Lecture 03: Simple, Marginal, and Interaction Effects

More about general linear model

Author
Affiliation

Jihong Zhang*, Ph.D

Educational Statistics and Research Methods (ESRM) Program*

University of Arkansas

Published

September 9, 2024

Presentation Outline

  1. Homework 1 has been posted online

  2. Centering and Coding Predictors

  3. Interpreting Parameters in the Model for the Means

  4. Main Effects Within Interactions

  5. GLM Example 1: “Regression” vs. “ANOVA”

Today’s Plan

  1. Unit 1: Review dummy coding and centering + A short In-Class quiz
  2. Unit 2: Illustrate the GLM with interaction using on example
  3. Unit 3: Practice R code + Q&A

Unit 1: Dummy Coding and Centering

Today’s Example:

  • Study examining effect of new instruction method (New: 0=Old, 1=New) on test performance (Test: % correct) in college freshmen vs. senior (Senior: 0=Freshman, 1=Senior). “New by Senior” leads to 4 groups with n = 25 per group (Group: 1= Ctl, 2=T1, 3=T2, 4=T4).

\text{Test}_p =\beta_0+\beta_1T1_p + \beta_2T2_p + \beta_3T3_p+ e_p \tag{1}

\text{Test}_p =\beta_0+\beta_1\text{Senior}_p+\beta_2\text{New}_p+ \beta_3 \text{Senior}_p\text{New}_p+ e_p \tag{2}

Note that Equation 1 (single group variable model) and Equation 2 (group interaction model) are equivalent models with exactly same sets of parameters.

⌘+C 
library(ESRM64503)
library(tidyverse)
library(kableExtra)
library(gt)
data("dataTestExperiment")
dataTestExperiment |> 
  group_by(Senior, New) |> 
  summarise(
    Mean = mean(Test),
    SD = sd(Test),
    SE = sd(Test) / sqrt(25)
  ) |> 
  mutate(
    Statistics = paste0(signif(Mean, 4), "(", round(SD, 2), ")", "[", round(SE, 2), "]")
  ) |> 
  dplyr::select(-c(Mean, SD, SE)) |> 
  mutate(
    Senior = factor(Senior, levels = 0:1, labels = c("Freshmen", "Senior")),
    New = factor(New, levels = 0:1, labels = c("Old", "New")),
         ) |> 
  pivot_wider(names_from = Senior, values_from = Statistics) |> 
  kable()
New Freshmen Senior
Old 80.2(2.6)[0.52] 82.36(2.93)[0.59]
New 87.96(2.24)[0.45] 87.08(2.9)[0.58]
Note

Test Mean (SD), [SE = \frac{SD}{\sqrt{n}}]

The Two Sides of a Model

\mathbf{\text{Test}_p =\color{tomato}{\beta_0+ \beta_1T1_p +\beta_2T2_p + \beta_3T3_p}+ \color{turquoise}{e_p}}

\mathbf{\text{Test}_p =\color{tomato}{\beta_0+\beta_1\text{Senior}_p+\beta_2\text{New}_p+ \beta_3 \text{Senior}_p\text{New}_p}+ \color{turquoise}{e_p}}

  • Model for the Means (Predicted Values)

    • Each person’s expected (predicted) outcome is a function of his/her values on x and z (and their interaction), each measured once person

    • Estimated parameters are called fixed effects (here, \beta_0, \beta_1, \beta_2, and \beta_3); although they have a sampling distribution, they are not random variables

    • The number of fixed effects will show up in formulas as k (so k = 4 here)

  • Model for the Variance

    • e_p \sim N(0, \sigma^2_e) \rightarrow ONE residual (unexplained) deviation
    • e_p has a mean of 0 with some estimated constant variance \sigma^2_e, is normally distributed, is unrelated across people
    • Estimated parameter is the residual variance only (in the model above)

General Form of Two Models

  • Model for the Means (Predicted Values): The Best Guess of Estimates

    \boldsymbol{\beta} = \begin{bmatrix}{\beta_0,\\ \beta_1,\\ ...,\\\beta_K}\end{bmatrix}

  • Model for the Variance: Stability of Estimates

    1. Variance-Covariance Matrix of Regression Coefficients (Diagnoal: variances of betas (squared standard errors of betas); Nondiagnoal: residual covariances between betas)

