Why Mathematics? | Akaike Information Criterion for Reliability Models: Log-Likelihood, Parameter Penalties, Delta AIC and Predictive Fit

Adding parameters cannot reduce the maximised likelihood when the simpler model is nested inside the larger one, so training fit has a built-in preference for flexibility. A two-parameter Weibull can fit at least as well as its one-parameter exponential special case. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. The improvement may follow random sample noise rather than a stable failure pattern. Separate fit reward from complexity cost before comparing models.

Keep the fitted likelihood, free-parameter list and candidate set together; an AIC column without those records cannot be audited. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use why maximum likelihood alone keeps rewarding complexity as the focus and test the worked illustration: A two-parameter Weibull can fit at least as well as its one-parameter exponential special case. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

Did You Know? Model-selection mathematics can be perfectly calculated and still be misleading when candidates use different observations, units or likelihood definitions. Fair comparison begins before the formula: one response, one analysis set, one observation model and a documented candidate set.

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AIC combines -2 times the maximised log-likelihood with a 2k penalty, where k is the number of independently estimated parameters counted under the chosen likelihood. With log L-hat = -120 and k = 3, -2 log L-hat = 240 and 2k = 6. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Alternative software may include different constants or dispersion parameters. Write the exact formula and parameter count used by the software.

Because the aim is predictive information rather than literal model truth, compare the winner with residual, calibration and mechanism evidence. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use aic as a penalised likelihood criterion as the focus and test the worked illustration: With log L-hat = -120 and k = 3, -2 log L-hat = 240 and 2k = 6. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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The worked calculation adds the fit term and the penalty rather than multiplying them or comparing raw likelihoods. AIC = 240 + 6 = 246. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. AIC has no useful absolute pass mark; 246 is interpreted relative to candidate models on the same data. Recalculate a second candidate before interpreting the number.

When a ranking changes after one candidate or observation is removed, report that sensitivity instead of hiding it behind a single minimum. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use completing the 246 calculation as the focus and test the worked illustration: AIC = 240 + 6 = 246. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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Exact failures contribute event densities and right-censored records contribute survival probabilities before the log-likelihood is maximised. A failure at t contributes f(t); a unit still operating at c contributes S(c). This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Treating censoring as failure changes both fit and AIC. Audit every observation's likelihood contribution.

A numerical penalty cannot decide whether a lifetime distribution makes physical sense in the system being studied. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use building the censored reliability likelihood as the focus and test the worked illustration: A failure at t contributes f(t); a unit still operating at c contributes S(c). Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

Primary research or authoritative source: https://doi.org/10.1109/TAC.1974.1100705. Use it to check the criterion's derivation or reliability procedure rather than copying a rule without its assumptions.

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Count every parameter estimated from the data that enters the fitted likelihood, including shape, scale, regression coefficients and an estimated dispersion or threshold when applicable. A Weibull regression with intercept, two covariates and shape may have four fitted parameters under one parameterisation. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Fixed offsets and externally known constants are not automatically counted like fitted coefficients. Match k to the actual optimisation vector.

Use the criterion as one layer in a transparent model-selection workflow, then test the selected model on information it did not fit. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use counting k without guessing as the focus and test the worked illustration: A Weibull regression with intercept, two covariates and shape may have four fitted parameters under one parameterisation. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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AIC differences are valid only when candidate criteria are based on the same observed response and comparable likelihood. Two lifetime distributions are fitted to the same 100 units and the same censoring records. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Dropping different rows or changing the time origin makes the AIC values answer different questions. Freeze the analysis set before fitting candidates.

Keep the fitted likelihood, free-parameter list and candidate set together; an AIC column without those records cannot be audited. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use the same-data rule as the focus and test the worked illustration: Two lifetime distributions are fitted to the same 100 units and the same censoring records. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

Did You Know? Model-selection mathematics can be perfectly calculated and still be misleading when candidates use different observations, units or likelihood definitions. Fair comparison begins before the formula: one response, one analysis set, one observation model and a documented candidate set.

