When a Prediction Changes What Happens Next
A prediction is often followed by an action. A model may identify policyholders who appear likely to miss a payment, after which staff contact some of them. The contact could help prevent missed payments. If it does, the model’s prediction may be followed by an outcome different from the one that would have occurred without contact.
This creates an important distinction: the outcome recorded in historical data is the outcome under the actions that were actually taken. It is not automatically the outcome that would have occurred if nobody had acted, nor does it necessarily predict outcomes under a new action process. To interpret performance, specify the action context as well as the outcome window and unit of prediction.
For example, if \(Y\) indicates whether a payment is missed during a defined period, then \(Y(1)\) is the indicator under contact and \(Y(0)\) is the indicator without contact. A person has only one observed outcome in that period, even though both potential outcomes are useful for defining the question. The unobserved potential outcome is counterfactual: it describes what would have happened to that same person under the other action.
The Observed Outcome Depends on the Action Process
Let \(X\) denote information available when the prediction is made. Define \(e(x)=P(A=1\mid X=x)\), the probability that a unit with information \(X=x\) receives the action. The historical mean outcome among units with \(X=x\) is a mixture of outcomes in the two action groups.
This expression uses the mean potential outcome among those actually receiving each action. It does not, in general, use the mean potential outcome for the entire \(X=x\) group. If staff choose whom to contact using risk information not represented in \(X\), the contacted and uncontacted groups may differ even after conditioning on \(X\).
The formula follows by splitting the observed outcome mean according to the action actually received. In the first group, consistency makes \(Y=Y(1)\); in the second, it makes \(Y=Y(0)\).
Write the conditional mean as the sum of the means in the treated and untreated groups, weighted by their conditional probabilities.
Among units with \(A=1\), the observed outcome is \(Y(1)\); among units with \(A=0\), it is \(Y(0)\).
The resulting means are \(E[Y(1)\mid A=1,X=x]\) and \(E[Y(0)\mid A=0,X=x]\), not necessarily the means for all units with \(X=x\).
The first equality is the law of total expectation applied to the two action groups. The second uses consistency. If one action is never observed for a particular value of \(X\), the corresponding group mean cannot be estimated from those data; this is a lack of action overlap for that value.
Worked Example: Historical Contact and Missed Payments
In a hypothetical historical cohort of 100 policyholders, 40 were contacted after a model flagged them and 60 were not contacted. During the stated follow-up window, 8 contacted policyholders and 18 uncontacted policyholders missed a payment. Calculate the observed event rate in each action group and overall. Explain what the overall rate does and does not tell an analyst.
Method. Calculate each group’s observed rate, then combine the groups using their shares of the cohort.
For contacted policyholders, the observed rate is \(8/40=0.20\), or 20%. For uncontacted policyholders, it is \(18/60=0.30\), or 30%.
The contacted share is \(40/100=0.40\), and the uncontacted share is \(60/100=0.60\). Therefore, the cohort’s observed rate is \(0.40(0.20)+0.60(0.30)=0.08+0.18=0.26\), or 26%. Checking directly, \(8+18=26\) missed payments out of 100, also 26%.
Interpretation. The 26% rate describes outcomes under the historical contact process, in which 40% of these policyholders were contacted. It does not reveal the rate that would have occurred if nobody had been contacted. Nor does the difference between 20% and 30% establish that contact reduced missed payments: the model may have directed contact toward policyholders with greater underlying risk.
Common mistake avoided. Treating the 26% historical rate as the no-contact rate assumes that contact could not affect the outcome. That assumption is precisely what is in question; the unobserved no-contact outcomes for contacted policyholders cannot be recovered by simply averaging the recorded outcomes.
Random Assignment Helps Separate the Action From Risk
When action assignment is randomized, the action groups are comparable in expectation with respect to both recorded and unrecorded pre-action characteristics. With consistency, adequate follow-up, and no interference between units, the difference in observed group means can estimate the average difference between the potential outcomes. Randomization does not make every group exactly identical, but it provides a basis for interpreting the difference as an action effect rather than merely a difference in who received the action.
Worked Example: A Randomized Reminder Pilot
A hypothetical pilot randomly assigns 200 account holders, 100 to receive a reminder and 100 to receive no reminder. During the same follow-up window, 12 in the reminder group and 20 in the no-reminder group miss a payment. Calculate the two observed rates, their difference, and the estimated number of missed payments that would be avoided if all 200 received reminders rather than none. State the assumptions behind the interpretation.
Method. Compare the randomized groups’ event rates. Because assignment is random, this difference estimates the average effect of the reminder under the assumptions stated below.
The reminder-group rate is \(12/100=0.12\), or 12%. The no-reminder rate is \(20/100=0.20\), or 20%. The estimated difference, reminder minus no reminder, is \(0.12-0.20=-0.08\), an 8-percentage-point lower event rate with reminders.
If the estimated rates applied to all 200 account holders, the expected number of missed payments with reminders would be \(200(0.12)=24\). With no reminders, it would be \(200(0.20)=40\). The estimated difference is \(40-24=16\) fewer missed payments. Equivalently, \(200(0.08)=16\).
Check. The group counts imply 12 observed missed payments in the reminder group and 20 in the no-reminder group. The calculation for all 200 in each scenario is a projection from the randomized group rates, not a count observed under both scenarios.
This interpretation relies on random assignment being implemented as intended, consistency between the assigned and received action being addressed, comparable outcome measurement and follow-up, and no material spillover from one person’s reminder to another person’s outcome. The estimated difference is not proof that every individual benefits. The common mistake is to claim that the 16 are known prevented events for particular people; they are an estimated aggregate difference between two scenarios.
