Linear Regression T Test For Coefficients

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regression t test
Photo By Clint Shelton
<- Part I: Coefficient Estimates
Part III: Regression F Test ->

Table of Contents

Complete Code for Regression Formulas in RDownload


Context


Introduction to the Regression T Test

The first step in regression is to estimate the model coefficients and use them to created predicted dependent values. The linear regression T test is a method of testing the statistical significance of estimated those coefficients.

In part 1 I showed how dealing with linear regression in matrix rather than scalar form makes the solutions easier to work with and more intuitive to derive. This is continued in this part 2p; the standard solutions given are impractical and not a good starting point for understadning the concepts.
\[\require{cancel}\large{
\stackrel{\normalsize{\text{NO! – Worthless – Never Use}}\\}{\xcancel{\sigma_{\widehat{b}_0}^2 = \frac{\sigma_Y^2 \sum x^2}{n\sum(x-\widehat{\mu}_x)^2}}}
}\]
It is dramatically easier to understand and to do the required math for degrees of freedom, standard error, etc. (as they relate to linear regression) if vectors and matrices, rather than scalar variables only, are used.

The Test Hypotheses

The null hypothesis for every individual coefficient in vector \(b\) is that it is equal to 0.
\[\large{
\begin{align}
&y = Xb+\varepsilon & && &\normalsize \text{(Model)}\\ \\
&\widehat{b} = (X^\text{T}X)^{-1}X^\text{T}y & && &\normalsize \text{(coefficients estimator)}\\ \\
&\text{H}_0:b_\text{[i]} = 0 & && &\normalsize \text{(Null Hypotheses)} \\ \\
&\text{H}_a:b_{[i]}\ne 0 & && &\normalsize \text{(Alternative Hypotheses)}
\end{align}
}\]

The Test

The test statistic, is considered to be an instance of a T distribution (due to the assumption of normally distributed errors – more detail in derivation section). If the null, hypothesis is rejected, then that is considered statistical proof that the factor being tested is significantly contributing to the model.
\[\large{
\begin{align}
&t_{df_\text{model}} = \Big[\frac{\widehat{b}-0}{\widehat{\sigma}_\widehat{b}^2}\Big] \sim T(df_\text{model}) & && &\normalsize\text{(Test Statistic)} \\ \\
&P_T(t,df_\text{model}) \begin{cases}
>= a && \small\text{accept $H_0$} \\
< a && \small\text{reject $H_0$}
\end{cases} & && &\normalsize\text{(Test)}
\end{align}
}\]

WARNING

Be cautious of two conditions that can confound the T test. The first is including collinear features in the analysis model (which violates the assumption of independent input features), which biases the coefficient estimates toward zero.

# example of highly collinear feature confounding T test

rm(list=ls())
n <- 30
df <- data.frame(x1 = runif(n,-100,100))
df$e <- rnorm(n,0,35)

# x2 is collinear with x1
df$x2 <- sapply(df$x1,function(x) x + rnorm(1,0,5))


# y is only dependent on x1
df$y <- 40 + 2.5 * df$x1 + df$e

lm1 <- lm(y ~ x1 + x2 + 1, df)
lm2 <- lm(y ~ x1 + 1, df)
summary(lm1)
summary(lm2)

The other potential pitfall comes from the nature of running multiple separate tests. Suppose you have 20 unrelated features (to the dependent variable) in a model, and choose a test confidence level of 95%, what is the probability of at least 1 type-1 error (false positive)?

Given we know the factors are unrelated, the probability of 0 significant results is \(P(0) = 0.95^{20}\), which means probability of at least 1 type-1 error is \(P(\ge 1) = 1 – 0.95^{20} = 64.15%\). The more unrelated factors added to an analysis model the greater the expected value of type-1 errors.


The T Distribution

Any normal distribution can be transformed into a “standard normal” or Z distribution (i.e. normal distribution with mean 0 and standard deviation 1) by subtracting its own mean and dividing by its own standard deviation. This makes the Z distribution an analogue for every other normal distribution.
\[
Z = \frac{X-\mu_X}{\sigma_X}\sim\text{Nor(0,1)}
\]
This principle is most often applied in relation to a sample mean, which is itself a random variable (discussed in greater detail in the next section on sample distributions). If we sampled correctly, then according to the central limit theorem, the sample mean should be approximately normally distributed. Additionally, from the basic properties of expected value and standard error for sample mean, both can be written in terms of a single random variable from the same population.
\[\large{
\begin{align}
\widehat{\mu}_X = \frac{1}{n}\sum_{i=1}^n X_i\sim RV && Z= \frac{\widehat{\mu}_X\;-\;\mu_{\widehat{\mu}_X}}{\sigma_{\widehat{\mu}_X}}
\end{align}
}\]
However, almost always the population standard deviation is unknown, which means that the standard deviation for the sample mean, which remember is also a random variable, needs to be estimated, and also that the sample mean itself is a parameter in that estimator. The formula for a Z value using an estimated standard deviation produces a T distribution, which approximates, but is not exactly equal to the Z distribution.
\[\large
t_{df} = \frac{\widehat{\mu}_X\;-\;\mu_{\widehat{\mu}_X}}{\widehat{\sigma}_{\widehat{\mu}_X}}\sim T(df)
\]
The proof for why using an estimated standard deviation transforms the Z distribution into a T distribution is covered in this post. The T distribution accepts a single parameter representing the degrees of freedom of the sample standard deviation and it becomes closer to the Z distribution as the degrees of freedom tend toward infinity.
\[\large
\lim_{df\rightarrow\infty} T(df) = Z
\]


