What is Regression line: Definition and 15 Discussions

In statistics, linear regression is a linear approach to modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis.
Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications. This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.
Linear regression has many practical uses. Most applications fall into one of the following two broad categories:

If the goal is prediction, forecasting, or error reduction, linear regression can be used to fit a predictive model to an observed data set of values of the response and explanatory variables. After developing such a model, if additional values of the explanatory variables are collected without an accompanying response value, the fitted model can be used to make a prediction of the response.
If the goal is to explain variation in the response variable that can be attributed to variation in the explanatory variables, linear regression analysis can be applied to quantify the strength of the relationship between the response and the explanatory variables, and in particular to determine whether some explanatory variables may have no linear relationship with the response at all, or to identify which subsets of explanatory variables may contain redundant information about the response.Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the "lack of fit" in some other norm (as with least absolute deviations regression), or by minimizing a penalized version of the least squares cost function as in ridge regression (L2-norm penalty) and lasso (L1-norm penalty). Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms "least squares" and "linear model" are closely linked, they are not synonymous.

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  1. hongseok

    I Stellar evolution path and Regression line

    I analyzed the relationship between the surface temperature and luminosity of stars of similar mass using a regression model. Through this, I was able to obtain a regression line. Since stars of similar mass show similar evolutionary paths, I believe this regression line can be viewed as a rough...
  2. F

    I Regression line with zero slope and average as best prediction

    Hello, I was considering some made up data ##(X,Y)## and a its best fit regression line. The outcome variable ##Y## is the number of likes and ##X## is the number of comments on a website. We have 100 data points which spread in such a way that the best fit line has zero slope. This implies...
  3. chwala

    Find the equation of the regression line of ##x## on ##y##

    The question is as shown below. ( Text book question). The textbook solution is indicated below. Discussion; Now they seemingly used ##r=1## to arrive at ##x=0.8+0.2y##. That is, ##y=-4+5x## then, since ##r=1##, ...implying perfect correlation therefore, ##5x=4+y## ##x=0.8+0.2y## My other...
  4. S

    B Question about regression line

    I have note that states regression line x on y is used when we want to calculate x for given y but in this case y is dependent variable. I am pretty sure I can use either line if the value of product moment correlation coefficient (r) is close to 1 but for the case, let say r = 0.6, can we use...
  5. M

    MHB Least squares regression line (I'm very lost)

    Hi! Basically this is the exercise: Given the covariance of x and y is -12 and the variance of x is 6,5, using the least squares line of best fit connecting x and y yo estimate the value of x when y=15 x 2 5 9 7 9 10 7 y 25 17 11 10 8 7 13 any help would mean everything, I'm desperate :(
  6. R

    Statistical weighting of data to improve fitting

    Homework Statement I am trying to perform a weighted fit of a data set ##(x,y)## shown below. The only information I have are the two vectors ##x## and ##y## and the uncertainty present in the ##y## values (##=0.001##). Homework Equations The Attempt at a Solution Statistical weighting of...
  7. F

    How can I use the expression for a in this problem

    Homework Statement A random sample of size ##n## from a bivariate distribution is denoted by ##(x_r,y_r), r=1,2,3,...,n##. Show that if the regression line of ##y## on ##x## passes through the origin of its scatter diagram then[/B] $$\bar y \sum^n_{r=1} x_r^2=\bar x\sum^n_{r=1} x_r y_r$$ where...
  8. pluviosilla

    I Slope of LS line = Cov(X,Y)/Var(X). Intuitive explanation?

    The slope of a fitted line = Cov(X,Y)/Var(X). I've seen the derivation of this, and it is pretty straightforward, but I am still having trouble getting an intuitive grasp. The formula is extremely suggestive and it is bothering me that I can't quite see its significance. Perhaps, my mental...
  9. phosgene

    Comparing SD of data with RMSE of regression line

    Homework Statement I'm being asked to compare the standard deviation of a data set with the root mean square error of the regression line used to model the data, in order to determine the reliability of the regression line. Homework Equations Mean squared error = variance + bias squared The...
  10. H

    Errors of the slope and intercept of a regression line

    I have set of date with error bars of different length on my y values. I want to know what the error is on the slope and intercept of my line of best fit through this data. Is there a numerical way to calculate this that takes into account the fit of the regression line and the y error bars?
  11. H

    Error on regression line slope

    I'm currently trying to determine the error on the slope of a regression line and the y-intercept. My y values are: My y error is: My x values are: 27.44535013 0.03928063 136 29.78207524 0.07836946 44 27.4482858 0.0385213 143 27.27481069...
  12. B

    Urgent Response Needed: Can Partial Means Predict a Regression Line?

    urgent reply needed Can we use partial means as a predictor for the response while fitting a regression line or curve?.
  13. D

    A question about uncertainty using a regression line and experimental data

    Homework Statement I'm new to uncertainty, so I needed a little assistance. The circuit in this experiment was simple, a dry cell connected to a variable resistor. I used an ammeter on the circuit and used a voltmeter across the resistor and took some measurments. The experiment was to...
  14. T

    Formula for closest distance to regression line

    i need to calculate the closest distance of the point that lies closest to the regression line for my programing but i am not sure what is the formula. maybe someone can help me out here? thanks in advance
  15. S

    Estimating error in slope of a regression line

    OK, I have a question I have no idea how to answer (and all my awful undergrad stats books are useless on the matter). Say I make a number of pairs of measurements (x,y). I plot the data, and it looks strongly positively correlated. I do a linear regression and get an equation for a line of best...