• Dr. Jian Qian
  • Department of Epidemiology and Biostatistics
  • School of Public Health and Health Sciences
  • University of Massachusetts Amherst
• Dr. Elizabeth Mormino
  • Department of Neurology
  • School of Medicine
  • Stanford University
• Dr. Rebecca Betensky
  • Department of Biostatistics
  • College of Global Public Health
  • New York University

• Motivation and background
• Existing works
• Proposed model
  • Conditional and unconditional permutation
  • Test statistics
  • \(p\)-value
• Simulation studies
• Cognitive and functional decline in aging study
• Conclusion
• Reference

• 490 cognitively normal older individuals (age \(\ge\) 66) from
  • Alzheimer's Disease Neuroimaging Initiative (ADNI, \(n\) = 198)
  • Australian Imaging Biomarkers and Lifestyle Study of Ageing (AIBL, \(n\) = 131)
  • Harvard Aging Brain Study (HABS, \(n\) = 161)
• Participants had a Clinical Dementia Rating (CDR) 0 at enrollment.
• Survival features:
  • Time to event time from baseline to progression to global CDR of 0.5
  • Truncation had a PET imaging scan with 1 year (\(n\) = 444)
  • Right-censoring global CDR \(<\) 0.5 by the end of study (8.4%)

• Standard survival approaches assume independence between time to event and truncation, these lead to substantial bias.
• Possible dependence:
  • Subjects appear to be completely cognitively normal may receive early PET scans as clean baselines.
• Non-monotone dependence:
  • Subjects have been declining slowly over extended follow-up receive late PET scans.

• \(X\) is the failure or event time
• \(T\) is the truncation time for \(X\)
• \(C\) is the right censoring time
• \(Y\) is the observed failure time: \(Y=\min(X, C)\)
• \(\delta\) is the censoring indicator: \(\delta = 1\) if \(X\leq C\) and 0 otherwise.
• The observed data are \((Y, T, \delta\mid Y \geq T)\)

• An example of a Left-truncated right-censored survival data

• Can we test the independence between failure time \((X)\) and truncation time \((T)\) nonparametrically?
• No information is observed when \(X\le T\)
• We can answer this question by testing for quasi-independence, \(X\perp_q T\)

• The knowledge of the distribution of a test statistics under null hypothesis is not required
• To illustrate the idea, we temporarily ignore right-censoring by letting \(C\to\infty\), the observed data is then \(\{(T_i, X_i); i = 1, \ldots, n\}\)

• In general, permutation test consists of the following procedures:

  1. Generate a large number of permuted data under null
  2. For each permuted data, compute a test statistics
  3. Compute a \(p\)-value

• When there is no truncation, all n! ways are equal likely under the null hypothesis of quasi-independence.
• Assume \(X_1\le \ldots \le X_n\), and define the permuted data as \(\{(T_i^\ast, X_i); i = 1, \ldots, n\}\)

• We consider two permutation algorithms in the presents of left truncation

  1. Conditional permutation Tsai (1990), Efron and Petrosian (1992)
  2. Unconditional permutation

• The conditional permutation procedure consists of the steps:

  1. Initialize with \(m = 1\)
  2. For \(X_m\), selects a \(T_m^\ast\) from \(\{i: T_i \leq X_i\}\)
  3. Remove \(T_m^*\) from \(\{T_1, \ldots, T_n\}\) and repeat step 2. with \(m= 2, \ldots, n\).

• Example Suppose the observed data consists of 4 observations: \(\{(X,T):(3,2), (5,1), (8,7), (9,6)\}\) then we have a total of four possible legal permutations:

• The unconditional permutation approach consists of the steps

  1. Permutes \(T\) across all subjects in the dataset
  2. Delete those with \(T_i^\ast>X_i, i = 1, \ldots, n\)

• Example Suppose Suppose the observed data consists of 4 observations: \(\{(X,T):(3,2), (5,1), (8,7), (9,6)\}\) then we have a total of \(4!=24\) possible legal permutations:

• Conditional permutation
  • Suffer from low power when test statistics are risk-set-based, but not fully determined by the sizes of risk sets.
• Unconditional permutation
  • Reduce sample sizes due to inadmissibility of some permuted observations but increase the size of the sample space
  • Easier to generate

1. Conditional Kendall's tau
   • A consistent estimator of \(\tau_c^*\) is \[\hat\tau_c^*=\frac{1}{M}\sum_{i = 1}^{n-1}\sum_{j=1}^n \mbox{sgn} [(Y_i - Y_j)(T_i-T_j)] I(\Lambda_{ij}),\]
   • Asymptotic properties are established via U-statistics Martin and Betensky (2005)
   • Powerful for monotone relationships, but may completely miss non-montone relationships
2. & 3. Minimally selected \(p\)-value: \(\mbox{minp}_1\), \(\mbox{minp}_2\)
   • Aim to detect non-monotone dependencies
   • Asymptotic variance is complicated

• We proposed to obtain \(\mbox{minp}_1\) \(p\)-value from:

  1. Partition the data into two groups: \(\{T<t\}\) or \(\{T>t\}\)
  2. Compute the log-rank statistic (\(p\) value) for the two groups
  3. Repeat 1. and 2. for \(t\in\{T_1, \ldots, T_n\}\)
  4. The \(\mbox{minp}_1\) test statistic (\(p\)-value) is the maximum (minimum) of these realization

• For every cut-point, require at least \(E\) events in the each group.

