QBUS2820 Chap.3 Likelihood and Regression Inference
Likelihood and Regression Inference
Likelihood ranks parameter values by how strongly each supports the observed sample under a probability model. Log-likelihood converts products into sums and makes differentiation practical. This chapter links maximum likelihood to regression inference through coefficient standard errors, confidence intervals, null-value tests and overall model tests.
It emphasises visible working: write the likelihood, take logs, differentiate, solve within the parameter space, then interpret the estimate without confusing a p-value with the probability that a hypothesis is true. Maximum likelihood is conditional on a complete probability specification, including support and independence assumptions.
A stationary point outside the allowed parameter space cannot be the estimate, and a boundary solution needs separate handling. Curvature near the maximum helps explain uncertainty: a sharply peaked log-likelihood supports a more precise estimate than a broad, flat profile. Regression standard errors play a related role by scaling how much estimates would vary across repeated samples under the model.
Confidence intervals describe a long-run procedure, not a probability statement about one fixed parameter after the interval is observed. Hypothesis tests require the null value and alternative direction to be declared before reading the statistic. An overall model test and an individual coefficient test answer different questions, especially when predictors overlap.
Practical importance must be assessed from effect size and units rather than a p-value alone. Prediction intervals are wider than mean-response intervals because a new observation contains both estimation uncertainty and fresh outcome noise.
What this chapter covers
- 01
Joint likelihood for independent cases
- 02
Log-likelihood transformations
- 03
Score equations and maximisation
- 04
Coefficient standard errors
- 05
Confidence interval construction
- 06
Tests against nonzero benchmarks
- 07
Individual and whole-model tests
- 08
Magnitude compared with uncertainty
Worked example · free
Estimate an exponential rate
- +1The marks shown here organise independent revision and are not an official University mark allocation. Write the likelihood as λ^n exp(−λΣx_i).
- +1Take logs to obtain n log λ − λΣx_i.
- +1Differentiate: n/λ − Σx_i, then set the score equal to zero.
- +1Solve λ̂ = n/Σx_i = 1/x̄ = 0.25 per minute.
- +1The second derivative −n/λ² is negative for positive λ, confirming a maximum.
Key terms
- Likelihood
- Observed-sample support for candidate parameter values under a specified probability model.
- Log-Likelihood
- The logarithm of likelihood, preserving the maximiser while turning products into sums.
- Score Function
- The derivative of log-likelihood with respect to a parameter.
- Standard Error
- An estimate of sampling variability for a fitted coefficient or statistic.
- Confidence Interval
- A procedure-based range combining an estimate, critical value and standard error.
- Null Hypothesis
- A specific population claim tested against sample evidence under a model.
- Likelihood Curvature
- The sharpness of the log-likelihood near its maximum, which reflects how precisely the data identify a parameter.
Likelihood and Regression Inference FAQ
Why use the log-likelihood?
The logarithm is increasing, so it preserves the parameter value that maximises likelihood. It also converts products over independent observations into sums and makes derivatives easier to calculate.
How do I test against a benchmark?
Subtract the stated benchmark from the estimate and divide by its standard error. Compare that statistic with the appropriate reference value, then report reject or do not reject with a contextual interpretation.
What does failure to reject mean?
It means the sample does not provide sufficient evidence against the specified null at the chosen threshold. It does not prove exact equality or show that the effect is practically unimportant.
Why check the parameter space?
A likelihood calculation may produce a stationary point that violates a rate, probability or variance constraint. The valid estimate must maximise the function over values the probability model actually permits.
How do prediction intervals differ from mean intervals?
A mean-response interval reflects uncertainty about the conditional mean. A prediction interval also includes new observation noise, so it is wider when both are built for the same predictor profile and confidence level.
Exam move
Reproduce maximum-likelihood solutions without skipping lines. Start from the density or probability mass function, form the joint likelihood, take the log, collect parameter-dependent terms, differentiate and solve. Check the parameter domain and the sign of the second derivative. Then practise regression inference with one interval and one nonzero benchmark test.
Write the null value in the numerator before inserting numbers. Use the solved problem set to compare your sequence: it exposes the likelihood, log-likelihood, derivative and estimate, then shows interval and t-statistic calculations with named standard errors.
Build a language checklist that blocks three overclaims: a p-value is not the probability the null is true, failure to reject is not proof, and an observational coefficient is not automatically causal. Finish each response with the parameter units and the business meaning.
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