Failure-Time Statistics and Data Quality

Problem Statement: **Primary relation:** `R(t)=1-F(t); h(t)=f(t)/R(t); H(t)=-ln R(t).` The relation should first be read in words and translated into a dimensional statement. Identify which quantities are probabilities, time or cycle measures, rates, costs, loads, engineering scores or empirical indicators. If an input is a probability, its feasible interval is from zero to one; if it is an event intensity, it must be nonnegative and accompanied by a unit such as failures per operating hour. The output should be interpreted at the same stated boundary and over the same mission or reporting period as its inputs. **Reasoning sequence.** (i) Define the asset and the function; (ii) write all inputs and their units; (iii) choose the simplest physically defensible model; (iv) perform the computation using an independent check; (v) compare the result with limiting cases; (vi) state uncertainty and model boundaries before recommending maintenance. For example, the limit of a no-failure event probability as an assumed event rate approaches zero should be consistent with ordinary experience. A result outside its physically possible range suggests either an arithmetic fault, incompatible units or a model used outside its assumptions. **Worked engineering example.** If 90 of 100 initially identical components survive to 1,000 hours in a complete uncensored study, empirical survival is 0.90 at that time; a hazard value cannot be obtained from this single proportion without additional temporal assumptions. After evaluating the arithmetic, report what the result means and—equally important—what it does not mean. Distinguish a hypothetical prediction from a measured population statistic. If a numerical example uses a notional fleet or synthetic data, it must not appear in an operator's work package as a measured site result. For any policy comparison, repeat the calculation using at least two plausible ranges for its most uncertain input and report whether the choice changes. ### Define the denominator Write down the reference population and the interval across which the observation has meaning. An event per machine-day differs from an event per operating hour if production schedules differ. A simulation that silently uses an assumed denominator produces a neat but potentially false comparison. ## 3. Where the model can fail A rising hazard indicates age-related selection within the modeled population, not proof that one physical wear mechanism has been identified. An important diagnostic practice is to seek evidence against the simplest explanation. A repair which temporarily restores output can disguise an uncorrected design, installation or operating error. If the apparent rate improves after a maintenance change, normalize against exposure and examine whether production load, weather, equipment population or reporting conventions changed in the same period. Correlation between a signal and a failure does not establish that intervening on the signal will prevent the failure.

Problem Statement: **Primary relation:** `R(t)=exp(-λt), MTTF=1/λ, λ>0.` The relation should first be read in words and translated into a dimensional statement. Identify which quantities are probabilities, time or cycle measures, rates, costs, loads, engineering scores or empirical indicators. If an input is a probability, its feasible interval is from zero to one; if it is an event intensity, it must be nonnegative and accompanied by a unit such as failures per operating hour. The output should be interpreted at the same stated boundary and over the same mission or reporting period as its inputs. **Reasoning sequence.** (i) Define the asset and the function; (ii) write all inputs and their units; (iii) choose the simplest physically defensible model; (iv) perform the computation using an independent check; (v) compare the result with limiting cases; (vi) state uncertainty and model boundaries before recommending maintenance. For example, the limit of a no-failure event probability as an assumed event rate approaches zero should be consistent with ordinary experience. A result outside its physically possible range suggests either an arithmetic fault, incompatible units or a model used outside its assumptions. **Worked engineering example.** For λ=2e-4 per hour and mission length 1,000 hours, R=exp(-0.2)=0.8187; the probability of first failure during the mission is 0.1813. After evaluating the arithmetic, report what the result means and—equally important—what it does not mean. Distinguish a hypothetical prediction from a measured population statistic. If a numerical example uses a notional fleet or synthetic data, it must not appear in an operator's work package as a measured site result. For any policy comparison, repeat the calculation using at least two plausible ranges for its most uncertain input and report whether the choice changes. ### Keep the physical mechanism visible An apparent trend in a chart is evidence about recorded variables, not automatically the explanation of a component failure. Ask which loads, contamination pathways, temperatures, controls, maintenance practices or environmental stresses could produce the observed pattern. ## 3. Where the model can fail Replacing an item solely due to calendar age does not improve its modeled exponential survival probability; repairs, infant mortality and imperfect replacement may violate the model. An important diagnostic practice is to seek evidence against the simplest explanation. A repair which temporarily restores output can disguise an uncorrected design, installation or operating error. If the apparent rate improves after a maintenance change, normalize against exposure and examine whether production load, weather, equipment population or reporting conventions changed in the same period. Correlation between a signal and a failure does not establish that intervening on the signal will prevent the failure.

