Chapter 2 Parameter Estimation

library(tidyverse)  # dplyr, tidyr, ggplot2

Learning Outcomes

  • Write simple regression or one-way ANOVA models as \(\boldsymbol{y} \sim N(\boldsymbol{X\beta} , \sigma^2 \textbf{I}_n)\)

  • Utilizing R and a sample of data, calculate the estimators

    \[\hat{\boldsymbol{\beta}} = (\textbf{X}^T\textbf{X})^{-1} \textbf{X}^T \textbf{y}\] \[\hat{\textbf{y}} = \textbf{X}\hat{\boldsymbol{\beta}}\] and \[\hat{\sigma}^2 = \textrm{MSE} = \frac{1}{n-p}\;\sum_{i=1}^n (y_i - \hat{y}_i)^2\]

  • Calculate the uncertainty of \(\hat{\beta_j}\) as \[\textrm{StdErr}\left(\hat{\beta}_{j}\right)=\sqrt{\hat{\sigma}^{2}\left[\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\right]_{jj}}\]

2.1 Introduction

We have previously looked at ANOVA and regression models and, in many ways, they felt very similar. In this chapter we will introduce the theory that allows us to understand both models as a particular flavor of a larger class of models known as linear models.

First we clarify what a linear model is. A linear model is a model where the data (which we will denote using roman letters as \(\boldsymbol{x}\) and \(\boldsymbol{y}\)) and parameters of interest (which we denote using Greek letters such as \(\boldsymbol{\alpha}\) and \(\boldsymbol{\beta}\)) interact only via addition and multiplication. The following are linear models:

Model Formula
ANOVA \(y_{ij}=\mu+\tau_{i}+\epsilon_{ij}\)
Simple Regression \(y_{i}=\beta_{0}+\beta_{1}x_{i}+\epsilon_{i}\)
Quadratic Term \(y_{i}=\beta_{0}+\beta_{1}x_{i}+\beta_{2}x_{i}^{2}+\epsilon_{i}\)
General Regression \(y_{i}=\beta_{0}+\beta_{1}x_{i,1}+\beta_{2}x_{i,2}+\dots+\beta_{p}x_{i,p}+\epsilon_{i}\)

Notice in the Quadratic model, the square is not a parameter and we can consider \(x_{i}^{2}\) as just another column of data. This leads to the second example of multiple regression where we just add more slopes for other covariates where the \(p\)th covariate is denoted \(\boldsymbol{x}_{\cdot,p}\) and might be some transformation (such as \(x^{2}\) or \(\log x\)) of another column of data. The critical point is that the transformation to the data \(\boldsymbol{x}\) does not depend on a parameter. Thus the following is not a linear model \[ y_{i}=\beta_{0}+\beta_{1}x_{i}^{\alpha}+\epsilon_{i} \]

2.2 Model Specifications

2.2.1 Simple Regression

We would like to represent all linear models in a similar compact matrix representation. This will allow us to make the transition between simple and multiple regression (and ANCOVA) painlessly.

To begin, lets consider the simple regression model.

Typically we’ll write the model as if we are specifying the \(i^{th}\) element of the data set \[y_i = \underbrace{\beta_0 + \beta_1 x_i}_{\textrm{signal}} + \underbrace{\epsilon_i}_{\textrm{noise}} \;\;\; \textrm{ where } \epsilon_i \stackrel{iid}{\sim} N(0, \sigma^2)\] Notice we have a data generation model where there is some relationship between the explanatory variable and the response which we can refer to as the “signal” part of the defined model and the noise term which represents unknown actions effecting each data point that move the response variable. We don’t know what those unknown or unmeasured effects are, but we do know the sum of those effects results in a vertical shift from the signal part of the model.