\mathbf{\Sigma_{\boldsymbol{\beta}}} =

\begin{bmatrix} \color{tomato}{Var(\beta_1)} & Cov(\beta_1,\beta_2) & Cov(\beta_1,\beta_3), \cdots & \cdots & Cov(\beta_1, \beta_K) \\ Cov(\beta_2, \beta_1)& \color{tomato}{Var(\beta_2)}& Cov(\beta_2,\beta_3), \cdots & \cdots & Cov(\beta_2, \beta_K) \\ \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ Cov(\beta_K, \beta_1) & Cov(\beta_K, \beta_2)& Cov(\beta_K,\beta_3)& \cdots & \color{tomato}{Var(\beta_K)} \end{bmatrix}
  1. Residual variances \mathbf{\sigma^2_e}

Representing the Effects of Predictor Variables

  • From now on, we will think carefully about how the predictor variables enter into the model for the means rather than the scales of predictors

  • Why don’t people always care about the scale of predictors (centering):

    1. Does NOT affect the amount of outcome variance account for (R^2)

    2. Does NOT affect the outcomes values predicted by the model for the means (so long as the same predictor fixed effects are included)

  • Why should this matter to us?

    1. Because the Intercept = expected outcome value when X = 0

    2. Can end up with nonsense values for intercept if X = isn’t in the data

    3. We will almost always need to deliberately adjust the scale of the predictor variables so that they have 0 values that could be observed in our data

    4. Is much bigger deal in models with random effects (MLM) or GLM once interactions are included

Adjusting the Scale of Predictor Variables

  • For continuous (quantitative) predictors, we will make the intercept interpretable by centering:

    • Centering = subtract as constant from each person’s variable value so that the 0 value falls within the range of the new centered predictor variable

    • Typical \rightarrow Center around predictor’s mean: X_1^\prime = X_1 - \bar{X_1}

    • Better \rightarrow Center around meaningful constant C: X_1^\prime = X_1 - C

x <- rnorm(100, mean = 1, sd = 1)
x_c <- x - mean(x)
mean(x)
mean(x_c)
[1] 1.163785
[1] 1.811051e-17

  • For categorical (grouping) predictors, either we or the program will make the intercept interpretable by creating a reference group:

    • Reference group is given a 0 value on all predictor variable created from the original group variable, such that the intercept is the expected outcome for that reference group specifically

    • Accomplished via “dummy coding” or “reference group coding”

  • For categorical predictors with more than two groups

    • We need to dummy coded the group variable using factor() in R
    • For example, I dummy coded group variable with group 1 as the reference
dataTestExperiment$Group <- factor(dataTestExperiment$Group, 
                                   levels = 1:4, 
                                   labels = c("Ctl", "T1", "T2", "T3"))
mod_e0 <- lm(Test ~ Group, data = dataTestExperiment)
summary(mod_e0)$coefficients |> kable(digits = 3)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 80.20 0.536 149.513 0.000
GroupT1 7.76 0.759 10.229 0.000
GroupT2 2.16 0.759 2.847 0.005
GroupT3 6.88 0.759 9.069 0.000

  • Variable Group: # dummy variables = # group -1

    • Dummy variables with four levels - {Control, Treatment1, Treatment2, Treatment3}:

      • d1 = GroupT1 (0, 1, 0, 0) \rightarrow \beta_1 = difference between Treatment1 vs. Control

      • d2 = GroupT2 (0, 0, 1, 0) \rightarrow \beta_2 = difference between Treatment2 vs. Control

      • d3 = GroupT3 (0, 0, 0, 1) \rightarrow \beta_3 = difference between Treatment3 vs. Control

