Yang J-S, Nam H-J, Seo M, Han SK, Choi Y, et al. 2011 OASIS: Online Application for the Survival Analysis of Lifespan Assays Performed in Aging Research. PLoS ONE 6(8): e23525.
Please use this paper to cite OASIS.

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Basic Survival Analysis

Statistical testing methods

Cox proportional hazards regression

Basic survival analysis

Input Format

• First column should be time (Required)
  
  
• Second column should be the number of deaths (Required)
  
  
• Other columns are the number of censored data withe different reasons (At least one column is required)

Output Format

Statistics for the estimation of observed survival

Statistical testing between samples

Input Format

Ouput Format

Test for the significance of difference in lifespan

1. Log-rank test

Mantel-Cox test, so-called log-rank test, is a kind of nonparametric test which is frequently used for the comparison of two survival functions through overall lifespan data. The log-rank statistics in two groups such as experiment and control is calculated as follows.

where d_i is the number of deaths in group 1, and e_i (estimated as (d_in_li)/n_i ) is the number of expected deaths in group 1. n_1i is the size of the population of group 1 at risk during the i^th interval, and ni is the total size of population at risk during the i^th interval.

P-value 0.00E+00 is provided when P < 1.0 * 10^-10.

2. Weighted log-rank test

One might want to put more emphasis on earlier deaths than the later ones or vice versa. To generalize log-rank test for these needs, Fleming and Harrington developed G(rho, gamma)-weighted log-rank test. The weighted test statistics is calculated by following equation.

where w_i represents the G(rho, gamma) weight defined as S(t)^rho(1-S(t))^gamma. Generally if rho > 0 and gamma = 0, the test is sensitive against early difference, whereas if rho = 0 and gamma > 0, the test is sensitive against later differences.

P-value 0.00E+00 is provided when P < 1.0 * 10^-10.

3. Fisher's exact test

Fisher's exact test is frequently used in survival analysis. To test different survival functions at specific time point instead of overall lifespan, the program can calculate the probability of observed data with Fisher's exact test at different time points as following formula.

where a and b are the number of living subjects in group 1 and group 2 respectively and c and d are that of dead subjects in group 1 and group 2 respectively at specific time t. P-value of Fisher's exact test was calculated with the sum of probabilities less than or equal to p_t of all combinations. Generally, 90% mortality is used for Fisher's exact test. However, in some cases, comparisons between two datasets showed no statistically significant difference because of several reasons, including drastic death at old age. This analysis suggests that one might want to put more emphasis on earlier deaths than the later ones because later deaths might result from causes unrelated to normal ageing.

4. Kolmogorov-Smirnov test

While the log-rank test is commonly used for comparing survival data between samples, it is optimized for special assumptions on the underlying distributions such that the hazard ratio or relative risk / is constant in time t. In that case, a log-rank test generally gives optimal results. However, for considering general situation, a generalized test that does not depend on a special underlying distribution is needed. The Kolmogorov-Smirnov test is an appropriate solution for this purpose so that it robustly works in the condition where the hazard functions and cross over through time t. The Kolmogorov-Smirnov test is based on the following equation.

where sup represents a supremum of a set which gives smallest real number that is greater than or equal to every number in the set and D represents the largest absolute vertical deviation. OASIS adopted surv2.ks function implemented in R packages (R Development Core Team and contributors worldwide, 2008) to provide Kolmogorov-Smirnov test. We note that the Kolmogorov-Smirnov test in OASIS is not applicable to survival data that contain tied observations [eg. multiple events (death or failure) during an observed time interval]. OASIS provides a warning message if there is any tied observation in survival data within or between samples.

5. Neyman's smooth test

Another test capable of detecting a wide spectrum of alternatives is Neyman's smooth test. It is developed to test the homogeneity of two different survival data by comparing a null model, S₁(t) = S₂(t), and various alternative models. The alternative models embedded the null model with Legendre polynomials based on Neyman's goodness-of-fit idea as following equation.

where is a parameter set of bounded functions which are modelling possible difference between S₁(t) and S₂(t). Therefore, if , then null hypothesis is accepted. Since the Neyman's smooth test selects optimal smooth model in Legendre polynomials with Schwarz's selection rule, it is different from Kolmogorov-Smirnov test in the respect of providing an idea of the types of difference between two survival data. The selected dimension represents type of difference between S₁(t) and S₂(t). If the selected dimension (d) is 1, it suggests that S₁(t) and S₂(t) are different from each other by the constant hazard ratio. If the selected d is 2, then the relationship between two samples is likely to be monotonic. If the selected d is 3, then relationship between two samples is likely to have convex or concave form. OASIS adopted surv2.neyman function implemented in R packages to provide the Neyman's smooth test. Similar to the Kolmogorov-Smirnov test, the Neyman's smooth test is not currently applicable when there are tied observations in survival data.