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Conditional, marginal, partial and full likelihoods are not generally interchangeable because their normalising terms and targets differ. A Cox partial-likelihood AIC and a parametric full-likelihood AIC need special justification before direct comparison. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Similar printed labels do not establish a shared probability model. Record the exact likelihood represented by each software output.

Because the aim is predictive information rather than literal model truth, compare the winner with residual, calibration and mechanism evidence. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use the same-likelihood rule as the focus and test the worked illustration: A Cox partial-likelihood AIC and a parametric full-likelihood AIC need special justification before direct comparison. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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Subtract the smallest candidate AIC from every AIC so the best observed candidate has delta zero and the remaining differences express relative information loss. AIC values 246, 249 and 258 produce deltas 0, 3 and 12. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Rules of thumb for deltas are guides rather than universal evidence thresholds. Show the full candidate table and deltas.

When a ranking changes after one candidate or observation is removed, report that sensitivity instead of hiding it behind a single minimum. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use delta aic as the focus and test the worked illustration: AIC values 246, 249 and 258 produce deltas 0, 3 and 12. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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For the usual AIC interpretation, exp(-Delta/2) converts an AIC difference into a relative likelihood-like support measure within the declared candidate set. Delta 3 gives exp(-1.5), about 0.223. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. This is not the probability that a model is true. Label it relative support and name the candidate set.

A numerical penalty cannot decide whether a lifetime distribution makes physical sense in the system being studied. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use relative likelihood from a delta as the focus and test the worked illustration: Delta 3 gives exp(-1.5), about 0.223. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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Normalising exp(-Delta_i/2) across the candidates gives Akaike weights that sum to one for that set. For deltas 0 and 3, the weights are about 0.818 and 0.182. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Adding or removing a plausible candidate changes every weight. Publish the candidate list beside the weights.

Use the criterion as one layer in a transparent model-selection workflow, then test the selected model on information it did not fit. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use akaike weights as the focus and test the worked illustration: For deltas 0 and 3, the weights are about 0.818 and 0.182. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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The ratio of two Akaike weights summarises their relative support under the same analysis. A weight ratio 0.818/0.182 is about 4.49. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. It does not measure causal evidence, engineering safety or posterior odds under arbitrary priors. Use plain language tied to predictive information.

Keep the fitted likelihood, free-parameter list and candidate set together; an AIC column without those records cannot be audited. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use evidence ratios as the focus and test the worked illustration: A weight ratio 0.818/0.182 is about 4.49. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

Did You Know? Model-selection mathematics can be perfectly calculated and still be misleading when candidates use different observations, units or likelihood definitions. Fair comparison begins before the formula: one response, one analysis set, one observation model and a documented candidate set.

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Akaike's derivation targets an estimate of expected relative information loss between a fitted model and the unknown data-generating process. The criterion ranks approximating models rather than declaring one literally true. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Finite samples and misspecified likelihoods can weaken the approximation. Keep the information-loss interpretation conditional on assumptions.

Because the aim is predictive information rather than literal model truth, compare the winner with residual, calibration and mechanism evidence. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use aic and kullback-leibler thinking as the focus and test the worked illustration: The criterion ranks approximating models rather than declaring one literally true. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

Second authoritative source: https://www.itl.nist.gov/div898/handbook/eda/section4/eda4292.htm. Compare its definitions, data structure and limitations with the worked reliability example here.

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AIC can compare non-nested candidates when they use the same observations and a comparable likelihood for the same outcome. Weibull and lognormal lifetime models may be compared even though neither is a restriction of the other. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Different tail shapes can agree in-sample and diverge far outside the observed range. Add probability plots and tail sensitivity checks.