model.matrix(mod_e0) |> cbind(dataTestExperiment) |> show_table(font_size = 33)
(Intercept) GroupT1 GroupT2 GroupT3 PersonID Senior New Group Test
1 0 0 0 1 0 0 Ctl 78
1 0 0 0 2 0 0 Ctl 77
1 0 0 0 3 0 0 Ctl 75
1 0 0 0 4 0 0 Ctl 86
1 0 0 0 5 0 0 Ctl 78
1 0 0 0 6 0 0 Ctl 79
1 0 0 0 7 0 0 Ctl 84
1 0 0 0 8 0 0 Ctl 78
1 0 0 0 9 0 0 Ctl 80
1 0 0 0 10 0 0 Ctl 78
1 0 0 0 11 0 0 Ctl 82
1 0 0 0 12 0 0 Ctl 83
1 0 0 0 13 0 0 Ctl 83
1 0 0 0 14 0 0 Ctl 82
1 0 0 0 15 0 0 Ctl 78
1 0 0 0 16 0 0 Ctl 81
1 0 0 0 17 0 0 Ctl 78
1 0 0 0 18 0 0 Ctl 82
1 0 0 0 19 0 0 Ctl 81
1 0 0 0 20 0 0 Ctl 79
1 0 0 0 21 0 0 Ctl 83
1 0 0 0 22 0 0 Ctl 82
1 0 0 0 23 0 0 Ctl 79
1 0 0 0 24 0 0 Ctl 81
1 0 0 0 25 0 0 Ctl 78
1 1 0 0 26 0 1 T1 87
1 1 0 0 27 0 1 T1 88
1 1 0 0 28 0 1 T1 89
1 1 0 0 29 0 1 T1 90
1 1 0 0 30 0 1 T1 86
1 1 0 0 31 0 1 T1 89
1 1 0 0 32 0 1 T1 85
1 1 0 0 33 0 1 T1 88
1 1 0 0 34 0 1 T1 83
1 1 0 0 35 0 1 T1 88
1 1 0 0 36 0 1 T1 87
1 1 0 0 37 0 1 T1 91
1 1 0 0 38 0 1 T1 86
1 1 0 0 39 0 1 T1 89
1 1 0 0 40 0 1 T1 86
1 1 0 0 41 0 1 T1 89
1 1 0 0 42 0 1 T1 87
1 1 0 0 43 0 1 T1 84
1 1 0 0 44 0 1 T1 90
1 1 0 0 45 0 1 T1 90
1 1 0 0 46 0 1 T1 90
1 1 0 0 47 0 1 T1 89
1 1 0 0 48 0 1 T1 87
1 1 0 0 49 0 1 T1 88
1 1 0 0 50 0 1 T1 93
1 0 1 0 51 1 0 T2 80
1 0 1 0 52 1 0 T2 89
1 0 1 0 53 1 0 T2 86
1 0 1 0 54 1 0 T2 85
1 0 1 0 55 1 0 T2 81
1 0 1 0 56 1 0 T2 81
1 0 1 0 57 1 0 T2 85
1 0 1 0 58 1 0 T2 84
1 0 1 0 59 1 0 T2 84
1 0 1 0 60 1 0 T2 82
1 0 1 0 61 1 0 T2 79
1 0 1 0 62 1 0 T2 82
1 0 1 0 63 1 0 T2 84
1 0 1 0 64 1 0 T2 82
1 0 1 0 65 1 0 T2 80
1 0 1 0 66 1 0 T2 83
1 0 1 0 67 1 0 T2 77
1 0 1 0 68 1 0 T2 85
1 0 1 0 69 1 0 T2 76
1 0 1 0 70 1 0 T2 83
1 0 1 0 71 1 0 T2 82
1 0 1 0 72 1 0 T2 81
1 0 1 0 73 1 0 T2 85
1 0 1 0 74 1 0 T2 79
1 0 1 0 75 1 0 T2 84
1 0 0 1 76 1 1 T3 90
1 0 0 1 77 1 1 T3 85
1 0 0 1 78 1 1 T3 85
1 0 0 1 79 1 1 T3 88
1 0 0 1 80 1 1 T3 88
1 0 0 1 81 1 1 T3 89
1 0 0 1 82 1 1 T3 92
1 0 0 1 83 1 1 T3 90
1 0 0 1 84 1 1 T3 82
1 0 0 1 85 1 1 T3 82
1 0 0 1 86 1 1 T3 89
1 0 0 1 87 1 1 T3 84
1 0 0 1 88 1 1 T3 90
1 0 0 1 89 1 1 T3 91
1 0 0 1 90 1 1 T3 88
1 0 0 1 91 1 1 T3 88
1 0 0 1 92 1 1 T3 81
1 0 0 1 93 1 1 T3 90
1 0 0 1 94 1 1 T3 85
1 0 0 1 95 1 1 T3 85
1 0 0 1 96 1 1 T3 88
1 0 0 1 97 1 1 T3 86
1 0 0 1 98 1 1 T3 87
1 0 0 1 99 1 1 T3 87
1 0 0 1 100 1 1 T3 87

  • Other examples of things people do to categorical predictors:

    • “Contrast/effect coding” \rightarrow Gender: -0.5 = Man, 0.5 = Women

      • Effect: Man vs. Average | Women vs. Average

    • Test other contrasts among multiple groups

      • Four-group variable: Control, Treatment1, Treatment2, Treatment3

      • Effect: contrast1 = {-1, .33, .33, .34}

      • Control vs. Any Treatment?