This representation of the model implicitly assumes that our data set has \(n\) observations and we could write the model using all the obsesrvations using matrices and vectors that correspond the the data and the parameters.

\[\begin{aligned} y_{1} & = \beta_{0}+\beta_{1}x_{1}+\epsilon_{1}\\ y_{2} & = \beta_{0}+\beta_{1}x_{2}+\epsilon_{2}\\ y_{3} & = \beta_{0}+\beta_{1}x_{3}+\epsilon_{3}\\ & \vdots\\ y_{n-1} & = \beta_{0}+\beta_{1}x_{n-1}+\epsilon_{n-1}\\ y_{n} & = \beta_{0}+\beta_{1}x_{n}+\epsilon_{n} \end{aligned}\]

where, as usual, \(\epsilon_{i}\stackrel{iid}{\sim}N\left(0,\sigma^{2}\right)\). These equations can be written using matrices as

\[ \underset{\boldsymbol{y}}{\underbrace{\left[\begin{array}{c} y_{1}\\ y_{2}\\ y_{3}\\ \vdots\\ y_{n-1}\\ y_{n} \end{array}\right]}}=\underset{\boldsymbol{X}}{\underbrace{\left[\begin{array}{cc} 1 & x_{1}\\ 1 & x_{2}\\ 1 & x_{3}\\ \vdots & \vdots\\ 1 & x_{n-1}\\ 1 & x_{n} \end{array}\right]}}\underset{\boldsymbol{\beta}}{\underbrace{\left[\begin{array}{c} \beta_{0}\\ \beta_{1} \end{array}\right]}}+\underset{\boldsymbol{\epsilon}}{\underbrace{\left[\begin{array}{c} \epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \vdots\\ \epsilon_{n-1}\\ \epsilon_{n} \end{array}\right]}} \]

and we compactly write the model as

\[ \boldsymbol{y}=\boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon} \;\; \textrm{ where } \; \boldsymbol{\epsilon} \sim N(\boldsymbol{0}, \sigma^2 \boldsymbol{I}_n) \] where \(\boldsymbol{X}\) is referred to as the design matrix and \(\boldsymbol{\beta}\) is the vector of location parameters we are interested in estimating. To be very general, a vector of random variables needs to describe how each of them varies in relation to each other, so we need to specify the variance matrix. However because \(\epsilon_i\) is independent of \(\epsilon_j\), for all \((i,j)\) pairs, the variance matrix can be written as \(\sigma^2\,\boldsymbol{I}\) because all of the covariances are zero.

2.2.2 ANOVA model

The anova model is also a linear model and all we must do is create a appropriate design matrix. Given the design matrix \(\boldsymbol{X}\), all the calculations are identical as in the simple regression case. Cell means representation

Recall the cell means representation is \[ y_{i,j}=\mu_{i}+\epsilon_{i,j} \] where \(y_{i,j}\) is the \(j\)th observation within the \(i\)th group. To clearly show the creation of the \(\boldsymbol{X}\) matrix, let the number of groups be \(p=3\) and the number of observations per group be \(n_{i}=4\). We now expand the formula to show all the data. \[\begin{aligned} y_{1,1} &= \mu_{1}+\epsilon_{1,1}\\ y_{1,2} &= \mu_{1}+\epsilon_{1,2}\\ y_{1,3} &= \mu_{1}+\epsilon_{1,3}\\ y_{1,4} &= \mu_{1}+\epsilon_{1,4}\\ y_{2,1} &= \mu_{2}+\epsilon_{2,1}\\ y_{2,2} &= \mu_{2}+\epsilon_{2,2}\\ y_{2,3} &= \mu_{2}+\epsilon_{2,3}\\ y_{2,4} &= \mu_{2}+\epsilon_{2,4}\\ y_{3,1} &= \mu_{3}+\epsilon_{3,1}\\ y_{3,2} &= \mu_{3}+\epsilon_{3,2}\\ y_{3,3} &= \mu_{3}+\epsilon_{3,3}\\ y_{3,4} &= \mu_{3}+\epsilon_{3,4} \end{aligned}\]

In an effort to write the model as \(\boldsymbol{y}=\boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}\) we will write the above as

\[\begin{aligned} y_{1,1} &= 1\mu_{1}+0\mu_{2}+0\mu_{3}+\epsilon_{1,1}\\ y_{1,2} &= 1\mu_{1}+0\mu_{2}+0\mu_{3}+\epsilon_{1,2}\\ y_{1,3} &= 1\mu_{1}+0\mu_{2}+0\mu_{3}+\epsilon_{1,3}\\ y_{1,4} &= 1\mu_{1}+0\mu_{2}+0\mu_{3}+\epsilon_{1,4}\\ y_{2,1} &= 0\mu+1\mu_{2}+0\mu_{3}+\epsilon_{2,1}\\ y_{2,2} &= 0\mu+1\mu_{2}+0\mu_{3}+\epsilon_{2,2}\\ y_{2,3} &= 0\mu+1\mu_{2}+0\mu_{3}+\epsilon_{2,3}\\ y_{2,4} &= 0\mu+1\mu_{2}+0\mu_{3}+\epsilon_{2,4}\\ y_{3,1} &= 0\mu+0\mu_{2}+1\mu_{3}+\epsilon_{3,1}\\ y_{3,2} &= 0\mu+0\mu_{2}+1\mu_{3}+\epsilon_{3,2}\\ y_{3,3} &= 0\mu+0\mu_{2}+1\mu_{3}+\epsilon_{3,3}\\ y_{3,4} &= 0\mu+0\mu_{2}+1\mu_{3}+\epsilon_{3,4} \end{aligned}\]