When a ranking changes after one candidate or observation is removed, report that sensitivity instead of hiding it behind a single minimum. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use non-nested models as the focus and test the worked illustration: Weibull and lognormal lifetime models may be compared even though neither is a restriction of the other. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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AIC can also rank nested models, but it asks a predictive-selection question rather than testing a null restriction at a fixed significance level. Exponential versus Weibull can be ranked by AIC. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. AIC selection and a likelihood-ratio test need not make the same choice. State whether the goal is prediction or hypothesis testing.

A numerical penalty cannot decide whether a lifetime distribution makes physical sense in the system being studied. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use nested models as the focus and test the worked illustration: Exponential versus Weibull can be ranked by AIC. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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A one-to-one reparameterisation changes labels but not the number of independent degrees of freedom or the maximised likelihood. Using log scale instead of positive scale leaves k unchanged. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Changing the data scale without a common Jacobian can change likelihood values. Distinguish parameter transformation from response transformation.

Use the criterion as one layer in a transparent model-selection workflow, then test the selected model on information it did not fit. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use parameter transformations do not change model dimension as the focus and test the worked illustration: Using log scale instead of positive scale leaves k unchanged. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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For continuous densities, changing hours to days changes log-density values; common changes may cancel in AIC differences only when every candidate uses the same transformed data and likelihood convention. All candidates are refitted after converting the same lifetimes. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Mixing units or omitted constants across software invalidates comparison. Use one time scale and one implementation.

Keep the fitted likelihood, free-parameter list and candidate set together; an AIC column without those records cannot be audited. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use time units and likelihood constants as the focus and test the worked illustration: All candidates are refitted after converting the same lifetimes. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

Did You Know? Model-selection mathematics can be perfectly calculated and still be misleading when candidates use different observations, units or likelihood definitions. Fair comparison begins before the formula: one response, one analysis set, one observation model and a documented candidate set.

Contents · Previous chapter · Next chapter

AIC chooses among supplied candidates even when every candidate fits poorly. The lowest AIC curve can still miss a bathtub-shaped hazard. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Relative best does not mean adequate. Inspect residuals, survival calibration and failure-mechanism plausibility.

Because the aim is predictive information rather than literal model truth, compare the winner with residual, calibration and mechanism evidence. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use aic is not a goodness-of-fit test as the focus and test the worked illustration: The lowest AIC curve can still miss a bathtub-shaped hazard. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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The ordinary AIC formula presumes a genuine likelihood; quasi-likelihood settings may require specialised criteria and variance adjustments. A count model shows more variability than its Poisson assumption. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Blindly reading a software AIC can under-penalise misspecification. Use a criterion derived for the fitted estimating framework.

When a ranking changes after one candidate or observation is removed, report that sensitivity instead of hiding it behind a single minimum. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use overdispersion and quasi-likelihood caution as the focus and test the worked illustration: A count model shows more variability than its Poisson assumption. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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When n is not large relative to k, the 2k penalty can be too optimistic and a justified corrected criterion may be preferable. A five-parameter model fitted to 40 units receives a noticeable AICc correction under the common formula. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. The correction and effective n depend on model assumptions. Check whether an AICc derivation applies.

A numerical penalty cannot decide whether a lifetime distribution makes physical sense in the system being studied. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use small-sample bias and aicc as the focus and test the worked illustration: A five-parameter model fitted to 40 units receives a noticeable AICc correction under the common formula. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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Predictions can be averaged using prespecified Akaike weights when uncertainty among plausible candidates matters. Weighted survival at 1,000 hours combines two model curves. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Averaging incompatible estimands or extrapolations can hide structural uncertainty. Average predictions on a common scale and show component curves.

Use the criterion as one layer in a transparent model-selection workflow, then test the selected model on information it did not fit. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use model averaging as the focus and test the worked illustration: Weighted survival at 1,000 hours combines two model curves. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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Searching many distributions, covariates and interactions on one sample creates selection optimism that the printed AIC table does not fully reveal. A wide candidate search finds a chance winner. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. AIC is not a substitute for resampling the whole selection pipeline. Bootstrap or cross-validate the declared search process.