Categorical Predictors: Main Effects and Group Means

  • The single group model:

  • \hat Y_p = \beta_0+\beta_1\text{GroupT1}_p+\beta_2\text{GroupT2}_p + \beta_3\text{GroupT3}_p

    • d1 = GroupT1 (0, 1, 0, 0) \rightarrow \beta_1 = mean difference between Treatment1 vs. Control
    • d2 = GroupT2 (0, 0, 1, 0) \rightarrow \beta_2 = mean difference between Treatment2 vs. Control
    • d3 = GroupT3 (0, 0, 0, 1) \rightarrow \beta_3 = mean difference between Treatment3 vs. Control

  • How does the model give us all possible group difference?

Control Mean Treatment 1 Mean Treatment 2 Mean Treatment 3 Mean
\beta_0 \beta_0+\beta_1 \beta_0+\beta_2 \beta_0+\beta_3
  • The model (coefficients with dummy coding) directly provides 3 differences (control vs. each treatment), and indirectly provides another 3 differences (differences between treatments)

Group differences from Dummy Codes

Set J = 4 as number of groups:

The total number of group differences is J * (J-1) / 2 = 4*3/2 = 6

Control Mean Treatment 1 Mean Treatment 2 Mean Treatment 3 Mean
\beta_0 \beta_0+\beta_1 \beta_0+\beta_2 \beta_0+\beta_3
  • All group differences
Alt Group Ref Group Difference
Control vs. T1 (\beta_0+\beta_1) - \beta_0 = \beta_1
Control vs. T2 (\beta_0+\beta_2) - \beta_0 = \beta_2
Control vs. T3 (\beta_0+\beta_3) - \beta_0 = \beta_3
T1 vs. T2 (\beta_0+\beta_2) -(\beta_0+\beta_1) = \beta_2-\beta_1
T1 vs. T3 (\beta_0+\beta_3) -(\beta_0+\beta_1) = \beta_3-\beta_1
T2 vs. T3 (\beta_0+\beta_3) -(\beta_0+\beta_2) = \beta_3 - \beta_2

R Code: Estimating All Group Differences in R

library(multcomp) # install.packages("multcomp")
mod_e0 <- lm(Test ~ Group, data = dataTestExperiment)
summary(mod_e0)
contrast_model_matrix = matrix(c(
  0,  1, 0, 0,  # Control vs. T1
  0,  0, 1, 0,  # Control vs. T2
  0,  0, 0, 1,  # Control vs. T3
  0, -1, 1, 0,  # T1 vs. T2
  0, -1, 0, 1,  # T1 vs. T3
  0,  0,-1, 1   # T2 vs. T3
), nrow = 6, byrow = T)
rownames(contrast_model_matrix) <- c( "Control vs. T1", "Control vs. T2", "Control vs. T3", "T1 vs. T2", "T1 vs. T3", "T2 vs. T3" ) 
contrast_value <- glht(mod_e0, linfct = contrast_model_matrix)
summary(contrast_value)

     Simultaneous Tests for General Linear Hypotheses

Fit: lm(formula = Test ~ Group, data = dataTestExperiment)

Linear Hypotheses:
                    Estimate Std. Error t value Pr(>|t|)    
Control vs. T1 == 0   7.7600     0.7586  10.229   <0.001 ***
Control vs. T2 == 0   2.1600     0.7586   2.847   0.0271 *  
Control vs. T3 == 0   6.8800     0.7586   9.069   <0.001 ***
T1 vs. T2 == 0       -5.6000     0.7586  -7.382   <0.001 ***
T1 vs. T3 == 0       -0.8800     0.7586  -1.160   0.6534    
T2 vs. T3 == 0        4.7200     0.7586   6.222   <0.001 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Adjusted p values reported -- single-step method)

What the intercept should mean to you

  • The model for the means will describe what happens to the predicted outcome Y “as X increases” or “as Z increases” and so forth

  • But you wont what Y is actually supposed to be unless you know where the predictor variables are starting from!

  • Therefor, the intercept is the “YOU ARE HERE” sign in the map of your data… so it should be somewhere in the map*!

Main Effects Within Interactions

Interaction Effects

  • Interaction = Moderation: the effect of a predictor depends on the value of the interacting predictor

    • Either predictor can be “the moderator” (interpretive distinction only)

  • Interaction can always be evaluated for any combination of categorical and continuous predictors, although

    1. In “ANOVA”: By default, all possible interactions are estimated

      • Software does this for you; oddly enough, nonsignificant interactions usually still are kept in the model (even if only significant interactions are interpreted)

    2. In “ANCOVA”: Continuous predictors (“covariates”) do not get to be part of interaction ➡️ make the “homogeneity of regression” assumption

      • There is no reason to assume this – it is a testable hypothesis!

