standard partial regression coefficient

Examples of standard 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.
  • When the purpose of multiple regression is understanding functional relationships, the important result is an equation containing standard partial regression coefficients, like this:
  • 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.

• 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.
  • A regression weight for standardized variables is called a "beta weight" and is designated by the Greek letter β.
  • As is typically the case, the partial slopes are smaller than the slopes in simple regression.
  • Clearly, a variable with a regression coefficient of zero would explain no variance.

• Network regression

  • The standard tool for this question is linear regression, and the approach may be extended to using more than one independent variable.
  • To estimate standard errors for R-squared and for the regression coefficients, we can use quadratic assignment.
  • We will run many trials with the rows and columns in the dependent matrix randomly shuffled, and recover the R-square and regression coefficients from these runs.
  • Figure 18.9 shows the results of the "full partialling" method.
  • QAP regression of information ties on money ties and governmental status by full partialling method

• 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.
  • 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. )
  • The standard deviations of the population $y$ values about the line are equal for each value of $x$.

• Model Assumptions

  • Standard linear regression models with standard estimation techniques make a number of assumptions.
  • Standard linear regression models with standard estimation techniques make a number of assumptions about the predictor variables, the response variables, and their relationship.
  • The following are the major assumptions made by standard linear regression models with standard estimation techniques (e.g. ordinary least squares):
  • This means that the mean of the response variable is a linear combination of the parameters (regression coefficients) and the predictor variables.
  • It can also happen if there is too little data available compared to the number of parameters to be estimated (e.g. fewer data points than regression coefficients).

• Checking the Model and Assumptions

  • There are a number of assumptions that must be made when using multiple regression models.
  • When working with multiple regression models, a number of assumptions must be made.
  • These assumptions are similar to those of standard linear regression models.
  • Error will not be evenly distributed across the regression line.
  • Independent variables should not be overly correlated with one another (they should have a regression coefficient less than 0.7).

• Assumptions in Testing the Significance of the Correlation Coefficient

  • Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied.
  • Our regression line from the sample is our best estimate of this line in the population. )
  • The standard deviations of the population y values about the line are equal for each value of x.
  • The y values for each x value are normally distributed about the line with the same standard deviation.
  • For each x value, the mean of the y values lies on the regression line.

• A Graph of Averages

  • A graph of averages and the least-square regression line are both good ways to summarize the data in a scatterplot.
  • The regression line drawn through the points describes how the dependent variable $y$ changes with the independent variable $x$.
  • A good line of regression makes the distances from the points to the line as small as possible.
  • The least-squares regression line is of the form $\hat{y} = a+bx$, with slope $b = \frac{rs_y}{s_x}$ ($r$ is the correlation coefficient, $s_y$ and $s_x$ are the standard deviations of $y$ and $x$).
  • The points on a graph of averages do not usually line up in a straight line, making it different from the least-squares regression line.

• Using the Model for Estimation and Prediction

  • Standard multiple regression involves several independent variables predicting the dependent variable.
  • Standard multiple regression is the same idea as simple linear regression, except now we have several independent variables predicting the dependent variable.
  • We would use standard multiple regression in which gender and weight would be the independent variables and height would be the dependent variable.
  • In addition to telling us the predictive value of the overall model, standard multiple regression tells us how well each independent variable predicts the dependent variable, controlling for each of the other independent variables.
  • (A negative relationship is present in the case in which the greater a person's weight, the shorter the height. ) We can determine the direction of the relationship between weight and height by looking at the regression coefficient associated with weight.

• Regressing position on attributes

  • Tools>Testing Hypotheses>Node-level>Regression will compute basic linear multiple regression statistics by OLS, and estimate standard errors and significance using the random permutations method for constructing sampling distributions of R-squared and slope coefficients.
  • Figure 18.15 shows the result of the multiple regression estimation.
  • As before, the coefficients are generated by standard OLS linear modeling techniques, and are based on comparing scores on independent and dependent attributes of individual actors.
  • What differs here is the recognition that the actors are not independent, so that estimation of standard errors by simulation, rather than by standard formula, is necessary.
  • Multiple regression of eigenvector centrality with permutation based significance tests