for the survival probability S(t) is not symmetric about the estimator Ŝ(t). Another way of getting approximate confidence intervals for S(t) =e Λ(t) is by using the delta method. This method guarantees symmetric confidence intervals. Hence a (1 α)th confidence interval for f(θ) isgivenby f( θ) ± z α/2 f ( θ) σ. In our case, Λ(t) takes on the role of θ, Λ(t) takes on the role of θ, f(θ) =e θ so that S(t) =f{λ(t)}. Since f (θ) = e θ =e θ, and Ŝ(t) =e Λ(t). PAGE 33
24 Consequently, using the delta method we get Ŝ(t) a N(S(t), [S(t)] 2 Var[ Λ(t)]), and a (1 α)th confidence interval for S(t) is given by Ŝ(t) ± z α/2 {Ŝ(t) se[ Λ(t)]}. Remark: Note that [S(t)] 2 Var[ Λ(t)] is an estimate of Var[Ŝ(t)], where Ŝ(t) =exp[ Λ(t)]. Previously, we showed that the Kaplan-Meier estimator KM(t) = was well approximated by Ŝ(t) =exp[ Λ(t)]. [ 1 ] Thus a reasonable estimator of Var(KM(t)) would be to use the estimator of Var[exp( Λ(t))], or (by using the delta method) [Ŝ(t)]2 Var[ Λ(t))] = [Ŝ(t)]2 Y 2 (x). This is very close (asymptotically the same) as the estimator for the variance of the Kaplan- Meier estimator given by Greenwood. Namely Var{KM(t)} = {KM(t)} 2 [ ]. [ w(x)/2][ w(x)/2] Note: SAS uses the above formula to calculate the estimated variance for the life-table estimate of the survival function, by replacing KM(t) on both sides by LT (t). Note: The summation in the above equation can be viewed as the variance estimate for the cumulative hazard estimator defined by Λ KM (t) = log[km(t)]. Namely, Var{ Λ KM (t)} = [ w(x)/2][ w(x)/2]. In the example shown in Figure 2.3, using the delta-method approximation for getting a confidence interval with the Nelson-Aalen estimator, we get that a 95% CI for S(t) (where t=17) PAGE 34
25 is e Λ(t) ± 1.96 e Λ(t) se[ Λ(t)] = e ± 1.96 e = [0.114, 0.784]. The estimated se[ŝ(t)] = If we use the Kaplan-Meier estimator, together with Greenwood s formula for estimating the variance, to construct a 95% confidence interval for S(t), we would get [ KM(t) = 1 1 ][ 1 1 ][ 1 1 ][ 1 1 ][ 1 1 ] = { 1 Var[KM(t)] = } = ŝe[km(t)] = = Var[ Λ KM (t)] = =0.182 se[ Λ KM (t)] = Thus a 95% confidence interval for S(t) is given by KM(t) ± 1.96 ŝe[km(t)] = ± = [0.068, 0.754], which is close to the confidence interval using delta method, considering the sample size is only 10. In fact the estimated standard errors for Ŝ(t) andkm(t) using delta method and Greenwood s formula are and respectively, which agree with each other very well. Note: If we want to use R function survfit() to construct a confidence interval for S(t)with the form KM(t) ± z α/2 ŝe[km(t)],wehavetospecifytheargumentconf.type=c("plain") in survfit(). The default constructs the confidence interval for S(t) by exponentiating the confidence interval for the cumulative hazard using the Kaplan-Meier estimator. For example, a 95% CI for S(t) iskm(t) [e 1.96 se[ ΛKM (t)],e 1.96 se[ ΛKM (t)] ]=0.411 [e , [e ]= [0.178, 0.949]. Comparison of confidence intervals for S(t) 1. exponentiating the 95% CI for cumulative hazard using Nelson-Aalen estimator: [0.212, 0.944]. PAGE 35
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On Wednesday, though, they won't be facing any old offense. The Irish's attack is 20th nationally, racking up 398 yards per contest. This includes the 12th best passing offense, led by quarterback Brady Quinn. Head coach Charlie Weis has been preaching to his players that LSU's defense resembles Tennessee from last year. They're big, fast, athletic and can rush the passer with their front four. It's probably no coincidence that Notre Dame beat the Volunteers in 2005 and it's the blueprint they're working from.
"It's easy to compare them to Tennessee last year," running back Darius Walker said. "They are a team with speed and very physical and in the SEC. You know what type of game it's going to be and what the flow will be. In the SEC, the games seem to go a lot faster because they have so much speed."
Walker's job description will have the utmost importance. The junior halfback not only will try to penetrate the nation's 14th best rushing defense but also pick up oncoming defenders in protecting Quinn. LSU can get pressure with their front four and Walker might have to help an offensive line that's allowed 2.5 per game, which ranks 95th in Division 1A. Blitz pickup has been an area that's improved for Walker.
"One of the things he's done is progress as a pass protector," offensive coordinator/running backs coach Mike Haywood said. "It was one of his deficiencies when the season started. But he's got significantly as the season has gone on."
"You still work on it as much as possible," Walker said. "It's important for a running back. Your duties as a running back are significant. You run the ball, run routes, catch the ball and all of that stuff. You have to be on top of your game."
The Irish offensive line will have to be on top of their game. The unit has faced some great groups in 2006. Michigan's LaMarr Woodley and Alan Branch wreaked havoc with pressure and sacks. UCLA's Bruce Davis and Justin Hickman combined for five sacks.
The LSU defensive line is as good as it gets. They're anchored by All-American defensive tackle Glenn Dorsey, who is contemplating skipping his senior year to go pro. The offensive line must open holes for Walker to run through and keep Tiger defenders off the back of Quinn. It's quite a task but one that must be achieved for Notre Dame to have a shot at the upset.
"They're a lot like the UCLA line," offensive lineman Ryan Harris said. "They have talent across the board. They are real talented and know their technique. They don't make a lot of mistakes. It's a talented group but we've seen talented groups before.
"We are definitely preparing for a physical contest. They have size, speed and ability. We expect a very full contact game."
LSU ranks third against the pass. The best group the Irish have seen this year is Army's, which is No. 7 nationally. A fierce Tiger pass rush, coupled with a lockdown secondary, contributes to this high ranking.
LSU has faced seven teams ranked 67th or worse in passing. Notre Dame's attack will be the second highest of the year. All-American safety LaRon Landry leads the group, who mix it up with tough, physical one-on-one press coverage and a zone here and there. Notre Dame is preparing for either coverage style.
"I'm sure there is going to be a little bit of both," wide receiver Jeff Samardzija said. "They like to bring pressure at certain times. If you bring pressure, you're going to be manned up on the outside. It's tough to say. They'll be mixing it up. They have confidence in their guys."
What type of game does Notre Dame need to play? A low-scoring affair is unlikely because of the Irish defense's vulnerability to the big play. A shootout would provide the best scenario. One problem: LSU is 5th in scoring defense, surrendering a meager 12 points per game. Notre Dame has been held under 20 points once and 30 points four times. They know how to put points up on the scoreboard.
"I hope we get into the end zone a couple of times," Samardzija said. "That's how you score points. We've been doing good during the year at scoring points. We'll stick to our strengths and hopefully we can put one more up than them."
One final feature of the Sugar Bowl is the venue. The Louisiana Superdome is the site for the contest. Usually, Notre Dame has no problem traveling and filling up the majority of the seats. It's expected that LSU fans will dominate the crowd. It's no secret that the Tiger faithful have no problem being heard or making noise during games. It's another daunting aspect that the Irish players must be ready for.
"We know there's going to be a heavy presence of LSU fans," Harris said. "What helps for them is the hard covered dome, which will be greatly in their favor. We've played at Tennessee but I could only imagine if they had a hard covered dome. We're definitely going to have some noise."