    3. In “Regression”: No default – effects of predictors are as you specify them

      • Requires most thought, but gets annoying because in regression programs you usually have to manually create the interaction as an observed variable:

      • e.g., XZ_interaction = centered_X * centered_Z

Simple / Conditional Main Effects in GLM with Interactions

Note

Main effects of predictors within interactions should remain in the model regardless of whether or not they are significant

😶: Y_p = \beta_0 + \beta_1 X_1 X_2

😄: Y_p = \beta_0 + \beta_1X_1+\beta_2X_2+\beta_3 X_1 X_2

  • Reason: the role of two-way interaction is to adjust its main effects

  • However, the original idea of a “main effect” no longer applied … each main effect is conditional on the interacting predictor as 0 (X_1X_2=0)

  • Conditional Main Effect = What it is + What modified it

    Simple Main Effect = The modified value of main effect

  • “Conditional” main effect is a general form of “Simple” main effect:

    • \beta_1 is the “simple” main effect of X1 when X2 = 0

      • \beta_1 + \beta_3 X_2 is the “conditional” main effect of X1 depending on X2 values

    • \beta_2 is the “simple” main effect of X2 when X1 = 0

      • \beta_2 + \beta_3 X_1 is the “conditional” main effect of X2 depending on X1 values

Model-Implied Simple Main Effects

To quickly compute simple main effects:

Tip

The trick is keeping track of what 0 means for every interacting predictor, which depends on the way each predictor is being represented, as determined by you, or by the software without you!

\text{GPA}_p = 30 + 1\times \text{Motiv}_p +2 \times\text{Exam}_p+ 0.5 \times \text{Motiv}\times \text{Exam}_p

  • GPA scores of anyone can be predicted by academic motivation, final exam scores, and their interaction by this model

  • Conditional Main Effect of Motiv : (1 + 0.5\times \text{Exam})

    • Simple main effect = 1 if Exam = 0; = 1.5 if Exam = 1; = 3 if Exam = 4

  • Conditional Main Effect of Exam : (2 + 0.5\times \text{Motiv})

    • Simple main effect = 2 if Motiv = 0; = 3 if Motiv = 2; = 4 if Motiv = 4

Interpretation

\begin{aligned} \text{GPA}_p = \beta_0 + \beta_1\times \text{Motiv}_p + \beta_2 \times\text{Exam}_p+ \beta_3 \times \text{Motiv}_p \times \text{Exam}_p \\= 30 + 1\times \text{Motiv}_p +2 \times\text{Exam}_p+ 0.5 \times \text{Motiv}_p \times \text{Exam}_p \end{aligned}

  • \beta_0: Expected GPA when motivation is 0 and exam score is 0

  • \beta_1: Increase in GPA per unit motivation when exam score is 0

  • \beta_2: Increase in GPA per unit exam score when motivation is 0

  • \beta_3: Two ways of interpretation

    • Motivation is moderator: Increase in effect of exam scores per unit motivation

      • One unit of motivation leads to the effect of exam score 2 \rightarrow 2.5

    • Exam Score is moderator: Increase in effect of motivation per unit exam score

      • One unit of exam score leads to the effect of motivation 1 \rightarrow 1.5
Note

If interaction effect is significant, we typically report that motivation significantly moderates the effect of exam score on GPA

Why centering matters

When we centered Exam Score with 3 as centering point, then intercept and the main effect of motivation will change:

\begin{align} \text{GPA}_p = \color{tomato}{30} + \color{tomato}{1}\times \text{Motiv}_p +2 \times\text{Exam}_p+ 0.5 \times \text{Motiv}\times \text{Exam}_p \\= \color{tomato}{\beta_0^\prime} + \color{tomato}{\beta_1\prime} \times \text{Motiv}_p + \beta_2 \times (\text{Exam}_p-3)+ \beta_3 \times \text{Motiv}\times (\text{Exam}_p-3) \\= \color{red}{36} + \color{red}{2.5}\times \text{Motiv}_p +2 \times (\text{Exam}_p-3) + 0.5 \times \text{Motiv}\times (\text{Exam}_p-3) \end{align}

  • Trick of new coefficients: Predicted value stay the same

    • Expected GPA score is 30 + 1*0 + 2*3 + 0.5*0 = 36 when Motiv = 0 and Exam = 3

    • Expected conditional main effect of Motiv is 1 + 0.5*3 = 2.5 when Exam = 3

  • Reason: \beta_0 and \beta_1 are conditional on exam score while \beta_2 and \beta_3 are unconditional on exam score

