Regression

Correlation and Regression Analysis, simple and multiple regression, regression, regression diagnostics, Assumptions of the regression model, model selection criteria, Linear Regression models, Relationship between Variables, Strength, and direction of the relationship

Simple Linear Regression Model (SLRM)

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A simple linear regression model (SLRM) is based on a single independent (explanatory) variable and it fits a straight line such that the sum of squared residuals of the regression model (or vertical distances between the fitted line and points of the data set) as small as possible. The simple linear regression model (usually known as a statistical or probabilistic model) is

\begin{align*}
y_i &= \alpha + \beta x_i +\varepsilon_i\\
\text{OR} \quad y_i&=b_0 + b_1 x_i + \varepsilon_i\\
\text{OR} \quad y_i&=\beta_0 + \beta x_i + \varepsilon_i
\end{align*}
where $y$ is the dependent variable, $x$ is the independent variable. In the regression context, $y$ is the regressand, and $x$ is the regressor. The epsilon ($\varepsilon$) is unobservable, denoting random error or the disturbance term of a regression model. $\varepsilon$ (random error) has some specific importance for its inclusion in the regression model:

Importance of Error Term in Simple Linear Regression Model

  1. Random error ($\varepsilon$) captures the effect on the dependent variable of all variables which are not included in the model under study, because the variable not included in the model may or may not be observable.
  2. Random error ($\varepsilon$) captures any specification error related to the assumed linear-functional form.
  3. Random error ($\varepsilon$) captures the effect of unpredictable random components present in the dependent variable.

We can say that $\varepsilon$ is the variation in variable$y$ not explained (unexplained) by the independent variable $x$ included in the model.

In the above equation or model $\hat{\beta_0}, \hat{\beta_1}$ are the parameters of the model and our main objective is to obtain the estimates of their numerical values i.e. $\hat{\beta_0}$ and $\hat{\beta_1}$, where $\beta_0$ is the intercept (regression constant), it passes through the ($\overline{x}, \overline{y}$) i.e. center of mass of the data points and $\beta_1$ is the slope or regression coefficient of the model and slope is the correlation between variable $x$ and $y$ corrected by the ratio of standard deviations of these variables.

The subscript $i$ denotes the ith value of the variable in the model.
\[y=\beta_0 + \beta_1 x_1\]
This is a mathematical model as all the variation in $y$ is due solely to change in $x$. There are no other factors affecting the dependent variable. If this is true then all the pairs $(x, y)$ will fall on a straight line if plotted on a two-dimensional plane. However, the plot may or may not be a straight line for observed values. A dimensional diagram with points plotted in pair form is called a scatter diagram.

Simple Linear Regression Model scatter with regression line

See Assumptions about Simple Linear Regression Model

FAQs about Simple Linear Regression Models

  1. What is a simple linear regression Model?
  2. What is a Probabilistic/ Statistical model?
  3. What is the equation of a simple linear regression model?
  4. Write about the importance of error terms in the regression model.
  5. What are the parameters in a simple linear regression model?
  6. What is the objective of estimating the parameters of a simple linear regression model?
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Role of Hat Matrix in Regression Analysis

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The post is about the importance and role of the Hat Matrix in Regression Analysis.

Hat matrix is a $n\times n$ symmetric and idempotent matrix with many special properties that play an important role in the diagnostics of regression analysis by transforming the vector of observed responses $Y$ into the vector of fitted responses $\hat{Y}$.

The model $Y=X\beta+\varepsilon$ with solution $b=(X’X)^{-1}X’Y$ provided that $(X’X)^{-1}$ is non-singular. The fitted values are ${\hat{Y}=Xb=X(X’X)^{-1} X’Y=HY}$.