and we will finally be able to write the matrix version \[ \underset{\boldsymbol{y}}{\underbrace{\left[\begin{array}{c} y_{1,1}\\ y_{1,2}\\ y_{1,3}\\ y_{1,4}\\ y_{2,1}\\ y_{2,2}\\ y_{2,3}\\ y_{2,4}\\ y_{3,1}\\ y_{3,2}\\ y_{3,3}\\ y_{3,4} \end{array}\right]}}=\underset{\mathbf{X}}{\underbrace{\left[\begin{array}{ccc} 1 & 0 & 0\\ 1 & 0 & 0\\ 1 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 1 & 0\\ 0 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 1\\ 0 & 0 & 1\\ 0 & 0 & 1 \end{array}\right]}}\underset{\boldsymbol{\beta}}{\underbrace{\left[\begin{array}{c} \mu_{1}\\ \mu_{2}\\ \mu_{3} \end{array}\right]}}+\underset{\boldsymbol{\epsilon}}{\underbrace{\left[\begin{array}{c} \epsilon_{1,1}\\ \epsilon_{1,2}\\ \epsilon_{1,3}\\ \epsilon_{1,4}\\ \epsilon_{2,1}\\ \epsilon_{2,2}\\ \epsilon_{2,3}\\ \epsilon_{2,4}\\ \epsilon_{3,1}\\ \epsilon_{3,2}\\ \epsilon_{3,3}\\ \epsilon_{3,4} \end{array}\right]}} \]

Notice that each column of the \(\boldsymbol{X}\) matrix is acting as an indicator if the observation is an element of the appropriate group. As such, these are often called indicator variables. Another term for these, which I find less helpful, is dummy variables. Offset from reference group

In this model representation of ANOVA, we have an overall mean and then offsets from the control group (which will be group one). The model is thus \[ y_{i,j}=\mu+\tau_{i}+\epsilon_{i,j} \] where \(\tau_{1}=0\). We can write this in matrix form as \[ \underset{\boldsymbol{y}}{\underbrace{\left[\begin{array}{c} y_{1,1}\\ y_{1,2}\\ y_{1,3}\\ y_{1,4}\\ y_{2,1}\\ y_{2,2}\\ y_{2,3}\\ y_{2,4}\\ y_{3,1}\\ y_{3,2}\\ y_{3,3}\\ y_{3,4} \end{array}\right]}}=\underset{\mathbf{X}}{\underbrace{\left[\begin{array}{ccc} 1 & 0 & 0\\ 1 & 0 & 0\\ 1 & 0 & 0\\ 1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 1 & 0\\ 1 & 1 & 0\\ 1 & 1 & 0\\ 1 & 0 & 1\\ 1 & 0 & 1\\ 1 & 0 & 1\\ 1 & 0 & 1 \end{array}\right]}}\underset{\boldsymbol{\beta}}{\underbrace{\left[\begin{array}{c} \mu\\ \tau_{2}\\ \tau_{3} \end{array}\right]}}+\underset{\boldsymbol{\epsilon}}{\underbrace{\left[\begin{array}{c} \epsilon_{1,1}\\ \epsilon_{1,2}\\ \epsilon_{1,3}\\ \epsilon_{1,4}\\ \epsilon_{2,1}\\ \epsilon_{2,2}\\ \epsilon_{2,3}\\ \epsilon_{2,4}\\ \epsilon_{3,1}\\ \epsilon_{3,2}\\ \epsilon_{3,3}\\ \epsilon_{3,4} \end{array}\right]}} \]

