--- title: "DecisionDrift: Auditing Temporal Drift in Repeated Decision Systems" author: "Subir Hait" date: "`r Sys.Date()`" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{DecisionDrift: Auditing Temporal Drift in Repeated Decision Systems} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r setup, include = FALSE} knitr::opts_chunk$set( collapse = TRUE, comment = "#>", fig.width = 7, fig.height = 4 ) library(DecisionDrift) set.seed(42) ``` ## Overview `DecisionDrift` provides tools for detecting, decomposing, and stress-testing **temporal drift** in repeated binary decision systems. It is designed for settings in which an institutional or algorithmic system issues a binary decision — approve or deny, flag or pass, intervene or not — repeatedly over time for the same or overlapping set of units. The package does not estimate causal effects and does not require full specification of the underlying decision rule. Instead, it treats the decision system as a **time-indexed stochastic process** and asks whether that process changed over time — and if so, how, when, and for whom. --- ## Formal framework ### 1. The decision system as a stochastic process Let: - $i = 1, \ldots, N$: units (individuals, cases, applications) - $t = 1, \ldots, T$: time points (waves, periods, rounds) - $D_{it} \in \{0, 1\}$: observed binary decision at time $t$ for unit $i$ - $G_i$: group membership (optional) The decision system is defined as the stochastic process: $$S = \{ D_{it} : i = 1, \ldots, N;\ t = 1, \ldots, T \}$$ Critically, $S$ is treated as a **generator of decisions** over time, not as a parametric model. This makes the framework applicable to hybrid human--AI systems where the decision rule $f_t$ is only partially observed. ### 2. The human--AI decision mechanism In many applied settings, decisions are produced by a joint human--AI process: $$D_{it} = f_t(X_{it},\ H_{it})$$ where $X_{it}$ are inputs (features, signals) available to the system, $H_{it}$ represents human influence (override, discretion, intervention), and $f_t$ is the time-varying decision mechanism emerging from the interaction of algorithmic and human components. Drift in this system means $f_t \neq f_{t'}$, or equivalently: $$\Pr(D_{it} = 1 \mid X_{it}, H_{it}) \text{ changes over time}$$ `DecisionDrift` audits changes in $f_t$ **without requiring its full specification**. This is the core methodological contribution. ### 3. Definition of temporal drift **Marginal drift** (level shift): a change in the marginal distribution of decisions across waves. $$\text{Drift exists if } \exists\ t_1 \neq t_2 \text{ such that } P_{t_1} \neq P_{t_2}$$ where $P_t(d) = \Pr(D_{it} = d)$. **Transition drift** (dynamic instability): a change in the conditional distribution given the prior decision. $$\text{Transition drift exists if } P_t(d_t \mid d_{t-1}) \text{ varies with } t$$ The distinction matters. A system can have **stable prevalence but unstable dynamics** (transition drift without marginal drift), or **shifting prevalence with stable dynamics** (marginal drift without transition drift). `DecisionDrift` separates these two phenomena. --- ## The four drift indices ### Decision Drift Index (DDI) The DDI estimates the linear trend in wave-level decision prevalence: $$p_t = \frac{1}{N} \sum_{i=1}^{N} D_{it}$$ $$\text{DDI} = \frac{\hat{\beta}}{\text{SD}(p_t)}$$ where $\hat{\beta}$ is the OLS slope of $p_t$ on $t$. Standardising by $\text{SD}(p_t)$ makes the DDI scale-invariant and comparable across datasets. **Properties:** - $\text{DDI} > 0$: system became more permissive over time - $\text{DDI} < 0$: system became more restrictive over time - $\text{DDI} = 0$: no linear trend in decision rate - Independent of $N$; stable under random reordering of units - Does not capture nonlinearity or abrupt shifts — this motivates the change-point module ### Transition Drift Index (TDI) For the step from wave $s$ to $s+1$, define uptake and reversal: $$u_s = \Pr(D_{i,s+1} = 1 \mid D_{is} = 0), \qquad r_s = \Pr(D_{i,s+1} = 0 \mid D_{is} = 1).