Problem Statement: **Primary relation:** `R(t)=exp[-(t/η)^β], h(t)=(β/η)(t/η)^(β-1).` The relation should first be read in words and translated into a dimensional statement. Identify which quantities are probabilities, time or cycle measures, rates, costs, loads, engineering scores or empirical indicators. If an input is a probability, its feasible interval is from zero to one; if it is an event intensity, it must be nonnegative and accompanied by a unit such as failures per operating hour. The output should be interpreted at the same stated boundary and over the same mission or reporting period as its inputs. **Reasoning sequence.** (i) Define the asset and the function; (ii) write all inputs and their units; (iii) choose the simplest physically defensible model; (iv) perform the computation using an independent check; (v) compare the result with limiting cases; (vi) state uncertainty and model boundaries before recommending maintenance. For example, the limit of a no-failure event probability as an assumed event rate approaches zero should be consistent with ordinary experience. A result outside its physically possible range suggests either an arithmetic fault, incompatible units or a model used outside its assumptions. **Worked engineering example.** For β=2 and η=4,000 hours, survival at 2,000 hours is exp(-0.25)=0.7788. At η it is exp(-1)=0.3679 regardless of β. After evaluating the arithmetic, report what the result means and—equally important—what it does not mean. Distinguish a hypothetical prediction from a measured population statistic. If a numerical example uses a notional fleet or synthetic data, it must not appear in an operator's work package as a measured site result. For any policy comparison, repeat the calculation using at least two plausible ranges for its most uncertain input and report whether the choice changes. ### Separate requirement from measurement An engineer should be able to state the acceptance requirement before seeing the measurement. Otherwise thresholds tend to move after results are known. Preserve the original requirement, report any exception and obtain authorized approval for changes. ## 3. Where the model can fail A mixed population with poor installation quality and a normal population can mimic a broad lifetime pattern; segment by operating condition before declaring a universal β. An important diagnostic practice is to seek evidence against the simplest explanation. A repair which temporarily restores output can disguise an uncorrected design, installation or operating error. If the apparent rate improves after a maintenance change, normalize against exposure and examine whether production load, weather, equipment population or reporting conventions changed in the same period. Correlation between a signal and a failure does not establish that intervening on the signal will prevent the failure.

Problem Statement: **Primary relation:** `For independent competing mechanisms with survivals R1(t),R2(t), overall R(t)=R1(t)R2(t).` The relation should first be read in words and translated into a dimensional statement. Identify which quantities are probabilities, time or cycle measures, rates, costs, loads, engineering scores or empirical indicators. If an input is a probability, its feasible interval is from zero to one; if it is an event intensity, it must be nonnegative and accompanied by a unit such as failures per operating hour. The output should be interpreted at the same stated boundary and over the same mission or reporting period as its inputs. **Reasoning sequence.** (i) Define the asset and the function; (ii) write all inputs and their units; (iii) choose the simplest physically defensible model; (iv) perform the computation using an independent check; (v) compare the result with limiting cases; (vi) state uncertainty and model boundaries before recommending maintenance. For example, the limit of a no-failure event probability as an assumed event rate approaches zero should be consistent with ordinary experience. A result outside its physically possible range suggests either an arithmetic fault, incompatible units or a model used outside its assumptions. **Worked engineering example.** If corrosion-related survival over a mission is 0.96 and bearing-related survival is 0.90, independence gives 0.864, with an important assumption that the mechanisms do not share stress drivers. After evaluating the arithmetic, report what the result means and—equally important—what it does not mean. Distinguish a hypothetical prediction from a measured population statistic. If a numerical example uses a notional fleet or synthetic data, it must not appear in an operator's work package as a measured site result. For any policy comparison, repeat the calculation using at least two plausible ranges for its most uncertain input and report whether the choice changes. ### Distinguish data from decisions A probability is an input to a decision rather than a decision by itself. Two alternatives with equal failure probability can lead to different responses when consequences, restoration time and regulatory obligations are different. ## 3. Where the model can fail A component replaced after corrosion before fatigue occurs produces competing-risk censoring; naïve failure tallies do not identify the underlying latent fatigue lifetime distribution. An important diagnostic practice is to seek evidence against the simplest explanation. A repair which temporarily restores output can disguise an uncorrected design, installation or operating error. If the apparent rate improves after a maintenance change, normalize against exposure and examine whether production load, weather, equipment population or reporting conventions changed in the same period. Correlation between a signal and a failure does not establish that intervening on the signal will prevent the failure.

Problem Statement: **Primary relation:** `Kaplan–Meier estimator: S(t)=Π_(event times <=t)(1-d_i/n_i), with n_i at risk just before each time.` The relation should first be read in words and translated into a dimensional statement. Identify which quantities are probabilities, time or cycle measures, rates, costs, loads, engineering scores or empirical indicators. If an input is a probability, its feasible interval is from zero to one; if it is an event intensity, it must be nonnegative and accompanied by a unit such as failures per operating hour. The output should be interpreted at the same stated boundary and over the same mission or reporting period as its inputs. **Reasoning sequence.** (i) Define the asset and the function; (ii) write all inputs and their units; (iii) choose the simplest physically defensible model; (iv) perform the computation using an independent check; (v) compare the result with limiting cases; (vi) state uncertainty and model boundaries before recommending maintenance. For example, the limit of a no-failure event probability as an assumed event rate approaches zero should be consistent with ordinary experience. A result outside its physically possible range suggests either an arithmetic fault, incompatible units or a model used outside its assumptions. **Worked engineering example.** Three units fail at 100, 200 and 400 hours; one more is censored at 300. The risk set at the 400-hour event excludes the censored unit. Treating the censor at 300 as a failure forces survival too low. After evaluating the arithmetic, report what the result means and—equally important—what it does not mean. Distinguish a hypothetical prediction from a measured population statistic. If a numerical example uses a notional fleet or synthetic data, it must not appear in an operator's work package as a measured site result. For any policy comparison, repeat the calculation using at least two plausible ranges for its most uncertain input and report whether the choice changes. ### Audit model assumptions Every model has a region in which its assumptions make sense. Before using an equation in a recommendation, check units, boundary, independence assumptions, the way failures are counted, and whether repair changes the modeled population. ## 3. Where the model can fail A confidence interval is not the probability that the unknown fixed parameter lies within one particular computed interval under frequentist interpretation. An important diagnostic practice is to seek evidence against the simplest explanation. A repair which temporarily restores output can disguise an uncorrected design, installation or operating error. If the apparent rate improves after a maintenance change, normalize against exposure and examine whether production load, weather, equipment population or reporting conventions changed in the same period. Correlation between a signal and a failure does not establish that intervening on the signal will prevent the failure.