Sampling Distributions, Estimators, and “Standard Error”

A ubiquitous term in statistics especially in regards to hypothesis testing is “standard error.” It is often defined in an esoteric way so I briefly reintroduce it here with a more functional definition.

Imagine that you want to measure the height of everyone in your town. For many this would be impractical, but you may be able to estimate the height of your town by creating a sampling distribution. A sampling distribution is exactly what it sounds like: a probability distribution created from a sample. Your sample estimates the population it was drawn from (if you did your sampling correctly).

The values from your sampling distribution can be used in formulas called estimators, which as the name suggests are intended to estimate unknown values of the population distribution (e.g. mean, variance, et.c). See example below.
\[\large
\begin{align}
&X\sim\text{R.V.}\;\;\text{E}[X]=\mu_X && \normalsize (X\text{ is a random variable with unknown mean }\mu_X) \\ \\
&x =\{x_1,x_2,\ldots,x_n\} && \normalsize (x\text{ is a set containing a sample drawn from }X) \\ \\
&\widehat{\mu}_X = \frac{1}{n}\sum_{i=1}^n x_i && \normalsize (\widehat{\mu}_X\text{ is an estimator for }\mu_X)
\end{align}
\]
Every value in an estimator can be considered an instance (i.e. value generated from) of the population random variable. The aggregation of many random variables is itself a random variable (e.g. if \(X\) and \(Y\) are random variables then \(C=X+Y\sim RV\)). Hence estimators are also random variables.

Estimators are random variables built on sampling distributions and “Standard Error” is simply the standard deviation of an estimator.
\[\large
\begin{align}
&\frac{1}{n}\sum_{i=1}^n x_i = \widehat{\mu}_X \sim RV &&\normalsize (\widehat{\mu}_X\text{ is a random variable}) \\ \\
&\text{Std.Err}[\widehat{\mu}_X] = \sqrt{\text{Var}[\widehat{\mu}_X]}
\end{align}
\]


Estimator Bias and Degrees of Freedom


Biased vs Unbiased Estimator

Estimators are both formulas and random variables and as such they have expected values. If the expected value of an estimator is NOT equal to the what it is estimating, then the estimator is “biased.”
\[\large
\begin{align}
&Y\sim RV & && &\text{E}[Y] = \mu_Y& \\ \\
&\widehat{\mu}_Y = \frac{1}{n}\sum_{i=1}^n y_i & && & \stackrel{\text{unbiased estimator}\\}{\text{E}[\widehat{\mu}_Y] = \mu_Y} \\ \\
&\stackrel{\text{unadjusted formula}\\}{\widehat{\sigma}_Y^2 = \frac{1}{n}\sum_{i=1}^n{(y-\widehat{\mu}_Y)^2}} & && &\stackrel{\text{biased estimator}\\}{\text{E}[\widehat{\sigma}_Y^2] = \frac{n-1}{n} \sigma_Y^2}
\end{align}
\]
As you can see above sample mean (\(\widehat{\mu}_Y\)) is an unbiased estimator. However, the standard sample variance (\(\widehat{\sigma}_Y^2\)) is biased; it estimates \(\sigma_Y^2\), but its expected value is not equal to \(\sigma_Y^2\). The formula requires an adjustment to be unbiased.

Degrees of Freedom – Functional Definition

Degrees of freedom can be a confusing concept. The most straight forward way to approach it is through its functional definition: Degrees of freedom is the expected value of the sum of squared errors from a variance estimator expressed in units of the variance being estimated.