subset: commoncrawl


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With the pending public offering of Facebook anticipated to be the largest tech IPO in history, it's an interesting time to think about where we go from here. Some say "social is done," Facebook is all the social media anyone would ever want or need.
Scooped by
Marylene Delbourg-Delphis
onto Entrepreneurship, Innovation


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Let us know when your receive great advice so we can thank our lawyers and point others with similar questions in the right direction.Got it!
I am planning to travel to Philippines this April . . . I just got my green card 3 months ago , and my passport will expire on Oct . 2013 . My passport has still has my Maiden Name , and Married name on Green Card , will I get trouble in immigration on this ? I'll be renewing my passport in the Philippines and will have my married name in it , so I should be OK when I get back ?
You should apply for a new passport on an expedited basis before you travel out of the United States to prevent issues at the border.Ask a similar question
If I can't do it on time, which name will I put for my airline ticket? Thanks for your advice
It is my opinion that you should not travel without changing your name on the passport. They are not going to just believe that you are the same person because you say so and it will probably lead to issues at the border. Passport applications can be expedited and you should start on the process right away.
And the name on your ticket should reflect your current legal name.
You should renew your passport under your married name, to match your name on your green card. Hopefully you also applied for, and obtained a state issued ID and a social security card under your married name.
Behar Intl. Counsel 619.234.5962 Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.Ask a similar question
Yes, I have a state I.D and Social Security Card with my married name on it. Thank you for your time.
You are very welcome. Don't forget to designate a "Best Answer", as well as a "Helpful" one.
I find both of colleagues answers sufficiently insightful. You will not get trouble with immigration on this. Our immigration system is complex and cumbersome. Sometimes overly bureaucratic too. But not on this particular issue.Ask a similar question
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Eir Stegalkin - Guildwars 2 by dinotisteWatch
©2013-2017 dinotiste
Eir Stegalkin from Guildwars 2
That looks really awesome, I love Eir, she's so cool, and probably one of my favorite members of Destiny's Edge She looks really great here, it looks a lot like the art style in the game too, which makes it really cool.
wonderful painting of my favourite gw2 npc
I adore this. Fantastic job. Her eye seems slightly high I think. But this is so beautiful. I love the antique ish background. I especially love the hair and how perfect it seems and the collar is just super. Overall I think you really excel at texture.
Wow. This is beautiful! I love your paint-like coloring style! The hair, the fur of her coat, her striking blue eye, this all has a really nice color scheme and very nice shading etc. Definately faving this!
And I'm so happy to see that you like Guild Wars 2 as well!
And I'm so happy to see that you like Guild Wars 2 as well!