• An alternative is the minp 2 test:

  1. Partition the data into two groups: \(\{T\in(t-\epsilon, t+\epsilon)\}\) or \(\{T\not\in(t-\epsilon, t+\epsilon)\}\)
  2. Compute the log-rank statistic (\(p\)-value) for the two groups
  3. Repeat 1. and 2. for \(t\in\{T_1, \ldots, T_n\}\)
  4. The \(\mbox{minp}_2\) test statistic (\(p\)-value) is the maximum (minimum) of these realization.
• This allows for \(X\) to be associated with moderate \(T\) differently from small or large \(T\).

• Choose \(\epsilon\) so that each group retains at least \(E\) events.

• Let \(z_{\mbox{obs}}\) be the observed test statistic
• The exact permutation \(p\)-value is defined as \[p_N = \frac{\sum_{i=1}^NI(|z_i| \geq |z_{\mbox{obs}}|)}{N},\] where \(z_1, z_2, \ldots, z_N\) are all possible test statistic computed from permuted dataset.
• \(p_N\) can be approximated by \[\hat{p}_N = \frac{\sum_{i=1}^{N^*}I(|z^*_i|\geq |z_{\mbox{obs}}|)}{N^*} \approx \frac{\sum_{i=1}^{N^*}I(|z^*_i|\geq |z_{\mbox{obs}}|)+1}{N^*+1},\] where \(z^\ast_1, \ldots, z^\ast_{N^\ast}\) are the simpled permutation test statistics.

• Generate \((X,T)\) from a bivariate normal copula
  • \(X\sim\mbox{Weibull}(3, 8.5)\)
  • \(T\sim\mbox{exp}(0.2)\)
• Nine levels of dependence %measured by Kendall's tau \((\tau)\):
  • \(\tau=0, \pm0.2, \pm0.4, \pm0.6, \pm0.8\)
• Sample size after truncation: 100 and 200.
• Censoring times follow an independent \(\mbox{Uniform}(0, c)\)
  • \(0\%, 25\%,\) and \(50\%\) after truncation
• 5000 permutations
• 1000 replications
• We compare the rejection proportions at a significant level of 0.05

Consistency of the rejection proportions (0% censoring)

Timing results in seconds (0% censoring)

Rejection proportion with \(n = 100\):

• Generate \((\mid T- 2.5\mid, X)\) from a bivariate normal copula
  • \(T\sim\mbox{Uniform}(0, 5)\)
  • \(X\sim\mbox{Weibull}(3, 8.5)\)
• Nine dependence level
  • \(0, \pm0.2, \pm0.4, \pm0.6, \pm0.8\)
• Sample size after truncation: 100 and 200
• Censoring times follow an independent \(\mbox{Uniform}(0, c)\)
  • 0%, 25%, and 50% after truncation
• 5000 permutations
• 1000 replications
• We compare the rejection proportions at a significant level of 0.05.

Rejection proportion with \(n = 100\):

• A large number of cognitively normal older individuals (\(n\) = 490) aggregated across three independent observational cohort

  1. Alzheimer's Disease Neuroimaging Initiative (ADNI, \(n\) = 198)
  2. Australian Imaging Biomarkers and Lifestyle Study of Ageing (AIBL, \(n\) = 131)
  3. Harvard Aging Brain Study (HABS, \(n\) = 161)

• Investigate relationship between the detection time of abnormal beta-amyloid accumulation and the time to show cognitive decline.

• Tests (minp\(_1\) and minp\(_2\)) for quasi-independence that are powerful against nonmonotone dependencies.
• Inversion of dependence models in the presence of dependent truncation:
  • estimation and evaluation of the survivor function
  • estimation and evaluation of Cox and accelerated failure time regression models
• R package is availabe at Chiou (2017)

> install.packages("devtools")
> devtools::install_github("stc04003/permDep")

> library(permDep)
> args(permDep)
function (trun, obs, permSize, cens, sampling = c("conditional", 
    "unconditional", "is.conditional", "is.unconditional"), kendallOnly = FALSE, 
    minp1Only = FALSE, minp2Only = FALSE, nc = ceiling(detectCores()/2), 
    seed = NULL) 
NULL