2.3 Parameter Estimation

For both simple regression and ANOVA, we can write the model in matrix form as \[\boldsymbol{y} = \boldsymbol{X\beta} + \boldsymbol{\epsilon} \;\; \textrm{ where } \boldsymbol{\epsilon} \sim N(\boldsymbol{0},\sigma^2\boldsymbol{I}_n)\] which could also be written as \[\boldsymbol{y} \sim N(\boldsymbol{X\beta},\sigma^2\boldsymbol{I}_n)\] and we could use the maximum-likelihood principle to find estimators for \(\beta\) and \(\sigma^2\). In this section, we will introduce the estimators \(\hat{\boldsymbol{\beta}}\) and \(\hat{\sigma}^2\).

2.3.1 Estimation of Location Paramters

Our goal is to find the best estimate of \(\boldsymbol{\beta}\) given the data. To justify the formula, consider the case where there is no error terms (i.e. \(\epsilon_{i}=0\) for all \(i\)). Thus we have \[ \boldsymbol{y}=\boldsymbol{X}\boldsymbol{\beta} \] and our goal is to solve for \(\boldsymbol{\beta}\). To do this, we must use a matrix inverse, but since inverses only exist for square matrices, we pre-multiple by \(\boldsymbol{X}^{T}\) (notice that \(\boldsymbol{X}^{T}\boldsymbol{X}\) is a symmetric \(2\times2\) matrix). \[ \boldsymbol{X}^{T}\boldsymbol{y}=\boldsymbol{X}^{T}\boldsymbol{X}\boldsymbol{\beta} \] and then pre-multiply by \(\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\).

\[\begin{aligned} \left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\boldsymbol{y} &= \left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\boldsymbol{X}\boldsymbol{\beta} \\ \left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\boldsymbol{y} &= \boldsymbol{\beta} \end{aligned}\]

This exercise suggests that \(\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\boldsymbol{y}\) is a good place to start when looking for the maximum-likelihood estimator for \(\boldsymbol{\beta}\).

Happily it turns out that this quantity is in fact the maximum-likelihood estimator for the data generation model \[\boldsymbol{y} \sim N(\boldsymbol{X\beta}, \sigma^2 \boldsymbol{I}_n)\] (and equivalently minimizes the sum-of-squared error). In this course we won’t prove these two facts, but we will use this as our estimate of \(\boldsymbol{\beta}\). \[\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\boldsymbol{y}\]

2.3.2 Estimation of Variance Parameter

Recall our simple regression model is \[y_{i}=\beta_{0}+\beta_{1}x_{i}+\epsilon_{i}\] where \(\epsilon_{i}\stackrel{iid}{\sim}N\left(0,\sigma^{2}\right)\).

Using our estimates \(\hat{\boldsymbol{\beta}}\) we can obtain predicted values for the regression line at any x-value. In particular we can find the predicted value for each \(x_i\) value in our dataset. \[\hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i\] Using matrix notation, I would write \(\hat{\boldsymbol{y}}=\boldsymbol{X}\hat{\boldsymbol{\beta}}\).

As usual we will find estimates of the noise terms (which we will call residuals or errors) via \[\begin{aligned} \hat{\epsilon}_{i} &= y_{i}-\hat{y}_{i}\\ &= y_{i}-\left(\hat{\beta}_{0}+\hat{\beta}_{1}x_{i}\right) \end{aligned}\]

Writing \(\hat{\boldsymbol{y}}\) in matrix terms we have \[\begin{aligned} \hat{\boldsymbol{y}} &= \boldsymbol{X}\hat{\boldsymbol{\beta}}\\ &= \boldsymbol{X}\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\boldsymbol{y}\\ &= \boldsymbol{H}\boldsymbol{y} \end{aligned}\] where \(\boldsymbol{H}=\boldsymbol{X}\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\) is often called the hat-matrix because it takes \(y\) to \(\hat{y}\) and has many interesting theoretical properties.