$$ Because each row of the transition matrix sums to one, $u_s$ and $r_s$ describe it completely. Transition drift between consecutive steps is $$D^T_s = \tfrac12\left(|u_s - u_{s-1}| + |r_s - r_{s-1}|\right),$$ and the TDI is its mean over the $T-2$ pairs of consecutive steps: $$\text{TDI} = \frac{1}{T-2}\sum_{s=2}^{T-1} D^T_s.$$ **Properties:** - $\text{TDI} = 0$ when uptake and reversal are constant - Sensitive to policy or model updates that alter switching dynamics - Can be large while prevalence is flat (see *Flow decomposition* below) - Descriptive: it is never exactly zero in finite samples, so use `dd_global_test()` for a formal test ### Group Differential Drift (GDD) Let $\text{DDI}^{(g)}$ denote the DDI estimated within group $g$. For groups $A$ and $B$: $$\text{GDD}_{AB} = \text{DDI}^{(A)} - \text{DDI}^{(B)}$$ GDD measures **inequality in system evolution**, not static disparity. A positive $\text{GDD}_{AB}$ indicates the system became more permissive for group $A$ faster than for group $B$. A value near zero indicates parallel drift regardless of any baseline disparity. ### Cumulative Drift Burden (CDB) $$\text{CDB} = \sum_{t=2}^{T} |p_t - p_{t-1}|$$ CDB captures total accumulated volatility in the decision rate, regardless of direction. Unlike the DDI, CDB is non-zero even when drift oscillates (e.g., the system alternates between permissive and restrictive periods). A high CDB with a near-zero DDI indicates instability without a directional trend. --- ## Modules The five core modules each estimate one aspect of drift: | Module | Stochastic object estimated | Index | |---|---|---| | `dd_prevalence()` | $\Pr(D_t = 1)$ over time | DDI | | `dd_transition()` | $\Pr(D_t \mid D_{t-1})$ over time | TDI | | `dd_entropy_trend()` | Complexity of path distribution | --- | | `dd_group_drift()` | Heterogeneity in $\Pr(D_t)$ across $G$ | GDD | | `dd_changepoint()` | Structural break in process | --- | | `dd_decompose()` | Inflow and outflow behind each change in $\Pr(D_t = 1)$ | NEC, CMB, CTB | | `dd_group_decompose()` | Within-group vs. composition change | $D^C$ | | `dd_global_test()` | Constancy of uptake, reversal, prevalence | bootstrap test | Together they answer: **Did the system change, and how?** The robustness and sensitivity modules then ask: **Is that conclusion credible?** `dd_simulate()` generates panels with known drift for checking methods. --- ## Worked example ```{r data} # Simulate a system that became more permissive after wave 4. # Group A drifts faster than Group B. n_units <- 80 n_waves <- 8 probs_A <- c(rep(0.20, 4), rep(0.60, 4)) # sharp increase at wave 5 probs_B <- c(rep(0.25, 4), rep(0.40, 4)) # modest increase at wave 5 dat <- data.frame( id = rep(seq_len(n_units), each = n_waves), time = rep(seq_len(n_waves), times = n_units), decision = c( rbinom(40 * n_waves, 1, rep(probs_A, 40)), rbinom(40 * n_waves, 1, rep(probs_B, 40)) ), group = rep(c("A", "B"), each = 40 * n_waves) ) dp <- dd_build(dat, id, time, decision, group = group, event_time = 5L) print(dp) ``` ### Module 1 — Prevalence drift ```{r prevalence} prev <- dd_prevalence(dp) print(prev) plot(prev) ``` The DDI is positive and the slope is statistically significant, confirming that the overall decision rate increased over time. ### Module 2 — Transition drift ```{r transition} tr <- dd_transition(dp) print(tr) plot(tr) ``` The TDI captures the instability in switching and persistence dynamics that accompanies the structural break at wave 5. ### Module 3 — Entropy and stability trend ```{r entropy} et <- dd_entropy_trend(dp, window = 3L) print(et) plot(et) ``` ### Module 4 — Group-differential drift ```{r group} gd <- dd_group_drift(dp) print(gd) plot(gd) ``` Group A experienced faster