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Center for East Asian Studies
KU's Center for East Asian Studies is a Title VI National Resource Center dedicated to teaching people of all ages about East Asia. We provide an array of resources for educators at all levels, sponsor East Asia themed events and lectures at the University of Kansas and at partner institutions, and assist KU students to develop expertise in East Asian languages and cultures.
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text: The fourth annual Baseball Hack Day is a fun, friendly one-day hacking competition for baseball fans. This year, O3 World CXO Mike Gadsby and Developer Courtney Wilburn are teaming up with Thoughtbot Chief Design Officer Kyle Fiedler to organize Philadelphia area baseball minds to create amazing baseball-oriented projects. Sound like a home run to you? Register to join the fun at Philadelphia's Baseball Hack Day on March 19 at Huntsman Hall at the University of Pennsylvania!


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Offshore Engineering and Production
This book includes every aspect of today's offshore exploration and production: virtually every element of equipment used and each phase of the procedures,various specifications and relevant regulations, etc are covered in understandable terms.
It should be a permanent part of every engineer's baggage, and it will be as valuable to the experienced engineer as to the new recruit. Today's offshore engineer must carry out an enormous range of tasks and so a full appreciation of the industry is an essential prerequisite to success.
- 9781905331987
- Witherby Seamanship International
- 852686750232


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Previous Challenge Entry (Level 1 – Beginner)
Topic: Extra (08/29/13)
TITLE: Reflection of God
By nicole davis
My husband had worked out of town all week. I was eager to see him. I always miss him terribly when he is away. He would be rushing to get to the field from the airport so the kids and I had agreed to meet him at the game.
It is Friday night in Texas. Expecting an exciting game and having two sons on the field, this was just one of the many activities we did as a family. Getting all three boys and our daughter headed to the entrance gait, I heard my name being called from a distance. There he is, coming across the parking lot. All these years and I still get butterflies. It is a wonderful feeling.
Those little moments are nurtured from all the extra work we put into our relationship and our family. We want to set a good example for our children. We want them to be able to watch us love and have a marriage that honors God. To be honest, it is really fun for all of us. Following God's plan is always rewarding.
These days our lives are so busy with work, school, and church, that we nearly have to schedule a time to steal a kiss or a soothing glance from our husbands before we all rush out the door in the mornings. Once we find that time, we must treasure it and never take it for granted. This takes a conscious effort. It is becoming self-sacrificial. Giving more physically, emotionally, and spiritually of ourselves. With us sacrificing ourselves, we begin to experience love the way God intended.
This could be making an extra effort to show how much you love them by just being considerate of their needs. I always like slipping little "I love you" notes in places where they will find them throughout their day. Sometimes it could be the smallest gestures that speak to the heart the loudest.
I try to remind myself, that together, my husband and I reflect the image of God on earth. (Eggerichs, 2010) It is important that we strive to make it a true reflection. It takes a lot of hard work to make this happen.
A family is a wonderful gift from God and we must do all we can to protect it. So, start tomorrow with a little extra effort to show your spouse that you love them. Like I had said, those small gestures will speak loud to their hearts.
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text: 6 hours on word to come up with this. What do you guys think of it? I'm going to do a test run in one of my target areas. If it gets results I'll use it as inspiration for my post cards I'll be doing in march. Ben. Winter Flyer


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Midnapur, Nov 12 (UNI) The Sub-Divisional Headquarter of Jhargram town in Midnapur district tonight became unreachable from the rest of State after agitating tribals fell trees on the State Highway No 9.
Hundreds of tribals congregated at around 2200 hrs and started felling big Sal trees standing on both sides on the road near Sarada Vidyapith point, about 200 meters ahead of the town, SDO Tanmoy Chakraborty said.
Earlier yesterday, the road towards Jhargram was cut off from Dahijuri side and today it was cut off from the other side of the National Highway 6 Mumbai Road, he added.
Police remained a silent spectator as the road link towards Purulia, Bankura, Durgapur and Belpahari was completely cut off.
Hundreds of trucks and buses have already stopped on the road, he alleged.
The demonstrators used chalk and red ink to inscribe their demands on the black-metalled Highway and walls.


