Outline

• Motivation
• Quantifying Discriminability
• Real Data
• Discussion

Additional Content

Outline

• Motivation
• Quantifying Discriminability
• Real Data
• Discussion

Additional Content

What is Reproducibility?

• Reproducibility: ability to replicate, or reproduce, a conclusion
• serves as a "first-pass" check for scientific utility
• currently in a "reproducibility crisis"

How do we address the Reproducibility Crisis?

• fix post hoc analyses (e.g., $p$-values)?
• fix measurements (e.g., measurement reproducibility)?

Proposal: design experiments to maximize inter-item discriminability, rather than simply checking reproducibility after conducting the experiment

Outline

• Motivation
• Quantifying Discriminability
• Real Data
• Discussion

Additional Content

What do we want of our data?

If we measure a sample multiple times, then each measurement of that sample is closer to all the other measurements of that sample, as compared to any of the measurements of other samples.

Perfect discriminability

What do we want of our data?

Imperfect discriminability

What do we want of our statistic?

Discriminability is the probability of a measurement from the same item being closer than a measurement from a different item.

Discriminability Statistic: Step 1

• Compute $N \times N$ pairwise distance matrix between all measurements

Discriminability Statistic: Step 2

• For each measurement, identify which measurements are from the same individual (green boxes)
• let $\color{green}g$ be the total number of green boxes = 20

Discriminability Statistic: Step 3

• For each measurement, identify measurements from other individuals that are more similar than the measurement from the same individual (orange boxes)
• let $\color{orange}f$ be the total number of orange boxes = 84

Discriminability Statistic

• Discr = $1 - \frac{\color{orange}f}{N(N-1) - \color{green}g} = 1 - \frac{\color{orange}{84}}{20\cdot 19 - \color{green}{20}} \approx .77$

High discriminability: same-item measurements are more similar than across-item measurements

Discriminability is Construct Valid

Outline

• Motivation
• Quantifying Discriminability
• Real Data
• Discussion

Additional Content

What data will we be using?

• CoRR metadataset
• $N>1,700$ individuals imaged across $26$ different datasets
  • anatomical MRI and fMRI scans for each
  • Individuals are measured at least twice

Analysis Procedure

Process each measurement using $192$ different pipelines

1. Brain alignment (ANTs/FSL)
2. Frequency filtering (Y/N)
3. Scrubbing (Y/N)
4. Global Signal Regression (Y/N)
5. Parcellation (4 options)
6. Rescaling connectomes (Raw, Log, Pass-to-Rank)

$192 = 2 \times 2 \times 2 \times 2 \times 4 \times 3$

All options represent strategies experts consider useful

Pipeline impacts discriminability

Marginally most discriminabile options tend to be best global options

• Each point is the pairwise difference holding other options fixed (e.g., FNNGCP - ANNGCP)
• Best pipeline marginally (FNNGCP) is second best pipeline overall, and not much worse (2-sample test, p=.14) than the best pipeline FNNNCP
• We may not need to always try every pre-processing strategy every time

Selection via Discriminability improves inference

For each pre-processing strategy, for each dataset, compute:

1. Within-dataset Discr.
2. Demographic effects (sex and age) within the dataset via Distance Correlation (DCorr)
3. Within a single dataset, regress demographic effect on Discr.

Question: does a higher discriminability tend to yield larger effects for known biological signals?

Selection via Discriminability improves inference

Outline

• Motivation
• Quantifying Discriminability
• Real Data
• Discussion

Additional Content

Contributions

1. Discriminability quantifies the contributions of systematic and accidental deviations
2. Provide theoretical motivation for discriminability in connection with predictive accuracy
3. Formalize tests for assessing and comparing discriminabilities within and between collection strategies
4. Illustrate the value of discriminability for neuroscience and genomics (not discussed) data
5. Code implementations in python and R

Acknowledgements

• BioRxiv manuscript
• Code implementations in python and R

Outline

Additional Content

• Theory
• Other Reproducibility Statistics
• Limitations
• Extension: Discriminability Decomposition

Outline

Additional Content

• Theory
• Other Reproducibility Statistics
• Limitations
• Extension: Discriminability Decomposition

Population Discriminability

• population discriminability $D$ is a property of the distribution of measurements $D = \mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}))$
• Probability of within-individual measurements being more similar than between-individual measurements

