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: Model Selection (2012)

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$R^2$ pronounced R-squared (Coefficient of determination) is a useful statistic to check the regression fit value. $R^2$ measures the proportion of total variation about the mean $\bar{Y}$ explained by the regression. R is the correlation between $Y$ and $\hat{Y}$ and is usually the multiple correlation coefficient. The coefficient of determination ($R^2$) can take values as high as 1 or  (100%) when all the values are different i.e. $0\le R^2\le 1$.

Coefficient of Determination

When repeat runs exist in the data the value of $R^2$ cannot attain 1, no matter how well the model fits, because no model can explain the variation in the data due to the pure error. A perfect fit to data for which $\hat{Y}_i=Y_i$, $R^2=1$. If $\hat{Y}_i=\bar{Y}$, that is if $\beta_1=\beta_2=\cdots=\beta_{p-1}=0$ or if a model $Y=\beta_0 +\varepsilon$ alone has been fitted, then $R^2=0$. Therefore we can say that $R^2$ is a measure of the usefulness of the terms other than $\beta_0$ in the model.

Note that we must be sure that an improvement/ increase in $R^2$ value due to adding a new term (variable) to the model under study should have some real significance and is not because the number of parameters in the model is getting else to saturation point. If there is no pure error $R^2$ can be made unity.

\begin{align*}
R^2 &= \frac{\text {SS due to regression given}\, b_0}{\text{Total SS corrected for mean} \, \bar{Y}} \\
&= \frac{SS \, (b_1 | b_0)}{S_{YY}} \\
&= \frac{\sum(\hat{Y_i}-\bar{Y})^2} {\sum(Y_i-\bar{Y})^2}r \\
&= \frac{S^2_{XY}}{(S_{XY})(S_{YY})}
\end{align*}

where summation are over $i=1,2,\cdots, n$.

Coefficient of Determination
Coefficient of Determination

Interpreting R-Square $R^2$ does not indicate whether:

  • the independent variables (explanatory variables) are a cause of the changes in the dependent variable;
  • omitted-variable bias exists;
  • the correct regression was used;
  • the most appropriate set of explanatory variables has been selected;
  • there is collinearity (or multicollinearity) present in the data;
  • the model might be improved using transformed versions of the existing explanatory variables.

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Interpreting Regression Coefficients

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Interpreting Regression Coefficients in Multiple Regression

In multiple regression models, for the interpreting regression coefficients, case, the unstandardized multiple regression coefficient is interpreted as the predicted change in $Y$ (i.e., the dependent variable abbreviated as DV) given a one-unit change in $X$ (i.e., the independent variable abbreviated as IV) while controlling for the other independent variables included in the equation.

Interpreting Regression Coefficients in Multiple Regression
  • The regression coefficient in multiple regression is called the partial regression coefficient because the effects of the other independent variables have been statistically removed or taken out (“partially out”) of the relationship.
  • If the standardized partial regression coefficient is being used, the coefficients can be compared for an indicator of the relative importance of the independent variables (i.e., the coefficient with the largest absolute value is the most important variable, the second is the second most important, and so on.)
SPSS Output: Interpreting Regression Coefficients

Interpreting regression coefficients involves understanding the relationship between the IV(s) and the DV in a regression model.

  • Magnitude: The coefficient tells about the change in the DV associated with a one-unit change in the IV, holding all other variables constant. For example, if the regression coefficient for IV (regressor) is 0.5, then it means that for every one-unit increase in that predictor, the DV is expected to increase by 0.5 units while keeping all else equal.
  • Direction: The sign of the regression coefficient (+ or -) indicates the direction of the relationship between the IV and DV. A positive coefficient means that as the IV increases, the DV is expected to increase as well. A negative coefficient means that as the IV increases, the DV is expected to decrease.
  • Statistical Significance: The statistical significance of the coefficient is important to consider. The significance of a regression coefficient tells about whether the relationship between the IV and the DV is likely to be due to chance or if it’s statistically meaningful. Generally, if the p-value of a regression coefficient is less than a chosen significance level (say 0.05), then that coefficient will be considered to be statistically significant.
  • Interaction Effects: The relationship between an IV and the DV may depend on the value of another variable. In such cases, the interpretation of regression coefficients may involve the interaction effects, where the effect of one variable on the DV varies depending on the value of another variable.
  • Context: Always interpret coefficients in the context of the specific problem being investigated. It is quite possible that a coefficient might not make practical sense without considering the nature of the data and the underlying phenomenon being studied.

Therefore, the interpretation of regression coefficients should be done carefully. The assumptions of the regression model, and the limitations of the data, should be considered. On the other hand, interpretation may differ based on the type of regression model being used (e.g., linear regression, logistic regression) and the specific research question being addressed.

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How to interpret Coefficients of Simple Linear Regression Model

Performing Linear Regression Analysis in R Language

Interpreting Regression Coefficients in Simple Regression

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How are the regression coefficients interpreted in simple regression?

The simple regression model is

Simple Regression Coefficients

The formula for Regression Coefficients in Simple Regression Models is:

$$b = \frac{n\Sigma XY – \Sigma X \Sigma Y}{n \Sigma X^2 – (\Sigma X)^2}$$

$$a = \bar{Y} – b \bar{X}$$

The basic or unstandardized regression coefficient is interpreted as the predicted change in $Y$ (i.e., the dependent variable abbreviated as DV) given a one-unit change in $X$ (i.e., the independent variable abbreviated as IV). It is in the same units as the dependent variable.

Interpreting Regression Coefficients

Interpreting regression coefficients involves understanding the relationship between the IV(s) and the DV in a regression model.

  • Magnitude: For simple linear regression models, the coefficient (slope) tells about the change in the DV associated with a one-unit change in the IV. For example, if the regression coefficient for IV (regressor) is 0.5, then it means that for every one-unit increase in that predictor, the DV is expected to increase by 0.5 units while keeping all else equal.
  • Direction: The sign of the regression coefficient (+ or -) indicates the direction of the relationship between the IV and DV. A positive coefficient means that as the IV increases, the DV is expected to increase as well. A negative coefficient means that as the IV increases, the DV is expected to decrease.
  • Statistical Significance: The statistical significance of the coefficient is important to consider. The significance of a regression coefficient tells whether the relationship between the IV and the DV is likely to be due to chance or if it’s statistically meaningful. Generally, if the p-value of a regression coefficient is less than a chosen significance level (say 0.05), then that coefficient will be considered to be statistically significant.
  • Interaction Effects: The relationship between an IV and the DV may depend on the value of another variable. In such cases, the interpretation of regression coefficients may involve the interaction effects, where the effect of one variable on the DV varies depending on the value of another variable.
  • Context: Always interpret coefficients in the context of the specific problem being investigated. It is quite possible that a coefficient might not make practical sense without considering the nature of the data and the underlying phenomenon being studied.

Therefore, the interpretation of regression coefficients should be done carefully. The assumptions of the regression model, and the limitations of the data, should be considered. On the other hand, interpretation may differ based on the type of regression model being used (e.g., linear regression, logistic regression) and the specific research question being addressed.

  • Note that there is another form of the regression coefficient that is important: the standardized regression coefficient. The standardized coefficient varies from –1.00 to +1.00 just like a simple correlation coefficient;
  • If the regression coefficient is in standardized units, then in simple regression the regression coefficient is the same thing as the correlation coefficient.
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How to interpret the Regression Coefficients in Multiple Linear Regression Models

How to Perform Linear Regression Analysis in R Language