Why a Risk-Based Action Can Mislead Evaluation
In routine practice, actions are often assigned using a prediction or staff judgment. The action group may then have a higher or lower observed event rate because it had different underlying risk before the action. Comparing observed outcomes across groups mixes the action’s possible effect with the way people were selected for action.
Worked Example: Selection Produces a Misleading Comparison
Consider 100 hypothetical customers with the same recorded value of \(X\). An unrecorded risk indicator divides them into 50 higher-risk customers, all contacted, and 50 lower-risk customers, none contacted. Suppose the potential event rates are as follows: among the higher-risk customers, the rate is 30% with contact and 50% without contact; among the lower-risk customers, it is 30% with contact and 10% without contact. Calculate the observed mean, the action-group means, and the overall mean of each potential outcome. Explain why the observed action-group difference does not identify the average action effect.
Method. First use the action assignment to identify which potential outcome is observed. Then average potential outcomes across both risk groups, giving each group weight \(50/100=0.5\).
The higher-risk customers are contacted, so their observed rate is their contact rate, 30%. The lower-risk customers are not contacted, so their observed rate is their no-contact rate, 10%. Thus the observed action-group means are \(E[Y\mid A=1,X]=0.30\) and \(E[Y\mid A=0,X]=0.10\). Since the groups are the same size, the overall observed mean is \(0.5(0.30)+0.5(0.10)=0.20\), or 20%.
Across all 100 customers, the mean potential outcome under contact is \(E[Y(1)\mid X]=0.5(0.30)+0.5(0.30)=0.30\). The mean under no contact is \(E[Y(0)\mid X]=0.5(0.50)+0.5(0.10)=0.30\). Their difference is \(0.30-0.30=0\), so the average effect in this constructed population is zero.
The correct observed-outcome mixture is \(0.5E[Y(1)\mid A=1,X]+0.5E[Y(0)\mid A=0,X]=0.5(0.30)+0.5(0.10)=0.20\), matching the observed mean. Substituting the overall potential-outcome means instead gives \(0.5(0.30)+0.5(0.30)=0.30\), which does not match the observed mean.
The discrepancy occurs because contact status is determined by the unrecorded risk group. The observed difference, \(0.30-0.10=0.20\), is not the average action effect, which is zero here. Common mistake avoided: using the overall potential-outcome means in the mixture without conditional exchangeability. The example’s potential-outcome rates are supplied to demonstrate the distinction; routine observed data alone do not reveal both potential outcomes for each person.
Implications for Predictive Evaluation
A model evaluated on historical outcomes is evaluated against labels generated under the historical action process. If contact, review, or another intervention changed outcomes, those labels reflect both the underlying circumstances and the action received. A model can therefore appear to overpredict risk among people who were contacted if the action helped prevent the event. That pattern does not, by itself, show that the model was poorly calibrated for the risk that would have existed without contact.
A change to the action process can also change the outcomes used to assess the model. For instance, if a larger share of flagged cases is contacted, the event rate among flagged cases may fall even if the model’s ranking of underlying risk is unchanged. Conversely, withdrawing an effective action could raise observed event rates. Evaluation should therefore say which process generated the outcomes and avoid treating a performance result under one process as automatically transferable to another.
This does not mean every model evaluation needs a randomized trial. It means the interpretation must match the evidence. A historical outcome rate can describe what happened under the historical process. To estimate what would happen under a different action policy requires additional design or assumptions; a simple comparison of groups selected for different actions is not enough. In particular, when assignment depends on information absent from the analysis, the observed data may not support a fair comparison of the potential outcomes.
- State what action, if any, follows the prediction.
- Define the outcome and its time window in relation to that action.
- Clarify whether the evaluation target is the outcome under the observed process or under a specified alternative.
- When comparing action groups, ask how assignment occurred and whether relevant selection information was recorded.
Check Your Understanding
For each question, distinguish what the observed outcomes establish from what would require additional assumptions.
- A hypothetical cohort has 30 contacted members, 6 of whom experience an event, and 70 uncontacted members, 21 of whom experience it. Calculate the rate in each group and overall. What does the overall rate describe? (Answer: 20.0% contacted, 30.0% uncontacted, 27.0% overall; the overall rate is under the observed contact process.)
- A randomized pilot has 80 units in each group. There are 8 events among units receiving an intervention and 16 among controls. Calculate the rate difference, intervention minus control, and the estimated event-count difference if applied to 160 units in both scenarios. (Answer: \(0.10-0.20=-0.10\), or 10 percentage points lower; an estimated 16 fewer events with intervention.)
- For a fixed \(X=x\), half the units receive an action, with observed mean \(E[Y\mid A=1,X=x]=0.24\); the other half do not, with observed mean \(E[Y\mid A=0,X=x]=0.36\). Calculate the observed mean. Can these figures alone establish that the action reduces the average outcome? (Answer: \(0.5(0.24)+0.5(0.36)=0.30\); no, not if action assignment may depend on risk beyond \(X\).)
- For a fixed \(X=x\), suppose \(e(x)=0.25\), \(E[Y(1)\mid A=1,X=x]=0.08\), and \(E[Y(0)\mid A=0,X=x]=0.20\). Calculate the observed mean using the correct mixture. (Answer: \(0.25(0.08)+0.75(0.20)=0.02+0.15=0.17\).)
- In a historical cohort, actions were assigned deterministically by a risk flag that was not retained in the data. Explain why comparing the observed outcome rates by action may not estimate the action effect, even after conditioning on the recorded predictors. (Answer: The omitted flag may influence both action assignment and outcomes, so the action groups can differ in underlying risk conditional on the recorded predictors; observed rate differences can combine selection and action effects.)