To understand this look at the example for the standard sample variance formula.
\[\large{
\begin{align}
&\stackrel{\text{unadjusted formula}\\}{\widehat{\sigma}_Y^2 = \frac{1}{n}\sum_{i=1}^n (y_i-\widehat{\mu}_Y)^2} & && & \stackrel{\text{unadjusted formula}\\}{\widehat{\sigma}_Y^2 = \frac{1}{n}\sum_{i=1}^n SSE} \\ \\
&\text{E}[SSE_{\widehat{\sigma}_Y^2}] = df\;\sigma_Y^2 & && & df = (n-1) \\ \\
\end{align}
}\]
The degrees of freedom for the standard variance estimator is \((n-1)\). The DF for any variance estimator are used to adjust the estimator formula so that it is unbiased (how to find the degrees of freedom for a variance estimator is shown in the fromula section).
\[\require{cancel}\large{
\begin{align}
&\stackrel{\text{unadjusted formula}\\}{\widehat{\sigma}_Y^2 = \frac{1}{n}\sum_{i=1}^n (y_i-\widehat{\mu}_Y)^2} \;\longrightarrow\; \stackrel{\text{bias adjusted formula}\\}{\widehat{\sigma}_Y^2 = \frac{\cancel{n}}{df}\frac{1}{\cancel{n}}\sum_{i=1}^n (y_i-\widehat{\mu}_Y)^2} \\ \\
&\longrightarrow\;\stackrel{\text{unbiased estimator}\\}{\text{E}[\widehat{\sigma}_Y^2] = \text{E}\Big[\frac{1}{df}\sum_{i=1}^n(y_i-\widehat{\mu}_Y)^2\Big] = \frac{1}{\cancel{df}}\cancel{df}\sigma_Y^2}
\end{align}
}\]

Degrees of Freedom – Intuitive Explanation

The standard sample variance estimator formula uses a 1-dimensional mean estimator. Picture the observed values and the mean as points on a line. With only one actual point the mean estimator will have that same value and variance will be zero. Variance is only possible with two or more points.

A 2-dimensional mean estimator can be conceptualized as line drawn on a plane. With only two points the line will be touching both, which again would produce zero variance. In 2 dimensions variance from an estimator can only occur with 3 or more points. A 3-dimensional mean produces a plane. With only 3 points the plane touches all three points, which again makes zero variance. In 3 dimensions 4 or greater points are required for variance from an estimator.

Only the first value after the mean has been established is free to produce variance. And the expected squared deviation from mean is the same for every value. If 4 values exist, but only the 4th contributed any squared distance a sum (which is distributed among all 4 in the actual calculation) then the expected value of variance is 1/4th what it would be with 4 values and a non-estimated mean.


Formulas and Derivation


Regression T Test Formulas

\[\large
\begin{align}
\widehat{y} = \stackrel{n\times q\;q\times 1}{X\;\;\widehat{b}} = \widehat{\mu}_{Y|X},\;\;n>q && &\normalsize \text{(model estimate)} \\ \\
\text{Var}[\widehat{b}] = (X^\mathrm{T}X)^{-1} \sigma^2_Y && &\normalsize \text{(covariance matrix for coefficient estimate)} \\ \\
\text{Var}[\widehat{b}] = (R_X^\mathrm{T}R_X)^{-1} \sigma^2_Y && &\normalsize \text{(more efficient version of cov matrix)} \\ \\
df_\text{model}= (n\;-\;q)&& &\normalsize \text{(degrees of freedom for model)} \\ \\
\widehat{\sigma}^2_\text{model} = \frac{1}{df_\text{model}}\sum_{i=1}^n(y_i-\widehat{y}_i)^2 && &\normalsize \text{(model variance estimator)} \\ \\
t_{df_\text{model}} = \Bigg[\frac{\widehat{b}\;-\;0}{\sqrt{\text{diag}((X^\text{T}X)^{-1}\widehat{\sigma}_Y^2)}}\Bigg] && &\normalsize \text{(t statistic vector)}
\end{align}
\]


R Code For Regression T Test

The R code below manually implements the formulas from above, also uses the standard R functionality to achieve the same results, and then compares the two.

If you are new to R I suggest R-Studio as and IDE.

######################################
## Generate Data, Declare Variables ##
######################################

rm(list = ls())
`%+%` <- function(a, b) paste(a, b, sep="")

IsConstFactor <- T    # control if constant factor in model
IsSigFactors <- T     # control if significant factors in model
IsNonSigFactor <- T   # control if non-significant factor in model

n <- 100              # sample size
sigma.model <- 40     # error standard deviation

# independent factors aka design matrix
X <- cbind(                         
  if(IsConstFactor == T){rep(1,n)}else{NULL}
  ,if(IsSigFactors == T){runif(n,-100,100)}
  ,if(IsSigFactors == T){rpois(n,10)}
  ,if(IsNonSigFactor == T){rexp(n,0.1)}else{NULL}
)

# coefficient vector
b <- rbind(
  if(IsConstFactor == T){40}else{NULL}  
  ,if(IsSigFactors == T){2.5}
  ,if(IsSigFactors == T){4}
  ,if(IsNonSigFactor == T){0}else{NULL}
)   

# error, linear regression model, baseline estimate
e <- cbind(rnorm(n,0,sigma.model))
y <- X %*% b + e
baseline <-                       
  if(IsConstFactor == T) {
    mean(y)
  } else {0}