Discriminability: unbiased and consistent

• Sample Discr. $=$fraction of times $\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''})$
• $i, j = 1, ..., n$ and $i \neq j$ for $n$ individuals
• $k, k', k'' = 1, ..., s$ and $k \neq k'$ for $s$ sessions
• Sample Discr. is an unbiased estimator of $D$
• Sample Discr. converges to $D$ (asymptotically consistent in $n$)

Connecting Discriminability to Downstream Inference

Assumption: Multivariate Additive Noise Setting

• $y_i \sim Bern(\pi)\;i.i.d.$,
• $\theta_i \sim \mathcal N(\mu(y_i), \Sigma_t)\;ind.$,
• (the individual means have a center which depends on the class)

Connecting Discriminability to Downstream Inference

Assumption: Multivariate Additive Noise Setting

• $y_i \sim Bern(\pi)\;i.i.d.$,
• $\theta_i \sim \mathcal N(\mu(y_i), \Sigma_t)\;ind.$,
• (the individual means have a center which depends on the class)
• $\epsilon_{i}^k \sim \mathcal N(c, \Sigma_e)\;i.i.d.$ and $ind.$ of $\theta_i$,
• $x_{i}^k = \theta_i + \epsilon_i^k$.
• (the measurements $x_i^k$ are normally dispersed about the individual means)

Connecting Discriminability to Downstream Inference

Suppose $(x_i^k, y_i)$ follow the Multivar. Additive Noise Setting, where $i=1, ..., n$ and $k=1,...,s$.

Theorem 1

There exists an increasing function of $D$, $f(D)$, which provides a lower bound on the predictive accuracy of a subsequent classification task

• $f(D) \leq A$, where $A$ is the Bayes Accuracy of the classification task

Consequence

• $D \uparrow \Rightarrow f(D) \uparrow$

Corollary 2

A strategy with a higher $D$ provably provides a higher bound on predictive accuracy than a strategy with a lower $D$

Consequence

Suppose $D_1 < D_2$, then since $f$ is increasing, $f(D_1) < f(D_2)$

Implication

We should use strategies with higher discriminability, as the worst-case for subsequent inference is better than a generic strategy with a lower discriminability

Simulation Setup

Discriminability and Accuracy

Discr. decreases proportionally with accuracy

Are data discriminable?

Is one dataset more discriminable than another?

Outline

Additional Content

• Theory
• Other Reproducibility Statistics
• Limitations
• Extension: Discriminability Decomposition

Intraclass Correlation Coefficient (ICC)

• can be thought of as looking at the "relative size" of the within-group vs total variance
• $y_i^k = \mu + \mu_i + \epsilon_i^k$
• let $\mu_i \sim \mathcal N(0, \sigma_b^2)$, and $\epsilon_i^k \sim \mathcal N(0, \sigma_e^2)$
• $ICC = \frac{\sigma_b^2}{\sigma_e^2 + \sigma_b^2}$
• $ICC \uparrow \Rightarrow$ between-group variance "contains" most of the total variance
• negative ICC? mean squared error-based estimator

Image Intraclass Correlation Coefficient (I2C2)

• simplest "multivariate extension" of ICC
• $y_i^k = \mu + \mu_i + \epsilon_i^k$
• let $\mu \sim \mathcal N(0, \Sigma_b)$ and $\epsilon_i^k \sim \mathcal N(0, \Sigma_e)$

• Wilk's $\Lambda = \frac{\det(\Sigma_b)}{\det(\Sigma_b) + \det(\Sigma_e)}$

• $I2C2 = \frac{tr(\Sigma_b)}{tr(\Sigma_b) + tr(\Sigma_e)}$

• "ratio of total variability accounted for between groups"

• Why I2C2 over Wilk's $\Lambda$? Ease-of-use for high-dimensional data

Fingerprinting Index (Finger.)

