Essential Concepts from Lecture 1

• Decomposition of the association between X and Y into cross-sectional and longitudinal components:

This model implies two key identities:

1. Cross-sectional effect (C):

2. Longitudinal effect (L):    

• What’s Intraclass Correlation Coefficient (ICC), aka

We model the outcome as:

: fixed effect (overall intercept)
: random effect (random intercept)
: residual error

Variance of the Outcome:

Covariance Within a Cluster:

Intraclass Correlation Coefficient (ICC)

• Consequences of ignoring clustering in data

  • Regression estimates remain unbiased if missing data is minimal
  • Standard errors, confidence intervals, and tests may be incorrect
  • Estimates may be inefficient (i.e., more variable than necessary)

Autocorrelation

• “Past is prologue”
• “No two people are alike”
• Repeated measurements on a person tend to be more similar to each other than to observations from another person!

Empirical Correlation Matrix for Residuals

	2003	2004	2005	2006
2004	0.983
2005	0.971	0.986
2006	0.970	0.977	0.983
2007	0.970	0.965	0.963	0.984

• Lag Formula:
  

• Autocorrelation function:
  

• The standard error of is roughly where is the number of pairs of observations at lag

• Lag 1 correlation estimates: 0.983, 0.986, 0.983, 0.984 -> Pooled Lag 1 correlation estimate: 0.984 (average)
  

• Lag 2 correlation estimates: 0.971, 0.977, 0.963 -> Pooled Lag 2 correlation estimate: 0.970 (average)
  

• Lag 3 correlation estimates: 0.970, 0.965 -> Pooled Lag 3 correlation estimate: 0.966 (average)
  

• Lag 4 correlation estimates: 0.970

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Common Covariance Structures

2. Compound Symmetry (Exchangeable)

• Same variance, same correlation
• Only 2 parameters needed
• Assuming autocorrelation is not fading away by time

	Time1	Time2	Time3	Time4
Time1	Var	Var*	Var*	Var*
Time2	Var*	Var	Var*	Var*
Time3	Var*	Var*	Var	Var*
Time4	Var*	Var*	Var*	Var

---

3. Autoregressive (AR-1)

• The correlation at lag is
• Only 2 parameters needed
• Suitable for discrete time

	Time1	Time2	Time3	Time4
Time1	Var	Var*	Var*	Var*
Time2	Var*	Var	Var*	Var*
Time3	Var*	Var*	Var	Var*
Time4	Var*	Var*	Var*	Var

---

4. Toeplitz

• One variance; Different correlation per lag
• Requires parameters

	Time1	Time2	Time3	Time4
Time1	Var	Var*	Var*	Var*
Time2	Var*	Var	Var*	Var*
Time3	Var*	Var*	Var	Var*
Time4	Var*	Var*	Var*	Var

---

6. Banded Structures

• Set correlations to zero beyond lag (e.g., Banded Toeplitz 2)

	Time1	Time2	Time3	Time4
Time1	Var	Var*	0	0
Time2	Var*	Var	Var*	0
Time3	0	Var*	Var	Var*
Time4	0	0	Var*	Var

---

7. Heteroskedastic Models

• Allow variance to vary by time
• Combine with AR, Toeplitz, or CS structures
• Heteroskedastic toeplitz: n + (n-1) estimates

	Time1	Time2	Time3	Time4
Time1	Var1
Time2		Var2
Time3			Var3
Time4				Var4

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Overview

This lecture focuses on the principles and applications of design-based regression analysis using complex survey data in R. It introduces the correct use of lm(), glm(), and svyglm() functions with appropriate survey design using the survey package. It emphasizes best practices in model fitting, diagnostics, and interpretation, and highlights common pitfalls such as misuse of stepwise regression and pseudo R².

