91Ó°ÊÓ

A study was conducted attempting to relate home ownership to family income. Twenty households were selected and family income was estimated, along with information concerning home ownership \((y=1\) indicates yes and \(y=0\) indicates \(n o\) ). The data are shown below. $$\begin{array}{ccc}\hline & & \text { Home } \\\& & \text { Ownership } \\\\\text { Household } & \text { Income } & \text { Status } \\\\\hline 1 & 38,000 & 0 \\\2 & 51,200 & 1 \\\3 & 39,600 & 0 \\\4 & 43,400 & 1 \\\5 & 47,700 & 0 \\\6 & 53,000 & 0 \\\7 & 41,500 & 1 \\\8 & 40,800 & 0 \\ 9 & 45,400 & 1 \\\10 & 52,400 & 1 \\\11 & 38,700 & 1 \\\12 & 40,100 & 0 \\\13 & 49,500 & 1 \\\14 & 38,000 & 0 \\\15 & 42,000 & 1 \\\16 & 54,000 & 1 \\\17 & 51,700 & 1 \\\18 & 39,400 & 0 \\\19 & 40,900 & 0 \\\20 & 52,800 & 1 \\\\\hline\end{array}$$ (a) Fit a logistic regression model to the response variable \(y\). Use a simple linear regression model as the structure for the linear predictor. (b) Is the logistic regression model in part (a) adequate? (c) Provide an interpretation of the parameter \(\beta_{1}\) in this model.

Step by step solution

01

Understanding the Problem

We are asked to fit a logistic regression model to binary data where the outcome is home ownership based on family income. The model should follow the form of a simple linear regression as the linear predictor.

02

Fitting the Logistic Regression Model

To fit a logistic regression model, we use the form: \[\log \left(\frac{p}{1-p}\right) = \beta_0 + \beta_1X\]where \(p\) is the probability of owning a home (\(y=1\)), and \(X\) is the family income. Using statistical software or calculations, estimate \(\beta_0\) and \(\beta_1\) using the given data.

03

Evaluating Model Adequacy

Evaluate the adequacy of the model using a goodness-of-fit test, such as the Hosmer-Lemeshow test, which compares observed and predicted outcomes. Examine the p-value to determine if the model fits well; a high p-value (usually above 0.05) suggests an adequate fit.

04

Interpreting the Parameter \(\beta_1\)

The parameter \(\beta_1\) represents the change in the log odds of home ownership for a one-unit increase in income. In other words, the exponential of \(\beta_1\) indicates how the odds of owning a home are multiplied with every additional dollar of income.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binary Data

In statistics, binary data refers to outcomes or variables that can only take two possible values. For the exercise at hand, these values are represented as 1 or 0. In the context of home ownership, 1 signifies that a household owns a home, while 0 means they do not. This format is ideal for logistic regression, a type of analysis suited for predicting binary outcomes.
Logistic regression works by modeling the probability that a given instance falls into one of these two categories. The goal is to find a relationship between one or more explanatory variables—like family income in this study—and the binary outcome variable. By doing so, analysts can estimate the likelihood of events, such as home ownership, based on input variables.

Family Income

Family income serves as the key explanatory variable in the logistic regression model used in the exercise. It acts as a predictor to ascertain the probability of home ownership. When performing the analysis, family income is considered a continuous variable, meaning it can take on a range of numerical values.

• Family income levels vary significantly between households, from low to high amounts, reflecting different financial capabilities.
• In modeling terms, family income is represented by the notation 'X' and contributes to determining the logistic regression's linear predictor.

These variations in income provide insights into how likely a household is to own a home, capturing economic power's effect on purchasing decisions. Thus, analyzing family income can reveal patterns and associations with home ownership.

Home Ownership

Home ownership status in the logistic regression model is the response variable, identified as 'y.' This variable is binary, only taking two outcomes: 1 for homeowners and 0 for non-homeowners. Understanding this variable is crucial, as the goal is to predict this particular outcome.
Home ownership is often considered a sign of financial stability and can be influenced by several factors, such as income, employment status, and market conditions. In logistic regression, these factors translate into predictor variables that impact the probability of home ownership. By analyzing these connections, one can gauge how these determinants affect one's likelihood to own a home, making it a valuable area of study for fields like economics and urban development.

Goodness-of-Fit Test

To determine if the logistic regression model fits the data well, a Goodness-of-Fit Test is employed. This statistical test evaluates how well observed outcomes match the outcomes predicted by the model. One common test for logistic regression is the Hosmer-Lemeshow test.

• The Hosmer-Lemeshow test divides data into groups based on predicted probabilities and assesses discrepancies between observed and predicted event rates within these groups.
• The outcome is a p-value, which indicates model adequacy. A high p-value (typically greater than 0.05) suggests that the model fits the data well, implying that any observed discrepancies could be due to random chance.

This step is vital for ensuring the reliability of the logistic regression model's predictions, ultimately enhancing the credibility of the analysis.