Logistic Regression

Concept

---

Theoretically, Logistic Regression is a generalised linear modelling technique used to estimate the relationship between one or more explanatory variables and a binary outcome. Unlike linear regression, logistic regression models the probability of an event by transforming probabilities into log-odds using the logistic function. The method exists to estimate risk, identify predictors of binary outcomes and quantify associations when the dependent variable represents the occurrence or absence of an event.

Mathematically, logistic regression represents the logarithm of the odds of the outcome as a linear function of predictor variables. Model parameters are estimated using Maximum Likelihood Estimation rather than Ordinary Least Squares because the outcome follows a binomial distribution. Estimated regression coefficients describe changes in the log-odds of the outcome, while exponentiation of the coefficients yields odds ratios that are widely used for interpretation.

In practice, logistic regression is applied to datasets containing binary outcomes such as disease occurrence, hospital admission, treatment response or mortality. In health economics, it is commonly used to model treatment success, healthcare utilisation, adverse events, screening outcomes and patient behaviour. Estimated probabilities are frequently incorporated into decision models, cost-effectiveness analyses and risk prediction tools.

Purpose

---

Used to model binary outcomes, estimate event probabilities, quantify associations through odds ratios, identify significant predictors and generate risk predictions for health economic and clinical decision-making.

---

Mathematical Formulae

Primary Formula

Logistic regression model:

log(p / (1 ? p)) = ?? + ??X? + ??X? + ? + ??X?

Probability form:

p = 1 / (1 + e???? ? ???? ? ? ? ?????)

where:

• p = probability of the outcome
• ?? = intercept
• ????? = regression coefficients
• X??X? = explanatory variables

Supporting Formulae

Odds:

Odds = p / (1 ? p)

Odds Ratio:

OR = e?

Log-likelihood:

ln(L) = ?[y?ln(p?) + (1 ? y?)ln(1 ? p?)]

Related Mathematical Methods

• Generalised Linear Models
• Maximum Likelihood Estimation
• Probit Regression
• Multinomial Logistic Regression
• Ordinal Logistic Regression
• Penalised Logistic Regression
• Receiver Operating Characteristic Analysis

---

Example

A health economist models the probability of hospital readmission within 30 days using age, number of chronic conditions and treatment group.

The estimated model is:

log(p / (1 ? p)) = ?2.40 + 0.03(Age) + 0.45(Chronic Conditions) ? 0.60(Treatment)

For a patient aged 70 years with two chronic conditions receiving the treatment:

Linear predictor:

? = ?2.40 + (0.03 ? 70) + (0.45 ? 2) ? 0.60

? = 0.00

Predicted probability:

p = 1 / (1 + e?)

p = 0.50

The model estimates a 50% probability of hospital readmission for this patient. The treatment odds ratio is:

OR = e??�?? = 0.55

indicating that treatment is associated with approximately a 45% reduction in the odds of readmission.

---

Excel Implementation

Function	Example Formula	Health Economics Application
EXP	=1/(1+EXP(-(B0+B1*A2+B2*C2)))	Calculate predicted probabilities from the fitted logistic model.
LN	=LN(D2/(1-D2))	Calculate the log-odds of an event.
EXP	=EXP(B1)	Convert regression coefficients into odds ratios.
Solver	Maximise the log-likelihood function by varying regression coefficients.	Estimate logistic regression parameters using maximum likelihood.
IF	=IF(E2>=0.5,1,0)	Classify observations according to predicted probability.

---

VBA (Optional)

A VBA routine can automate maximum likelihood estimation, calculate predicted probabilities and odds ratios, and generate classification and diagnostic summaries for logistic regression models.

---

Sources

• Hosmer DW, Lemeshow S, Sturdivant RX. Applied Logistic Regression. Wiley.
• Agresti A. Categorical Data Analysis. Wiley.
• McCullagh P, Nelder JA. Generalized Linear Models. Chapman & Hall.
• Harrell FE. Regression Modeling Strategies. Springer.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
• ISPOR Good Practice Reports.