Example: Main Effect for continuous centered predictors

  • Conditional main effect for Motiv depending on value of Exam:

    • (1 + 0.5\times\text{Exam}) or [2.5 + 0.5 \times(\text{Exam}-3)]

    • “Simple” main effect is 1 when Exam = 0; 2.5 when Exam = 3

    • Assume centering point of Exam is T, we can get the general form of simple main effect of Motivation as:

    • f(\beta_1,\beta_3,\text{Exam})= (\beta_1 + \beta_3*T) + \beta_3*(\text{Exam} - T) = \beta_1^{new}+\beta_3*\text{Exam}_C

    • where new conditional main effect \beta_1^{new} = \beta_1+\beta_3*T

  • Similarly, we can get the general form of conditional main effect of Exam as

    • f(\beta_2,\beta_3,\text{Motiv})= (\beta_2 + \beta_3*T) + \beta_3*(\text{Motiv} - T)

    • where new conditional main effect \beta_2^{new} = \beta_2 + \beta_3 * T

Quiz

Testing the Significance of Model-Implied Fixed Effects

  • We now know how to calculate any conditional main effect:

    • Effect of interest (“conditional” main effect) = what it is + what modifies it

    • Output Effect (“simple” main effect) = what it is + what modifies it is 0

  • But if we want to test whether that new effect is \neq 0, we also need its standard error (SE needed to get Wald test T-value ➡️p-value)

  • Even if the conditional main effect is not directly given by the model, its estimate and SE are still implied by the model

  • 3 options to get the new conditional main effect estimates and SE

    • Method 1: Ask the software to give it to you using your original model

      • e.g., glht function in R package multicomp, ESTIMATE in SAS, TEST in SPSS, NEW in Mplus

    • Model 2: Re-center your predictors to the interacting value of interest (e.g., make Exam = 3 the new 0 for \text{Exam}_C) and re-estimate your model; repeat as needed for each value of interest

    • Method 3: Hand calculations (what the program is doing for you in option #1)

  • Method 3 for example: Effect of Motiv = \beta_1 + \beta_3 * \text{Exam}

    • We have following formula to calculate the sampling error variance of “conditional” main effect as

    • \mathbf{SE_{\beta_1^{New}}^2 = Var(\beta_1) + Var(\beta_3) * Exam + 2Cov(\beta_1, \beta_3)*Exam}

      • Values come from “asymptotic (sampling) covariance matrix”

      • Variance of a sum of terms always includes covariance among them

      • Here, this is because what each main effect estimate could be is related to what the other main effect estimates could be

      • Note that if a main effect is unconditional, its SE^2 = Var(\beta) only

Unit 2: GLM Example 1

GLM via Dummy-Coding in “Regression”

# Model 1: 2 X 2 predictors with 0/1 coding
model1 <- lm(Test ~ Senior + New + Senior * New, data = dataTestExperiment)
# Alternative
formular_mod1 <- as.formula("Test ~ Senior + New + Senior * New")
model1 <- lm(formular_mod1, data = dataTestExperiment)
summary(model1)

Call:
lm(formula = formular_mod1, data = dataTestExperiment)

Residuals:
   Min     1Q Median     3Q    Max 
 -6.36  -2.08   0.04   1.83   6.64 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  80.2000     0.5364 149.513  < 2e-16 ***
Senior        2.1600     0.7586   2.847  0.00539 ** 
New           7.7600     0.7586  10.229  < 2e-16 ***
Senior:New   -3.0400     1.0728  -2.834  0.00561 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.682 on 96 degrees of freedom
Multiple R-squared:  0.6013,    Adjusted R-squared:  0.5888 
F-statistic: 48.26 on 3 and 96 DF,  p-value: < 2.2e-16
anova(model1)
Analysis of Variance Table

Response: Test
           Df Sum Sq Mean Sq  F value    Pr(>F)    
Senior      1  10.24   10.24   1.4235  0.235762    
New         1 973.44  973.44 135.3253 < 2.2e-16 ***
Senior:New  1  57.76   57.76   8.0297  0.005609 ** 
Residuals  96 690.56    7.19                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note

These ANOVA table is displaying marginal tests (F-test) for the main effects. Marginal tests are for the main effect only and are not conditional on any interacting variables.

