Lecture 02: General Linear Model

Descriptive Statistics and Basic Statistics

Author

Affiliation

Jihong Zhang*, Ph.D

Educational Statistics and Research Methods (ESRM) Program*

University of Arkansas

Published

August 26, 2024

0.1 Some notes before our class

Vote for delaying class time (to 5:15?) for free parking
The R code I’ll show in the slides is just for illustration. Don’t be worried about hardly following up. We will practice together at the end of each unit.

0.2 Learning Objectives

Review Homework 0
Review previous lecture
Unit 1: Descriptive Statistics
1. Introduce R package – ESRM64503
2. Univariate Descriptive Statistics
  - Central tendency: Mean, Median, Mode
  - Variation/Spread: Standard deviation (SD), Variance, Range
3. Bivariate descriptive statistics
  - Correlation
  - Covariance
4. Types of variable distributions
  - Marginal
  - Joint
  - Conditional
5. Bias in estimators
Unit 2: General Linear Model

1 Unit 1: Descriptive Statistics

1.1 Homework 0

library(tidyverse); library(nycflights13) # install.packages(c("nycflights13", "tidyverse"))
flights
glimpse(flights)
View(flights)
print(flights, width = Inf)

1.2 Installation of ESRM64503 package

⌘+C

install.packages("devtools")
devtools::install_github("JihongZ/ESRM64503") # Method 1
pak::pak("JihongZ/ESRM64503") # Method 2: Install the Github package
pak::pak("JihongZ/ESRM64503", upgrade = TRUE) # If you already install the package, try to upgrade

1.3 Test ESTM64503 Package

package version

library(ESRM64503)
devtools::package_info("ESRM64503") # Make sure the version number is 2024.08.20

 package   * version    date (UTC) lib source
 ESRM64503 * 2024.08.20 2024-08-20 [1] Github (JihongZ/ESRM64503@ed78e3d)

 [1] /Users/jihong/Rlibs
 [2] /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library

homework information

homework() # You can call "homework()" function to access homework info


There will be four homeworks in total for ESRM 64503.
You can work on HW0 now, please click the following link to have access:
https://jihongzhang.org/posts/Lectures/2024-07-21-applied-multivariate-statistics-esrm64503/HWs/HW_demo.html
Next homework (Homework 1) will be posted on 09/09/2024.
For more details, please refer to
https://jihongzhang.org/posts/Lectures/2024-07-21-applied-multivariate-statistics-esrm64503/

1.4 Data Files of ESRM64503

Data dataSexHeightWeight automate loaded

dataSexHeightWeight # You should find data named "dataSexHeightWeight" already loaded

   id sex heightIN weightLB
1   1   F       56      117
2   2   F       60      125
3   3   F       64      133
4   4   F       68      141
5   5   F       72      149
6   6   F       54      109
7   7   F       62      128
8   8   F       65      131
9   9   F       65      131
10 10   F       70      145
11 11   M       64      211
12 12   M       68      223
13 13   M       72      235
14 14   M       76      247
15 15   M       80      259
16 16   M       62      201
17 17   M       69      228
18 18   M       74      245
19 19   M       75      241
20 20   M       82      269

Type ? to check variable information of data

?dataSexHeightWeight

1.5 Simulated Data for Today’s Lecture - dataSexHeightWeight

To help demonstrate the concepts of today’s lecture, we will be using a toy data set with three variables
- Female (Gender): Coded as Male (= 0) or Female (= 1)
  - In R, factor variable with two levels: FALSE, TRUE
- Height: in inches
- Weight: in pounds
The goal of lecture 02 will be to build a general linear model that predicts a person’s weight
- Linear (regression) model: a statistical model for an outcome that uses a linear combination (a weighted sum) of one or more predictor variables to produce an estimate of an observation’s predicted value
- \mathbb{y} = \beta_0+\beta_1 \mathbf{X}
All models we learnt today will follow this framework.

1.6 Recoding Variables

library(ESRM64503) # INSTALL: pak::pak("JihongZ/ESRM64504")
library(kableExtra) # INSTALL: pak::pak("JihongZ/ESRM64504")

## First checking sample size
## N = 20

## Recode Sex as Female
dataSexHeightWeight$female = dataSexHeightWeight$sex == "F"
kable(dataSexHeightWeight,
      caption = 'toy data set') |> 
  kable_styling(font_size = 30)

toy data set
id	sex	heightIN	weightLB	female
1	F	56	117	TRUE
2	F	60	125	TRUE
3	F	64	133	TRUE
4	F	68	141	TRUE
5	F	72	149	TRUE
6	F	54	109	TRUE
7	F	62	128	TRUE
8	F	65	131	TRUE
9	F	65	131	TRUE
10	F	70	145	TRUE
11	M	64	211	FALSE
12	M	68	223	FALSE
13	M	72	235	FALSE
14	M	76	247	FALSE
15	M	80	259	FALSE
16	M	62	201	FALSE
17	M	69	228	FALSE
18	M	74	245	FALSE
19	M	75	241	FALSE
20	M	82	269	FALSE

1.7 Descriptive Statistics

First, we can inspect each variable individually (marginal distribution) through a set of descriptive statistics
- Visual way: histogram plot or density plot
- Statistical way: Central tendency and Variability
  - Mean, Median, Mode
  - SD, Range
Second, we can also summarize the joint (bivariate) distribution of two variables through a set of descriptive statistics:
- Joint vs. Marginal: joint distribution describes more than one variable simultaneously
- Common bivariate descriptive statistics:
  - Correlation and covariance

1.8 Quick inspection: Missing data rate

Check (1) if any case has missing value (2) distributions for continuous and categorical variables

⌘+C

table(complete.cases(dataSexHeightWeight))


TRUE 
  20

⌘+C

dataWithMissing <- dataSexHeightWeight
dataWithMissing[11, 2] <- NA
table(complete.cases(dataWithMissing))


FALSE  TRUE 
    1    19

psych::describe(dataSexHeightWeight[,-1], omit = TRUE, trim = 0) |> kable(digits = 3)

	vars	n	mean	sd	median	trimmed	mad	min	max	range	skew	kurtosis	se
heightIN	2	20	67.9	7.440	68	67.9	7.413	54	82	28	0.048	-0.812	1.664
weightLB	3	20	183.4	56.383	175	183.4	72.647	109	269	160	0.110	-1.804	12.608

1.9 Quick Inspect: Distribution of Categorical Variable

table(dataSexHeightWeight$female)


FALSE  TRUE 
   10    10

hist(as.numeric(dataSexHeightWeight$female), main = "Histogram of Female", xlab = "Female", )

1.10 Visualization: Pairwise Scatterplot

pairs(dataSexHeightWeight[, c('female', 'heightIN', 'weightLB')])

1.11 Histograms of Height and Weight

hist(dataSexHeightWeight$weightLB, main = 'Weight', xlab = 'Pounds')

hist(dataSexHeightWeight$heightIN, main = 'Height', xlab = 'Inches')

1.12 Descriptive Statistics for Toy Data: Marginal

library(dplyr)
## wide-format marginal description
wide_marginal_desp <- dataSexHeightWeight |> 
  summarise(across(c(heightIN, weightLB, female), list(mean = mean, sd = sd, var = var)))

library(tidyr)
wide_marginal_desp |> 
  pivot_longer(everything(), names_to = "Variable", values_to = "Value") |> 
  separate(Variable, sep = "_", into = c("Variable", "Stats")) |> 
  pivot_wider(names_from = Stats, values_from = Value) |> 
  kable(digits = 3)

Variable	mean	sd	var
heightIN	67.9	7.440	55.358
weightLB	183.4	56.383	3179.095
female	0.5	0.513	0.263

1.13 Descriptive Statistics for Toy Data: Joint

Correlation-Covariance Matrix:
- Diagonal: Variance
- Above Diagonal (upper triangle): Covaraince
- Below Diagonal (lower triangle): Correlation
Question: What we can tell regarding the relationships among three variables?

cor_cov_mat <- matrix(nrow = 3, ncol = 3)
colnames(cor_cov_mat) <- rownames(cor_cov_mat) <- c("heightIN", "weightLB", "female")
cov_mat <- cov(dataSexHeightWeight[, c("heightIN", "weightLB", "female")])
cor_mat <- cor(dataSexHeightWeight[, c("heightIN", "weightLB", "female")])
## Assign values
cor_cov_mat[lower.tri(cor_cov_mat)] <- cor_mat[lower.tri(cor_mat)]
cor_cov_mat[upper.tri(cor_cov_mat)] <- cov_mat[upper.tri(cov_mat)]
diag(cor_cov_mat) <- diag(cov_mat)
kable(cor_cov_mat, digits = 3)

	heightIN	weightLB	female
heightIN	55.358	334.832	-2.263
weightLB	0.798	3179.095	-27.632
female	-0.593	-0.955	0.263

2 Variance, Correlation, Covariance

2.1 Re-examining the Concept of Variance

Variability is a central concept in advanced statistics
- In multivariate statistic, covariance is also central
Two formulas for the variance (about the same when N is larger):

S^2_Y= \frac{\Sigma_{p=1}^{N}(Y_p-\bar Y)}{N-1} \tag{1}

S^2_Y= \frac{\Sigma_{p=1}^{N}(Y_p-\bar Y)}{N} \tag{2}
Equation 1: Unbiased or “sample”
Equation 2: Biased/ML or “population”

Here: p = person;

2.2 Biased VS. Unbiased Variability

⌘+C

set.seed(1234)
sample_sizes = seq(10, 200, 10) # N = 10, 20, 30, ..., 200
population_var = 100 # True variance for population
variance_mat <- matrix(NA, nrow = length(sample_sizes), ncol = 4)
iter = 0
for (sample_size in sample_sizes) {
  iter = iter + 1
  sample_points <- rnorm(n = sample_size, mean = 0, sd = sqrt(population_var))
  sample_var_biased <- var(sample_points) * (sample_size-1) / sample_size
  sample_var_unbiased <- var(sample_points)
  variance_mat[iter, 1] <- sample_size
  variance_mat[iter, 2] <- sample_var_biased
  variance_mat[iter, 3] <- sample_var_unbiased
  variance_mat[iter, 4] <- population_var
}
colnames(variance_mat) <- c(
  "sample_size",
  "sample_var_biased",
  "sample_var_unbiased",
  "population_var"
)

Take home note: Unbiased variance estimators can get more accurate estimate of variance than the biased one.