Keep the fitted likelihood, free-parameter list and candidate set together; an AIC column without those records cannot be audited. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use internal validation after candidate search as the focus and test the worked illustration: A wide candidate search finds a chance winner. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

Did You Know? Model-selection mathematics can be perfectly calculated and still be misleading when candidates use different observations, units or likelihood definitions. Fair comparison begins before the formula: one response, one analysis set, one observation model and a documented candidate set.

Contents · Previous chapter · Next chapter

A selected reliability model should be frozen and tested on later units, another site or another production period. The AIC winner from a laboratory fleet is evaluated in field use. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Case mix, censoring and failure mechanisms may shift. Report calibration and predictive loss in the new setting.

Because the aim is predictive information rather than literal model truth, compare the winner with residual, calibration and mechanism evidence. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use external validation as the focus and test the worked illustration: The AIC winner from a laboratory fleet is evaluated in field use. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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AIC contains no maintenance cost, downtime penalty, safety consequence or capacity constraint. A lower-AIC model may not change the preferred replacement policy. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Predictive information and operational utility are different objectives. Connect validated probabilities to a separate decision model.

When a ranking changes after one candidate or observation is removed, report that sensitivity instead of hiding it behind a single minimum. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use decision consequences as the focus and test the worked illustration: A lower-AIC model may not change the preferred replacement policy. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

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AIC connects logarithms, functions, optimisation, probability and disciplined comparison in one compact calculation. Students can compute 246, then compare a four-parameter model with log L-hat = -118.5. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Arithmetic alone does not justify the candidate models. Ask what each parameter and likelihood contribution represents.

A numerical penalty cannot decide whether a lifetime distribution makes physical sense in the system being studied. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use parent and student learning route as the focus and test the worked illustration: Students can compute 246, then compare a four-parameter model with log L-hat = -118.5. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

Useful next reading: Mathematics Learning Hub, eduKate Mathematics Learning System, right-censored log-likelihood guide, Bukit Timah Mathematics curriculum overview, cross-subject learning approach. Follow the link that matches the next question: subject navigation, learning method, school-to-advanced progression or continuity with the preceding reliability score.

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AIC is not an absolute quality score, not a posterior model probability and not permission to ignore physics or diagnostics. The smallest AIC wins only within the declared comparable set. This chapter isolates that step because information loss, predictive fit and fair comparison depend on keeping the mathematical object and the engineering question aligned. Close values signal model-selection uncertainty rather than a guaranteed tie-break. Read Akaike's paper and the NIST fatigue-model example, then reproduce the table.

Use the criterion as one layer in a transparent model-selection workflow, then test the selected model on information it did not fit. Write the variables, units, data subset and likelihood convention before opening software. Then reproduce the displayed number from the saved inputs. If the result cannot be rebuilt, the comparison is not ready to guide a conclusion.

For a practical check, Create a two-row candidate table with log-likelihood, k, -2 log-likelihood, penalty, AIC and delta AIC. Predict which change will improve fit and which will increase the penalty before recalculating. In this chapter, use common misconceptions and next reading as the focus and test the worked illustration: The smallest AIC wins only within the declared comparable set. Change one defensible input, predict the direction first, and finish by stating what the calculation supports and what remains uncertain.

Quick answers

What should be checked before the calculation? Confirm the same observations, outcome, time scale, likelihood convention and independently estimated parameters.

Does the smallest criterion or p-value prove the model is true? No. It is conditional on the candidates, data and assumptions.

Which school mathematics is doing the work? Algebra organises the formula, logarithms transform products, functions describe distributions, probability represents uncertainty and graphs expose fit.

How can a student practise safely? Use a fictional candidate table, preserve every intermediate value and change one input at a time.

What should follow model selection? Diagnostics, uncertainty analysis, external validation where possible, and a separate decision model when action costs matter.

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