We can now estimate the error terms via

\[\begin{aligned} \hat{\boldsymbol{\epsilon}} &= \boldsymbol{y}-\hat{\boldsymbol{y}}\\ &= \boldsymbol{y}-\boldsymbol{H}\boldsymbol{y}\\ &= \left(\boldsymbol{I}_{n}-\boldsymbol{H}\right)\boldsymbol{y} \end{aligned}\]

As usual we estimate \(\sigma^{2}\) using the mean-squared error, but the general formula is \[\begin{aligned} \hat{\sigma}^{2} &= \textrm{MSE} \\ \\ &= \frac{1}{n-p}\;\sum_{i=1}^{n}\hat{\epsilon}_{i}^{2}\\ \\ &= \frac{1}{n-p}\;\hat{\boldsymbol{\epsilon}}^{T}\hat{\boldsymbol{\epsilon}} \end{aligned}\] where \(\boldsymbol{\beta}\) has \(p\) elements, and thus we have \(n-p\) degrees of freedom.

2.4 Standard Errors

Because our \(\hat{\boldsymbol{\beta}}\) estimates vary from sample to sample, we need to estimate how much they vary from sample to sample. This will eventually allow us to create confidence intervals for and perform hypothesis test relating to the \(\boldsymbol{\beta}\) parameter vector.

2.4.1 Expectation and variance of a random vector

Just as we needed to derive the expected value and variance of \(\bar{x}\) in the previous semester, we must now do the same for \(\hat{\boldsymbol{\beta}}\). But to do this, we need some properties of expectations and variances.

In the following, let \(\boldsymbol{A}_{n\times p}\) and \(\boldsymbol{b}_{n\times1}\) be constants and \(\boldsymbol{\epsilon}_{n\times1}\) be a random vector.

Expectations are very similar to the scalar case where \[E\left[\boldsymbol{\epsilon}\right]=\left[\begin{array}{c} E\left[\epsilon_{1}\right]\\ E\left[\epsilon_{2}\right]\\ \vdots\\ E\left[\epsilon_{n}\right] \end{array}\right]\] and any constants are pulled through the expectation \[E\left[\boldsymbol{A}^{T}\boldsymbol{\epsilon}+\boldsymbol{b}\right]=\boldsymbol{A}^{T}\,E\left[\boldsymbol{\epsilon}\right]+\boldsymbol{b}\]

Variances are a little different. The variance of the vector \(\boldsymbol{\epsilon}\) is \[ Var\left(\boldsymbol{\epsilon}\right)=\left[\begin{array}{cccc} Var\left(\epsilon_{1}\right) & Cov\left(\epsilon_{1},\epsilon_{2}\right) & \dots & Cov\left(\epsilon_{1},\epsilon_{n}\right)\\ Cov\left(\epsilon_{2},\epsilon_{1}\right) & Var\left(\epsilon_{2}\right) & \dots & Cov\left(\epsilon_{2},\epsilon_{n}\right)\\ \vdots & \vdots & \ddots & \vdots\\ Cov\left(\epsilon_{n},\epsilon_{1}\right) & Cov\left(\epsilon_{n},\epsilon_{2}\right) & \dots & Var\left(\epsilon_{1}\right) \end{array}\right] \]

and additive constants are ignored, but multiplicative constants are pulled out as follows:

\[ Var\left(\boldsymbol{A}^{T}\boldsymbol{\epsilon}+\boldsymbol{b}\right)=Var\left(\boldsymbol{A}^{T}\boldsymbol{\epsilon}\right)=\boldsymbol{A}^{T}\,Var\left(\boldsymbol{\epsilon}\right)\,\boldsymbol{A} \]

2.4.2 Variance of Location Parameters

We next derive the sampling variance of our estimator \(\hat{\boldsymbol{\beta}}\) by first noting that \(\boldsymbol{X}\) and \(\boldsymbol{\beta}\) are constants and therefore \[\begin{aligned} Var\left(\boldsymbol{y}\right) &= Var\left(\boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}\right)\\ &= Var\left(\boldsymbol{\epsilon}\right)\\ &= \sigma^{2}\boldsymbol{I}_{n} \end{aligned}\]

because the error terms are independent and therefore \(Cov\left(\epsilon_{i},\epsilon_{j}\right)=0\) when \(i\ne j\) and \(Var\left(\epsilon_{i}\right)=\sigma^{2}\). Recalling that constants come out of the variance operator as the constant squared.

\[\begin{aligned} Var\left(\hat{\boldsymbol{\beta}}\right) &= Var\left(\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\boldsymbol{y}\right)\\ &= \left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\,Var\left(\boldsymbol{y}\right)\,\boldsymbol{X}\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\\ &= \left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\,\sigma^{2}\boldsymbol{I}_{n}\,\boldsymbol{X}\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\\ &= \sigma^{2}\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\boldsymbol{X}\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\\ &= \sigma^{2}\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1} \end{aligned}\]