2.3 Biased VS. Unbiased Estimator of Variance (Cont.)

⌘+C

library(ggplot2)
variance_mat |> 
  as.data.frame() |> 
  pivot_longer(-sample_size) |> 
  ggplot() +
  geom_point(aes(x = sample_size, y = value, color = name), linewidth = 1.1) +
  geom_line(aes(x = sample_size, y = value, color = name), linewidth = 1.5) +
  labs(x = "Sample Size (N)", y = "Estimates of Variance") +
  scale_color_manual(values = 1:3, labels = c("Population Variance", "Sample Biased Variance (ML)", "Sample Unbiased Variance"), name = "Estimator") +
  scale_x_continuous(breaks = seq(10, 200, 10)) + 
  theme_classic() +
  theme(text = element_text(size = 25))

Take home note: When sample size is small, unbiased variance estimators can get the estimate of variance closer to the population variance than the biased one.

2.4 Interpretation of Variance

The variance (\sigma^2 and S^2) describes the spread of a variable in squared units (which come from (Y_p - \bar Y)^2 term in the equation)
Variance: the average squared distance of an observation from the mean
- For the toy sample, the variance of height is 55.358 inches squared
- For the toy sample, the variance of weight is 3179.095 pounds squared
- The variance of female — not applicable in the same way!
  - How is the sample equally distributed across different groups: 50/50 -> largest variance
Because squared units are difficult to work with, we typically use the standard deviation – which is reported in units
Standard deviation: the average distance of an observation from the mean
- SD of Height: 7.44 inches
- SD of Weight: 56.383 pounds

2.5 Variance/SD as a More General Statistical Concept

Variance (and the standard deviation) is a concept that is applied across statistics – not just for data
- Statistical parameters (slope, intercept) have variance
  - e.g., The sample mean \bar Y has a “standard error” (SE) of S_{\bar Y} = S_Y / \sqrt{N}
  - How accurately we can estimate the sample mean \neq How dispersed the samples are
The standard error is another name for standard deviation
- So “standard error of the mean” is equivalent to “standard deviation of the mean”
- Usually “error” refers to parameters; “deviation” refers to data
- Variance of the mean would be S_{\bar Y}^2 = S^2_Y / N

2.6 Example: Table of Descriptive Statistics

Key information that be reprted
- All variables that you think relevant to the study: (1) demographic (2) context factors (3) outcomes or predictors
- Categorical Variables: (1) Percentage of each level (2) Sample size of each level (3) Range
- Continuous Variables: Mean + SD + Range

2.7 Correlation of Variables

Moving from marginal summaries of each variable to joint (bivariate) summaries, we can use the Pearson Correlation to describe the association between a pair of continuous variables:

r_{Y_1, Y_2} = \frac{\frac{1}{N-1}\sum_{p=1}^N (Y_{1p}-\bar Y_1) (Y_{2p}-\bar Y_2)}{S_{Y_1}S_{Y_2}} \\ = \frac{\sum_{p=1}^N (Y_{1p}-\bar Y_1) (Y_{2p}-\bar Y_2)}{\sqrt{\sum_{p=1}^{N} (Y_{1p}-\bar Y)^2}\sqrt{\sum_{p=1}^{N} (Y_{2p}-\bar Y)^2}}

Human words: how much Variable 1 vary with Variable 2 relative to their own variability.
Note that pearson correlation does not take other variables into account

2.8 R: Pearson Correlation

Method 1: Use R function

cor() function with the argument method = “pearson”

X = 1:10
Y = 10:1
Z = 1:10
cor(X, Y, method = "pearson")
cor(X, Z, method = "pearson")
cor(cbind(X, Y, Z), method = "pearson")

[1] -1
[1] 1
   X  Y  Z
X  1 -1  1
Y -1  1 -1
Z  1 -1  1

Method 2: Manual way

cor_X_Y = sum((X-mean(X))*(Y-mean(Y)))/(length(X)-1)/(sd(X)*sd(Y))
cor_X_Y

[1] -1

sum() adds up all numbers of vector within the parenthesis
X-mean(X) returns the mean centered values of X
Two vectors can be multiplied with each other and return a new vector: X*Y
length() return the number of values of X

2.9 More on the Correlation Coefficient

The Pearson correlation is a biased estimator
- Biased estimator: the expected value differs from the true value for a statistic
The unbiased version of correlation estimation would be:

r_{Y_1, Y_2}^U= r_{Y_1,Y_2}(1+\frac{1-r^2_{Y_1,Y_2}}{2N})

Properties of biased estimator:
- As N gets large bias goes away;
- Bias is largest when r_{Y_1, Y_2} =0; Pearson Corr. is unbiased when r_{Y_1, Y_2} =1
- Pearson correlation is an underestimate of true correlation

2.10 Covariance: Association with Units

The numerator of the Pearson correlation (r_{Y_1Y_2}) is the Covariance
- Human words: “Unscaled (Unstandardized)” Correlation
S^2_{Y_1, Y_2}= \frac{\sum_{p=1}^{N}(Y_{1p}-\bar Y_1)(Y_{2p}-\bar Y_2)}{N-1}

S^2_{Y_1, Y_2}= \frac{\sum_{p=1}^{N}(Y_{1p}-\bar Y_1)(Y_{2p}-\bar Y_2)}{N}
Variance is a special Covariance so that the variable covary with itself
The covariance uses the units of the original variables (but now they are multiples):
- Covariance of height and weight: 334.832 inch-pounds
The covariance of a variable with itself is the variance
The covariance if often used in multivariate analyses because it ties directly into multivariate distribution
- Covariance and correlation are easy to switch between

2.11 Going from Covariance to Correlation

If you have the covariance matrix (variances and covariances):

r_{Y_1,Y_2}=\frac{S_{Y_1,Y_2}}{S_{Y_1}S_{Y_2}}
If you have the correlation matrix and the standard deviations:

S_{Y_1, Y_2} = r_{Y_1, Y_2} S_{Y_1} S_{Y_2}

2.12 Summary of Unit 1

Descriptive statistics: describe central tendency and variability of variables in a visual or statistical way.
- Visual way: Histogram, Scatter plot, Density plot
- Statistical way: mean/mode/median; variance/standard deviation
Alternatively, we cab describe the relationships between two or more variables using their joint distributions
- Visual way: Scatter plot, Group-level Histogram
- Statistical way: Covariance, Pearson Correlation, Chi-square (\chi^2) test

3 Unit 2: General Linear Model

3.1 Learning Objectives

Types of distributions:
- Conditional distribution: a special joint distribution condition on other variable
The General Linear Model
- Regression
- Analysis of Variance (ANOVA)
- Analysis of Covariance (ANCOVA)
- Beyond – Interactions

3.2 Taxonomy of GLM

The general linear model (GLM) incorporates many different labels of analysis under one unifying umbrella:

	Categorical Xs	Continuous Xs	Both Types of Xs
Univariate Y	ANOVA	Regression	ANCOVA
Multivariate Ys	MANOVA	Multivariate Regression	MANCOVA

The typical assumption is that error term (residual or \epsilon) is normally distribution – meaning that the data are conditionally normally distributed
Models for non-normal outcomes (e.g., dichotomous, categorical, count) fall under the Generalized Linear Model, of which general linear model is a special case

3.3 Property of GLM: Conditional Normality of Outcome

The general form of GLM with two predictors:

Y_p = {\color{blue}\beta_0+\beta_1X_p+\beta_2Z_p+\beta_3X_pZ_p} + {\color{red}e_p}

Model for the Means (Predicted Values):
- Each person’s expected (predicted) outcome is a function of his/her values x and z (and their interaction)
- y, x, and z are each measured only once per person (p subscript)
Model for the Variance:
- e_p \sim N(0, \sigma_e^2) \rightarrow One residual (unexplained) deviation
- e_p has a mean of 0 and variance of \sigma^2_e and is normally distributed, unrelated to x and z, unrelated across observation
- Model for the variance is important for model evaluation