drift than Group B (positive GDD). The gap trajectory shows divergence over time. ### Module 5 — Change-point detection ```{r changepoint} cp <- dd_changepoint(dp) print(cp) plot(cp) ``` Both the CUSUM and segmented regression methods identify a structural break near wave 5, consistent with the known event time. ### All four indices in one call ```{r indices} idx <- dd_indices(dp) print(idx) ``` ### Full audit The audit runs every module, one shared cluster bootstrap, and the global stability test. It does not attach labels such as "strong drift". Its conclusion states the test result, and any practical cut-offs are supplied by the user through `thresholds`. ```{r audit} aud <- dd_audit(dp, include_robustness = TRUE, R = 199, seed = 1, thresholds = c(TDI = 0.05, NEC = 0.10), verbose = FALSE) print(aud) ``` --- ## Flow decomposition and masking Prevalence can stay flat while the process underneath changes. The marginal-transition identity makes this exact: $$\pi_{s+1} - \pi_s = \underbrace{(1-\pi_s)\,u_s}_{\text{inflow}} - \underbrace{\pi_s\,r_s}_{\text{outflow}}.$$ If inflow and outflow rise together, prevalence does not move. The `"masked"` scenario in `dd_simulate()` does exactly this: uptake and reversal both rise from 0.08 to 0.32 over five steps while the positive rate stays near 0.5. ```{r masked-data} masked <- dd_simulate(n = 600, waves = 6, scenario = "masked", seed = 11) dpm <- dd_build(masked, id, time, decision) round(tapply(masked$decision, masked$time, mean), 3) ``` ```{r masked-flow} fl <- dd_decompose(dpm, R = 199, seed = 2) print(fl) plot(fl) ``` Net change (`net`) stays near zero at every step while turnover more than triples. In the summary, net endpoint change (NEC) and cumulative marginal burden (CMB) are small, but total turnover and cumulative transition burden (CTB) are large. Intervals come from a unit-level cluster bootstrap, which keeps each unit's full history together. Intervals for quantities defined as absolute values (NEC, CMB, CTB, TDI) cannot include zero and are biased upward when the true value is near zero; use the global test, not these intervals, to decide whether the process changed. ### Global stability test `dd_global_test()` tests whether uptake and reversal were constant over all steps. It uses the same centred cluster bootstrap and reports familywise-adjusted p-values and simultaneous bands for each consecutive change, so it also shows where the process changed. ```{r masked-test} gt <- dd_global_test(dpm, R = 199, seed = 3) print(gt) plot(gt) ``` Compare the prevalence trend test from `dd_prevalence()`: ```{r masked-prev} dd_prevalence(dpm)$trend$p_value ``` Expected prevalence is constant in this scenario, so a prevalence test has nothing to find; the transition test targets the change that did happen. A rejection says the decision-generating process was not constant. It does not say why, and it is not a causal effect. By default each step is compared with the one before it, which localises abrupt change. For gradual drift, `contrast = "baseline"` compares each step with the first one; small step-to-step changes then add up, so this version has more power against steady trends. ```{r masked-baseline} dd_global_test(dpm, contrast = "baseline", R = 199, seed = 3)$p_value ``` --- ## Within-group and composition change The overall rate is a weighted average of group rates, $\pi_t = \sum_g w_{gt}\pi_{gt}$. A change in $\pi_t$ can come from the group rates (within-group change) or from the group shares (composition change). `dd_group_decompose()` uses a symmetric midpoint decomposition, $$\Delta\pi_t = \sum_g \bar w_{gt}\,\Delta\pi_{gt} + \sum_g \bar\pi_{gt}\,\Delta w_{gt},$$ which is exact and does not depend on which wave is the baseline. In the `"composition"` scenario, both groups follow stable processes, but the higher-rate group A enters the panel over time, so its share