Introduction to Modeling in R

Joshua Wiley, M.A.

Senior Analyst — Elkhart Group


Examples using linear regression and linear regression with smooths

ggplot2 for graphics and visualization

Synthesize models and visualization to


(p <- ggplot(d, aes(mentalHealth)) + geom_histogram(binwidth = 2))
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How would you summarize the distribution?


You could use the expectation \[E(y) = 52.92 \]

p + geom_vline(xintercept = mean(d$mentalHealth), size = 2, colour = "red")
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(p <- ggplot(d, aes(mastery, mentalHealth)) + geom_point())
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How would you summarize this distribution?


You could use the conditional expectation \( E(y | f(x)) \) where \(f(\cdot)\) is some function, say linear

m <- lm(mentalHealth ~ mastery, data = d)

\[ \widehat{mentalHealth} = 36.9 + 5.14 mastery \]

p + geom_abline(intercept = coef(m)[1], slope = coef(m)[2])
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Multiple dimensions

The conditional expectation generalizes to \(\mathbb{R}^{N}\), but graphs do not.

coef(lm(mentalHealth ~ mastery + neuroticism, d))[["mastery"]]
## [1] -1

Under certain conditions, \( E(y | f(x_1, x_2)) \approx E( y | f(x_1) ~~ | ~~ f(x_2 | f(x_1))~) \)

m1 <- lm(cbind(mentalHealth, mastery) ~ neuroticism, d)
coef(lm(mentalHealth ~ mastery, rd <-[["mastery"]]
## [1] -1

Multiple dimensions

We can take advantage of this for data and model visualization

p <- ggplot(d, aes(mastery, mentalHealth)) + geom_point() + stat_smooth(method = "lm")
grid.arrange(p, p %+% rd, ncol = 2)
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Multiple dimensions

This concept also generalizes: \( E( y | f(x_1, \ldots, x_{n - 1}) ~~ | ~~ f(x_n | f(x_1, \ldots, x_{n - 1}))~) \)

rd <-, mastery) ~ age + 
    hope + neuroticism, d)))
p %+% rd
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Multiple dimensions

ggpairs(d[, c(1, 2, 11, 12)], lower = list(continuous = "smooth"))
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Other Functional Forms

So far simple linear functional forms, but \(f(\cdot)\) can be any

When form is unknown, splines or smooth parameters are useful, especially for "nuissance" variables

Many options -- mgcv package has the gam function for generalized additive models with a general smooth function s() defaulting to thin-plate

Also the splines package for b-splines bs() or natural cubic splines ns() you can use in any modeling function


Other Functional Forms

# fit and store models
m.linear <- lm(positiveAffect ~ deltaTime, d)
m.quadratic <- lm(positiveAffect ~ deltaTime + I(deltaTime^2), d)
m.smooth <- gam(positiveAffect ~ s(deltaTime, k = 5), data = d)

# data set of raw and predicted values
pdat <- cbind(d[, c("deltaTime", "positiveAffect")],
          linear = fitted(m.linear),
          quadratic = fitted(m.quadratic),
          smooth = fitted(m.smooth))

# plot of raw data with separate coloured lines by model
p <- ggplot(pdat, aes(deltaTime, positiveAffect)) + geom_point() +
  geom_line(aes(y = linear), colour = "green", size=1.2) +
  geom_line(aes(y = quadratic), colour = "blue", size=1.2) +
  geom_line(aes(y = smooth), colour = "red", size=1.2)

Other Functional Forms

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Other Functional Forms

Days since treatment, visual acuity (logMAR), and age are nuissance variables; we care about mastery & positive affect

ldat <- melt(d[, c("deltaTime", "logMAR1", "age", "mastery",
  "positiveAffect")], id.vars="positiveAffect")

ggplot(ldat, aes(value, positiveAffect)) + geom_point() +
  stat_smooth(se=FALSE) + facet_wrap(~variable, scales="free_x")
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Other Functional Forms

# smooth model - k = 10 for high flexibility
m.smooth <- gam(positiveAffect ~ s(deltaTime, k = 10) +
  s(logMAR1, k = 10) + s(age, k = 10), data = d)

# linear model only
m.linear <- lm(positiveAffect ~ deltaTime + logMAR1 + age, data = d)

# pull out residuals in a data set
pdat <- data.frame(mastery = d$mastery,
  residualPositiveAffect = c(resid(m.smooth), resid(m.linear)),
  Type = rep(c("smooth", "linear"), each = nrow(d)))

# make the graph (separate by model type)
p <- ggplot(pdat, aes(mastery, residualPositiveAffect)) +
  geom_point() + stat_smooth(method="lm") +
  facet_wrap(~ Type)

Other Functional Forms

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# correlation between residuals
cor(resid(m.smooth), resid(m.linear))
## [1] 0.96



Graphics Functions

Model Functions



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