3.4 Example: Building a GLM for Weight Prediction

Goal: build a GLM for predicting a person’s weight, using height and gender as predictors
Plan: we proposed several models for didactic reasons — to show how regression and ANOVA work with GLM
- In practice, you wouldn’t necessarily run these models in this sequence
Beginning model (Model 1): An empty model – no predictors for weight (an unconditional model)
Final model (Model 5): A full model – include all predictors and their interaction model

3.5 Example: All models

Model 1

\hat{ \text{Weight}_p }= \beta_0

Model 2:

\hat{\text{Weight}_p} =\beta_0 + \beta_1\text{Height}_p

Model 2a:

\hat{\text{Weight}_p} =\beta_0 + \beta_1\text{HeightC}_p

Model 3:

\hat{\text{Weight}_p}=\beta_0+\beta_2\text{Female}_p

Model 4:

\hat{\text{Weight}_p}=\beta_0+\beta_1\text{HeightC}_p+\beta_2\text{Female}_p

Model 5:

\hat{\text{Weight}_p}=\beta_0+\beta_1\text{HeightC}_p+\beta_2\text{Female}_p \\ + \beta_3\text{HeigthCxFemale}_p

3.6 Model 1: Empty Model

\text{Weight}_p = \beta_0 + e_p e_p \sim N(0, \sigma^2_e)

Estimated parameters:
- \beta_0: Overall intercept - the “grand” mean of weight across all people
- e_p: Residual error - how each one’s observes deviate from \beta_0
- \sigma^2_e: Residual variance - variability of residual error across all people

3.6.1 R Code

library(ESRM64503)
library(kableExtra)
model1 <- lm(weightLB ~ 1, data = dataSexHeightWeight) # model formula
summary(model1)$coefficients |> kable() # regression cofficients table

	Estimate	Std. Error	t value	Pr(>\|t\|)
(Intercept)	183.4	12.60773	14.54664	0

anova(model1) |> kable() # F-statistic table

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Residuals	19	60402.8	3179.095	NA	NA

Interpretation
- \beta_0 = 183.4 is the predicted value of a Weight for all people is 183.4 pound
  - Just the mean of weight
- SE(\beta_0) = 12.60 is the standard error of the mean for weight with higher value suggesting more inaccuracy
- t = 14.55, p < .001 is t-test of the parameter suggesting the mean of weight significantly deviate from 0
- Error term / Residual: \sigma^2_e = 3179.095 (variance of the residuals)
  - Equal to the unbiased variance of weight in empty model
  - Also know as Mean Square Error in F-table

var(dataSexHeightWeight$weightLB)

[1] 3179.095

var(residuals(model1))

[1] 3179.095

3.7 Model 2: Predicting Weight from Height

\text{Weight}_p = \beta_0 + \beta_1 \text{Height}_p + e_p e_p \sim N(0, \sigma^2_e)

Estimated parameters:
- \beta_0: Intercept - is the predicted value of a Weight for a people with Height is 0
- \beta_1: Slope of Height - the predicted increase of Weight for one-unit increase in Height
- e_p: Residual error - how each one’s observes deviate from \beta_0
- \sigma^2_e: Residual variance - variability of residual error across all people

3.7.1 R code

model2 <- lm(weightLB ~ heightIN, data = dataSexHeightWeight) # model formula
summary(model2)$coefficients |> kable(digit = 3) # regression cofficients table

	Estimate	Std. Error	t value	Pr(>\|t\|)
(Intercept)	-227.292	73.483	-3.093	0.006
heightIN	6.048	1.076	5.621	0.000

anova(model2) |> kable(digit = 3) # F-statistic table

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
heightIN	1	38479.27	38479.273	31.593	0
Residuals	18	21923.53	1217.974	NA	NA

Interpretation
- \beta_0 = -227.3 is the predicted value of a Weight for a people with Height is 0
  - Nonsensical – but we can fix it by centering Height
- \beta_1 = 6.048: Weight goes up 6.048 per inch
- SE(\beta_1) = 1.076: the inaccuracy of \beta_1; CI = 6.048 \pm 1.96\times 1.076
- t = 5.621, p < .001 is t-test of the parameter suggesting the slope of height significantly deviate from 0
- F(1, 18) = 31.593 = \frac{38479.273}{1217.974} = t^2
- Error term / Residual: \sigma^2_e = 1217.974 (variance of the residuals)
  - Height explains \frac{3179.095-1217.974}{3179.095}=61.7\% of variance of weight

3.8 Model 2a: Predicting Weight from Centered Height

\text{Weight}_p = \beta_0 + \beta_1 (\text{Height}_p - \bar{Height}) + e_p \\ = \beta_0 + \beta_1 \text{HeightC}_p + e_p e_p \sim N(0, \sigma^2_e)

Estimated parameters:
- \beta_0: Intercept - is the predicted value of a Weight for a people with Height is Mean Height
- \beta_1: Slope of Height - the predicted increase of Weight for one-unit increase in Height
- e_p: Residual error - how each one’s observes deviate from \beta_0
- \sigma^2_e: Residual variance - variability of residual error across all people

3.8.1 R code

dataSexHeightWeight$heightC = dataSexHeightWeight$heightIN - mean(dataSexHeightWeight$heightIN)
model2a <- lm(weightLB ~ heightC, data = dataSexHeightWeight) # model formula
summary(model2a)$coefficients |> kable(digit = 3) # regression cofficients table

	Estimate	Std. Error	t value	Pr(>\|t\|)
(Intercept)	183.400	7.804	23.501	0
heightC	6.048	1.076	5.621	0

anova(model2a) |> kable(digit = 3) # F-statistic table

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
heightC	1	38479.27	38479.273	31.593	0
Residuals	18	21923.53	1217.974	NA	NA

Interpretation
- \beta_0 = 183.4 pounds (previously -227.3) is the predicted value of a Weight for a people with Height is 67.9 inches
- Everything except \beta_0, SE(\beta_0), and t-value of intercept should be same as the previous model

3.8.2 Visualization

ggplot(dataSexHeightWeight) +
  geom_point(aes(x = heightC, y = weightLB)) +
  geom_smooth(aes(x = heightC, y = weightLB), method = "lm") +
  geom_hline(aes(yintercept = 183.4), color = "red") +
  labs(title = "Model 2a: Fit Plot for weight")

3.9 Wrap up: Hypothesis Tests for Parameters

To determine if the regression slope is significantly different from zero, we must use a hypothesis test:

H_0: \beta_1 = 0

H_1: \beta_1 \neq 0

We have two options for this test (both are same here)
- Use ANOVA table: sums of squares – F-test
- Use “Wald” test for parameter: t = \frac{\beta_1}{SE(\beta_1)}
- Here t^2 = F
p < 0.001 \rightarrow reject null (H_0) \rightarrow Conclusion: slope is significant

3.10 Model 3: Predicting Weight from Gender

\text{Weight}_p = \beta_0 + \beta_1 \text{Female}_p + e_p e_p \sim N(0, \sigma^2_e)

Note: because gender is a categorical predictor, we must first code it into a number before entering it into model (typically done automatically in software)
- Here we code the variable as Female = 1 for females; Female = 0 for males

Estimated parameters:
- \beta_0: Intercept - predicted value of Weight for a person with Female = 0 (males)
- \beta_2: Slope of Female - Change in predicted value of Weight between Males and Famales

3.10.1 R Code

model3 <- lm(weightLB ~ female, data = dataSexHeightWeight) # model formula
summary(model3)$coefficients |> kable(digit = 3) # regression cofficients table

	Estimate	Std. Error	t value	Pr(>\|t\|)
(Intercept)	235.9	5.415	43.565	0
femaleTRUE	-105.0	7.658	-13.711	0

anova(model3) |> kable(digit = 3) # F-statistic table

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
female	1	55125.0	55125.000	188.004	0
Residuals	18	5277.8	293.211	NA	NA

Interpretation
- \beta_0 = 235.9 pounds is the predicted value of a Weight for a male
- \beta_0 +\beta_1 = 235.9 - 105.0 = 130.9 pounds is the predicted value of a Weight for a female
- \beta_1 = 105 expected difference between gender, which is significant based on t-test and F-test

3.10.2 Visualization:

Line Plot (not frequently used in this case)

ggplot(dataSexHeightWeight) +
  geom_point(aes(x = as.numeric(female), y = weightLB)) +
  geom_abline(intercept = coef(model3)[1], slope = coef(model3)[2]) +
  scale_x_continuous(breaks = 0:1) +
  labs(title = "Model 3: Fit Plot for Female",
       x = "Female")

3.10.3 Visualization: box-and-whisker plot (Basic Way)

boxplot(weightLB ~ female, data = dataSexHeightWeight, 
        names = c("Male", "Female"), ylab = "Weight", xlab = "",boxwex = .3)    # Basic boxplot in R

3.10.4 Visualization: box-and-whisker plot (Fancy Way)

ggplot2 package

ggplot(dataSexHeightWeight, aes(x = female, y = weightLB)) +
  geom_dotplot(aes(fill = female, color = female), binaxis='y', stackdir='center') +
  stat_summary(fun.data= "mean_cl_normal", fun.args = list(conf.int=.75), geom="crossbar", width=0.3, fill = "yellow", alpha = .5) +
  stat_summary(fun.data= "median_hilow", geom="errorbar", fun.args = list(conf.int=1), width = .1, color="black") + # Mean +- 2SD 
  stat_summary(fun.data= "mean_sdl",  geom="point", shape = 5, size = 3) +
  scale_x_discrete(labels = c("Male", "Female")) +
  labs(x = "", y = "Weight") +
  theme_bw() +
  theme(legend.position = "none", text = element_text(size = 20))