grows. ```{r composition} comp <- dd_simulate(n = 600, waves = 6, scenario = "composition", seed = 12) dpc <- dd_build(comp, id, time, decision, group = group) gdc <- dd_group_decompose(dpc, R = 199, seed = 4) print(gdc) plot(gdc) ``` Almost all of the rise in the overall rate is composition change. A prevalence-only audit would attribute it to a change in the decision rule. The reverse also happens: group rates can move in opposite directions while the overall rate stays flat. Testing group-specific parameters with `by_group = TRUE` checks for that. Because the divergence is gradual, baseline contrasts are used. ```{r group-masked} gm <- dd_simulate(n = 800, waves = 6, scenario = "group_masked", seed = 13) dpg <- dd_build(gm, id, time, decision, group = group) round(tapply(gm$decision, gm$time, mean), 3) # pooled rate gtg <- dd_global_test(dpg, components = "prevalence", by_group = TRUE, contrast = "baseline", R = 199, seed = 5) print(gtg) ``` --- ## Simulation scenarios `dd_simulate()` provides stable, marginal, masked, abrupt, group-masked, and composition-change panels, plus custom uptake and reversal sequences. The script in `system.file("simulation", "mc_study.R", package = "DecisionDrift")` runs a Monte Carlo study comparing the prevalence trend test with the global stability test across the first four scenarios. --- ## Robustness and sensitivity A key claim of the `DecisionDrift` framework is that observed drift is not an artefact of analytic choices or data problems. The robustness and sensitivity modules test this claim directly. **Robustness** (`dd_robustness()`) tests whether the DDI sign and magnitude are stable across: - Balanced vs. unbalanced panels - Leave-one-period-out (LOPO) - Leave-one-group-out (LOGO) - Alternative minimum follow-up requirements A **fragility index** summarises the proportion of analytic variants in which the slope changes sign. Values near zero indicate robust conclusions. **Sensitivity** (`dd_sensitivity()`) probes vulnerability to plausible data problems: - Decision miscoding - Missing-wave attrition - Threshold shifts (when decisions derive from scores) - Subgroup composition shifts A **tipping-point estimate** identifies the smallest perturbation level at which more than 50 per cent of resampled conclusions flip sign. ```{r robustness} rob <- dd_robustness(dp, variants = c("lopo", "min_waves")) print(rob) plot(rob) ``` --- ## Conceptual positioning `DecisionDrift` is designed to occupy a specific and currently underserved position in the audit toolbox: > We define temporal drift as a change in the decision-generating process > over time and introduce a set of summary indices that quantify level > shifts, dynamic instability, subgroup divergence, and cumulative change. > By implicitly treating decision systems as evolving stochastic processes, > the framework enables longitudinal auditing of hybrid human--AI systems > without requiring full specification of the underlying decision rule. This positions the package as **semi-parametric auditing of dynamic systems** — distinct from full Markov modelling, time-series estimation, and cross-sectional fairness testing. --- ## Relationship to `decisionpaths` `decisionpaths` constructs and summarises longitudinal decision paths, answering *what happened*. `DecisionDrift` tests whether the system that generated those paths changed over time, answering *did the system change*. A third package in this ecosystem, `AIBias`, will ask whether inequality accumulated or amplified over repeated decisions. --- ## References Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. *Psychometrika*, 16(3), 297--334. Hait, S. (2026). Artificial intelligence as decision infrastructure: Rethinking institutional decision processes. Unpublished manuscript. Shannon, C. E. (1948). A mathematical theory of communication. *Bell System Technical Journal*, 27(3), 379--423.