| age | hrs_wk | method | R_AGE |
|---|---|---|---|
| 9.982 | 5.137 | phonics | 14.090 |
| 8.006 | 4.353 | phonics | 11.762 |
| 9.349 | 5.808 | phonics | 13.838 |
| 8.620 | 3.612 | word | 6.620 |
| 8.060 | 4.447 | word | 7.524 |
| 6.117 | 5.085 | word | 5.502 |
Interactions
Univariate Statistics and Methodology using R
Two Weeks Ago
Before we Reach for R
our model says that
ageandhrs_wkhave orthogonal effects- whatever your age, an extra hour of reading/wk improves your reading age by 0.95 years
- however much you read, an extra year of age improves your reading age by 0.58 years
what if practice affects people of different ages differently?
- possibly our model isn’t a good fit to the data?
How Good is the Model?
plot(mod.m, which=1:4)
Model with Interaction
\[\color{red}{y_i}=\color{blue}{b_0}\cdot{}\color{orange}{1}+\color{blue}{b_1}\cdot{}\color{orange}{x_{1i}}+\color{blue}{b_2}\cdot{}\color{orange}{x_{2i}}+\color{blue}{b_3}\cdot{}\color{orange}{x_{1i}x_{2i}}+\epsilon_i\]
\[\color{red}{\hat{y_i}}=\color{blue}{b_0}\cdot{}\color{orange}{1}+\color{blue}{b_1}\cdot{}\color{orange}{x_{1i}}+\color{blue}{b_2}\cdot{}\color{orange}{x_{2i}}+\color{blue}{b_3}\cdot{}\color{orange}{x_{1i}x_{2i}}\]
mod.i <- lm(R_AGE ~ 1 + hrs_wk + age + hrs_wk:age, data = reading)\[\color{red}{\hat{y_i}}=\color{blue}{b_0}\cdot{}\color{orange}{1}+\color{blue}{b_1}\cdot{}\color{orange}{x_{1i}}+\color{blue}{b_2}\cdot{}\color{orange}{x_{2i}}+\color{blue}{b_3}\cdot{}\color{orange}{x_{1i}x_{2i}}\]
\(\color{blue}{b_0=1}\)
\(\color{blue}{b_1=1.5}\)
\(\color{blue}{b_2=2}\)
\(\color{blue}{b_3=0.5}\)
\(\color{red}{x_{1i}=2}\)
\[\color{red}{\hat{y_i}}=\color{blue}{b_0}\cdot{}\color{orange}{1}+\color{blue}{b_1}\cdot{}\color{orange}{x_{1i}}+\color{blue}{b_2}\cdot{}\color{orange}{x_{2i}}+\color{blue}{b_3}\cdot{}\color{orange}{x_{1i}x_{2i}}\]
\(\color{blue}{b_0=1}\)
\(\color{blue}{b_1=1.5}\)
\(\color{blue}{b_2=2}\)
\(\color{blue}{b_3=0.5}\)
\(\color{red}{x_{1i}=2}\)
\(\color{green}{x_{1i}=4}\)
Model with Interaction
A Two-Predictor Model
note that the lines are parallel
an hour of practice has the same effect, however you’re taught
No More Parallel Lines
A Nice Graph
reading |>
ggplot(aes(x = hrs_wk,
y = R_AGE, colour = method)) +
xlab("practice (hrs/wk)") +
ylab("reading age") +
geom_point(size = 3) +
geom_smooth(method = "lm")
geom_smooth(method="lm")effectively runs a linear model, to make a graphit’s not an analysis
Category x Category
| age | hrs_wk | method | school | R_AGE |
|---|---|---|---|---|
| 9.982 | 5.137 | phonics | private | 14.090 |
| 8.006 | 4.353 | phonics | state | 11.762 |
| 9.349 | 5.808 | phonics | state | 13.838 |
| 8.620 | 3.612 | word | private | 6.620 |
| 8.060 | 4.447 | word | private | 7.524 |
| 6.117 | 5.085 | word | state | 5.502 |
shoehorning in one more reading example
does where it’s taught affect the efficacy of a method?
Where to Start
- when looking at continuous predictors, we started with a model criticism
- when looking at mixed predictors, we looked at model improvement
Analysis of Variance Table
Response: R_AGE
Df Sum Sq Mean Sq F value Pr(>F)
hrs_wk 1 59.4 59.4 29.5 2.0e-06
method 1 245.6 245.6 122.2 1.5e-14
hrs_wk:method 1 11.3 11.3 5.6 0.022
Residuals 46 92.5 2.0 Reading Age by School and Method
A Much Better Graph
What the Model Tells Us
summary(mod.s)
Call:
lm(formula = R_AGE ~ method + school + method:school, data = reading)
Residuals:
Min 1Q Median 3Q Max
-2.620 -1.132 -0.030 0.926 3.877
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 11.258 0.428 26.31 < 2e-16 ***
methodword -3.038 0.618 -4.92 0.000012 ***
schoolstate 2.209 0.618 3.58 0.00083 ***
methodword:schoolstate -3.152 0.873 -3.61 0.00076 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.54 on 46 degrees of freedom
Multiple R-squared: 0.732, Adjusted R-squared: 0.715
F-statistic: 41.9 on 3 and 46 DF, p-value: 3.32e-13
F(3,46)=41.9
What the Model Doesn’t Tell Us
we can’t say anything general about the value of phonics
we can’t say anything general about the effectiveness of schools
no main effects
possibly the only useful information we have is the interaction
we can fix this…
Running the Model
- NB., we’ve changed the contrast coding in the previous slide
mod.sC <- lm(R_AGE ~ method * school, data = reading)
summary(mod.sC)
Call:
lm(formula = R_AGE ~ method * school, data = reading)
Residuals:
Min 1Q Median 3Q Max
-2.620 -1.132 -0.030 0.926 3.877
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 10.056 0.218 46.05 < 2e-16 ***
method1 -4.614 0.437 -10.56 6.9e-14 ***
school1 0.634 0.437 1.45 0.15357
method1:school1 -3.152 0.873 -3.61 0.00076 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.54 on 46 degrees of freedom
Multiple R-squared: 0.732, Adjusted R-squared: 0.715
F-statistic: 41.9 on 3 and 46 DF, p-value: 3.32e-13
F(3,46)=41.9
What’s Good About This?
- effect of
schoolwith (-.5, .5) contrast coding
What’s Good About This? (2)
- effect of
methodwith (-.5, .5) contrast coding
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Sum Coding
Sum Coding- R provides a quick and dirty way of setting similar contrasts
contrasts(reading$method) <- c(-0.5, 0.5)
contrasts(reading$method) [,1]
phonics -0.5
word 0.5mod.a <- lm(R_AGE~method,data=reading)
summary(mod.a) Estimate Std. Error t value Pr(>|t|)
(Intercept) 10.025 0.246 40.71 < 2e-16 ***
method1 -4.589 0.492 -9.32 0.0000000000024 ***contrasts(reading$method) <- contr.sum(2)
contrasts(reading$method) [,1]
phonics 1
word -1mod.b <- lm(R_AGE~method,data=reading)
summary(mod.b) Estimate Std. Error t value Pr(>|t|)
(Intercept) 10.025 0.246 40.71 < 2e-16 ***
method1 2.294 0.246 9.32 0.0000000000024 ***