01 · Define the event and denominator
Define the event and denominator
Logistic regression models the probability of a binary outcome, such as missing a dispatch cutoff, shipment damage, or a picking error. Define the event precisely and record one outcome for each eligible independent unit. State whether one means failure or success. Reversing event coding reverses coefficient signs and reciprocates odds ratios, so this is an interpretation requirement rather than a software detail.
Record the denominator and exposure conditions. Twenty late shipments among one hundred eligible shipments differs from twenty late shipments among one thousand. If outcomes are grouped, preserve event counts and trials. Repeated shipments on the same truck or shift share conditions and may need cluster aware analysis. A changed defect detection process can alter the recorded event probability even when physical quality is unchanged.
02 · Understand probability, odds, and the link
Understand probability, odds, and the link
For event probability p, odds are p/(1−p). The logistic model is log[p/(1−p)] = η = β0 + Σβjxj. Here η is the linear predictor, β coefficients describe conditional log odds changes, and x are predictor values. Convert back with p = 1/(1+exp(−η)). This ensures fitted probabilities remain between zero and one.
For a one unit predictor change with other terms fixed and no relevant interaction, exp(βj) is the odds ratio. It is not a risk ratio or a percentage point change. The probability change depends on starting probability. An odds ratio of two turns probability 0.10 into odds 0.2222 and probability 0.1818, whereas doubling odds from probability 0.50 gives probability 0.6667. Report probability predictions at representative process conditions when managers need an operational interpretation.
03 · Worked two method cutoff example
Worked two method cutoff example
In fictional comparable independent shipments, the old method has 20 late and 80 on time; the new method has 10 late and 90 on time. Use a predictor x = 0 for old and 1 for new. Old odds are 20/80 = 0.25 and new odds are 10/90 = 0.1111. The intercept is ln(0.25) = −1.3863 and the method coefficient is ln(0.1111/0.25) = −0.8109.
The fitted probabilities are 0.20 and 0.10, matching the two group proportions in this saturated two group model. The odds ratio is exp(−0.8109) = 0.4444. Thus estimated new method odds are about 44% of old method odds, while the estimated risk ratio is 0.50 and the absolute reduction is ten percentage points. These are distinct summaries of the same data.
An approximate coefficient standard error is √(1/10 + 1/90 + 1/20 + 1/80) = 0.4167. A Wald 95% log odds interval is −0.8109 ± 1.96 × 0.4167, approximately −1.6276 to 0.0057. Exponentiating gives an odds ratio interval of approximately 0.1964 to 1.0058. It includes one. The data suggest a potentially useful improvement but this particular approximate interval does not exclude no odds change. Likelihood based inference can differ near a borderline result, so state the procedure rather than shopping for a preferred conclusion.
04 · Fit a model that reflects the process
Fit a model that reflects the process
Coefficients are typically estimated by maximizing a binomial likelihood, rather than ordinary least squares on zero and one values. Numeric predictors imply linearity in log odds, not in probability. Check whether order size needs nonlinear terms and whether method interacts with congestion. Categorical factors need clear reference coding. Keep meaningful lower order terms when including interactions.
Confounders remain important. If the new method was used only on easier shipments, its coefficient may capture workload composition. Adjustment can help for measured variables with adequate overlap, but cannot make an observational comparison randomized. Rare events need enough information for the number and complexity of terms. An empty event cell can create separation, where conventional maximum likelihood coefficients diverge. Penalized or specialist methods may be required; do not interpret a huge unstable coefficient as decisive evidence.
05 · Evaluate fit, calibration, and discrimination
Evaluate fit, calibration, and discrimination
Likelihood ratio tests compare nested models using changes in log likelihood under their assumptions. They quantify evidence, not operational benefit. Deviance and Pearson residuals can identify poorly represented covariate patterns, though individual binary residual plots are less intuitive than continuous residual plots. Inspect influential records and groups, then verify whether unusual outcomes reflect errors or real process conditions.
Calibration asks whether predicted risks agree with observed frequencies. Assess calibration on data not used to build the model, with uncertainty especially in rare event groups. Discrimination asks whether higher risk shipments are ranked above lower risk shipments; ROC area is one summary. A model can rank well while overpredicting every probability. Classification accuracy can be misleading when failures are rare because always predicting on time may appear successful.
06 · Make decisions with costs and uncertainty
Make decisions with costs and uncertainty
A probability threshold should follow action capacity and consequences. If extra inspection prevents costly damage but consumes labor, assess expected costs across plausible thresholds. Report sensitivity and false alarm tradeoffs for meaningful operating ranges. A default threshold of 0.50 is rarely justified solely because it is familiar. A cutoff model may need a much lower action threshold if missed failures are costly.
Predict only where the joint input patterns resemble the training population. Changes in base rate, carrier mix, or detection can degrade calibration without obvious coefficient instability. Account for clustered shipments when estimating uncertainty. Report adjusted probabilities with intervals at realistic settings rather than relying on odds ratios alone. A p value below a threshold does not validate the probability model or establish a causal intervention effect.
07 · Apply through DMAIC
Apply through DMAIC
A strong Analyze application defines event eligibility, models meaningful predictors, examines calibration and dependence, and distinguishes association from causal evidence. A weak application reports an odds ratio as an identical proportional risk change or judges a rare event model only by percent correct. Another weak application selects dozens of predictors from few failures and trusts precise looking probabilities.
Improve confirms controllable factor changes using an appropriate randomized or carefully justified design. Control monitors event rates, calibration, denominator definitions, and model drift. Preserve the event coding, model terms, inference method, operating range, and action threshold rationale. A probability score is useful only when its interpretation and consequences remain tied to the real warehouse process.
08 · Distinguish model inference from deployment value
Distinguish model inference from deployment value
The example's estimated ten percentage point reduction can be operationally interesting even though the approximate odds ratio interval includes one. A nonsignificant coefficient is not evidence that the methods are equal. Conversely, a significant coefficient in a much larger dataset could represent a trivial absolute improvement. Evaluate expected prevented events per eligible shipment and collect enough independent information to support the practical threshold.
For an adjusted model, report representative probability contrasts alongside odds ratios. The same odds ratio can imply very different absolute risks at quiet and congested conditions. If an interaction is included, a single method odds ratio no longer applies everywhere. State the conditions used for the calculation and propagate coefficient uncertainty into predicted probabilities or contrasts. Validation should use the intended deployment population, because calibration from one carrier or season need not transfer to another.
For grouped outcomes, retain both event counts and trials in the data record. Reporting only percentages hides unequal information: ten percent based on ten orders is less precise than ten percent based on a thousand. Use binomial likelihood with the actual trial counts and assess whether shared conditions create more variability than the independent binomial model allows.
Connections
Related methods and phase use
Placed under Analyze because of the outcome it supports. Use the method in other phases when its purpose and assumptions fit the work; the phase list is a guide to selection, not a compulsory sequence.
Sources
Professional foundations and further reading
The explanation and fictional teaching examples are written for LeanTrainingLab. Linked primary sources provide professional foundations; example values and local conventions are stated in the guide. The accepted toolset draws on Six Sigma+Lean Toolset, second edition (Lunau, editor; Springer, 2013). No book text or examples are reproduced here.