6.3: Logistic Regression Model, Explanation, and Numeric Example
- Page ID
- 141408
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Logistic Regression Model
The basic logistic regression model is:
P(Y = 1 | x) = 1 / (1 + e^-(β₀ + β₁x))
Alternatively, in log-odds form:
log(P / (1 - P)) = β₀ + β₁x
Explanation of Each Term
- P(Y = 1 | x): The predicted probability that the dependent variable Y equals 1 (the “success” category), given the predictor variable x.
- β₀ (intercept): The log-odds of the outcome when x = 0; shifts the curve left or right.
- β₁ (slope): Represents the change in the log-odds of the outcome for a one-unit increase in x. It determines the steepness and direction of the S-shaped curve.
- e: Exponential number, approximately 2.71828. It is the base of the natural logarithm used in the model.
What the Model Does
Logistic regression is used to predict the probability of a binary outcome—i.e., whether an event occurs (1) or does not occur (0)—based on one or more predictor variables. Unlike linear regression, which predicts a continuous number, logistic regression maps input values to a probability between 0 and 1, using a sigmoid (S-shaped) curve.
Numeric Example
Suppose we are modeling the probability that a customer will purchase a product (Y = 1) based on the number of emails they've received (x).
Let’s say the estimated model is:
P(Y = 1 | x) = 1 / (1 + e^-( -1.5 + 0.8x ))
Predict for x = 3 (customer received 3 emails):
P = 1 / (1 + e^(-(-1.5 + 0.8 * 3)))
= 1 / (1 + e^(-(-1.5 + 2.4)))
= 1 / (1 + e^(-0.9))
e^(-0.9) ≈ 0.4066
→ P = 1 / (1 + 0.4066) ≈ 1 / 1.4066 ≈ 0.711
Interpretation: When a customer has received 3 emails, the model predicts a 71.1% probability that they will purchase the product.
What can marketing do with this information?
- Segment and target customers: Marketers can prioritize customers with high predicted probabilities (for example, greater than 50 percent) for follow-up efforts such as promotions, personalized offers, or sales calls.
- Optimize email campaigns: If 3 emails result in a 71 percent chance of purchase, this insight helps fine-tune the email frequency to maximize conversions without spamming.
- Allocate resources efficiently: Marketing teams can invest more in prospects above a certain likelihood of conversion and avoid spending excessively on low-probability leads.
- Design A/B tests or campaign triggers: The model enables creation of rule-based automation—for instance, triggering a special discount once a customer's predicted probability exceeds a certain level.