6. Chow test

Chow test, a variant of F-test, was invented by economist Gregory Chow to test whether the coefficients in two linear regressions on different data sets are same or not. This test is generally used for detecting structural break that is an unexpected shift in time series data. In the OASIS, we used this analysis for detecting structural differences between two different log cumulative hazard functions by using the following equation.

where RSS_p represents the sum of squared residuals from the pooled log cumulative hazard data. RSS₁ and RSS₂ represent the sum of squared residuals from two different log cumulative hazard data respectively. N₁ and N₂ are the number of observation in each data and k is 3 which is the total number of parameters of linear regression model. The test statistic follows the F distribution with k and N₁ + N₂ - 2k degrees of freedom. OASIS adopted chow.py function implemented by Dr. Ernesto P. Adorio in http://adorio-research.org/wordpress/?p=1789.

7. Survival time F-test

We provide survival time F-test, which is used to examine whether two normal populations have same variance or not. Because censored data are generally used in survival analysis, one can estimate the number of dead animals using survival function S(t) and then perform F-test for the comparison of variance between two different survival data.

The F-test is used under the condition that the survival times of individual follow normal distribution. As a normality check method for the given dataset, we provide the Shapiro-Wilk test in the OASIS website. If a P-value from the Shapiro-Wilk test is smaller than 0.01, then the chance of survival data following the normal distribution is less than 0.01, and therefore the results of F-test are not applicable. Thus, we provide warning message in this case.

8. Partial slopes Ranksum test

We devised another statistical test method for comparing the difference of slope of two log cumulative hazard plots. We calculated partial slopes of the log cumulative hazard plot. With null hypothesis that two different log cumulative hazard plots have same slopes, we conducted rank-sum tests with set of partial slopes as following definitions.

The partial slopes rank-sum test is based on non-parametric statistics that requires sufficient number of samples (in this case, partial slopes) for the reliable analysis. Since a partial slope is defined as the changes in log cumulative hazard divided by the corresponding change in survival time between two neighbouring time points, the number of observed time points rather than the total sample size is important for the non-parametric analysis. To obtain statistically meaningful results, at least six observed time points are needed.

,where D₁ and D₂ are set of partial slopes of each group. These sets are compared with ranksum test.

Cox proportional hazards regression

Input Format

• First column should be time (Required)
  
  
• Second column should be the status of events (Required)
  
  
• Other columns are considered as the values of risk factors (At least one column is required)
  
  

Ouput Format

Evaluating the effect of several risk factors

1. Partial likelihood estimator

OASIS provides two kinds of regression methods using partial likelihood estimator (PLE) and robust estimator through coxr function in R package. The PLE is generally used to estimate the regression coefficients by maximizing the partial likelihood

where is a k x 1 vector of regression coefficients, X_i is a k x 1 vector of risk factors, and ti is a observed time of event. To estimate , which maximizes the partial likelihood, Cox's estimator solves the following score function that is the derivative with respect to of the log partial likelihood function.

where t_i is a observed time of event and equals 1 for the deaths observed and 0 otherwise.

2. Robust estimator

While the PLE is commonly used for Cox proportional hazards regression, it is known to be sensitive to outliers and the deflection of underlying assumption. Especially the PLE method is strongly influenced by large values of t exp(^TX). To reduce the influence of large values of t exp(^TX), Minder and Bednarski introduced smooth modification to the score function of the partial likelihood method,

where smooth function A(t_i, X_i) is defined as A(t_i, X_i) = M-min( M, t_i exp(^TX_i)) and M is a 95th percentile of the samples {t₁ exp(^TX₁), ..., t_n exp(^TX_n)}. The smooth functions in the outer and inside sum have an reducing effect on large values of t exp(^TX) and ^TX respectively. OASIS adopted coxr function implemented in R packages (R Development Core Team and contributors worldwide, 2009) to provide the robust estimation. In the coxr function, a better solution for smooth function A(t_i, X_i) is used by using (t)exp(^TX) instead of heuristic t exp(^TX), where (t) is -logS(t).