Getting Group Means as a Contrast

We can get group means of test scores with Senior/Freshman by New/Old combination

glht() requests predicted outcomes from model for the means:

\mathbf{\hat{Test} =\beta_0+\beta_1Senior+\beta_2New+\beta_3SeniorNew}

  • Freshmen-Old: \mathbf{\hat{Test_1}=\beta_0*1+\beta_1*0+\beta_2*0 +\beta_3*0}

  • Freshmen-New: \mathbf{\hat{Test_2}=\beta_0*1+\beta_1*0+\beta_2*1 +\beta_3*0}

  • Senior-Old: \mathbf{\hat{Test_3}=\beta_0*1+\beta_1*1+\beta_2*0 +\beta_3*0}

  • Senior-New: \mathbf{\hat{Test_4}=\beta_0*1+\beta_1*1+\beta_2*1 +\beta_3*1}

contrast_model1_matrix = matrix(c(
  1, 0, 0, 0,  # Freshman-Old
  1, 0, 1, 0,  # Freshman-New
  1, 1, 0, 0,  # Senior-Old
  1, 1, 1, 1   # Senior-New
), nrow = 4, byrow = T)
rownames(contrast_model1_matrix) <- c( "Freshman-Old", "Freshman-New", "Senior-Old", "Senior-New") 
means_model1 <- glht(model1, linfct = contrast_model1_matrix)
summary(means_model1)

     Simultaneous Tests for General Linear Hypotheses

Fit: lm(formula = formular_mod1, data = dataTestExperiment)

Linear Hypotheses:
                  Estimate Std. Error t value Pr(>|t|)    
Freshman-Old == 0  80.2000     0.5364   149.5   <2e-16 ***
Freshman-New == 0  87.9600     0.5364   164.0   <2e-16 ***
Senior-Old == 0    82.3600     0.5364   153.5   <2e-16 ***
Senior-New == 0    87.0800     0.5364   162.3   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Adjusted p values reported -- single-step method)

Standard Errors of Group Means

  • Freshman-Old group mean’s standard error:

\mathbf{SE^2(\hat{Test_1}) = \mathbf{SE^2(\beta_0)} = Var(\beta_0)}

  • Freshman-New group mean’s standard error:

\mathbf{SE^2(\hat{Test_2}) = \mathbf{SE^2(\beta_0+\beta_2)} = Var(\beta_0) + Var(\beta_2) + 2Cov(\beta_0,\beta_2)}

  • Senior-Old group mean’s standard error:

    \mathbf{SE^2(\hat{Test}_3)} = \mathbf{SE^2(\beta_0+\beta_1)} = ???

  • Senior-New group mean’s standard error:

    \mathbf{SE^2(\hat{Test}_4)} = \mathbf{SE^2(\beta_0+\beta_1+\beta_2+\beta_3)} = ??? = \sum(\Sigma_\beta) 

vcov(model1)# Variance-covariance of betas
            (Intercept)     Senior        New Senior:New
(Intercept)   0.2877333 -0.2877333 -0.2877333  0.2877333
Senior       -0.2877333  0.5754667  0.2877333 -0.5754667
New          -0.2877333  0.2877333  0.5754667 -0.5754667
Senior:New    0.2877333 -0.5754667 -0.5754667  1.1509333
# Standard error of freshman-old group: SE(beta_0)
Var_Test_1 <- Var_beta_0 <- vcov(model1)[1,1]
(SE_Test_1 <- sqrt(Var_Test_1))
[1] 0.5364078
# Standard error of freshman-new group: SE(beta_0+beta_1)
Var_beta_0 <- vcov(model1)[1,1]
Var_beta_2 <- vcov(model1)[3,3]
Cov_beta_0_2 <- vcov(model1)[1,3]
Var_Test_2 <- Var_beta_0 + Var_beta_2 + 2*Cov_beta_0_2
(SE_Test_2 <- sqrt(Var_Test_2))
[1] 0.5364078
Your Homework 1 - Question 1

Copy and paste your R syntax abd output that calculates the Senior-Old’s standard errors of means (use the data and model we use in class). For example, our class syntax for Freshman-New is like:

# R syntax

library(ESRM64503)
library(tidyverse)
model1 <- lm(Test ~ Senior + New + Senior * New, data = dataTestExperiment)
data(“dataTestExperiment”)
Var_beta_0 <- vcov(model1)[1,1]
Var_beta_2 <- vcov(model1)[3,3]
Cov_beta_0_2 <- vcov(model1)[1,3]
Var_Test_2 <- Var_beta_0 + Var_beta_2 + 2*Cov_beta_0_2
(SE_Test_2 <- sqrt(Var_Test_2)) 

# R Output
[1] 0.5364078

Standard errors of Main Effects

We can use glht() to automatically calculate conditional main effects and their standard errors. Or, we can manually calculate them using vcov(model) function, which represents the variance-covariance matrix of main effects.