# QR factorization of X for more
# efficient processing
qr <- qr(X)
Q <- qr.Q(qr)
R <- qr.R(qr)
rm(qr)

# labels
colnames(X) <- c("X" %+% seq(as.numeric(!IsConstFactor),
  ncol(X) - as.numeric(IsConstFactor))) 
rownames(b) <- c("b" %+% seq(as.numeric(!IsConstFactor),
  nrow(b) - as.numeric(IsConstFactor)))


###############################
## Linear Regression Using R ##
###############################

model.formula <- if(IsConstFactor == T) {
  "y ~ 1" %+% paste(" + " %+% colnames(X)[2:ncol(X)], collapse='')
} else {"y ~ 0 " %+% paste(" + " %+% colnames(X), collapse='')}
linear.model <- lm(model.formula,as.data.frame(X))


#######################################
## Perform Liner Regression Manually ##
#######################################

b_ <- solve(R) %*% t(Q) %*% y     # estimated coefficients   
#b_ <- solve(t(X) %*% X) %*% t(X) %*% y
rownames(b_) <- rownames(b)
y_ <- X %*% b_                    # estimated model                   

# degrees of freedom
df.model <- n - nrow(b_)

# residuals
res <- cbind(
  c(y - y_)          # model/"unexplained" error
); colnames(res) <- c("model")

# variance
var_.model <- sum(res[,"model"]^2) / df.model

# covariance matrix for coefficient estimators 
covar_.b_ <- chol2inv(R) * var_.model
#covar_.b_ <- solve(t(X) %*% X) * var_.model
colnames(covar_.b_) <- rownames(b_)
rownames(covar_.b_) <- rownames(b_)

# T-tests
t.stat <- b_ / sqrt(diag(covar_.b_))
pt <- 2 * pt(-abs(t.stat),df.model)
ret.coef <- cbind(b_,sqrt(diag(covar_.b_)),t.stat,df.model,pt)
colnames(ret.coef) <- c("Coef.","Std. Error","T-stat","df","p-value")
rownames(ret.coef) <- rownames(b_)


#############
## Compare ##
#############

summary(linear.model)
ret.coef


Step 1: Find the Standard Error of the Coefficients Estimator

In order to calculate the T statistic of the coefficients their standard errors must be calculated. Again we have a situation where the scalar form of the solution is difficult to intuit and explodes in complexity as the number of variables in the model grows.

Below is the scalar formula for the standard deviation of the first coefficient estimate in a 3 factor model (it’s so large I had to display using tinyscript).
\[\scriptsize
\text{Std.Error}[\widehat{b}_1] = \frac{\sum x_3^2 \sigma_Y^2}{\sum x_1^2(\sum x_2^2\sum x_3^2 – (\sum x_2 x_3)^2) -\sum x_1 x_2(\sum x_1 x_2 \sum x_3^2 -\sum x_1 x_3 \sum x_2 x_3) +\sum x_1 x_3(\sum x_1 x_2 \sum x_2 x_3 -\sum x_2^2 \sum x_1 x_3)}
\]
And again the path to avoid this is to use matrices and vectors to create expressions that are more easily read and manipulated.

“Standard Error” of Coefficients in Matrix Form

To find the standard deviation of an estimator using matrices, we have to expand some basic random variable properties from scalar to matrix form.

In scalar algebra variance and covariance are presented as separate concepts / formulas.
\[\large
\begin{align}
X\sim\text{R.V.}\;\;&\;\;Y\sim\text{R.V.} && \normalsize (X\text{ and }Y\text{ are random variables}) \\ \\
\text{Var}[X] &= \text{E}[X^2]\;-\;\text{E}[X]^2 && \normalsize (\text{variance of }X) \\ \\
\text{Cov}[X,Y] &= \text{E}[XY]\;-\;\text{E}[X]\text{E}[Y] && \normalsize (\text{covariance of $X$ and }Y) \\ \\
\text{Cov}[X,X] &= \text{Var}[X] && \normalsize (\text{cov of $X$ and $X$ is var of }X)
\end{align}
\]
However, generalizing the concept of variance and covariance to vectors containing random variables produces a 2-dimensional array called a covariance matrix, which contains the variances and/or covariances for all the elements in the original vector(s).
\[\large
\begin{align}
v &= \begin{bmatrix} X \\ Y \end{bmatrix} && \normalsize (\text{$X$ and $Y$ are R.V.’s}) \\ \\
\text{Var}[v] &= \text{E}[v^2]\;-\;\text{E}[v]^2 && \normalsize (v^2 = vv^\mathrm{T})\\ \\
&= \normalsize{\begin{bmatrix}
\text{E}[X^2] & \text{E}[XY] \\
\text{E}[XY] & \text{E}[Y^2]
\end{bmatrix}\;-\;
\begin{bmatrix}
\text{E}[X]^2 & \text{E}[X]\text{E}[Y] \\
\text{E}[X]\text{E}[Y] & \text{E}[Y]^2
\end{bmatrix}} \\ \\