Using this, the standard error (i.e. the estimated standard deviation) of \(\hat{\beta}_{j}\) (for any \(j\) in \(1,\dots,p\)) is \[StdErr\left(\hat{\beta}_{j}\right)=\sqrt{\hat{\sigma}^{2}\left[\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\right]_{jj}}\]

2.4.3 Summary of pertinent results

The statistic \(\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\boldsymbol{y}\) is the unbiased maximum-likelihood estimator of \(\boldsymbol{\beta}\).

The Central Limit Theorem applies to each element of \(\boldsymbol{\beta}\). That is, as \(n\to\infty\), the distribution of \(\hat{\beta}_{j}\to N\left(\beta_{j},\left[\sigma^{2}\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\right]_{jj}\right)\).

The error terms can be calculated via \[\begin{aligned} \hat{\boldsymbol{y}} &= \boldsymbol{X}\hat{\boldsymbol{\beta}}\\ \hat{\boldsymbol{\epsilon}} &= \boldsymbol{y}-\hat{\boldsymbol{y}} \end{aligned}\]

The estimate of \(\sigma^{2}\) is \[\begin{aligned} \hat{\sigma}^{2} = \textrm{MSE} = \frac{1}{n-p}\;\sum_{i=1}^{n}\hat{\epsilon}_{i}^{2} = \frac{1}{n-p}\;\hat{\boldsymbol{\epsilon}}^{T}\hat{\boldsymbol{\epsilon}} \end{aligned}\]

and is the typical squared distance between an observation and the model prediction.

The standard error (i.e. the estimated standard deviation) of \(\hat{\beta}_{j}\) (for any \(j\) in \(1,\dots,p\)) is \[StdErr\left(\hat{\beta}_{j}\right)=\sqrt{\hat{\sigma}^{2}\left[\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\right]_{jj}}\]

2.5 R example

Here we will work an example in R and do both the “hand” calculation as well as using the lm() function to obtain the same information.

Consider the following data in a simple regression problem:

n <- 20
x <- seq(0,10, length=n)
y <- -3 + 2*x + rnorm(n, sd=2) <- data.frame(x=x, y=y)
ggplot( + geom_point(aes(x=x,y=y))

First we must create the design matrix \(\boldsymbol{X}\). Recall \[ \boldsymbol{X}=\left[\begin{array}{cc} 1 & x_{1}\\ 1 & x_{2}\\ 1 & x_{3}\\ \vdots & \vdots\\ 1 & x_{n-1}\\ 1 & x_{n} \end{array}\right] \] and can be created in R via the following:

X <- cbind( rep(1,n), x)

Given \(\boldsymbol{X}\) and \(\boldsymbol{y}\) we can calculate \[ \hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{T}\boldsymbol{X}\right)^{-1}\boldsymbol{X}^{T}\boldsymbol{y} \] in R using the following code:

XtXinv <- solve( t(X) %*% X )           # solve() is the inverse function
beta.hat <- XtXinv %*% t(X) %*% y %>%   # Do the calculations
  as.vector()                           # make sure the result is a vector

Our next step is to calculate the predicted values \(\hat{\boldsymbol{y}}\) and the residuals \(\hat{\boldsymbol{\epsilon}}\)

\[\begin{aligned} \hat{\boldsymbol{y}} &= \boldsymbol{X}\hat{\boldsymbol{\beta}}\\ \hat{\boldsymbol{\epsilon}} &= \boldsymbol{y}-\hat{\boldsymbol{y}} \end{aligned}\]

y.hat <- X %*% beta.hat
residuals <- y - y.hat

Now that we have the residuals, we can calculate \(\hat{\sigma}^{2}\) and the standard errors of \(\hat{\beta}_{j}\)



sigma2.hat <- 1/(n-2) * sum( residuals^2 )    # p = 2 
sigma.hat <- sqrt( sigma2.hat )
std.errs <- sqrt( sigma2.hat * diag(XtXinv) )

We now print out the important values and compare them to the summary output given by the lm() function in R.