Like fitted values ($\hat{Y}$), the residual can be expressed as linear combinations of the response variable $Y_i$.

\begin{align*}
e&=Y-\hat{Y}\\
&=Y-HY\\&=(I-H)Y
\end{align*}

The role of hat matrix in Regression Analysis and Regression Diagnostics is:

  • The hat matrix only involves the observation in the predictor variable X  as $H=X(X’X)^{-1}X’$. It plays an important role in diagnostics for regression analysis.
  • The hat matrix plays an important role in determining the magnitude of a studentized deleted residual and identifying outlying Y observations.
  • The hat matrix is also helpful in directly identifying outlying $X$ observations.
  • In particular, the diagonal elements of the hat matrix are indicators in a multi-variable setting of whether or not a case is outlying concerning $X$ values.
  • The elements of the “Hat matrix” have their values between 0 and 1 always and their sum is p i.e. $0 \le h_{ii}\le 1$  and  $\sum _{i=1}^{n}h_{ii} =p $
    where p is the number of regression parameters with intercept term.
  • $h_{ii}$ is a measure of the distance between the $X$ values for the ith case and the means of the $X$ values for all $n$ cases.

Mathematical Properties of Hat Matrix

  • $HX=X$
  • $(I-H)X=0$
  • $HH=H^2 = H H^p$
  • $H(I-H)=0$
  • $Cov(\hat{e},\hat{Y})=Cov\left\{HY,(I-H)Y\right\}=\sigma ^{2} H(I-H)=0$
  • $1-H$ is also symmetric and idempotent.
  • $H1=1$ with intercept term. i.e. every row of $H$ adds up to $1. 1’=1H’=1’H$  & $1’H1=n$
  • The elements of $H$ are denoted by $h_{ii}$ i.e.
    \[H=\begin{pmatrix}{h_{11} } & {h_{12} } & {\cdots } & {h_{1n} } \\ {h_{21} } & {h_{22} } & {\cdots } & {h_{2n} } \\ {\vdots } & {\vdots } & {\ddots } & {\vdots } \\ {h_{n1} } & {h_{n2} } & {\vdots } & {h_{nn} }\end{pmatrix}\]
    The large value of $h_{ii}$ indicates that the ith case is distant from the center for all $n$ cases. In this context, the diagonal element $h_{ii}$ is called leverage of the ith case. $h_{ii}$ is a function of only the $X$ values, so $h_{ii}$ measures the role of the $X$ values in determining how important $Y_i$ is affecting the fitted $\hat{Y}_{i} $ values.
    The larger the $h_{ii}$ the smaller the variance of the residuals $e_i$ for $h_{ii}=1$, $\sigma^2(ei)=0$.
  • Variance, Covariance of $e$
    \begin{align*}
    e-E(e)&=(I-H)Y(Y-X\beta )=(I-H)\varepsilon \\
    E(\varepsilon \varepsilon ‘)&=V(\varepsilon )=I\sigma ^{2} \,\,\text{and} \,\, E(\varepsilon )=0\\
    (I-H)’&=(I-H’)=(I-H)\\
    V(e) & =  E\left[e-E(e_{i} )\right]\left[e-E(e_{i} )\right]^{{‘} } \\
    & = (I-H)E(\varepsilon \varepsilon ‘)(I-H)’ \\
    & = (I-H)I\sigma ^{2} (I-H)’ \\
    & =(I-H)(I-H)I\sigma ^{2} =(I-H)\sigma ^{2}
    \end{align*}
    $V(e_i)$ is given by the ith diagonal element $1-h_{ii}$ and $Cov(e_i, e_j)$ is given by the $(i, j)$th  element of $-h_{ij}$ of the matrix $(I-H)\sigma^2$.
    \begin{align*}
    \rho _{ij} &=\frac{Cov(e_{i} ,e_{j} )}{\sqrt{V(e_{i} )V(e_{j} )} } \\
    &=\frac{-h_{ij} }{\sqrt{(1-h_{ii} )(1-h_{jj} )} }\\
    SS(b) & = SS({\rm all\; parameter)=}b’X’Y \\
    & = \hat{Y}’Y=Y’H’Y=Y’HY=Y’H^{2} Y=\hat{Y}’\hat{Y}
    \end{align*}
    The average $V(\hat{Y}_{i} )$ to all data points is
    \begin{align*}
    \sum _{i=1}^{n}\frac{V(\hat{Y}_{i} )}{n} &=\frac{trace(H\sigma ^{2} )}{n}=\frac{p\sigma ^{2} }{n} \\
    \hat{Y}_{i} &=h_{ii} Y_{i} +\sum _{j\ne 1}h_{ij} Y_{j}
    \end{align*}

Role of Hat Matrix in Regression Diagnostic

Internally Studentized Residuals

$V(e_i)=(1-h_{ii})\sigma^2$ where $\sigma^2$ is estimated by $s^2$

i.e. $s^{2} =\frac{e’e}{n-p} =\frac{\Sigma e_{i}^{2} }{n-p} $  (RMS)

we can studentized the residual as $s_{i} =\frac{e_{i} }{s\sqrt{(1-h_{ii} )} } $

These studentized residuals are said to be internally studentized because $s$ has within it $e_i$ itself.

