Inference

2025-04-08

Motivation

  • Motivation

  • Hypothesis Testing

  • Example 1

  • Example 2

Motivation

The ncbirths data set provides information on whether different predictors. The outcome of interest will be premie (“full term” or “premie”).

Motivation 1

Model the relationship between the outcome premie and habit (“smoker” and “nonsmoker”).

Motivation 2

Model the relationship between the outcome premie and visits (“smoker” and “nonsmoker”).

Logistic Model 1

Use a logistic regression to model the outcome premie and habit.

Code 
glm(premie ~ habit, data = ncbirths, family = binomial())

Odds Ratio 1

Code 
b(glm(premie ~ habit, data = ncbirths, family = binomial()), 1)
exp(b(glm(premie ~ habit, data = ncbirths, family = binomial()), 1))

Odds Ratio 1

Code 
b(glm(premie ~ habit, data = ncbirths, family = binomial()), 1)
#> [1] -0.01344105
Code 
exp(b(glm(premie ~ habit, data = ncbirths, family = binomial()), 1))
#> [1] 0.9866489

The odds of a mother who smokes having a premature infant is 0.98 times lower than the odds of a mother who does not smoke.

Logistic Model 2

Use a logistic regression to model the outcome premie and visits.

Code 
glm(premie ~ visits, data = ncbirths, family = binomial())

Odds Ratio 2

Code 
b(glm(premie ~ visits, data = ncbirths, family = binomial()), 1)
exp(b(glm(premie ~ visits, data = ncbirths, family = binomial()), 1))

Odds Ratio 2

Code 
b(glm(premie ~ visits, data = ncbirths, family = binomial()), 1)
#> [1] -0.1063097
Code 
exp(b(glm(premie ~ visits, data = ncbirths, family = binomial()), 1))
#> [1] 0.8991461

As the number of hospital visits increases by 1, the odds of having a premature infant decreases by a factor 0.89.

Real or Random

Does smoking have a protective effect on having a premature infant?

Does the number of hospital visits have a protective effect?

What is real and what is random?

Hypothesis Testing

  • Motivation

  • Hypothesis Testing

  • Example 1

  • Example 2

Hypothesis Tests

Hypothesis tests are used to test whether claims are valid or not. This is conducted by collecting data, setting the Null and Alternative Hypothesis.

Null Hypothesis \(H_0\)

The null hypothesis is the claim that is initially believed to be true. For the most part, it is always equal to the hypothesized value.

Alternative Hypothesis \(H_a\)

The alternative hypothesis contradicts the null hypothesis.

Example of Null and Alternative Hypothesis

We want to see if \(\beta\) is different from \(\beta^*\)

Null Hypothesis Alternative Hypothesis
\(H_0: \beta=\beta^*\) \(H_a: \beta\ne\beta^*\)
\(H_0: \beta\le\beta^*\) \(H_a: \beta>\beta^*\)
\(H_0: \beta\ge\beta^*\) \(H_0: \beta<\beta^*\)

One-Side vs Two-Side Hypothesis Tests

Notice how there are 3 types of null and alternative hypothesis, The first type of hypothesis (\(H_a:\beta\ne\beta^*\)) is considered a 2-sided hypothesis because the rejection region is located in 2 regions. The remaining two hypotheses are considered 1-sided because the rejection region is located on one side of the distribution.

Null Hypothesis Alternative Hypothesis Side
\(H_0: \beta=\beta^*\) \(H_a: \beta\ne\beta^*\) 2-Sided
\(H_0: \beta\le\beta^*\) \(H_a: \beta>\beta^*\) 1-Sided
\(H_0: \beta\ge\beta^*\) \(H_0: \beta<\beta^*\) 1-Sided

Decision Making

Hypothesis Testing will force you to make a decision: Reject \(H_0\) OR Fail to Reject \(H_0\)

Reject \(H_0\): The effect seen is not due to random chance, there is a process making contributing to the effect.

Fail to Reject \(H_0\): The effect seen is due to random chance. Random sampling is the reason why an effect is displayed, not an underlying process.

Decision Making: P-Value

The p-value approach is one of the most common methods to report significant results. It is easier to interpret the p-value because it provides the probability of observing our test statistics, or something more extreme, given that the null hypothesis is true.

If \(p < \alpha\), then you reject \(H_0\); otherwise, you will fail to reject \(H_0\).

Decision Making: Confidence Interval Approach

The confidence interval approach can evaluate a hypothesis test where the alternative hypothesis is \(\beta\ne\beta^*\). The bootstrapping approach will result in a lower and upper bound denoted as: \((LB, UB)\).

If \(\beta^*\) is in \((LB, UB)\), then you fail to reject \(H_0\). If \(\beta^*\) is not in \((LB,UB)\), then you reject \(H_0\).

Significance Level \(\alpha\)

The significance level \(\alpha\) is the probability you will reject the null hypothesis given that it was true.

In other words, \(\alpha\) is the error rate that a research controls.

Typically, we want this error rate to be small (\(\alpha = 0.05\)).

Hypothesis Test Process

  1. Set up the Null and Alternative Hypothesis.

    • \(H_0: \beta = 0\) and \(H_a: \beta \ne 0\)

  2. Compute p-value or confidence interval

  3. Make a decision

  4. Interpret the results

Example 1

  • Motivation

  • Hypothesis Testing

  • Example 1

  • Example 2

Example

Use a logistic regression to model the outcome premie and habit.

Code 
glm(premie ~ habit, data = ncbirths, family = binomial())
b(glm(premie ~ habit, data = ncbirths, family = binomial()), 1)
exp(b(glm(premie ~ habit, data = ncbirths, family = binomial()), 1))

Example 2

  • Motivation

  • Hypothesis Testing

  • Example 1

  • Example 2

Example

Logistic Model 2

Code 
glm(premie ~ visits, data = ncbirths, family = binomial())

b(glm(premie ~ visits, data = ncbirths, family = binomial()), 1)
exp(b(glm(premie ~ visits, data = ncbirths, family = binomial()), 1))