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SlashGear Week in Review – August 3rd
Week in Review time, and it seems like there's plenty to look back on over the past seven days. Continuing our fresh tradition, in the fast-moving netbook segment we've seen confirmation of ASUS' Eee PC plans – 23 models in all, the next of which being the 701SD, together with dual-core and bigger SSD versions – as well as Sylvania's new G Netbook MESO, the target $299 tag of which got plenty of people hot under the collar. Price is a sore subject over at MSI, who saw their Wind netbook rise in cost uncontrollably, while E-Lead are likely celebrating their Noahpad – with its distinctive dual-touchpad keyboard – finally reaching customers.
There's plenty to look forward to, as well, with Intel confirming a refresh of their Classmate PC, a video showing Fujitsu's upcoming Amilo Mini playing with interchangeable casings, and Lenovo tipped for a new netbook to be released in September. If you can afford a bit more, and can't wait until September, Lenovo's ThinkPad X200 is a proper, grown-up ultraportable set to launch imminently; going by the first hands-on reports it looks to be a winner, with battery life getting particular credit.


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Much Awaited Kannada Movies Of 2014
After ending 2013 on a good note, Sandalwood is likely to have another blissful year ahead. The Kannada movie makers have scheduled their release dates, by avoiding clash with other movies. Click on images to know about the much awaited Kannada movies of 2014...
Ninnindale starring Puneet Rajkumar and Erica Fernandes in leads is likely to release by the end of January. The actor will be seen on silver screen after an year. The movie also features Telugu ace comedian Brahmanandam, Thilak Shekar and Avinash in the significant roles. It is directed by Jayanth C Paranji.
Brahma The Leader
Directed by R Chandru, Brahma The Leader features Superstar Upendra and Pranitha in leads. The bilingual flick, which is being simultaneously made in Kannada and Telugu will also be dubbed into Tamil. The movie is slated to release in February.
Huchudugaru, which was slated to release in December 2013 was delayed due to unknown reason. Now, RJ Pradeepa, who is making his directorial debut with the movie has revealed that the film will be released in the mid-week of February. It has Chetan Chandra and Adithi Rao in leads.
Crazy Star
The successful pair of Malla - Ravichandran and Priyanka Upendra is again back with Crazy Star. The movie is likely to hit screen in the last week of February.
Rocking Star Yash is playing the dual role in Gajakesari. The movie, which is likely to hit screen in first week of March has Amoolya in the female lead. The making video of Gajakesari has already caught the attention of the moviegoers.
Ugramm has already caught the attention of the cinegoers with its slick and sharp trailer. Actor Srimurali is all set to make his comeback with the movie in which he is sharing screen space with Haripriya. The film is likely to hit screen in the first week of March.
Karunada Chakravarthy Shivaraj Kumar, who is basking in the success of Bhajarangi is all set to give a Blockbuster in 2014 with Aryan! Sandalwood King Shivanna is paired opposite Sandalwood queen Ramya in the movie, which is being directed by Chi Gurudutt. The shooting of the movie was halted due to sudden death of director Rajendra Babu, who had nearly finished 80% of the shooting. Now, the movie is slated to release in March.
Duniya Vijay, who has become successful in psychological dark fantasy Jayammana Maga will be back with Shivajinagara. Viji, who plays don in the movie is paired opposite Parul Yadav. The movie is likely to hit screens in last week of March.
Maanikya is the remake of Telugu movie Mirchi. It has Sudeep, Ravichandran, Ramya Krishna in the lead roles. Directed by Sudeep is presently busy with the final shooting schedule of the movie. The movie is likely to release in mid-week of April.
After wowing Kannada audience with Bulbul and Brindavana in 2013, Challenging Star Darshan is all set to rock in another multi-starrer movie Ambareesha. Apart from Darshan, the movie also features Rebel Star Ambareesh, Priyamani, Rachita Ram, Umashree and Sumalatha Ambareesh in the significant roles. Presently, the crew is busy with the third shooting schedule of the movie.
Neer Dose
After having the Twitter war lead actor Jaggesh and Ramya has finally decided to complete the shooting of Neer Dose. The movie directed by Vijay Prasad is busy with the last shooting schedule.