&= \normalsize{\begin{bmatrix}
\text{Var}[X] & \text{Cov}[X,Y] \\
\text{Cov}[X,Y] & \text{Var}[Y]
\end{bmatrix}} && \normalsize (\text{covariance matrix of $v$}) \\ \\

\end{align}
\]
By calculating the covariance matrix for the estimator vector \(\widehat{b}\) we can obtain the variance estimates for every factor in a single equation. Since every factor is assumed to be independent the covariances are assumed to be 0.
\[\large
\begin{align}
\widehat{b} = \begin{bmatrix}
\widehat{b}_1 \\
\widehat{b}_2 \\
\vdots \\
\widehat{b}_q
\end{bmatrix} &&
\text{Var}[\widehat{b}] = \begin{bmatrix}
\text{Var}[\widehat{b}_1] & \text{Cov}[\widehat{b}_1,\widehat{b}_2] & \ldots & \text{Cov}[\widehat{b}_1,\widehat{b}_q] \\
\text{Cov}[\widehat{b}_1,\widehat{b}_2] & \text{Var}[\widehat{b}_2] & \ldots & \text{Cov}[\widehat{b}_2,\widehat{b}_q] \\
\vdots & \vdots & \ddots & \vdots \\
\text{Cov}[\widehat{b}_1,\widehat{b}_q] & \text{Cov}[\widehat{b}_2,\widehat{b}_q] & \ldots & \text{Var}[\widehat{b}_q]
\end{bmatrix}
\end{align}
\]
Remember that an assumption of the linear regression model is that the design matrix \(X\) can be treated as a fixed constant as opposed to the dependent vector \(y\), which is a random vector.
\[\large
\begin{align}
\widehat{b} &= (X^\mathrm{T}X)^{-1}X^\mathrm{T}y \\ \\
\text{Var}[\widehat{b}] &= \text{Var}[(X^\mathrm{T}X)^{-1}X^\mathrm{T}y] && \normalsize (\text{Var}[cy] = c\text{Var}[y]c^\mathrm{T}) \\ \\

&= (X^\mathrm{T}X)^{-1}X^\mathrm{T}\;\stackrel{n\times n}{\text{Var}[y]}\;X(X^\mathrm{T}X)^{-1} && \normalsize (\text{every element of $\stackrel{n\times n}{\text{Var}[y]}$ is $\sigma_Y^2$}) \\ \\

&= (X^\mathrm{T}X)^{-1}X^\mathrm{T}X(X^\mathrm{T}X)^{-1}\stackrel{1\times 1}{\sigma_Y^2} && \normalsize (X^\mathrm{T}X(X^\mathrm{T}X)^{-1}=I) \\ \\ \\

&= (X^\mathrm{T}X)^{-1}\sigma_Y^2 && \normalsize (\text{covariance matrix of $\widehat{b}$})
\end{align}
\]
The diagonal of the covariance matrix contains the variances so the standard error would be the square root of the elements in that vector.
\[\large
\text{Std.Err}[\widehat{b}] = \bigg[\sqrt{\text{diag}((X^\mathrm{T}X)^{-1}\sigma_Y^2)}\bigg]
\]


QR Decomposition

Just like the coefficient estimator formula, the standard error formula can be factored into its \(QR\) components an then reduced into a more computationally efficient solution.
\[\large
\begin{align}
\stackrel{n\times q}{X} &= \stackrel{n\times q}{Q_X} \stackrel{q\times q}{R_X} && \normalsize (n>q) \\ \\
\text{Var}[\widehat{b}] &= (X^\mathrm{T}X)^{-1}\sigma_Y^2 && \normalsize (X = Q_X R_X)\\ \\
&= (R_X^\mathrm{T}Q_X^\mathrm{T}Q_X R_X)^{-1}\sigma_Y^2 && \normalsize (\text{$Q_X$ is semi-orthogonal})\\ \\ \\
&= (R_X^\mathrm{T}R_X)^{-1}\sigma_Y^2
\end{align}
\]
\(R_X\) being a square upper triangle matrix it has elements that are known to be 0 and therefore the producing \((R_X^\mathrm{T}R_X)^{-1}\) is more computationally efficient than \((X^\mathrm{T}X)^{-1}\).


Step 2: Estimate “Model” Variance


What is “Model” Variance?

Model variance is the variance of the dependent random variable \(Y\) from a linear regression model. It is equal to the variance of the error term \(\varepsilon\) because that term is the only random component of the model. Adding constants to a random variable does not change the variance: \(\text{Var}[c + Y] = \text{Var}[Y]\).