Extra Sum of Squares attributable to $e_i$

\begin{align*}
e&=(1-H)Y\\
e_{i} &=-h_{i1} Y_{1} -h_{i2} Y_{2} -\cdots +(1-h_{ii} )Y_{i} -h_{in} Y_{n} =c’Y\\
c’&=(-h_{i1} ,-h_{i2} ,\cdots ,(1-h_{ii} )\cdots -h_{in} )\\
c’c&=\sum _{i=1}^{n}h_{i1}^{2}  +(1-2h_{ii} )=(1-h_{ii} )\\
SS(e_{i})&=\frac{e_{i}^{2} }{(1-h_{ii} )}\\
S_{(i)}^{2}&=\frac{(n-p)s^{2} -\frac{e_{i}^{2}}{e_{i}^{2}  (1-h_{ii} )}}{n-p-1}
\end{align*}
provides an estimate of $\sigma^2$ after deletion of the contribution of $e_i$.

Externally Studentized Residuals

$t_{i} =\frac{e_{i} }{s(i)\sqrt{(1-h_{ii} )} }$ are externally studentized residuals. Here if $e_i$ is large, it is emphasized even more by the fact that $s_i$ has excluded it. The $t_i$ follows a $t_{n-p-1}$ distribution under the usual normality of error assumptions.

Hat Matrix in Regression itfeature.com

Read more about the Role of the Hat Matrix in Regression Analysis https://en.wikipedia.org/wiki/Hat_matrix

Read about Regression Diagnostics

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Outliers and Influential Observations

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Here we will focus on the difference between the outliers and influential observations.

Outliers

The cases (observations or data points) that do not follow the model as the rest of the data are called outliers. In Regression, the cases with large residuals are a candidate for outliers. So an outlier is a data point that diverges from an overall pattern in a sample. Therefore, an outlier can certainly influence the relationship between the variables and may also exert an influence on the slope of the regression line.

An outlier can be created by a shift in the location (mean) or in the scale (variability) of the process. An outlier may be due to recording errors (may be correctable), or due to the sample not being entirely from the same population. This may also be due to the values from the same population but from the non-normal (heavy-tailed) population. That is, outliers may be due to incorrect specifications that are based on the wrong distributional assumptions.

Outliers and Influential Observations

Inferential Observations

An influential observation is often an outlier in the x-direction. Influential observation may arise from

  1. observations that are unusually large or otherwise deviate in unusually extreme forms from the center of a reference distribution,
  2. the observation may be associated with a unit that has a low probability and thus has a high probability weight.
  3. the observation may have a very large weight (relative to the weights of other units in the specified sub-population) due to problems with stratum jumping; sampling of birth units or highly seasonal units; large nonresponse adjustment factors arising from unusually low response rates within a given adjustment cell; unusual calibration-weighting effects; or other factors.

Importance of Outliers and Influential Observations

Outliers and Influential observations are important because:

  • Both outliers and influential observations can potentially mislead the interpretation of the regression model.
  • Outliers might indicate errors in the data or a non-linear relationship that the model cannot capture.
  • Influential observations can make the model seem more accurate than it is, masking underlying issues.

How to Identify Outliers and Influential Observations

Both outliers and influential observations can be identified by using:

  • Visual inspection: Scatterplots can reveal outliers as distant points.
  • Residual plots: Plotting residuals against predicted values or independent variables can show patterns indicative of influential observations.
  • Statistical diagnostics: Measures like Cook’s distance or leverage can quantify the influence of each data point.

By being aware of outliers and influential observations, one can ensure that the regression analysis provides a more reliable picture of the relationship between variables.

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