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My friend Ally is hands-down the most warm and welcoming person I know. When meeting new people, she's the first to smile and shake hands. Ally's also the first person to introduce me to people I don't know, which I've always appreciated.
However, her greetings have recently taken on a new tag line, and I've been trying to come to terms with what it means to people and how it makes me feel. Ever since April, Ally's traditional "This is my friend Erika" has become:
"This is my friend Erika-she's a Tiger-Cat cheerleader."
The entire introduction is said with a glowing smile. As she emphasizes 'Tiger-Cat,' her chin dips down and her eyebrows raise slightly. When it's over, she turns and grins at me the way my sister did when she told my parents I'd been hiding jars of Nutella under my bed and devouring them with a spoon.
It must be very satisfying to divulge information for the sole purpose of getting a reaction at my expense.
And the reactions certainly vary.
Reaction 1: Cheerleaders are funny.
If the people we meet are our age, I either get a request to do a cheer, or I'm subjected to their imitation of cheerleading. I know it's not a typical part-time job, but it seems to be the only one that people feel the need to enact. I wonder what would happen if every time someone told me what they did with their time, I turned it into a game of charades.
"Oh, you play in a band?" I'd say. "Do you play drums [I'd wave my fists as if holding drumsticks], guitar [I'd strum the air Guitar Hero-style] or maybe the accordion [insert your vision of playing the accordion here]?"
I would be socially ostracized.
But apparently when guys jump up and flail their arms, yelling 'Go Ti-Cats,' it's comedic genius.
Reaction 2: Cheerleaders are anti-feminist.
A co-worker who I both admire and respect told me that she thought it was a great gig for a dancer, but she had a problem with the female image it perpetuates. The gist of her opinion was that cheerleading is a performance exclusively for men and encourages the kind of verbal assault commonly associated with objectifying women.
Not wanting to abandon my feminist ideals, I used a defense that doesn't solve the argument, but still made a point.
I was walking downtown Burlington after a job interview. I was wearing black heels and a dress to match that was high-collared, hemmed below the knee, and fit loosely. I passed a group of men unloading furniture from a truck and was met with whistles and various hollers.
Later that week, Ally and her mom took their dog for a walk. Ally was wearing dance pants and an exercise top. Her mom wore a knee-length summer dress. A car drove by, and a man yelled "SLUTS" out of the window.
I've concluded that it doesn't matter if you're wearing a half-top on a football field or a business dress on Brant Street. All you have to do is look like a woman to be publicly degraded. Regardless of this defense, I still get uncomfortable waiting to hear how people react. Is it funny that I'm a cheerleader? Am I contributing to the oppression of women?
But a walk through the forest behind my house made me realize that sometimes I should take myself less seriously.
Reaction 3: Cheerleaders exist-at the most basic level-as a representation of the team for the fans.
I was returning from a promotional event wearing black dance pants and a white Tiger-Cats t-shirt with a logo about the size of a ping-pong ball on the front. As I approached a small bridge, two boys rode past me on their bicycles and stopped at the top. They looked about fifteen years old.
"Are you a Tiger-Cat cheerleader?" one of them asked. Too bad Ally wasn't there to introduce me.
"Um…yes," I said awkwardly as I paused on the bridge.
"Could I get a picture with you?"
I felt my face turn red as I agreed to pose for a photo. The guy handed his cell phone to his friend, who fumbled for at least thirty seconds before figuring out how to work the camera. I've never felt as strange as I did in the middle of the forest trying to smile for this kid's photo-op.
I stuck out my hand and introduced myself, thinking that I would make conversation for the rest of the walk home. But as soon as we shook hands, the boy turned his back to me. The two of them started talking to each other while I was still standing there, completely ignoring me.
The guys didn't care what my name was. They weren't laughing at me, and they certainly didn't judge me to be anti-feminist. The Tiger-Cat fans probably don't give us a second thought; they just accept us as part of the team that they love.
And so I briefly became a backwoods celebrity not because of who I am or what it means to be a cheerleader, but because of what I stand for.
All they wanted was a picture with someone who was part of the Tiger-Cats team. And there's nothing wrong with that.


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LANDOVER, Md. (AP) Tight end Kellen Winslow was on the Cleveland Browns' active roster Sunday against the Washington Redskins after being hospitalized with an undisclosed illness.
Winslow had played in 36 straight games before sitting out Monday night against the New York Giants, then missing practice most of the past week.
The Browns were without starting strong safety Mike Adams (hamstring). Sean Jones was to start in his place.
Starting cornerback Shawn Springs was inactive for the Redskins because of a calf injury, and Fred Smoot was inserted into the lineup despite a nagging groin injury. As expected, running back Ladell Betts missed the game with a sprained knee.
The Associated Press
Guidelines: You share in the USA TODAY community, so please keep your comments smart and civil. Don't attack other readers personally, and keep your language decent. Use the "Report Abuse" button to make a difference. Read more.