3.11 Model 4: Predicting Weight from Height and Gender

\text{Weight}_p = \beta_0 + \beta_1 \text{HeightC}_p + \beta_2 \text{Female}_p + e_p e_p \sim N(0, \sigma^2_e)

model4 <- lm(weightLB ~ heightC + female, data = dataSexHeightWeight) # model formula
summary(model4)$coefficients |> kable(digit = 3) # regression cofficients table

	Estimate	Std. Error	t value
(Intercept)	224.256	1.439	155.876
heightC	2.708	0.155	17.525
femaleTRUE	-81.712	2.241	-36.461

anova(model4) |> kable(digit = 3) # F-statistic table

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
heightC	1	38479.273	38479.273	2363.103	0
female	1	21646.710	21646.710	1329.376	0
Residuals	17	276.817	16.283	NA	NA

Interpretation
- \beta_0 = 224.256 (SE = 1.439)
  - The predicted weight is 224.256 pounds for a person with Female = 0 (males) and has Height as Mean Height 67.9 inches
- \beta_1 = 2.708 (SE = 0.155)
  - t = \frac{2.708}{0.155}=17.525; p <.001
  - The expected change in weights for every one-unit increase in height holding gender constant
- \beta_2=-81.713 (SE = 2.241)
  - The expected difference between the mean of weights for males and the mean for females holding height constant
- \sigma_e^2 = 16.283
  - The residual variance of weight

3.12 Model 4: By-Gender Regression Lines

Model 4 assumes identical regression slopes for both genders but has different intercepts

This assumption of different slopes will be tested statistically by model 5

Predicted Weight for Females with the height as 68.9 inch:

W_p=224.256+2.708\times\color{blue}{(68.9-67.9)} - 81.713*\color{red}{1} \\ = 224.256+2.708-81.712 = 145.252

Predicted Weight for Males with the height as 68.9 inch:

W_p=224.256+2.708\times\color{blue}{(68.9-67.9)} - 81.713*\color{red}{0} \\ = 224.256+2.708 = 226.964

predict(model4, data.frame(heightC = 1, female = TRUE))
predict(model4, data.frame(heightC = 1, female = FALSE))

3.12.1 Visualization: Different intercept and Same slope across groups

ggplot2 package

dataSexHeightWeight |> 
  mutate(predictedWeight = predict(model4, dataSexHeightWeight)) |> 
  ggplot(aes(x = heightC, y = predictedWeight, color = female)) +
  geom_line() +
  geom_point(aes(y = weightLB, shape = female), size = 2) +
  theme_classic()

3.13 Model 5: By-Gender Regression Lines

\text{Weight}_p = \beta_0 + \beta_1 \text{HeightC}_p + \beta_2 \text{Female}_p +\beta_3 \text{HeightCxFemale}_p+ e_p

model5 <- lm(weightLB ~ heightC * female, data = dataSexHeightWeight) # model formula
summary(model5)$coefficients |> kable(digit = 3) # regression cofficients table

	Estimate	Std. Error	t value
(Intercept)	222.184	0.838	265.107
heightC	3.190	0.111	28.646
femaleTRUE	-82.272	1.211	-67.932
heightC:femaleTRUE	-1.094	0.168	-6.520

anova(model5) |> kable(digit = 3) # F-statistic table

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
heightC	1	38479.273	38479.273	8132.723	0
female	1	21646.710	21646.710	4575.104	0
heightC:female	1	201.115	201.115	42.506	0
Residuals	16	75.703	4.731	NA	NA

Interpretation
- \beta_0 = 222.184 (SE = 0.838)
  - SE gets smaller compared to Model 4.
  - The predicted weight is 224.184 pounds for a person with Female = 0 (males) and has Height as Mean Height 67.9 inches
- \beta_1 = 3.189 (SE = 0.111)
  - SE gets smaller compared to Model 4.
  - Simple main effect of Height: the expected change in weights for every one-unit increase in height for males only
- \beta_2=-82.271 (SE = 1.211)
  - SE gets smaller compared to Model 4.
  - Simple main effect of gender: The expected difference between the mean of weights for males and the mean for females for people with mean height
- \beta_3 = -1.093 (SE = 1.211) , t = -6.52, p < .001
  - Gender-by-Height Interaction: Additional change in slope of height for change in gender.
  - The slope of heights for males is 3.189; the slope of heights decreases 1.093 (2.096) inches/pound for females
- \sigma_e^2 = 5
  - Compared to 16.283 in model 4

3.13.1 Visualization: Different slopes and different intercepts

Model 5 does not assume identical regression slopes

Because \beta_3 was significantly different from zero, the data supports different slopes for the gender

⌘+C

ggplot(dataSexHeightWeight, aes(x = heightC, y = weightLB, color = female)) +
  geom_point() +
  geom_smooth(method = "lm") +
  theme_bw()

3.14 Model Comparison

In practice, the empty model (Model 0) and the full model (Model 5) would be the only models to run
- The aim of model comparison is to decide on whether we want more complex model or the simple one
- The trick is to describe the impact of all and each of the predictors – typically using variance accounted for
Residual Variance and R^2

⌘+C

get_res_var <- function(model) {
  anova(model)$`Mean Sq` |>  tail(1)
}

data.frame(
  Model = paste("Model", c(1, 2, "2a", 3, 4, 5)),
  ResidualVariance = sapply(list(model1, model2, model2a, model3, model4, model5), get_res_var)
) |> 
  ggplot(aes(x = Model, y = ResidualVariance))  +
  geom_point() +
  geom_path(group = 1) +
  geom_label(aes(label = round(ResidualVariance, 2)),nudge_y = 200, nudge_x = 0.2) +
  labs(title = "Estimated Residual Variances across models") +
  theme(text = element_text(size = 20))

⌘+C

get_explained_var <- function(model) {
  tot_err_var <- anova(model1)$`Mean Sq` |>  tail(1)
  current_err_var <- anova(model)$`Mean Sq` |>  tail(1)
  prop <- (tot_err_var - current_err_var)/ tot_err_var
  prop
}

data.frame(
  Model = paste("Model", c(1, 2, "2a", 3, 4, 5)),
  ExplainedVariance = sapply(list(model1, model2, model2a, model3, model4, model5), get_explained_var)
) |> 
  ggplot(aes(x = Model, y = ExplainedVariance))  +
  geom_point() +
  geom_path(group = 1) +
  geom_label(aes(label = paste0(round(ExplainedVariance*100, 2), "%")),nudge_y = .05) +
  labs(title = "Proportions of Explained Variance across Models")

3.15 Comparison Across Models

Are height and gender are good predictors?

Total explained variances in weight by height and gender: Multiple R^2 of Model 5
- (3179.09-4.73)/3179.09 = 0.9985 \rightarrow 99.85% variances in weights can be explained by height and gender

F-test comparing Model 5 to Model 1

F_{3, 16} = 4250.1, p < .001

⌘+C

summary(model5)


Call:
lm(formula = weightLB ~ heightC * female, data = dataSexHeightWeight)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.8312 -1.7797  0.4958  1.3575  3.3585 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)        222.1842     0.8381  265.11  < 2e-16 ***
heightC              3.1897     0.1114   28.65 3.55e-15 ***
femaleTRUE         -82.2719     1.2111  -67.93  < 2e-16 ***
heightC:femaleTRUE  -1.0939     0.1678   -6.52 7.07e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.175 on 16 degrees of freedom
Multiple R-squared:  0.9987,    Adjusted R-squared:  0.9985 
F-statistic:  4250 on 3 and 16 DF,  p-value: < 2.2e-16

Are Height alone a good predictor?
- Explained remaining variances in weight by height: Multiple R^2 of Model 5 to Model 3
  - (293.21-4.73)/293.21 = 0.9839 \rightarrow 98.39% variances in weights remaining after gender can be explained by the main and interaction effects of height
- F-test comparing Model 5 to Model 3
  - F_{2, 16} = 548.74, p < .001 suggests the effect of hight on weight is significant after controlling the effect of gender
    ⌘+C
    
    anova(model3, model5) ((5277.8 - 75.7)/ (18-16))/(75.7 / 16)
- True explained variances out of total variances in weight: Unique contribution of adding Height into the model
  - Check model 3, we can found that gender explained 90.78% variance of weight
  - 0.9839 * (1 - 0.9078) = 0.0907 \rightarrow 9.07% more variances of weights can be explained by height after gender is already in the model
  - 90.78% + 9.07% = 99.85% is the total variance explained by height and gender

Are gender alone a good predictor?