The model variance functions by using the conditional expected value of \(Y\) ie expected value given a specific row vector \(x\). The estimated model’s predicted values are the estimators of the model’s conditional expected value.
\[\large{
\begin{align}
\sigma_Y^2 = \sigma_\varepsilon^2 = \text{E}\Big[(Y – \mu_{Y|x})^2\Big] &&
\mu_{Y|x} = \text{E}[Y|x] &&
\widehat{\mu}_{Y|x} = \widehat{y}
\end{align}
}\]
Obviously accurately estimating the model variance depends on identifying all of the model factors. If factors are missing then the estimate will be greater than the actual model variance.

Degrees of Freedom for the Model Variance Estimator

Degrees of freedom is used to adjust variances estimators for bias. The expected value of the sum of squared errors from a variance estimator is equal to degrees of freedom times the variance being estimated.
\[\large{
\text{E}[SSE_\text{model}] = df_\text{model}\;\sigma_Y^2
}\]
The sum of squared errors for the model variance estimator uses the regression model estimator.
\[\large
\begin{align}
\text{E}[SSE_\text{model}] &= \text{E}[\sum_{i=1}^n (y_i\;-\;\widehat{y}_i)^2] && \small (1) \\ \\

&= \text{E}[\sum_{i=1}^n y_i^2 \;-\; 2y_i \widehat{y}_i + \widehat{y}_i^2]
\end{align}
\]
It is easier to cancel and combine terms by representing these sums in vector form. If you need a refresher see the matrix quick reference.
\[\require{color}\large
\begin{align}
\definecolor{nick_green}{RGB}{0,69,57}
= \text{E}\Big[y^\mathrm{T}y \;-\; 2\widehat{y}^\mathrm{T}y + \widehat{y}^\mathrm{T}\widehat{y}\Big] && \small (2)
\end{align}
\]
It can be shown that the middle term and the final term in the sum are equal by expanding the final term and canceling: \(\widehat{y}^\text{T}\widehat{y} = y^\text{T}X(X^\text{T}X)^{-1}\cancel{X^\text{T}X(X^\text{T}X)^{-1}}X^\text{T}y = \widehat{y}^\text{T}y\).
\[\large
\begin{align}
&= \text{E}\Big[y^\mathrm{T}y \;-\; \cancel{2}\widehat{y}^\mathrm{T}\widehat{y} + \cancel{\widehat{y}^\mathrm{T}\widehat{y}}\Big] && \small (3)\\ \\
&= \text{E}[y^\mathrm{T}y] \;-\; \text{E}[\widehat{y}^\mathrm{T}\widehat{y}]
\end{align}
\]
Two basic properties of random variables are \(\text{Var}[Y] = \text{E}[Y^2] – \text{E}[Y]^2\), which can be rearranged into \(\text{E}[Y^2] = \text{Var}[Y] + \text{E}[Y]^2\), and \(\text{E}[X + Y] = \text{E}[X] + \text{E}[Y]\) given \(X\) and \(Y\) are independent. These can be generalized to vectors. Consider that \(y^\mathrm{T}y = \sum{y^2}\) and also every element of \(y\) and \(\widehat{y}\) are assumed independent.
\[\large
\begin{align}
= \text{Var}[\sum_{i=1}^ny_i] + \text{E}[y]^\mathrm{T}\text{E}[y] \;-\; \text{Var}[\sum_{i=1}^n\widehat{y}_i] – \text{E}[\widehat{y}]^\mathrm{T}\text{E}[\widehat{y}] && \small (4)
\end{align}
\]
It might not be immediately clear, but \(\text{E}[y]^\mathrm{T}\text{E}[y] = \text{E}[\widehat{y}]^\mathrm{T}\text{E}[\widehat{y}]\). Recall that the linear regression model can be rearranged into \(\varepsilon = y – Xb\) and also that \(\varepsilon\) is assumed to be a normally distributed random variable with mean 0.

The first step in linear regression is to pick the “best” estimate for \(b\), which will yield \(\widehat{y} = X\widehat{b}\). The choice of \(\widehat{b}\) will not change the expected value of \(\varepsilon\). Therefore, \(\text{E}[\varepsilon] = \text{E}[y – \widehat{y}]\) implies \(\text{E}[y]=\text{E}[\widehat{y}]\).