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14 Jailed For Norfolk Cannabis Operation
14 men have been jailed for a total of 82 years for a cannabis operation in Norfolk and Kent worth £25 million a year.
A couple who kidnapped and then robbed a quadriplegic man have been jailed for a total of 10 years.
Sarah Harcourt, 35 of Gypsy Close and Adam Webster, 38 of William Kett Close took Stephen Dooley from the Vauxhall day centre before wheeling him to Chapelfield Park against his will and robbing him of £7.
Mr Dooley was then abandoned at the park and left to find his way back to the centre alone where staff contacted police.
Harcourt and Webster were arrested 40 minutes after and were identified by CCTV footage.
This afternoon at Norwich Crown Court, Webster was jailed fof six years and Harcourt was sentenced to four. Harcourt was also given another 12 weeks for a separate theft offence.
Stephen, 38, is wheelchair bound and only has function of one arm and can't speak. He gave evidence in court communicating through an interpreter and using a computer.
Speaking after sentence, Stephen's father, Rev Gerald Dooley, said he struggled to understand why two people would do this and said the crime was "way outside humanity".
He said: "You hear things all the time but you never think something like this will happen to your family."
His sister Vicky Shucksmith said she admired her brother's bravery to give evidence in court.
She said: "The trial lasted for three weeks and Stephen gave evidence over three days. It was hard for us and distressing to see but he wanted to do it. He didn't want them to get away with it."
Investigating officer Detective Constable Jim Starling, from Norwich CID, welcomed the sentences.
He said: "Harcourt and Webster are the lowest of the low. For two people to target a man who clearly has no means of defending himself or even calling for help is beyond belief.
"Stephen would have been visibly distressed during the incident, but this didn't stop them pushing him to the park and robbing him of a few pounds before abandoning him on the street.
"Stephen needs constant care and is never alone and, understandably, has been left very distressed by what happened. We've got to know Steven through the course of the investigation and it's incredible to see the improvement he's made.
"Harcourt and Webster are prolific offenders and it's pleasing to see that both will be behind bars for several years where they belong."
Detective Sergeant Richard Dickinson also worked on the case and praised Stephen and his family for their patience and perseverance during the enquiry and subsequent trial.
He said: "Giving evidence in court is a traumatic experience for any victim of crime, but it's particularly poignant in Steve's case as he has limited means of communicating. He did extremely well and I believe his evidence was crucial in securing a guilty verdict."
14 men have been jailed for a total of 82 years for a cannabis operation in Norfolk and Kent worth £25 million a year.
Police say they're treating damage at a butchers in Norwich as a hate crime.
A seven-year-old girl drowned at her best friend's birthday pool party while there was no lifeguard on duty, an inquest heard.
Police say search of the Milton landfill site cannot continue after officers conduct fingertip search since March
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C-Level Reporting of Your B2B Marketing Success
C-Level executives want to invest more in Growth Marketing, but have traditionally lacked the confidence to do this. Today, Marketing is becoming more data-driven and that means less guessing about where to invest your time and money to accelerate growth. Your CEO and CFO want to invest in growth and are willing to spend more in marketing as long as the plan is well thought through and measurable. By leveraging Captora's reporting application and big-data platform, you can now report trended results across your entire funnel so your success is clear across your executive team.
Now you have the data to support your future growth investment plans. And think about it - an additional dollar of revenue is worth much more than an additional dollar of expense. Most growth companies' market capitalization (enterprise value) is 3-to-10 times their revenue level. That means if you spend $3m in additional marketing expense to generate $1m of new revenue, it should be an easy "yes" as long as your plan and reporting are trusted by your executive team.
To learn more about how Captora's Marketing Platform and how to report C-Level results, click on the offer and read our eBook!


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Company: Public Profit
Project Overview: logo, letterhead, envelope, postcard
An Oakland company that specializes in helping nonprofits, government, and social ventures measure and manage what matters by providing consulting services. They needed an identity that would work for their range of clients (nonprofits, school districts and funders) and reflect their strength in helping their clients use data well.
We wanted to stray away from any icons that would narrowly define Public Profit's work, so we went with the idea that less really is more. An elegant typeface simply bracketed together provided a way for the breadth of their work to be able to show. We continued this simplicity with open and clean design, which allowed us to go with strong complimentary colors that stand out.


