SMHS GLM - Revision history

Dinov: /* Gamma GLM vs. Beta Regression */

2026-03-13T21:09:50Z

‎Gamma GLM vs. Beta Regression

Dinov: /* Advanced Topics */

2026-03-13T21:08:19Z

‎Advanced Topics

Dinov at 17:01, 10 March 2026

2026-03-10T17:01:50Z

Dinov: /* Example 1: Logistic Regression for Contraceptive Use */

2026-03-10T16:56:49Z

‎Example 1: Logistic Regression for Contraceptive Use

Dinov: /* Example 1: Logistic Regression for Contraceptive Use */

2026-03-10T16:55:13Z

‎Example 1: Logistic Regression for Contraceptive Use

Dinov: /* 1) GLM Components */

2025-12-26T17:46:20Z

‎1) GLM Components

Dinov: /* 1) GLM Components */

2025-12-26T17:45:54Z

‎1) GLM Components

Dinov: /* 1) GLM Components */

2025-12-26T17:45:09Z

‎1) GLM Components

Dinov: /* Problems */

2025-12-08T22:15:03Z

‎Problems

Dinov: /* Advanced Topics */

2025-12-08T22:14:10Z

‎Advanced Topics

@@ Line 571: / Line 571: @@
 '''variance structure''' of the data they expect.
-* The Gamma Model (''glm''): The Gamma distribution is used for ''strictly positive, continuous data'' <math>(0, \infty)</math>. It is appropriate for data that is "skewed" to the right, where the variance increases as the mean increases.
+===== The Gamma Model (''glm'')=====
 If a variable, like ''qol_score'' is a raw score (e.g., 0 to 100), the link, ''link = "log"'',
 ensures the predicted values remain positive.
 In a Gamma GLM with a log link, we model the mean <math>\mu</math> as
 <math>\ln(\mu_i) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_k X_k.</math>
@@ Line 584: / Line 585: @@
 <math>f(y; \alpha, \beta) = \frac{\beta^\alpha y^{\alpha-1} e^{-\beta y}}{\Gamma(\alpha)} \quad \text{for } y > 0.</math>
-* The Beta Model (''betareg''), with Beta distribution, is used specifically for '''rates, proportions, or scores constrained between 0 and 1''', <math>(0, 1)</math>. Unlike the Gamma model, which can go off to infinity, the Beta distribution is "boxed in" this interval.
+===== The Beta Model (''betareg'')=====
 For instance, if ''qol_scaled'' is normalized, e.g., by dividing QoL score by its maximum value,
 it's values are forced into <math>(0, 1)</math> interval.
@@ Line 599: / Line 602: @@
 Key Differences between ''Gamma GLM'' and ''Beta Regression''
-| Feature | Gamma Model (model4) | Beta Model (`model_beta`) |
+{| class="wikitable"
-| --- | --- | --- |
+! Feature
-| Outcome Range | <math>(0, \infty)</math> (Positive numbers) | <math>(0, 1)</math> (Proportions/Scaled) |
+! Gamma Model (model4)
-| Typical Use | Costs, rainfall, raw test scores | Percentages, scaled QoL indices |
+! Beta Model (model_beta)
-| Interpretation | Log-link: Coefficients are multiplicative | Logit-link: Coefficients are odds ratios |
+|-
-| Boundaries | Can handle very large values | Naturally handles "ceiling effects" |
+| '''Outcome Range'''
 If the outcome dependent variable (DV) qol_score is a raw sum that could theoretically be much higher, stick with Gamma. But when ''qol_score'' is a bounded instrument (like a visual analog scale from 0–1) with many observations scoring near the very top or bottom, ''Beta'' may be superior because it understands the "walls" of the scale.

@@ Line 562: / Line 562: @@
    ) +
    theme_minimal()
 </pre>

@@ Line 208: / Line 208: @@
 :: An estimate of 0 means the variable has no effect on the odds (Odds Ratio = 1).
-:: The z-value tells you how many standard errors the estimate is from zero. Since all your p-values are small (<math>< 0.05</math>), all these predictors are statistically significant.
+:: The z-value tells us how many standard errors the estimate is from zero. Since all p-values are small (<math>< 0.05</math>), all these predictors are statistically significant.
 ====Example 2: Poisson Regression for Count Data====

@@ Line 199: / Line 199: @@
 * Education:
 :: Estimate: -0.3250
-:: Meaning: Being in the ''low education'' group is associated with a decrease in the log-odds of using contraception compared to the ''high'' group. Specifically, their odds are about 28\% lower (<math>e^{-0.325} \approx 0.72<\math>).
+:: Meaning: Being in the ''low education'' group is associated with a decrease in the log-odds of using contraception compared to the ''high'' group. Specifically, their odds are about 28\% lower (<math>e^{-0.325} \approx 0.72</math>).
 * Wants More Children:

@@ Line 132: / Line 132: @@
 plot(model1, which = 1:4)
 </pre>
 ====Example 2: Poisson Regression for Count Data====

@@ Line 19: / Line 19: @@
 . Random Component: The response variable \(Y\) follows a distribution from the exponential family:
     <math>f_Y(y|\theta,\phi) = \exp\left\{\frac{y\theta - b(\theta)}{a(\phi)} + c(y,\phi)\right\}</math>
-   where <math>\theta</math> is the natural parameter and \(\phi\) is the dispersion parameter.
+where <math>\theta</math> is the natural parameter and \(\phi\) is the dispersion parameter.
 . Systematic Component: The linear predictor \(\eta\):
@@ Line 27: / Line 27: @@
     <math>g(\mu) = \eta \quad \text{or equivalently} \quad \mu = g^{-1}(\eta)</math>
-The variance function relates the variance to the mean:
+The variance function relates the variance to the mean: <math>\text{Var}(Y) = a(\phi)V(\mu)</math>
 where <math>V(\mu)</math> is the variance function specific to the distribution.

@@ Line 500: / Line 500: @@
 . Conceptual Exercises:
     a) Derive the score equations for a Poisson GLM with log link
     b) Show that the binomial distribution with logit link is the canonical link
@@ Line 505: / Line 506: @@
 . Applied Problems:
     a) Analyze the [http://wiki.socr.umich.edu/index.php/SOCR_Data_Dinov_021808_ConsumerPriceIndex3Way Consumer Price Index] data using appropriate GLM
     b) Model the [http://wiki.socr.umich.edu/index.php/SOCR_Data_MonetaryBase1959_2009 Monetary Base] data considering temporal autocorrelation
     c) Using the <code>iris</code> dataset, build a multinomial logistic regression to classify species
-. Simulation Study:
+. Practice Simulation Study:
     <pre>
     # Simulate data from a logistic regression model

@@ Line 339: / Line 339: @@
 ===Advanced Topics===
-* Mixed Effects GLM (GLMM): Extension incorporating random effects:
+==== Mixed Effects GLM (GLMM)====
 <math>
@@ Line 389: / Line 391: @@
 </pre>
-* Bayesian GLM: Using Markov Chain Monte Carlo (MCMC) methods:
+====Bayesian GLM====
 <pre>
@@ Line 399: / Line 403: @@
                          prior = normal(0, 2.5),
                          prior_intercept = normal(0, 5))
 </pre>