cbind(Est=beta.hat, StdErr=std.errs) 
##         Est    StdErr
##   -1.448250 0.7438740
## x  1.681485 0.1271802
## [1] 1.726143
model <- lm(y~x)
## Call:
## lm(formula = y ~ x)
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.5127 -1.2134  0.1469  1.2610  3.2358 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -1.4482     0.7439  -1.947   0.0673 .  
## x             1.6815     0.1272  13.221 1.04e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Residual standard error: 1.726 on 18 degrees of freedom
## Multiple R-squared:  0.9066, Adjusted R-squared:  0.9015 
## F-statistic: 174.8 on 1 and 18 DF,  p-value: 1.044e-10

2.6 Exercises

  1. We will do a simple ANOVA analysis on example 8.2 from Ott & Longnecker using the matrix representation of the model. A clinical psychologist wished to compare three methods for reducing hostility levels in university students, and used a certain test (HLT) to measure the degree of hostility. A high score on the test indicated great hostility. The psychologist used 24 students who obtained high and nearly equal scores in the experiment. eight were selected at random from among the 24 problem cases and were treated with method 1. Seven of the remaining 16 students were selected at random and treated with method 2. The remaining nine students were treated with method 3. All treatments were continued for a one-semester period. Each student was given the HLT test at the end of the semester, with the results show in the following table. (This analysis was done in section 8.3 of my STA 570 notes)

    Method Values
    1 96, 79, 91, 85, 83, 91, 82, 87
    2 77, 76, 74, 73, 78, 71, 80
    3 66, 73, 69, 66, 77, 73, 71, 70, 74

    We will be using the cell means model of ANOVA \[ y_{ij}=\beta_{i}+\epsilon_{ij} \] where \(\beta_{i}\) is the mean of group \(i\) and \(\epsilon_{ij}\stackrel{iid}{\sim}N\left(0,\sigma^{2}\right)\).

    1. Create one vector of all 24 hostility test scores y. (Use the c() function.)

    2. Create a design matrix X with dummy variables for columns that code for what group an observation belongs to. Notice that X will be a \(24\) rows by \(3\) column matrix. Hint: An R function that might be handy is cbind(a,b) which will bind two vectors or matrices together along the columns. There is also a corresponding rbind() function that binds vectors/matrices along rows. Furthermore, the repeat command rep() could be handy.

    1. Find \(\hat{\boldsymbol{\beta}}\) using the matrix formula given in class. Hint: The R function t(A) computes the matrix transpose \(\mathbf{A}^{T}\), solve(A) computes \(\mathbf{A}^{-1}\), and the operator %*% does matrix multiplication (used as A %*% B).

    2. Examine the matrix \(\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\mathbf{X}^{T}\). What do you notice about it? In particular, think about the result when you right multiply by \(\mathbf{y}\). How does this matrix calculate the appropriate group means and using the appropriate group sizes \(n_i\)?

  2. We will calculate the y-intercept and slope estimates in a simple linear model using matrix notation. We will use a data set that gives the diameter at breast height (DBH) versus tree height for a randomly selected set of trees. In addition, for each tree, a ground measurement of crown closure (CC) was taken. Larger values of crown closure indicate more shading and is often associated with taller tree morphology (possibly). We will be interested in creating a regression model that predicts height based on DBH and CC. In the interest of reduced copying, we will only use 10 observations. (Note: I made this data up and the DBH values might be unrealistic. Don’t make fun of me.)

    DBH 30.5 31.5 31.7 32.3 33.3 35 35.4 35.6 36.3 37.8
    CC 0.74 0.69 0.65 0.72 0.58 0.5 0.6 0.7 0.52 0.6
    Height 58 64 65 70 68 63 78 80 74 76

    We are interested in fitting the regression model \[y_{i}=\beta_{0}+\beta_{1}x_{i,1}+\beta_{2}x_{i,2}+\epsilon_{i}\] where \(\beta_{0}\) is the y-intercept and \(\beta_{1}\) is the slope parameter associated with DBH and \(\beta_{2}\) is the slope parameter associated with Crown Closure.

    1. Create a vector of all 10 heights \(\mathbf{y}\).
    2. Create the design matrix \(\mathbf{X}\).
    3. Find \(\hat{\boldsymbol{\beta}}\) using the matrix formula given in class.
    4. Compare your results to the estimated coefficients you get using the lm() function. To add the second predictor to the model, your call to lm() should look something like lm(Height ~ DBH + CrownClosure).