Question

In linear regression, the rate of change of the dependent variable is determined by

Answer 1

Step-by-step explanation:

Interpreting a Coefficient as a Rate of Change in Y Instead of as a Rate of Change in the Conditional Mean of Y.

As pointed out in the discussion of overfitting, the computed regression equation estimates the true conditional mean function. How well it estimates the behavior of actual values of the random variable depends on the variability of the response variable Y. Thus, interpreting the computed coefficients in terms of the response variable is often misleading.

Illustration: In the graph shown below, the data are marked in green, the true line of conditional means is in violet, and the fitted (computed) regression line is in blue. Note that the fitted regression line is close to the true line of conditional means. The equation of the fitted regression line is (with coefficients rounded to a reasonable degree) ŷ = 0.56 + 2.18x.1 Thus it is accurate to say, "For each change of one unit in x, the average change in the mean of Y is about 2.18 units." It is not accurate to say, "For each change of one unit in x, Y changes about 2.18 units." For example, we can see from the graph that when x is 2, Y might be anywhere between a little below 4 to a little above 5.5; when x is 3, Y might be anywhere from a little more than 5.5 to a little more than 9. So when going from x = 2 to x = 3, the change in Y might be almost zero, or it might be as large as 5.5 units.

Graph showing data with some scatter, true mean line, fitted mean line

Notes:

1. The true line of means in this constructed example is E(Y|X = x) = 1 +

Answer 2

To Find:

The factor that determines the rate of change of the dependent variable in linear regression.

Solution:

• Linear regression is a way to find out the best possible linear relationship between the dependent variable (target variable) and one or more explanatory variables (independent variables) using a straight line.
• A linear regression line is of the form $y=bx+c$, where $x$ is the independent variable and $y$ is the dependent variable.

• The line $y=bx+c$ is called the regression line of $y$ on $x$. The slope of the line is $b$ and $c$ is the intercept i.e., the value of $y$ when $x=0$.
• The rate of change of the dependent variable $y$ with respect to $x$ is determined by the slope of the regression line i.e., $b$.
• Thus, the value of $b$ determines how much $y$ changes for each one-unit change in $x$.

Therefore, in linear regression, the rate of change of the dependent variable is determined by the slope of the regression line.

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