Method 1: glht method

effect_model1_matrix = matrix(c(
  0, 1, 0, 0,  # Senior vs. Freshmen | Old
  0, 1, 0, 1,  # Senior vs. Freshmen | New
  0, 0, 1, 0,  # New vs. Old | Freshmen
  0, 0, 1, 1   # New vs. Old | Senior
), nrow = 4, byrow = T)
rownames(effect_model1_matrix) <- c( "beta_Senior_New0", # Conditional Main Effect of Senior given New = 0
                                     "beta_Senior_New1", # Conditional Main Effect of Senior given New = 1
                                     "beta_New_Senior0", # Conditional Main Effect of New given Senior = 0
                                     "beta_New_Senior1") # Conditional Main Effect of New given Senior = 1
effect_model1 <- glht(model1, linfct = effect_model1_matrix)
summary(effect_model1)

     Simultaneous Tests for General Linear Hypotheses

Fit: lm(formula = formular_mod1, data = dataTestExperiment)

Linear Hypotheses:
                      Estimate Std. Error t value Pr(>|t|)    
beta_Senior_New0 == 0   2.1600     0.7586   2.847   0.0196 *  
beta_Senior_New1 == 0  -0.8800     0.7586  -1.160   0.5940    
beta_New_Senior0 == 0   7.7600     0.7586  10.229   <0.001 ***
beta_New_Senior1 == 0   4.7200     0.7586   6.222   <0.001 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Adjusted p values reported -- single-step method)

Method 2: vcov method

Example: Conditional main effect of Senior \mathbf{\beta_1^{Senior}} (Senior vs. Freshmen) when using the new method (New = 1)

  • \mathbf{\beta_1^{Senior} = \beta_1+\beta_3*New}

\begin{align} \mathbf{SE^2(\beta_1^{Senior})} &= \mathbf{SE^2(\beta_1+\beta_3*New)} \\ &= \mathbf{Var(\beta_1) + Var(\beta_3) * New^2 + 2Cov(\beta_1, \beta_3)*New} \\ &= \mathbf{Var(\beta_1) + Var(\beta_3) + 2Cov(\beta_1, \beta_3)} \end{align}

## conditional main effect of senior when new
fixed_effects <- summary(model1)$coefficients
fixed_effects
            Estimate Std. Error    t value      Pr(>|t|)
(Intercept)    80.20  0.5364078 149.513112 1.588771e-115
Senior          2.16  0.7585952   2.847368  5.391733e-03
New             7.76  0.7585952  10.229435  4.785144e-17
Senior:New     -3.04  1.0728156  -2.833665  5.609415e-03
beta_1 <- fixed_effects[2, 1]
beta_3 <- fixed_effects[4, 1]
(beta_1_new <- beta_1+beta_3) # conditional main effect of Senior when New = 1
[1] -0.88
vcov(model1) # Variance-covariance of betas
            (Intercept)     Senior        New Senior:New
(Intercept)   0.2877333 -0.2877333 -0.2877333  0.2877333
Senior       -0.2877333  0.5754667  0.2877333 -0.5754667
New          -0.2877333  0.2877333  0.5754667 -0.5754667
Senior:New    0.2877333 -0.5754667 -0.5754667  1.1509333
Var_beta_1 = vcov(model1)[2,2] # squared SE of beta1
Var_beta_3 = vcov(model1)[4,4] # squared SE of beta3
Cov_beta_1_3 = vcov(model1)[4,2] # residual covariance of beta1 and beta3
Var_beta_1_new <- Var_beta_1 + Var_beta_3 * 1 + 2 * Cov_beta_1_3 * 1
(SE_beta_1_new <- sqrt(Var_beta_1_new))
[1] 0.7585952
Your Homework 1 - Question 2

Copy and paste your R syntax and output that calculates the standard error of conditional main effect of New (New vs. Old) when Senior = 1. For example, standard error of conditional main effect of Senior when New = 1 is like:

# R syntax

library(ESRM64503)
library(tidyverse)
model1 <- lm(Test ~ Senior + New + Senior * New, data = dataTestExperiment)
data(“dataTestExperiment”)
Var_beta_1 = vcov(model1)[2,2]
Var_beta_3 = vcov(model1)[4,4]
Cov_beta_1_3 = vcov(model1)[4,2]
Var_beta_1_new <- Var_beta_1 + Var_beta_3 * 1 + 2 * Cov_beta_1_3 * 1
(SE_beta_1_new <- sqrt(Var_beta_1_new)) 

# R output
[1] 0.7585952

Unit 3: R practice and Q&A

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