4 Steps for Fitting a Model

library(survey)
# Define the survey design
des <- svydesign(ids = ~psu, strata = ~strata, weights = ~weight, data = df)

# Linear regression
model_lm <- svyglm(gfr ~ cpr, design = des)
summary(model_lm)

# Binary outcome model
model_logit <- svyglm(mcu ~ age + education, design = des, family = quasibinomial())
summary(model_logit)

model_logbin <- svyglm(mcu ~ age + education, design = des, family = binomial(link = "log"))
summary(model_logbin)

model_poisson <- svyglm(mcu ~ age + education, design = des, family = poisson(link = "log"))
summary(model_poisson)

plot(model_lm, which = 1)

plot(model_lm, which = 2)

bptest(model_lm)

    studentized Breusch-Pagan test

data:  model_lm
BP = 0.040438, df = 1, p-value = 0.8406

plot(model_lm, which = 4)

model_logit <- glm(am ~ wt + hp, data = mtcars, family = binomial)
summary(model_logit)

Call:
glm(formula = am ~ wt + hp, family = binomial, data = mtcars)

Coefficients:
            Estimate Std. Error z value Pr(>|z|)   
(Intercept) 18.86630    7.44356   2.535  0.01126 * 
wt          -8.08348    3.06868  -2.634  0.00843 **
hp           0.03626    0.01773   2.044  0.04091 * 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 43.230  on 31  degrees of freedom
Residual deviance: 10.059  on 29  degrees of freedom
AIC: 16.059

Number of Fisher Scoring iterations: 8

# Run Hosmer-Lemeshow test
hoslem.test(mtcars$am, fitted(model_logit))

    Hosmer and Lemeshow goodness of fit (GOF) test

data:  mtcars$am, fitted(model_logit)
X-squared = 4.9517, df = 8, p-value = 0.7627

AIC(model_logit)
BIC(model_logit)

exp(coef(model_logit))

exp(coef(model_poisson))

Key Takeaways

• Always use svydesign() and svyglm() when analyzing complex survey data in R
• Avoid stepwise regression—prefer theory-driven model building
• Validate model assumptions with residual plots and Goodness-of-Fit tests
• Use log-binomial or Poisson models to estimate risk ratios when outcomes are common
• AIC and BIC help compare model fit, especially when choosing between linear vs. polynomial or nested models

References

• Lecture slides by Dr. Saifuddin Ahmed, PFRH 712 (2025)
• Lumley, T. (2010). Complex Surveys: A Guide to Analysis Using R. Wiley.
• Harrell, F. (2015). Regression Modeling Strategies. Springer.
• Hosmer, D., Lemeshow, S., & Sturdivant, R. (2013). Applied Logistic Regression. Wiley.
• R packages: survey, lmtest, ResourceSelection, ggplot2, srvyr

Overview

This course introduces statistical methods for analyzing:

• Multilevel data: Observations nested within higher-level units (e.g., students within schools)
• Longitudinal data: Repeated measures over time from the same individuals

Statistical Foundations

Multilevel Model Basics

• Names: Mixed models, hierarchical models, random-effects models
• Structure: Level-1 (individuals) nested within Level-2 (e.g., families), etc.
• Applications: Health outcomes vary across multiple levels (e.g., person, family, neighborhood)

Longitudinal = Multilevel

• Longitudinal data = Time nested within individuals
• Repeated measures are correlated, requiring special models

Core Questions

1. What questions do multilevel/longitudinal models answer?
   • Population averages
   • Individual differences
   • Relationships at multiple levels
2. Why are repeated measures correlated?
   • Same person → similar outcomes across time
3. What happens if we ignore that correlation?
   • Invalid inferences
   • Poor confidence intervals
   • Biased results (especially with missing data)
4. How can we model it correctly?
   • Use appropriate models (marginal or mixed)

Types of Models

What Happens If We Ignore Correlation?

• Biased or misleading inferences
• Under/overestimated standard errors
• Less statistical power
• Invalid estimates, especially with missing data

Key Takeaways

• Multilevel models account for:
  • Nested structure in real-world data
  • Correlation within clusters or individuals
• Essential for accurate inference in public health and longitudinal research
• Ignoring structure leads to:
  • Bias
  • Invalid inference
  • Poor policy or scientific decisions

References

• Diggle et al. (2002). Analysis of Longitudinal Data
  
• Rabe-Hesketh & Skrondal. Multilevel and Longitudinal Modeling Using Stata
  
• Lecture slides by Dr. Zheyu Wang and Dr. Ji Soo Kim