1. Welcome to the class! set.seed(843) df <- tibble::data_frame(
x = rnorm(100),
y = x + rnorm(100),
group = ifelse(x > 0, 1, 0),
xs = scale(x) %>% as.numeric,
ys = scale(y) %>% as.numeric ) df
library(tidyverse)
df %>%
t.test(y ~ group,
data = .,
var.equal = TRUE)
lm(y ~ group,
data = .) %>%
summary()
aov(y ~ group,
cor.test(~ ys + xs,
data = .)
lm(ys ~ xs,

2. Statistical Control & Linear Models
EDUC 7610
Statistical Control & Linear Models
Chapter 1
Fall 2018
Tyson S. Barrett, PhD
𝒀 𝒊 = 𝜷 𝟎 + 𝜷 𝟏 𝑿 𝟏𝒊 + 𝝐 𝒊

3. Linear Models
We will discuss how to use linear models to understand relationships in data and how this may generalize to the population
Everything you learned in EDUC 6600 applies here, but we will extend it (and probably simplify it somewhat)

4. Linear Models
Most everything you’ve learned are specific approaches of linear models
T-tests
Two Sample t-test
data: y by group
t = , df = 98, p-value = 2.068e-09
95 percent confidence interval:
sample estimates:
mean in group 0 mean in group 1
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-06 ***
group e-09 ***
---
Residual standard error: on 98 degrees of freedom
Multiple R-squared: , Adjusted R-squared:
F-statistic: on 1 and 98 DF, p-value: 2.068e-09

5. Linear Models
Most everything you’ve learned are specific approaches of linear models
T-tests
ANOVA
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-06 ***
group e-09 ***
---
Residual standard error: on 98 degrees of freedom
Multiple R-squared: , Adjusted R-squared:
F-statistic: on 1 and 98 DF, p-value: 2.068e-09
Df Sum Sq Mean Sq F value Pr(>F)
group e-09
Residuals

6. Linear Models
Most everything you’ve learned are specific approaches of linear models
T-tests
ANOVA
Correlation
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e e
xs e e <2e-16 ***
---
Residual standard error: on 98 degrees of freedom
Multiple R-squared:
F-statistic: on 1 and 98 DF, p-value: < 2.2e-16
t = , df = 98, p-value < 2.2e-16
95 percent confidence interval:
sample estimates:
cor

7. So what are linear models?
“Model” is a simplification of how something works
“Linear” says the relationships are linear
Here, we use regression analysis (Ordinary Least Squares) to build linear models

8. So what are linear models?
Outcome
Variable
Covariates
𝒀 𝒊 = 𝜷 𝟎 + 𝜷 𝟏 𝑿 𝟏𝒊 + … + 𝝐 𝒊
Error (random) term
Intercept (constant)
Regression Slope (coefficient, weight)

9. Requirements for linear models
Participants (rows in a rectangular data matrix)
Measures (in rectangular data matrix form)
Each variable a single column of numbers
A single dependent variable
Dependent variable is numeric (otherwise, GLMs)
Other assumptions for statistical inference
Tidy
Data
Page 9

10. Tidy Data
The idea of having rows be observations and columns are variables
The ideal for data prep for all analyses (especially in R)
Pre-print:

11. Tidy Data Some pragmatic guidelines to have tidy data
1 Be Consistent 2 Choose good names for things 3 Write dates as YYYY-MM-DD 4 Put just one thing in a cell 5 Make it a rectangle 6 Create a data dictionary

12. Linear models are flexible
Predictor can be experimentally manipulated or naturally occurring
A dependent variable in one model can be a predictor in another
Predictors may be dichotomous, multi-categorical or continuous
Predictors can be correlated
Predictors may interact
Can be extended to handle curvilinear relations
Assumptions are not all that limiting
Page 10

13. Some Vocabulary Predictor = the variable that predicts the outcome
Independent Variable (IV) = the predictor(s) of interest
Covariates = the predictors that are not considered IV
Control = a class of methods that remove (or control for) the effect of another variable
Page 10

14. Controlling for Confounders
Five Types of Control
Random assignment on the independent variable
Manipulation of covariates
Other types of randomization
Exclusion of cases
Statistical control
Experimental Control

15. Statistical Control
i.e., “Adjust for”, “Correct for”, “Hold constant”, “Partial out”
Linear models allow us to control for the effects of covariates
We’ll discuss more about how this works, but essentially makes everyone statistically equal on each covariate
Can assess how much of the differences in the outcome are uniquely attributable to the IV (once we take out the effect of the covariates)

16. https://calpolystat2.shinyapps.io/3d_regression/