March 23, 2019, at UTRGV

Our team

  • 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

Outline

  • 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

Aging study

  • 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%)

Challenges

  • 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.

Notations

  • \(X\) 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 \((Y, T, \delta\mid Y \geq T)\)

LTRC survival data

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

Quasi-independence

  • 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, \(X\perp_q T\)

Permutation approach

  • 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

Permutation approach

  • 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

Conditional permutation method

  • 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:

{(3, 1), (5, 2), (8, 6), (9, 7)}

{(3, 1), (5, 2), (8, 7), (9, 6)}

{(3, 2), (5, 1), (8, 6), (9, 7)}

{(3, 2), (5, 1), (8, 7), (9, 6)}

Unconditional permutation method

  • 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:

{(3, 1), (5, 2), (8, 6), (9, 7)}

{(3, 1), (5, 2), (8, 7), (9, 6)}

{(3, 1), (5, 6), (8, 2), (9, 7)}

{(3, 1), (5, 6), (8, 7), (9, 2)}

The two 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

Test statistics

  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

Minimally selected \(p\)-value: \(\mbox{minp}_1\)

  • 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.

Minimally selected \(p\)-value: \(\mbox{minp}_2\)

  • 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.

Computing \(p\)-values

  • 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.

Simulation 1: setups

  • 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

Simulation 1: convergence

Consistency of the rejection proportions (0% censoring)

Simulation 1: Timing results

Timing results in seconds (0% censoring)

Simulation 1: Rejection proportion

Rejection proportion with \(n = 100\):

Simulation 2: setups

  • 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.

Simulation 2: data

Simulation 2: rejection proportion

Rejection proportion with \(n = 100\):