Explained remaining variances in weight by gender: Multiple R^2 of Model 5 to Model 2a
- (1217.97-4.73)/1217.97 = 0.9961 \rightarrow 99.61% variances in weights remaining after height can be explained by the main and interaction effects of gender
F-test comparing Model 5 to Model 2a
- F_{2, 16} = 2308.8, p < .001 suggests the effect of gender on weight is significant after controlling the effect of height
  ⌘+C
  anova(model2a, model5) ((21923.5 - 75.7)/ (18-16))/(75.7 / 16)
True explained variances out of total variances in weight: Unique contribution of adding Height into the model
- Check model 3, we can found that gender explained 90.78% variance of weight
- 0.9839 * (1 - 0.9078) = 0.0907 \rightarrow 9.07% more variances of weights can be explained by height after gender is already in the model
- 90.78% + 9.07% = 99.85% is the total variance explained by height and gender

3.16 Summary of Unit 2

We learned GLM using the 2-factor model
For specific effect, we examine the t-test of coefficient
For the total contribution of a bunch of effects (main + interaction), we look at R-square and F-test
We sometime need to report predicted regression lines, which can be done in R with ggplot2

4 Unit 3: Let’s Look at R code together

--- title: "Lecture 02: General Linear Model" subtitle: "Descriptive Statistics and Basic Statistics" author: "Jihong Zhang*, Ph.D" institute: | Educational Statistics and Research Methods (ESRM) Program* University of Arkansas date: "2024-08-26" sidebar: false execute: echo: true warning: false output-location: column format: html: page-layout: full toc: true toc-depth: 2 toc-expand: true lightbox: true code-fold: false uark-revealjs: scrollable: true chalkboard: true embed-resources: false code-fold: false number-sections: false footer: "ESRM 64503: Lecture 02 - Descriptive Statistics" slide-number: c/t tbl-colwidths: auto output-file: slides-index.html include-in-header: - file: mathjax-color.html --- ## Some notes before our class 1. Vote for delaying class time (to 5:15?) for free parking 2. The R code I'll show in the slides is just for illustration. Don't be worried about hardly following up. We will practice together at the end of each unit. ## Learning Objectives 1. Review Homework 0 2. Review previous lecture 3. Unit 1: Descriptive Statistics 1. Introduce R package – **ESRM64503** 2. Univariate Descriptive Statistics - Central tendency: Mean, Median, Mode - Variation/Spread: Standard deviation (SD), Variance, Range 3. Bivariate descriptive statistics - Correlation - Covariance 4. Types of variable distributions - Marginal - Joint - Conditional 5. Bias in estimators 4. Unit 2: General Linear Model # Unit 1: Descriptive Statistics ## Homework 0 ![Question 3](figures/homework0.png) ```{r} #| eval: false #| output-location: default library(tidyverse); library(nycflights13) # install.packages(c("nycflights13", "tidyverse")) flights glimpse(flights) View(flights) print(flights, width = Inf) ``` ## Installation of ESRM64503 package ```{r} #| code-fold: show #| eval: false #| output-location: default install.packages("devtools") devtools::install_github("JihongZ/ESRM64503") # Method 1 pak::pak("JihongZ/ESRM64503") # Method 2: Install the Github package pak::pak("JihongZ/ESRM64503", upgrade = TRUE) # If you already install the package, try to upgrade ``` ## Test ESTM64503 Package ```{r} #| message: true #| output-location: column #| code-fold: show #| code-summary: 'package version' library(ESRM64503) devtools::package_info("ESRM64503") # Make sure the version number is 2024.08.20 ``` ```{r} #| message: true #| output-location: column #| code-fold: show #| code-summary: 'homework information' homework() # You can call "homework()" function to access homework info ``` ## Data Files of *ESRM64503* ```{r} #| output-location: column #| code-fold: show #| code-summary: 'Data dataSexHeightWeight automate loaded' dataSexHeightWeight # You should find data named "dataSexHeightWeight" already loaded ``` ```{r} #| code-fold: show #| code-summary: 'Type ? to check variable information of data' ?dataSexHeightWeight ``` ## Simulated Data for Today's Lecture - dataSexHeightWeight - To help demonstrate the concepts of today's lecture, we will be using a toy data set with three variables - **Female (Gender)**: Coded as Male (= 0) or Female (= 1) - In R, factor variable with two levels: FALSE, TRUE - **Height**: in inches - **Weight**: in pounds - The goal of lecture 02 will be to build a general **linear model** that predicts a person's weight - **Linear (regression) model**: a statistical model for an outcome that uses a linear combination (a weighted sum) of one or more predictor variables to produce an estimate of an observation's predicted value - $$ \mathbb{y} = \beta_0+\beta_1 \mathbf{X} $$ - All models we learnt today will follow this framework. ## Recoding Variables ```{r} library(ESRM64503) # INSTALL: pak::pak("JihongZ/ESRM64504") library(kableExtra) # INSTALL: pak::pak("JihongZ/ESRM64504") ## First checking sample size ## N = 20 ## Recode Sex as Female dataSexHeightWeight$female = dataSexHeightWeight$sex == "F" kable(dataSexHeightWeight, caption = 'toy data set') |> kable_styling(font_size = 30) ``` ## Descriptive Statistics - First, we can inspect each variable individually (**marginal distribution**) through a set of descriptive statistics - Visual way: histogram plot or density plot - Statistical way: Central tendency and Variability - Mean, Median, Mode - SD, Range - Second, we can also summarize the **joint (bivariate) distribution** of two variables through a set of descriptive statistics: - **Joint vs. Marginal**: joint distribution describes more than one variable simultaneously - Common bivariate descriptive statistics: - Correlation and covariance ## Quick inspection: Missing data rate - Check (1) if any case has **missing** value (2) **distributions** for continuous and categorical variables ```{r} #| code-fold: show #| output-location: column table(complete.cases(dataSexHeightWeight)) ``` ```{r} #| code-fold: show #| output-location: column dataWithMissing <- dataSexHeightWeight dataWithMissing[11, 2] <- NA table(complete.cases(dataWithMissing)) ``` ```{r} #| code-summary: "Distribution of continous variables" psych::describe(dataSexHeightWeight[,-1], omit = TRUE, trim = 0) |> kable(digits = 3) ``` ## Quick Inspect: Distribution of Categorical Variable ```{r} #| code-summary: "Distribution of categorical variables" #| output-location: default table(dataSexHeightWeight$female) hist(as.numeric(dataSexHeightWeight$female), main = "Histogram of Female", xlab = "Female", ) ``` ## Visualization: Pairwise Scatterplot ```{r} #| output-location: default pairs(dataSexHeightWeight[, c('female', 'heightIN', 'weightLB')]) ``` ## Histograms of Height and Weight ```{r} #| output-location: default #| fig-align: center hist(dataSexHeightWeight$weightLB, main = 'Weight', xlab = 'Pounds') hist(dataSexHeightWeight$heightIN, main = 'Height', xlab = 'Inches') ``` ## Descriptive Statistics for Toy Data: Marginal ```{r} library(dplyr) ## wide-format marginal description wide_marginal_desp <- dataSexHeightWeight |> summarise(across(c(heightIN, weightLB, female), list(mean = mean, sd = sd, var = var))) ``` ```{r} library(tidyr) wide_marginal_desp |> pivot_longer(everything(), names_to = "Variable", values_to = "Value") |> separate(Variable, sep = "_", into = c("Variable", "Stats")) |> pivot_wider(names_from = Stats, values_from = Value) |> kable(digits = 3) ``` ## Descriptive Statistics for Toy Data: Joint - **Correlation-Covariance Matrix**: - Diagonal: Variance - Above Diagonal (upper triangle): Covaraince - Below Diagonal (lower triangle): Correlation - **Question:** What we can tell regarding the relationships among three variables? ```{r} cor_cov_mat <- matrix(nrow = 3, ncol = 3) colnames(cor_cov_mat) <- rownames(cor_cov_mat) <- c("heightIN", "weightLB", "female") cov_mat <- cov(dataSexHeightWeight[, c("heightIN", "weightLB", "female")]) cor_mat <- cor(dataSexHeightWeight[, c("heightIN", "weightLB", "female")]) ## Assign values cor_cov_mat[lower.tri(cor_cov_mat)] <- cor_mat[lower.tri(cor_mat)] cor_cov_mat[upper.tri(cor_cov_mat)] <- cov_mat[upper.tri(cov_mat)] diag(cor_cov_mat) <- diag(cov_mat) kable(cor_cov_mat, digits = 3) ``` # Variance, Correlation, Covariance ## Re-examining the Concept of Variance - Variability is a central concept in advanced statistics - In multivariate statistic, covariance is also central - Two formulas for the variance (about the same when $N$ is larger): ::: columns ::: {.column width="50%"} $$ S^2_Y= \frac{\Sigma_{p=1}^{N}(Y_p-\bar Y)}{N-1} $$ {#eq-unbiased-var} ::: ::: {.column width="50%"} $$ S^2_Y= \frac{\Sigma_{p=1}^{N}(Y_p-\bar Y)}{N}$$ {#eq-biased-var} ::: ::: - @eq-unbiased-var: **Unbiased** or "sample" - @eq-biased-var: **Biased/ML** or "population" Here: $p$ = person; ## Biased VS. Unbiased Variability ```{r} #| message: false #| code-fold: show #| output-location: default set.seed(1234) sample_sizes = seq(10, 200, 10) # N = 10, 20, 30, ..., 200 population_var = 100 # True variance for population variance_mat <- matrix(NA, nrow = length(sample_sizes), ncol = 4) iter = 0 for (sample_size in sample_sizes) { iter = iter + 1 sample_points <- rnorm(n = sample_size, mean = 0, sd = sqrt(population_var)) sample_var_biased <- var(sample_points) * (sample_size-1) / sample_size sample_var_unbiased <- var(sample_points) variance_mat[iter, 1] <- sample_size variance_mat[iter, 2] <- sample_var_biased variance_mat[iter, 3] <- sample_var_unbiased variance_mat[iter, 4] <- population_var } colnames(variance_mat) <- c( "sample_size", "sample_var_biased", "sample_var_unbiased", "population_var" ) ``` **Take home note**: Unbiased variance estimators can get more accurate estimate of variance than the biased one. ## Biased VS. Unbiased Estimator of Variance (Cont.) ```{r} #| message: false #| code-fold: true #| fig-align: center #| fig-width: 20 #| output-location: default library(ggplot2) variance_mat |> as.data.frame() |> pivot_longer(-sample_size) |> ggplot() + geom_point(aes(x = sample_size, y = value, color = name), linewidth = 1.1) + geom_line(aes(x = sample_size, y = value, color = name), linewidth = 1.5) + labs(x = "Sample Size (N)", y = "Estimates of Variance") + scale_color_manual(values = 1:3, labels = c("Population Variance", "Sample Biased Variance (ML)", "Sample Unbiased Variance"), name = "Estimator") + scale_x_continuous(breaks = seq(10, 200, 10)) + theme_classic() + theme(text = element_text(size = 25)) ``` **Take home note**: When sample size is small, unbiased variance estimators can get the estimate of variance closer to the population variance than the biased one. ## Interpretation of Variance - The variance ($\sigma^2$ and $S^2$) describes the spread of a variable in **squared units** (which come from $(Y_p - \bar Y)^2$ term in the equation) - Variance: **the average squared distance of an observation from the mean** - For the toy sample, the **variance of height** is 55.358 inches squared - For the toy sample, the **variance of weight** is 3179.095 pounds squared - The **variance of female** — not applicable in the same way! - How is the sample equally distributed across different groups: 50/50 -\> largest variance - Because squared units are difficult to work with, we typically use the standard deviation – which is reported in units - Standard deviation: the average distance of an observation from the mean - SD of Height: 7.44 inches - SD of Weight: 56.383 pounds ## Variance/SD as a More General Statistical Concept - Variance (and the standard deviation) is a concept that is applied across statistics – not just for data - Statistical parameters (slope, intercept) have variance - e.g., The sample mean $\bar Y$ has a "standard error" (SE) of $S_{\bar Y} = S_Y / \sqrt{N}$ - How accurately we can estimate the sample mean $\neq$ How dispersed the samples are - The standard error is another name for standard deviation - So "standard error of the mean" is equivalent to "standard deviation of the mean" - Usually "**error**" refers to **parameters**; "**deviation**" refers to **data** - Variance of the mean would be $S_{\bar Y}^2 = S^2_Y / N$ ## Example: Table of Descriptive Statistics ![Table 1 (Xiong et al., 2023)](figures/Exp_Descrip_Stats_Table.png){fig-align="center"} ------------------------------------------------------------------------ - Key information that be reprted - All variables that you think relevant to the study: (1) demographic (2) context factors (3) outcomes or predictors - Categorical Variables: (1) Percentage of each level (2) Sample size of each level (3) Range - Continuous Variables: Mean + SD + Range ## Correlation of Variables - Moving from marginal summaries of each variable to joint (bivariate) summaries, we can use the **Pearson Correlation** to describe the association between a pair of **continuous** variables: $$ r_{Y_1, Y_2} = \frac{\frac{1}{N-1}\sum_{p=1}^N (Y_{1p}-\bar Y_1) (Y_{2p}-\bar Y_2)}{S_{Y_1}S_{Y_2}} \\ = \frac{\sum_{p=1}^N (Y_{1p}-\bar Y_1) (Y_{2p}-\bar Y_2)}{\sqrt{\sum_{p=1}^{N} (Y_{1p}-\bar Y)^2}\sqrt{\sum_{p=1}^{N} (Y_{2p}-\bar Y)^2}} $$ - **Human words**: how much Variable 1 vary with Variable 2 relative to their own variability. - Note that pearson correlation does not take other variables into account ## R: Pearson Correlation #### Method 1: Use R function `cor()` function with the argument method = "pearson" ```{r} #| results: hold #| code-fold: false X = 1:10 Y = 10:1 Z = 1:10 cor(X, Y, method = "pearson") cor(X, Z, method = "pearson") cor(cbind(X, Y, Z), method = "pearson") ``` #### Method 2: Manual way ```{r} #| code-fold: false cor_X_Y = sum((X-mean(X))*(Y-mean(Y)))/(length(X)-1)/(sd(X)*sd(Y)) cor_X_Y ``` - *sum*() adds up all numbers of vector within the parenthesis - X-mean(X) returns the mean centered values of X - Two vectors can be multiplied with each other and return a new vector: X\*Y - *length*() return the number of values of X ## More on the Correlation Coefficient - The Pearson correlation is a **biased** estimator - **Biased estimator**: the expected value differs from the true value for a statistic - The unbiased version of correlation estimation would be: $$ r_{Y_1, Y_2}^U= r_{Y_1,Y_2}(1+\frac{1-r^2_{Y_1,Y_2}}{2N}) $$ - Properties of biased estimator: - As N gets large bias goes away; - Bias is largest when $r_{Y_1, Y_2} =0$; Pearson Corr. is unbiased when $r_{Y_1, Y_2} =1$ - Pearson correlation is an underestimate of true correlation ## Covariance: Association with Units - The **numerator** of the Pearson correlation ($r_{Y_1Y_2}$) is the **Covariance** - **Human words**: "Unscaled (Unstandardized)" Correlation ::: columns ::: {.column width="50%"} $$ S^2_{Y_1, Y_2}= \frac{\sum_{p=1}^{N}(Y_{1p}-\bar Y_1)(Y_{2p}-\bar Y_2)}{N-1} $$ ::: ::: {.column width="50%"} $$ S^2_{Y_1, Y_2}= \frac{\sum_{p=1}^{N}(Y_{1p}-\bar Y_1)(Y_{2p}-\bar Y_2)}{N}$$ ::: ::: - **Variance is a special Covariance so that the variable covary with itself** - The covariance uses the units of the original variables (but now they are multiples): - Covariance of height and weight: 334.832 *inch-pounds* - The covariance of a variable with itself is the variance - The covariance if often used in multivariate analyses because it ties directly into multivariate distribution - Covariance and correlation are easy to switch between ## Going from Covariance to Correlation - If you have the covariance matrix (variances and covariances): $$ r_{Y_1,Y_2}=\frac{S_{Y_1,Y_2}}{S_{Y_1}S_{Y_2}} $$ - If you have the correlation matrix and the standard deviations: $$ S_{Y_1, Y_2} = r_{Y_1, Y_2} S_{Y_1} S_{Y_2} $$ ## Summary of Unit 1 1. **Descriptive statistics**: describe central tendency and variability of variables in a visual or statistical way. - Visual way: Histogram, Scatter plot, Density plot - Statistical way: mean/mode/median; variance/standard deviation 2. **Alternatively**, we cab describe the relationships between two or more variables using their joint distributions - Visual way: Scatter plot, Group-level Histogram - Statistical way: Covariance, Pearson Correlation, Chi-square ($\chi^2$) test # Unit 2: General Linear Model ## Learning Objectives - Types of distributions: - Conditional distribution: a special joint distribution condition on other variable - The General Linear Model - Regression - Analysis of Variance (ANOVA) - Analysis of Covariance (ANCOVA) - Beyond – Interactions ## Taxonomy of GLM - The general linear model (GLM) incorporates many different labels of analysis under one unifying umbrella: | | Categorical Xs | Continuous Xs | Both Types of Xs | |-----------------|----------------|-------------------------|------------------| | Univariate Y | ANOVA | Regression | ANCOVA | | Multivariate Ys | MANOVA | Multivariate Regression | MANCOVA | - The typical assumption is that error term (residual or $\epsilon$) is normally distribution – meaning that the data are conditionally normally distributed - Models for non-normal outcomes (e.g., dichotomous, categorical, count) fall under the *Generalized Linear Model*, of which general linear model is a special case ## Property of GLM: Conditional Normality of Outcome The general form of GLM with two predictors: $$ Y_p = {\color{blue}\beta_0+\beta_1X_p+\beta_2Z_p+\beta_3X_pZ_p} + {\color{red}e_p} $$ - ::: {style="color: blue;"} Model for the Means (Predicted Values): ::: - Each person's expected (predicted) outcome is a function of his/her values x and z (and their interaction) - y, x, and z are each measured only once per person (*p* subscript) - ::: {style="color: red;"} Model for the Variance: ::: - $e_p \sim N(0, \sigma_e^2)$ $\rightarrow$ One residual (unexplained) deviation - $e_p$ has a mean of 0 and variance of $\sigma^2_e$ and is normally distributed, unrelated to x and z, unrelated across observation - Model for the variance is important for **model evaluation** ## Example: Building a GLM for Weight Prediction - **Goal**: build a GLM for predicting