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

Coefficient of Determination Formula: Quick Guide 2019

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In this post, we will discuss not only the coefficient of determination formula but also the use and computation of the coefficient of determination. Coefficient of Determination as a Link between Regression and Correlation Analysis.

Coefficient of Determination $R^2$ in Statistics

The R squared ($r^2$; the square of the correlation coefficient) shows the percentage of the total variation of the dependent variable ($Y$) that can be explained by the independent (explanatory) variable ($X$). For this reason, $r^2$ (r-squared) is sometimes called the coefficient of determination.

The coefficient of Determination (R-squared is commonly used in various fields like Social Science, Finance, and Economics to evaluate the performance of the regression models. It helps the researchers to understand how well their models capture the relationship between the variables being studied.

Since

\[r=\frac{\sum x_i y_y}{\sqrt{\sum x_i^2} \sqrt{\sum y_i^2}},\]

Coefficient of Determination Formula

\begin{align*}
r^2&=\frac{(\sum x_iy_i)^2}{(\sum x_i^2)(\sum y_i^2)}=\frac{\sum \hat{y}^2}{\sum y^2}\\
&=\frac{\text{Explained Variation}}{\text{Total Variation}}
\end{align*}

where $r$ shows the degree of covariability of $X$ and $Y$. Note that the formula used here is in deviation form, that is, $x=X-\mu$ and $y=Y-\mu$.

The link of $r^2$ between regression and correlation analysis can be considered from these points.

Coefficient of Determination Formula
  • If all the observations lie on the regression line then there will be no scattered points. In other words, the total variation of variable $Y$ is explained completely by the estimated regression line, which shows that there would be no scatterness in the data points(or no unexplained variation). That is
    \[\frac{\sum e^2}{\sum y^2}=\frac{\text{Unexplained Variation}}{\text{Total Variation}}=0\]
    Hence, $r^2=r=1$.
  • If the regression line explains only part of the variation in variable $Y$ then there will be some explained variation, that is,
    \[\frac{\sum e^2}{\sum y^2}=\frac{\text{Unexplained Variation}}{\text{Total Variation}}>0\]
    then, $r^2$ will be smaller than 1.
  • If the regression line does not explain any part of the variation of variable $Y$, that is,
    \[\frac{\sum e^2}{\sum y^2}=\frac{\text{Unexplained Variation}}{\text{Total Variation}}=1\Rightarrow=\sum y^2 = \sum e^2\]
    then, $r^2=0$.

Because $r^2=1-\frac{\text{unexlained variation}}{\text{total variation}}$

Key Points about Coefficient of Determination

  • Overfitting: A model can achieve a high $R^2$ value by simply memorizing the training data, but the model might not perform well on unseen data.
  • Number of Predictors: Adding more independent variables to a model will tend to increase the $R^2$ value, but it does not necessarily mean the additional variables are statistically significant.
  • Alternative Metrics: To assess the nuance of the model fit, use other metrics like adjusted R-squared or residual analysis.

Keeping in mind the limitations of R-squared, the data analysts can use the coefficient of determination as a valuable tool to assess how well their models capture real-world relationships between variables.

Note that there are two main ways to calculate R-squared value:

  1. Squared Correlation Coefficient: R-squared is the square of the correlation coefficient ($r$) between the predicted values ($\hat{y}$) from the model and the actual values of the dependent variable ($y$).
  2. Analysis of Variance (ANOVA): R-squared can also be calculated using the ratio of the explained variance to the total variance (variance in the dependent variable).
Coefficient of determination formula

FAQs about Coefficient of Determination

  1. For a simple linear regression model, what is the link between the coefficient of correlation and the coefficient of determination?
  2. How Coefficient of Determination is interpreted?
  3. How Coefficient of determination can be obtained from the ANOVA table?
  4. How overfitting can be identified from the value of $R^2$?
  5. What are alternatives to $R^2$?
  6. What is the link between total variation, explained variation, and unexplained variation?
  7. What is the impact of adding extra/ more explanatory variables in the linear regression model?
  8. What is the link between explained and unexplained variation?
  9. Give real-life examples of coefficient of determination in which it is high enough.

Learn more about the Coefficient of Determination Formula and Definition in Statistics

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Checking Normality of Error Term (2019)

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Normality of Error Term

In multiple linear regression models, the sum of squared residuals (SSR) is divided by $n-p$ (degrees of freedom, where $n$ is the total number of observations, and $p$ is the number of the parameter in the model) is a good estimate of the error variance. In the multiple linear regression model, the residual vector is

\begin{align*}
e &=(I-H)y\\
&=(I-H)(X\beta+e)\\
&=(I-H)\varepsilon
\end{align*}

where $H$ is the hat matrix for the regression model.