This is one example of why it is so important to validate the assumptions of the linear model. I’ll discuss in a future post how to interrogate the assumptions, which can be incorrect for your factors or made incorrect through bad sampling practices.
\[\large
\begin{align}
&= \text{Var}[\sum_{i=1}^ny_i] \;-\; \text{Var}[\sum_{i=1}^n\widehat{y}_i] && \small (5)
\end{align}
\]

Because every element of \(y\) is assumed to be independent, the variance of the sum of elements is the same as the sum of the individual variances. So \(\text{Var}[y^\mathrm{T}y] = \text{Var}[\sum{y_i}] = \sum{\text{Var}[y_i]} = n\sigma_Y^2\).
\[\large
\begin{align}
&= n\sigma_Y^2 \;-\; \sum_{i=1}^n{\text{Var}[\widehat{y}_i]} && \small (6) \\ \\

&= n\sigma_Y^2 \;-\; \sum_{i=1}^n{\text{Var}[X_{[i,]}\widehat{b}]} && \small (\text{$X_{[i,]}$ is the $i$th row from $X$}) \\ \\

&= n\sigma_Y^2 \;-\; \sum_{i=1}^n{\text{Var}[X_{[i,]}(X^\mathrm{T}X)^{-1}X^\mathrm{T}y]} && \small (\widehat{b} = (X^\mathrm{T}X)^{-1}X^\mathrm{T}y) \\ \\

&= n\sigma_Y^2 \;-\; \sum_{i=1}^n{X_{[i,]}(X^\mathrm{T}X)^{-1}X^\mathrm{T}\stackrel{n\times n}{\text{Var}[y]}X(X^\mathrm{T}X)^{-1}X_{[i,]}^\mathrm{T}} && \small (\text{Var}[cy] = c\text{Var}[y]c^\mathrm{T}) \\ \\

&= n\sigma_Y^2 \;-\; \sum_{i=1}^n{X_{[i,]}(X^\mathrm{T}X)^{-1}X^\mathrm{T}X(X^\mathrm{T}X)^{-1}X_{[i,]}^\mathrm{T}}\cdot\stackrel{1\times 1}{\sigma_Y^2} && \small (\text{every element of $\text{Var}[y]$ is $\sigma_Y^2$}) \\ \\

&= n\sigma_Y^2 \;-\; \sum_{i=1}^n{X_{[i,]}(X^\mathrm{T}X)^{-1}X_{[i,]}^\mathrm{T}}\cdot\stackrel{1\times 1}{\sigma_Y^2}
\end{align}
\]
The sum \(\sum_{i=1}^n{X_{[i,]}(X^\mathrm{T}X)^{-1}X_{[i,]}^\mathrm{T}}\) can be expressed as \(\sum_{j=1}^q\text{diag}\big(X^\mathrm{T}X(X^\mathrm{T})^{-1}\big)_{[j]}\).

To clarify why this is look at the example of a single iteration of the a similar sum (i.e. the first row vector from a matrix, multiplied by a square matrix, and then multiplied by the transposition of the row vector). Let the data matrix for this example be \(Q\) and the square matrix be \(O\), with column labels for \(Q\): \(X\), \(Y\), and \(Z\).

Remember, matrix multiplication is ROW to COLUMN. So the first multiplication will be the row vector used in 3 dot products, one for each column in the square matrix.
\[\require{color}\large
\begin{align}
&\stackrel{\Large Q_{[1,]}}{\begin{bmatrix}
\color{gold}{\overrightarrow{\color{aqua}{\overrightarrow{\color{darkorange}{\overrightarrow{\color{white}{X_1}}}}}}} &
\color{gold}{\overrightarrow{\color{aqua}{\overrightarrow{\color{darkorange}{\overrightarrow{\color{white}{Y_1}}}}}}} &
\color{gold}{\overrightarrow{\color{aqua}{\overrightarrow{\color{darkorange}{\overrightarrow{\color{white}{Z_1}}}}}}}
\end{bmatrix}}
\stackrel{\Large O}{
\begin{bmatrix}
\color{gold}{\begin{array}
\\ A^2 \\ AB \\ AC
\end{array}} &
\color{aqua}{\begin{array}
\\ AB \\ B^2 \\ BC
\end{array}} &
\color{darkorange}{\begin{array}
\\ AC \\ BC \\ C^2
\end{array}}
\end{bmatrix}}
\stackrel{\Large Q_{[1,]}^\mathrm{T}}{
\begin{bmatrix}
X_1 \\ Y_1 \\ Z_1
\end{bmatrix}} = \\ \\

&\small{\begin{bmatrix}
\color{gold}{A^2 X_1+ AB Y_1+ AC Z_1} & \color{aqua}{AB X_1+ B^2 Y_1+ BC Z_1} & \color{darkorange}{ACX_{1,1} + BCY_1 + C^2Z_1}
\end{bmatrix}
\begin{bmatrix}
\color{gold}{X_1} \\ \color{aqua}{Y_1} \\ \color{darkorange}{Z_1}
\end{bmatrix}=} \\ \\

