partial regression coefficient

Examples of partial regression coefficient in the following topics:

• Estimating and Making Inferences About the Slope

  • When the purpose of multiple regression is prediction, the important result is an equation containing partial regression coefficients (slopes).
  • The magnitude of the partial regression coefficient depends on the unit used for each variable.
  • Where $b'_1$ is the standard partial regression coefficient of $y$ on $X_1$.
  • The magnitude of the standard partial regression coefficients tells you something about the relative importance of different variables; $X$ variables with bigger standard partial regression coefficients have a stronger relationship with the $Y$ variable.
  • Discuss how partial regression coefficients (slopes) allow us to predict the value of $Y$ given measured $X$ values.

• Introduction to Multiple Regression

  • This slope is the regression coefficient for HSGPA.
  • Thus the regression coefficient of 0.541 for HSGPA and the regression coefficient of 0.008 for SAT are partial slopes.
  • As is typically the case, the partial slopes are smaller than the slopes in simple regression.
  • If the variance explained uniquely by a variable is not zero, then the regression coefficient cannot be zero.
  • Clearly, a variable with a regression coefficient of zero would explain no variance.

• The Coefficient of Determination

  • r2 is called the coefficient of determination. r2 is the square of the correlation coefficient , but is usually stated as a percent, rather than in decimal form. r2 has an interpretation in the context of the data:
  • r2 , when expressed as a percent, represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression (best fit) line.
  • 1-r2 , when expressed as a percent, represents the percent of variation in y that is NOT explained by variation in x using the regression line.
  • This can be seen as the scattering of the observed data points about the regression line.
  • Approximately 44% of the variation (0.4397 is approximately 0.44) in the final exam grades can be ex- plained by the variation in the grades on the third exam, using the best fit regression line.

• Polynomial Regression

  • For this reason, polynomial regression is considered to be a special case of multiple linear regression.
  • The least-squares method minimizes the variance of the unbiased estimators of the coefficients, under the conditions of the Gauss–Markov theorem.
  • Although polynomial regression is technically a special case of multiple linear regression, the interpretation of a fitted polynomial regression model requires a somewhat different perspective.
  • It is often difficult to interpret the individual coefficients in a polynomial regression fit, since the underlying monomials can be highly correlated.
  • This is similar to the goal of non-parametric regression, which aims to capture non-linear regression relationships.

• Slope and Intercept

  • The mathematical function is expressed in terms of a number of parameters that are the coefficients of the equation, and the values of the independent variable.
  • The coefficients are numeric constants by which variable values in the equation are multiplied or which are added to a variable value to determine the unknown.
  • A simple example is the equation for the regression line which follows:
  • So, $m$ and $b$ are the coefficients of the equation.
  • The case of one explanatory variable is called simple linear regression.

• Student Learning Outcomes

• Qualitative Variable Models

  • Dummy, or qualitative variables, often act as independent variables in regression and affect the results of the dependent variables.
  • Dummy variables are "proxy" variables, or numeric stand-ins for qualitative facts in a regression model.
  • A dummy independent variable (also called a dummy explanatory variable), which for some observation has a value of 0 will cause that variable's coefficient to have no role in influencing the dependent variable, while when the dummy takes on a value 1 its coefficient acts to alter the intercept.
  • The intercept (the value of the dependent variable if all other explanatory variables hypothetically took on the value zero) would be the constant term for males but would be the constant term plus the coefficient of the gender dummy in the case of females.
  • Graph showing the regression results of the ANOVA model example: Average annual salaries of public school teachers in 3 regions of a country.

• Making Inferences About the Slope

  • The slope of the regression line describes how changes in the variables are related.
  • Specifically, the interpretation of $m$ is the expected change in $y$ for a one-unit change in $x$ when the other covariates are held fixed—that is, the expected value of the partial derivative of $y$ with respect to $x$.
  • In contrast, the marginal effect of $x$ on $y$ can be assessed using a correlation coefficient or simple linear regression model relating $x$ to $y$; this effect is the total derivative of $y$ with respect to $x$.
  • Care must be taken when interpreting regression results, as some of the regressors may not allow for marginal changes (such as dummy variables, or the intercept term), while others cannot be held fixed.
  • Infer how variables are related based on the slope of a regression line

• Hypothesis Tests with the Pearson Correlation

  • Pearson's correlation coefficient, $r$, tells us about the strength of the linear relationship between $x$ and $y$ points on a regression plot.
  • If the test concludes that the correlation coefficient is significantly different from 0, we say that the correlation coefficient is "significant."
  • We can use the regression line to model the linear relationship between $x$ and $y$ in the population.
  • Therefore we can NOT use the regression line to model a linear relationship between $x$ and $y$ in the population.
  • Our regression line from the sample is our best estimate of this line in the population. )

• Coefficient of Determination

  • the regression sum of squares (also called the explained sum of squares); and
  • The coefficient of determination is actually the square of the correlation coefficient.
  • In regression, the $r^2$ coefficient of determination is a statistical measure of how well the regression line approximates the real data points.
  • An $r^2$ of 1 indicates that the regression line perfectly fits the data.
  • The correlation coefficient is $r=0.6631$.