Each component $e_i=\varepsilon – \sum\limits_{i=1}^n h_{ij} \varepsilon_i$. Therefore, In multiple linear regression models, the normality of the residual is not simply the normality of the error term.

Note that:

\[Cov(\mathbf{e})=(I-H)\sigma^2 (I-H)’ = (I-H)\sigma^2\]

We can write $Var(e_i)=(1-h_{ii})\sigma^2$.

If the sample size ($n$) is much larger than the number of the parameters ($p$) in the model (i.e. $n > > p$), in other words, if sample size ($n$) is large enough, $h_{ii}$ will be small as compared to 1, and $Var(e_i) \approx \sigma^2$.

In multiple regression models, a residual behaves like an error if the sample size is large. However, this is not true for a small sample size.

It is unreliable to check the normality of error term assumption using residuals from multiple linear regression models when the sample size is small.

Normality of the Error Term

Learn more about Hat matrix: Role of Hat matrix in Diagnostics of Regression Analysis.

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Binary Logistic Regression Minitab Tutorial

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Binary Logistic Regression is used to perform logistic regression on a binary response (dependent) variable (a variable only that has two possible values, such as the presence or absence of a particular disease, this kind of variable is known as a dichotomous variable i.e. binary in nature).

Binary Logistic Regression

Binary Logistic Regression can classify observations into one of two categories. These classifications can give fewer classification errors than discriminant analysis for some cases.

The default model contains the variables that you enter in Continuous Predictors and Categorical Predictors. You can also add interaction and/or polynomial terms by using the tools available in the model sub-dialog box.

Minitab stores the last model that you fit for each response variable. These stored models can be used to quickly generate predictions, contour plots, surface plots, overlaid contour plots, factorial plots, and optimized responses.

Minitab Tutorial for Binary Logistic Regression

To perform a Binary Logistic Regression Analysis in Minitab, follow the steps given below. It is assumed that you have already launched the Minitab software.

Step 1:  Choose Stat > Regression > Binary Logistic Regression > Fit Binary Logistic Model.

Binary Logistic Regression Minitab Tutorial

Step 2:  Do one of the following:

If your data is in raw or frequency form, follow these steps:

Binary Logistic Regression 2
  1. Choose Response in binary response/frequency format, from the combo box on top
  2. In the Response text box, enter the column that contains the response variable.
  3. In the Frequency text box, enter the optional column that contains the count or frequency variable.

If you have summarized data, then follow these steps:

Binary Logistic Regression 3
  1. Choose Response in event/trial format, from the combo box on top of the dialog box.
  2. In the Number of events, enter the column that contains the number of times the event occurred in your sample at each combination of the predictor values.
  3. In the Number of trials, enter the column that contains the corresponding number of trials.

Step 4:  In Continuous predictors, enter the columns that contain continuous predictors. In Categorical predictors, enter the columns that contain categorical predictors. You can add interactions and other higher-order terms to the model.

Step 5:  If you like, use one or more of the dialog box options, then click OK.

Minitab Binary Logistic Regression Options

The following are options available in the main dialog box of Minitab Binary Logistic Regression:

  • The response in binary response/frequency format: Choose if the response data has been entered as a column that contains 2 distinct values i.e. as a dichotomous variable.
  • Response: Enter the column that contains the response values.
  • Response event: Choose which event of interest the results of the analysis will describe.
  • Frequency (optional): If the data are in two columns i.e. one column that contains the response values and the other column that contains their frequencies then enter the column that contains the frequencies.
  • Response in event/trial format: Choose if the response data are two columns – one column that contains the number of successes or events of interest and one column that contains the number of trials.
  • Event name: Enter a name for the event in the data.
  • Number of events: Enter the column that contains the number of events.
  • Number of trials: Enter the column that contains the number of nonevents.
  • Continuous predictors: Select the continuous variables that explain changes in the response. The predictor is also called the X variable.
  • Categorical predictors: Select the categorical classifications or group assignments, such as the type of raw material, that explain changes in the response. The predictor is also called the X variable.

Step 6: To store diagnostic measures and characteristics of the estimated equation click the Storage… button.

Binary Logistic Regression Minitab Tutorial

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