&\small{\begin{bmatrix}
\color{gold}{A^2 X_1^2+ AB X_1 Y_1+ AC X_1 Z_1} \;+\; \color{aqua}{AB X_1 Y_1+ B^2 Y_1^2+ BC Y_1 Z_1} \;+\; \color{darkorange}{AC X_1 Z_1+ BC Y_1 Z_1+ C^2 Z_1^2}
\end{bmatrix}}
\end{align}
\]
Evaluating this same expression for every row vector in \(Q\) and summing those results yields the following.
\[\large
\small{\begin{bmatrix}
\color{gold}{A^2 \sum X^2+ AB \sum X Y+ AC \sum X Z} \;+\; \color{aqua}{AB \sum XY+ B^2 \sum Y^2+ BC \sum YZ} \;+\; \color{darkorange}{AC \sum XZ+ BC \sum YZ+ C^2 \sum Z^2}
\end{bmatrix}}
\]
Compare the result above to a matrix multiplication of \(Q^\mathrm{T}QO\).
\[\require{color}\normalsize
\begin{align}
&\stackrel{\Large Q^\mathrm{T}Q}{\begin{bmatrix}
\color{gold}{\sum X^2} & \color{gold}{\sum XY} & \color{gold}{\sum XZ} \\
\color{aqua}{\sum XY} & \color{aqua}{\sum Y^2} & \color{aqua}{\sum YZ} \\
\color{darkorange}{\sum XZ} & \color{darkorange}{\sum YZ} & \color{darkorange}{\sum Z^2} \\
\end{bmatrix}}
\stackrel{\Large O}{\begin{bmatrix}
\color{gold}{\begin{array}
\\ A^2 \\ AB \\ AC
\end{array}} &
\color{aqua}{\begin{array}
\\ AB \\ B^2 \\ BC
\end{array}} &
\color{darkorange}{\begin{array}
\\ AC \\ BC \\ C^2
\end{array}}
\end{bmatrix}} = \\ \\
&\small{\begin{bmatrix}
\color{gold}{ A^2\sum X^2 + AB\sum XY + AC\sum XZ} & & \\
& \color{aqua}{ AB\sum XY + B^2\sum Y^2 + BC\sum YZ} & \\
& & \color{darkorange}{ AC\sum XZ + BC\sum YZ + C^2\sum Z^2}
\end{bmatrix}}
\end{align}
\]
The values in the diagonal of the matrix multiplication are the same as the terms added together in the original summation (i.e. \(\sum_{i=1}^n Q_{[i,]}OQ_{[i,]}^\mathrm{T} = \sum_{j=1}^q\text{diag}(Q^\mathrm{T}QO)_{[j]}\)).
\[\large
\begin{align}
&= n\sigma_Y^2 \;-\; \sum_{j=1}^q\text{diag}(X^\mathrm{T}X(X^\mathrm{T}X)^{-1})_{[j]}\cdot\sigma_Y^2 && \small (7) \\ \\
&= n\sigma_Y^2 \;-\; \sum_{j=1}^q\text{diag}(\stackrel{q\times q}{I})\cdot\sigma_Y^2 \\ \\
&= n\sigma_Y^2 \;-\; q\sigma_Y^2 \\ \\
&= (n-q)\sigma_Y^2\longrightarrow df_\text{model} = (n-q)
\end{align}
\]
With the degrees of freedom formula we can now construct an unbiased model variance estimator.
\[\large{
\widehat{\sigma}_\text{model}^2 = \frac{1}{df_\text{model}}SSE_\text{model} = \frac{1}{(n-q)}\sum_{i=1}^n(y_i-\widehat{y}_i)^2
}\]


Step 3: Construct the Test Statistic

The T test is a test of the individual coefficient estimates of the model: \(\widehat{b} = (X^\text{T}X)^{-1}X^\text{T}y\). The null hypothesis is that every coefficient is 0. This gives us an implicit assumption: \(\text{E}[\widehat{b}_i] = 0,\;i\in [1,q]\).

Since \(y\) is normally distributed and \(X\) can be treated as a matrix of constants, every element of vector \(\widehat{b}\) is normally distributed. According to the test null hypothesis the expected value of each estimate is 0. The standard errors (the standard deviation of the estimator random variables) is given by \((X^\text{T}X)^{-1}\sigma_Y^2\) and the model estimator for \(\sigma_Y^2\) is \(\widehat{\sigma}_Y^2\) making the estimated standard error \(\widehat{\text{Var}}[\widehat{b}] = (X^\text{T}X)^{-1}\widehat{\sigma}_Y^2\).

A T random variable can be constructed from a normally distributed random variable minus its own mean and divided by a bias adjusted variance estimator.
\[\large{
t_{df_\text{model}} = \Bigg[\frac{\widehat{b}\;-\;0}{\sqrt{\widehat{\text{Var}}[\widehat{b}]}}\Bigg] = \Bigg[\frac{\widehat{b}\;-\;0}{\sqrt{\text{diag}((X^\text{T}X)^{-1}\widehat{\sigma}_Y^2)}}\Bigg]
}\]
If it is not true that the mean for a coefficient is 0, then with enough data (how much is determined by the standard error) the T statistic produced for that coefficient will have an associated p-value that is sufficiently improbable given the null hypothesis and therefore disproving the null hypothesis.

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