a person's weight, using height and gender as predictors - **Plan**: we proposed several models for didactic reasons — to show how regression and ANOVA work with GLM - In practice, you wouldn't necessarily run these models in this sequence - **Beginning model** (Model 1): An **empty model –** no predictors for weight (an **unconditional** model) - **Final model** (Model 5): A **full model** – include all predictors and their interaction model ## Example: All models ::: columns ::: {.column width="50%"} - Model 1 $$ \hat{ \text{Weight}_p }= \beta_0 $$ - Model 2: $$ \hat{\text{Weight}_p} =\beta_0 + \beta_1\text{Height}_p $$ - Model 2a: $$ \hat{\text{Weight}_p} =\beta_0 + \beta_1\text{HeightC}_p $$ ::: ::: {.column width="50%"} - Model 3: $$ \hat{\text{Weight}_p}=\beta_0+\beta_2\text{Female}_p $$ - Model 4: $$\hat{\text{Weight}_p}=\beta_0+\beta_1\text{HeightC}_p+\beta_2\text{Female}_p$$ - Model 5: $$\hat{\text{Weight}_p}=\beta_0+\beta_1\text{HeightC}_p+\beta_2\text{Female}_p \\ + \beta_3\text{HeigthCxFemale}_p$$ ::: ::: ## Model 1: Empty Model $$ \text{Weight}_p = \beta_0 + e_p $$ $$ e_p \sim N(0, \sigma^2_e) $$ - Estimated parameters: - $\beta_0$: **Overall intercept** - the "grand" mean of weight across all people - $e_p$: **Residual error** - how each one's observes deviate from $\beta_0$ - $\sigma^2_e$: **Residual variance** - variability of residual error across all people ------------------------------------------------------------------------ ### R Code ```{r} #| output-location: default library(ESRM64503) library(kableExtra) model1 <- lm(weightLB ~ 1, data = dataSexHeightWeight) # model formula summary(model1)$coefficients |> kable() # regression cofficients table anova(model1) |> kable() # F-statistic table ``` - Interpretation - $\beta_0 = 183.4$ is the predicted value of a Weight for all people is 183.4 pound - Just the mean of weight - $SE(\beta_0) = 12.60$ is the standard error of the mean for weight with higher value suggesting more inaccuracy - $t = 14.55, p < .001$ is t-test of the parameter suggesting the mean of weight significantly deviate from 0 - **Error term / Residual**: $\sigma^2_e = 3179.095$ (variance of the residuals) - Equal to the unbiased variance of weight in empty model - Also know as Mean Square Error in F-table ```{r} #| output-location: default var(dataSexHeightWeight$weightLB) var(residuals(model1)) ``` ## Model 2: Predicting Weight from Height $$ \text{Weight}_p = \beta_0 + \beta_1 \text{Height}_p + e_p $$ $$ e_p \sim N(0, \sigma^2_e) $$ - Estimated parameters: - $\beta_0$: Intercept - is the predicted value of a Weight for a people with [Height is 0]{style="color: red;"} - $\beta_1$: [Slope of Height]{style="color: red;"} - the predicted increase of Weight for one-unit increase in Height - $e_p$: Residual error - how each one's observes deviate from $\beta_0$ - $\sigma^2_e$: Residual variance - variability of residual error across all people ------------------------------------------------------------------------ ### R code ```{r} #| output-location: default model2 <- lm(weightLB ~ heightIN, data = dataSexHeightWeight) # model formula summary(model2)$coefficients |> kable(digit = 3) # regression cofficients table anova(model2) |> kable(digit = 3) # F-statistic table ``` - Interpretation - $\beta_0 = -227.3$ is the predicted value of a Weight for a people with [Height is 0]{style="color: red;"} - Nonsensical – but we can fix it by [centering Height]{style="color: #ff0040;"} - $\beta_1 = 6.048$: Weight goes up 6.048 per inch - $SE(\beta_1) = 1.076$: the inaccuracy of $\beta_1$; $CI = 6.048 \pm 1.96\times 1.076$ - $t = 5.621, p < .001$ is t-test of the parameter suggesting the slope of height significantly deviate from 0 - $F(1, 18) = 31.593 = \frac{38479.273}{1217.974} = t^2$ - **Error term / Residual**: $\sigma^2_e = 1217.974$ (variance of the residuals) - Height explains $\frac{3179.095-1217.974}{3179.095}=61.7\%$ of variance of weight ## Model 2a: Predicting Weight from [Centered]{style="color: red;"} Height $$ \text{Weight}_p = \beta_0 + \beta_1 (\text{Height}_p - \bar{Height}) + e_p \\ = \beta_0 + \beta_1 \text{HeightC}_p + e_p $$ $$ e_p \sim N(0, \sigma^2_e) $$ - Estimated parameters: - $\beta_0$: Intercept - is the predicted value of a Weight for a people with [Height is Mean Height]{style="color: red;"} - $\beta_1$: Slope of Height - the predicted increase of Weight for one-unit increase in Height - $e_p$: Residual error - how each one's observes deviate from $\beta_0$ - $\sigma^2_e$: Residual variance - variability of residual error across all people ------------------------------------------------------------------------ ### R code ```{r} #| output-location: default dataSexHeightWeight$heightC = dataSexHeightWeight$heightIN - mean(dataSexHeightWeight$heightIN) model2a <- lm(weightLB ~ heightC, data = dataSexHeightWeight) # model formula summary(model2a)$coefficients |> kable(digit = 3) # regression cofficients table anova(model2a) |> kable(digit = 3) # F-statistic table ``` - Interpretation - $\beta_0 = 183.4$ pounds (previously -227.3) is the predicted value of a Weight for a people with [Height is 67.9]{style="color: red;"} inches - Everything except $\beta_0$, $SE(\beta_0)$, and t-value of intercept should be same as the previous model ------------------------------------------------------------------------ ### Visualization ```{r} #| output-location: default #| fig-align: center ggplot(dataSexHeightWeight) + geom_point(aes(x = heightC, y = weightLB)) + geom_smooth(aes(x = heightC, y = weightLB), method = "lm") + geom_hline(aes(yintercept = 183.4), color = "red") + labs(title = "Model 2a: Fit Plot for weight") ``` ## Wrap up: Hypothesis Tests for Parameters - To determine if the regression slope is significantly different from zero, we must use a hypothesis test: $$ H_0: \beta_1 = 0 $$ $$ H_1: \beta_1 \neq 0 $$ - We have two options for this test (both are same here) - Use ANOVA table: sums of squares – F-test - Use "Wald" test for parameter: $t = \frac{\beta_1}{SE(\beta_1)}$ - Here $t^2 = F$ - p \< 0.001 $\rightarrow$ reject null ($H_0$) $\rightarrow$ Conclusion: slope is significant ## Model 3: Predicting Weight from Gender $$ \text{Weight}_p = \beta_0 + \beta_1 \text{Female}_p + e_p $$ $$ e_p \sim N(0, \sigma^2_e) $$ - Note: because gender is a categorical predictor, we must first code it into a number before entering it into model (typically done automatically in software) - Here we code the variable as [Female = 1 for females; Female = 0 for males]{style="color: blue;"}  - Estimated parameters: - $\beta_0$: Intercept - predicted value of Weight for a person with [Female = 0 (males)]{style="color: red;"} - $\beta_2$: Slope of Female - Change in predicted value of Weight between Males and Famales ------------------------------------------------------------------------ ### R Code ```{r} #| output-location: default model3 <- lm(weightLB ~ female, data = dataSexHeightWeight) # model formula summary(model3)$coefficients |> kable(digit = 3) # regression cofficients table anova(model3) |> kable(digit = 3) # F-statistic table ``` - Interpretation - $\beta_0 = 235.9$ pounds is the predicted value of a Weight for a male - $\beta_0 +\beta_1 = 235.9 - 105.0 = 130.9$ pounds is the predicted value of a Weight for a female - $\beta_1 = 105$ expected difference between gender, which is significant based on t-test and F-test ------------------------------------------------------------------------ ### Visualization: Line Plot (not frequently used in this case) ```{r} #| output-location: default #| fig-align: center ggplot(dataSexHeightWeight) + geom_point(aes(x = as.numeric(female), y = weightLB)) + geom_abline(intercept = coef(model3)[1], slope = coef(model3)[2]) + scale_x_continuous(breaks = 0:1) + labs(title = "Model 3: Fit Plot for Female", x = "Female") ``` ------------------------------------------------------------------------ ### Visualization: **box-and-whisker plot (Basic Way)** ```{r} #| output-location: default #| fig-align: center boxplot(weightLB ~ female, data = dataSexHeightWeight, names = c("Male", "Female"), ylab = "Weight", xlab = "",boxwex = .3) # Basic boxplot in R ``` ------------------------------------------------------------------------ ### Visualization: **box-and-whisker plot (Fancy Way)** ```{r} #| output-location: default #| fig-align: center #| code-fold: true #| code-summary: "ggplot2 package" ggplot(dataSexHeightWeight, aes(x = female, y = weightLB)) + geom_dotplot(aes(fill = female, color = female), binaxis='y', stackdir='center') + stat_summary(fun.data= "mean_cl_normal", fun.args = list(conf.int=.75), geom="crossbar", width=0.3, fill = "yellow", alpha = .5) + stat_summary(fun.data= "median_hilow", geom="errorbar", fun.args = list(conf.int=1), width = .1, color="black") + # Mean +- 2SD stat_summary(fun.data= "mean_sdl", geom="point", shape = 5, size = 3) + scale_x_discrete(labels = c("Male", "Female")) + labs(x = "", y = "Weight") + theme_bw() + theme(legend.position = "none", text = element_text(size = 20)) ``` ## Model 4: Predicting Weight from Height and Gender $$ \text{Weight}_p = \beta_0 + \beta_1 \text{HeightC}_p + \beta_2 \text{Female}_p + e_p $$ $$ e_p \sim N(0, \sigma^2_e) $$ ```{r} #| output-location: default model4 <- lm(weightLB ~ heightC + female, data = dataSexHeightWeight) # model formula summary(model4)$coefficients |> kable(digit = 3) # regression cofficients table anova(model4) |> kable(digit = 3) # F-statistic table ``` - Interpretation - $\beta_0 = 224.256 (SE = 1.439)$ - The predicted weight is 224.256 pounds for a person with Female = 0 (males) and has Height as Mean Height 67.9 inches - $\beta_1 = 2.708 (SE = 0.155)$ - $t = \frac{2.708}{0.155}=17.525; p <.001$ - The expected change in weights for every one-unit increase in height holding gender constant - $\beta_2=-81.713 (SE = 2.241)$ - The expected difference between the mean of weights for males and the mean for females holding height constant - $\sigma_e^2 = 16.283$ - The residual variance of weight ## Model 4: By-Gender Regression Lines Model 4 assumes identical regression slopes for both genders but has different intercepts - This assumption of different slopes will be tested statistically by model 5 Predicted Weight for Females with the height as 68.9 inch: $$ W_p=224.256+2.708\times\color{blue}{(68.9-67.9)} - 81.713*\color{red}{1} \\ = 224.256+2.708-81.712 = 145.252 $$ Predicted Weight for Males with the height as 68.9 inch: $$ W_p=224.256+2.708\times\color{blue}{(68.9-67.9)} - 81.713*\color{red}{0} \\ = 224.256+2.708 = 226.964 $$ ```{r} #| results: hold predict(model4, data.frame(heightC = 1, female = TRUE)) predict(model4, data.frame(heightC = 1, female = FALSE)) ``` ### Visualization: Different intercept and Same slope across groups ```{r} #| output-location: default #| fig-align: center #| code-fold: true #| code-summary: "ggplot2 package" dataSexHeightWeight |> mutate(predictedWeight = predict(model4, dataSexHeightWeight)) |> ggplot(aes(x = heightC, y = predictedWeight, color = female)) + geom_line() + geom_point(aes(y = weightLB, shape = female), size = 2) + theme_classic() ``` ## Model 5: By-Gender Regression Lines $$\text{Weight}_p = \beta_0 + \beta_1 \text{HeightC}_p + \beta_2 \text{Female}_p +\beta_3 \text{HeightCxFemale}_p+ e_p$$ ```{r} #| output-location: default model5 <- lm(weightLB ~ heightC * female, data = dataSexHeightWeight) # model formula summary(model5)$coefficients |> kable(digit = 3) # regression cofficients table anova(model5) |> kable(digit = 3) # F-statistic table ``` - Interpretation - $\beta_0 = 222.184 (SE = 0.838)$ - SE gets smaller compared to Model 4. - The predicted weight is 224.184 pounds for a person with Female = 0 (males) and has Height as Mean Height 67.9 inches - $\beta_1 = 3.189 (SE = 0.111)$ - SE gets smaller compared to Model 4. - **Simple main effect of Height:** the expected change in weights for every one-unit increase in height [for males only]{style="color: red"} - $\beta_2=-82.271 (SE = 1.211)$ - SE gets smaller compared to Model 4. - **Simple main effect of gender:** The expected difference between the mean of weights for males and the mean for females for people with mean height - $\beta_3 = -1.093 (SE = 1.211)$ , $t = -6.52, p < .001$ - **Gender-by-Height Interaction**: Additional change in slope of height for change in gender. - The slope of heights for males is 3.189; the slope of heights decreases 1.093 (2.096) inches/pound for females - $\sigma_e^2 = 5$ - Compared to 16.283 in model 4 ------------------------------------------------------------------------ ### Visualization: Different slopes and different intercepts Model 5 does not assume identical regression slopes - Because $\beta_3$ was significantly different from zero, the data supports different slopes for the gender ```{r} #| code-fold: true #| output-location: default ggplot(dataSexHeightWeight, aes(x = heightC, y = weightLB, color = female)) + geom_point() + geom_smooth(method = "lm") + theme_bw() ``` ## Model Comparison - In practice, the empty model (Model 0) and the full model (Model 5) would be the only models to run - The aim of model comparison is to decide on whether we want more complex model or the simple one - The **trick** is to describe the impact of all and each of the predictors – typically using variance accounted for - Residual Variance and $R^2$ ------------------------------------------------------------------------ ::: columns ::: {.column width="20%"} <iframe src="https://giphy.com/embed/b8RfbQFaOs1rO10ren" width="480" height="398" style frameBorder="0" class="giphy-embed" allowFullScreen> </iframe> ::: ::: {.column width="80%"} ```{r} #| code-fold: true #| output-location: default get_res_var <- function(model) { anova(model)$`Mean Sq` |> tail(1) } data.frame( Model = paste("Model", c(1, 2, "2a", 3, 4, 5)), ResidualVariance = sapply(list(model1, model2, model2a, model3, model4, model5), get_res_var) ) |> ggplot(aes(x = Model, y = ResidualVariance)) + geom_point() + geom_path(group = 1) + geom_label(aes(label = round(ResidualVariance, 2)),nudge_y = 200, nudge_x = 0.2) + labs(title = "Estimated Residual Variances across models") + theme(text = element_text(size = 20)) ``` ::: ::: ------------------------------------------------------------------------ ```{r} #| code-fold: true #| output-location: default get_explained_var <- function(model) { tot_err_var <- anova(model1)$`Mean Sq` |> tail(1) current_err_var <- anova(model)$`Mean Sq` |> tail(1) prop <- (tot_err_var - current_err_var)/ tot_err_var prop } data.frame( Model = paste("Model", c(1, 2, "2a", 3, 4, 5)), ExplainedVariance = sapply(list(model1, model2, model2a, model3, model4, model5), get_explained_var) ) |> ggplot(aes(x = Model, y = ExplainedVariance)) + geom_point() + geom_path(group = 1) + geom_label(aes(label = paste0(round(ExplainedVariance*100, 2), "%")),nudge_y = .05) + labs(title = "Proportions of Explained Variance across Models") ``` ## Comparison Across Models 1. **Are height and gender are good predictors?** - Total explained variances in weight by **height and gender:** Multiple $R^2$ of Model 5 - $(3179.09-4.73)/3179.09 = 0.9985$ $\rightarrow$ 99.85% variances in weights can be explained by height and gender - F-test comparing Model 5 to Model 1 - $F_{3, 16} = 4250.1$, p \< .001 ```{r} #| code-fold: true #| output-location: default summary(model5) ``` ------------------------------------------------------------------------ 2. **Are Height alone a good predictor?** - Explained **remaining variances** in weight by **height:** Multiple $R^2$ of Model 5 to Model 3 - $(293.21-4.73)/293.21 = 0.9839$ $\rightarrow$ 98.39% variances in weights [remaining after gender]{style="color: orange"} can be explained by **the main and interaction effects of height** - F-test comparing Model 5 to Model 3 - $F_{2, 16} = 548.74$, p \< .001 suggests the effect of hight on weight is significant after controlling the effect of gender ```{r} #| code-fold: true #| results: hide #| output-location: default anova(model3, model5) ((5277.8 - 75.7)/ (18-16))/(75.7 / 16) ``` - True explained variances out of total variances in weight: **Unique contribution** of adding Height into the model - Check model 3, we can found that gender explained 90.78% variance of weight - $0.9839 * (1 - 0.9078) = 0.0907$ $\rightarrow$ 9.07% more variances of weights can be explained by height after gender is already in the model - 90.78% + 9.07% = 99.85% is the total variance explained by height and gender ------------------------------------------------------------------------ 3. **Are gender alone a good predictor?** - Explained **remaining variances** in weight by **gender:** Multiple $R^2$ of Model 5 to Model 2a - $(1217.97-4.73)/1217.97 = 0.9961$ $\rightarrow$ 99.61% variances in weights [remaining after height]{style="color: orange"} can be explained by **the main and interaction effects of gender** - F-test comparing Model 5 to Model 2a - $F_{2, 16} = 2308.8$, p \< .001 suggests the effect of gender on weight is significant after controlling the effect of height ```{r} #| code-fold: true #| results: hide #| output-location: default anova(model2a, model5) ((21923.5 - 75.7)/ (18-16))/(75.7 / 16) ``` - True explained variances out of total variances in weight: **Unique contribution** of adding Height into the model - Check model 3, we can found that gender explained 90.78% variance of weight - $0.9839 * (1 - 0.9078) = 0.0907$ $\rightarrow$ 9.07% more variances of weights can be explained by height after gender is already in the model - 90.78% + 9.07% = 99.85% is the total variance explained by height and gender ## Summary of Unit 2 1. We learned GLM using the 2-factor model 2. For specific effect, we examine the t-test of coefficient 3. For the total contribution of a bunch of effects (main + interaction), we look at R-square and F-test 4. We sometime need to report predicted regression lines, which can be done in R with ggplot2 # Unit 3: Let's Look at R code together