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|Message view
|« Date » · « Thread »
|« Date » · « Thread »
|"Len Popp" <len.p...@gmail.com>
|Re: [OT] Why can NOT run Tomcat on my Laptop
|Sat, 11 Oct 2008 21:45:38 GMT
On Sat, Oct 11, 2008 at 16:45, André Warnier <email@example.com> wrote: > Caldarale, Charles R wrote: >>> >>> From: IceManPat [mailto:firstname.lastname@example.com] >>> Subject: Re: Why can NOT run Tomcat on my Laptop >>> >>> C:\>set JAVA_HOME =D:\Program Files\Java\jdk1.6.0_07 >> >> The problem is you have a space before the equals sign; that makes the >> environment variable "JAVA_HOME " (note the trailing space), rather than >> "JAVA_HOME". Take the space out. >> >> Despite Martin G's dire warning, spaces in the value of JAVA_HOME are not >> a problem; spaces at the end of the environment variable name are. >> > Allowing spaces in directory and file names was a nutty idea which > has cost more over time than the Iraq war and current financial debacle > combined. It should be re-regulated. And yet, as Charles pointed out, it's not a problem for the genius Tomcat developers! (Re-read his message - the problem is a space in the *name* of the env variable, not the directory. All of us Windows users have no problem with Java being installed in the default "Program Files" directory.) -- Len
|View raw message


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Sahara Force India recorded its first double points finish of the season with Paul di Resta securing seventh place and Nico Hulkenberg ninth in the dramatic and rain-hit Malaysian Grand Prix, in Sepang on Sunday.
The 55-lap race, won by Ferrai's Fernando Alonso, was stopped after first eight laps due to heavy rain.
At the break, India's Narain Karthikeyan was in P10 which meant that Hispania had ended a lap within points for the first time. Karthikeyan had not gone into the pit lane at the time for wet tyres as most of the drivers had done.
However, Karthikeyan eventually finished 21st, just ahead of team-mate Pedro de la Rosa.
Sergio Perez brought about the best-ever finish for Sauber after finishing second, ahead of pole-sitter Lewis Hamilton in a McLaren.
Force India enjoyed a good result by picking up eight points after Resta started in the 14th position on the grid and Hulkenberg in the 16th.
Mercedes' Michael Schumacher was last among points in the 10th position while defending champion Sebastian Vettel of Red Bull was behind him, finishing outside the points bracket.
Di Resta said that had it not been for the clash with Williams' Pastor Maldonado, he could have done even better. He was, although, satisfied with the result.
"I did pretty well until Maldonado hit me just before the red flag, which spun me around and cost me about three places. At the restart our pace was very good on the intermediate tyres and we were able to get well up the order and stay with the leading group.
"We were a bit lucky with some of the issues for cars ahead, but in races like this you need a bit of luck and given where we started I think we can come away from here very happy with this result," the Scot said.
Hulkenberg too was happy about having won his first points of the season that too in treacherous conditions.
"I think today's performance with two cars in the points is a great team result and I'm very happy to pick up my first points of the season," he said.
"I have to say that the wet conditions today were some of the worst I've raced in and the visibility was almost zero before the race was stopped.
"It was much better when the race restarted, but I struggled with the balance of the car on the intermediates, so it was just a case of trying to stay in the points and catch Vergne ahead," the German said.


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This is my first blog in a series of blogs to help educate people on the true history of our political parties, government, as well as historical figures. To bring to light history that has been lost, forgotten, and purposely omitted in our schools to ensure our children do not know the full truth which in doing so prevent them from making truly informed decisions with politics, state rights, individual rights, our countries history, and the truth behind our different cultures and the lies that have been told or rather the truths that have not been told in order to sway our citizens thought processes towards history and matters of race and government. Furthermore, I want to stress that my political beliefs or bias, and what party I represent play no role in what is written in these forthcoming posts, they are purely for educational purposes only.
It is very apparent today that the Democratic and Republican parties are very much polar opposites with very few party lines that cross a middle ground, but they actually used to be one, single party called the Democratic-Republican Party. The party was organized by James Madison and Thomas Jefferson in 1791 to stand against the Federalist Party in upcoming elections. It supported states' rights and the literal and strict interpretation of the Constitution. Because of a heightened fear of anything resembling England's monarchy, they contested elitism. They hated the Federalists, the wealthy aristocrats who wanted a national bank and increased national governmental power over state governments. Because of these fears the popularity of the party increased throughout the 1790s.
In 1801, Thomas Jefferson's election into office successfully brought the Democratic-Republican Party to power. The Federalist Party lost the majority of support and disbanded after the War of 1812 leaving them without opposition. However from 1815 to 1832 they started to make mistakes. They had no pressure from anyone anymore and so States began nominating local electors, with their own personal interest in mind. The party eventual split up in 1828 into the modern Democratic Party and the Whig Party. Andrew Jackson and Martin Van Buren led the Democratic Party along with farmers, urban laborers, and Irish Catholics. They opposed aristocracy, a national bank and pushed to modernize American industry instead of family-based agriculture. They also supported the Mexican-American War and expansion into the western territories. They also opposed anti-immigration legislation and monopolies. While the Whig Party was not as popular it did have access to more wealth giving to more funding and influence.
In 1848, the Democratic National Committee or DNC was created and General Lewis Cass was nominated for candidacy however he lost to the Whig candidate General Zachary Taylor. The reason for this win was due to the Free Soil Party of New York. It was established for one reason and that was to oppose the expansion of slavery into the new western states. It was made up of both Whig and Democratic members who felt that free men on free soil should remain free. They also fought for the rights of free men in many Midwest and northern states and because of this division many democrats voted for General Taylor.
In 1850, Democratic members of Congress passed what is known as the Compromise of 1850, which was a set of bills that attempted to prevent civil war over slavery. The Compromise essentially outlawed slavery in the Western states; however, it also included a bill called the Fugitive Slave Act of 1850, which demanded that slaves who ran away to northern states be returned to their "masters" in the South.
Because of division over issues of slavery and anti-immigration in 1852 the Whig Party disbanded leaving the path for Democrat Franklin Pierce to be elected followed by James Buchanan in 1856. Those Democrats who opposed slavery eventually left the party and joined those from the Whig Party in the North to form the Republican Party in 1854.
In 1860, the anti-slavery Republican Party had become immensely popular and delivered their message of anti-slavery and how the Democrats were pro-slavery. This message got Abraham Lincoln elected as the first Republican President.
By the 1900s, the Democratic and Republican parties became the two polar opposite parties we have today. Despite this it is important to remember they started as the same party with the purpose to create a government that is run for and by its citizens.
This is only the beginning I have plenty more to talk about in regards to the origin and platform of the American Democratic Party. So stay tuned for future posts.