• "greedy discriminability"
• $Finger. = \mathbb P(\delta(x_i^1, x_i^2) < \delta(x_i^1, x_j^2) \;\forall\; i \neq j)$
• $\forall\; i \neq j$: this property must occur for every measurement in the second session

Distance Components (Kernel)

• "non-parametric ANOVA"

• total dispersion is the sum of between and within-sample dispersions ($B$ and $W$)

• $DISCO = \frac{\frac{B}{n - 1}}{\frac{W}{n\cdot s - n}}$

• "pseudo F" statistic

Outline

Additional Content

• Theory
• Other Reproducibility Statistics
• Limitations
• Extension: Discriminability Decomposition

Limitations

• experimental design is not "one-size-fits-all"
• Discriminability is not sufficient for practical utility
  • categorical covariates are meaningful but not discriminable
  • fingerprints are discriminable but not typically biological useful
• These statistics are not immune to sample characteristics
  • confounds such as age may inflate discriminability

Outline

Additional Content

• Theory
• Other Reproducibility Statistics
• Limitations
• Extension: Discriminability Decomposition

Extension: Discriminability Decomposition

Setting

$(x_{i}^k, y_i)$ $i=1, ..., n$, $k=1,...,s$, $y_i \in$ {$1, ..., Y$}

• associated with each individual, I have some other categorical covariate of interest, $y_i$, taking one of $Y$ possible values
• Can the population discriminability be decomposed as a sum of the within-group discriminabilities?

Within-Group Discriminability

• Let $D(y) = \mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) | y_i, y_j = y)$
• $D(y)$ is the group discriminability for group $y$
• "How discriminable are samples from group $y$?"

Within-Group Discriminability

• Let $D(y) = \mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) | y_i, y_j = y)$
• $D(y)$ is the group discriminability for group $y$
• "How discriminable are samples from group $y$?"
• Note that $W = \mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) | y_i= y_j)$= $\frac{\mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) , y_i = y_j)}{\mathbb P(y_i = y_j)}$ by def conditional probability

Within-Group Discriminability

• Let $D(y) = \mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) | y_i, y_j = y)$
• $D(y)$ is the group discriminability for group $y$
• "How discriminable are samples from group $y$?"

• Note that $W = \mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) | y_i= y_j)$= $\frac{\mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) , y_i = y_j)}{\mathbb P(y_i = y_j)}$ by def conditional probability

• Let $w(y) = \mathbb P(y_i=y_j = y)$ denote the within-group weights

• With $\omega = \sum_y w(y)$, then:

  $W = \frac{1}{\omega}\sum_y w(y) D(y)$ is the within-group Discriminability

Between-Group Discriminability

• Let $D(y, y') = P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) | y_i = y, y_j = y')$
• $D(y, y')$ is the between-group discriminability for groups $y$ and $y'$
• "How discriminable are samples from group $y$ vs group $y'$, and vice versa?"

Between-Group Discriminability

• Let $D(y, y') = P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) | y_i = y, y_j = y')$
• $D(y, y')$ is the between-group discriminability for groups $y$ and $y'$

• "How discriminable are samples from group $y$ vs group $y'$, and vice versa?"

• Note that $B = \mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) | y_i\neq y_j)$= $\frac{\mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) , y_i \neq y_j)}{\mathbb P(y_i \neq y_j)}$ by def conditional probability

Between-Group Discriminability

• Let $D(y, y') = P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) | y_i = y, y_j = y')$
• $D(y, y')$ is the between-group discriminability for groups $y$ and $y'$

• "How discriminable are samples from group $y$ vs group $y'$, and vice versa?"

• Note that $B = \mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) | y_i\neq y_j)$= $\frac{\mathbb P(\delta(x_i^k, x_i^{k'}) < \delta(x_i^k, x_j^{k''}) , y_i \neq y_j)}{\mathbb P(y_i \neq y_j)}$ by def conditional probability

• Let $b(y, y') = \mathbb P(y_i = y, y_j = y')$ denote the between group weights

• With $\beta = \sum_{y\neq y'} b(y,y')$, then:

$B = \frac{1}{\beta}\sum_{y \neq y'}b(y,y')D(y,y')$ is the between-group Discriminability

Discriminability Decomposition

• $D = \omega W + \beta B$
• Population discriminability is a weighted sum of within and between-group Discriminabilities
• Can look at how the within, or between, group discriminabilities compare
• $\frac{W}{D}$ ratio of within-group Discriminability and pop. discriminability
• $\frac{B}{D}$ ratio of between-group Discriminability and pop. discriminabillity
• are certain groups more discriminable than others?
• are certain between-group discriminabilities greater than others?
• "ANOVA-esque" or DISCO-esque"