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Nominees for the Metropolitan Fashion Awards, which honor outstanding achievements in fashion and costume design in motion pictures, TV and live performance, were revealed on Wednesday.
The winners will be revealed during Metropolitan Fashion Week's closing gala on Oct. 1 at Warner Bros. Studios in Hollywood. Categories include Costume Designer of the Year (Motion Pictures), Costume Designer of the Year (TV), Costume Designer of the Year (Live Performance), Celebrity Stylist of the Year, Fashion Program of the Year, Fashion Commentator of the Year, Fashion Blogger of the Year and Outstanding Ensemble for Make-Up Artists and Hair Stylists on a single program.
"We are so excited to receive a record number of entries from all over the world. The enthusiastic response of the entertainment and fashion community is remarkable. These awards are unique because they also embrace fashion bloggers, stylists, commentators and programs that cover the art and talents of both fashion and costume industries," Metropolitan Fashion Week founder Eduardo Khawam said in a statement.
Cirque du Soleil Las Vegas will receive the 2016 Metropolitan Fashion Excellence Award for their outstanding work in costume design during a special presentation.
In addition to the awards presentation, the closing gala will also feature a runway show with more than 150 one-of-a-kind, avant-garde costumes and couture evening gowns created by national and international celebrity costume and fashion designers.
Keep reading to see the full list of nominees.
Metropolitan Fashion Excellence Award
Cirque Du Soleil Las Vegas
Fashion Blogger of the Year
Blake Scott: The Scott Effect
Dulce Candy: Dulce Candy
Olivia Culpo: Swavy.com
Shea Marie: Peace Love Shea
Tommy Lei: MYBELONGING
Celebrity Stylist of the Year
Leslie Fremar
Micaela Erlanger
Rob Zangardi & Mariel Haenn
Samantha McMillen
Tara Swennen
Fashion Commentator of the Year
Jeannie Mai
Jomari Goyso
Giuliana Rancic
Melissa Rivers
Fashion Program of the Year
E!'s Live From The Red Carpet: The 88th Annual Oscars
Entertainment Tonight
Miss Universe
Nuestra Belleza Latina
Costume Designer of the Year — Live Performance
William Ivey Long: "Grease: Live"
Brandon Michael Currie: "The Wiz Live!"
Suttirat Anne Larlarb: "Macbeth"
Thierry Mugler: "Zumanity: Cirque Du Soleil"
Costume Designer of the Year — Motion Pictures
Tony Acosta Jr.: "Captain America: Civil War"
Angus Strathie: "Deadpool"
Consolata Boyle: "Florence Foster Jenkins"
Sanja Hays: "Star Trek Beyond"
Ruth Myers: "The Legend of Tarzan"
Costume Designer of the Year — TV
Lou Eyrich: "American Horror Story: Hotel"
Doctor Thorne - Designer: Colleen Kelsall
Paolo Nieddu: "Empire"
Roland Sanchez: "11.22.63"
Alejandro Gastelum: "El Señor de los Cielos"
Marie Schley: "Transparent"
Outstanding Ensemble for Makeup Artists and Hair Styling
"American Horror Story: Hotel"
"Dancing With the Stars"
"Grease: Live"
"The People v. O.J. Simpson: American Crime Story"
"Star Trek Beyond"
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