Multiple Linear Regression | R Tutorial


where is the coefficient of multiple determination resulting from regressing the th predictor variable, , on the remaining -1 predictor variables. Mean values of considerably greater than 1 indicate multicollinearity problems.A few methods of dealing with multicollinearity include increasing the number of observations in a way designed to break up dependencies among predictor variables, combining the linearly dependent predictor variables into one variable, eliminating variables from the model that are unimportant or using coded variables.

All multiple linear regression models can be expressed in the following general form:

Multiple regression analysis is almost the same as . The only difference between simple linear regression and multiple regression is in the number of (“x” variables) used in the regression.


Weighted Linear Regression | Real Statistics Using Excel

Consider a multiple linear regression model with  predictor variables:

Linear regression and correlation assume that the data points are of each other, meaning that the value of one data point does not depend on the value of any other data point. The most common violation of this assumption in regression and correlation is in time series data, where some Y variable has been measured at different times. For example, biologists have counted the number of moose on Isle Royale, a large island in Lake Superior, every year. Moose live a long time, so the number of moose in one year is not independent of the number of moose in the previous year, it is highly dependent on it; if the number of moose in one year is high, the number in the next year will probably be pretty high, and if the number of moose is low one year, the number will probably be low the next year as well. This kind of non-independence, or "autocorrelation," can give you a "significant" regression or correlation much more often than 5% of the time, even when the null hypothesis of no relationship between time and Y is true. If both X and Y are time series—for example, you analyze the number of wolves and the number of moose on Isle Royale—you can also get a "significant" relationship between them much too often.


Regression Analysis: Linear Regression, Scatter Plots


The fitted regression model can be used to obtain fitted values, , corresponding to an observed response value, . For example, the fitted value corresponding to the fifth observation is:

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“The answer to the sample size question appears to depend in part on the objectives
of the researcher, the research questions that are being addressed, and the type of
model being utilized. Although there are several research articles and textbooks giving
recommendations for minimum sample sizes for multiple regression, few agree
on how large is large enough and not many address the prediction side of MLR.” ~ Gregory T. Knofczynski

Linear Regression for Business Statistics | Coursera


The fitted regression model can also be used to predict response values. For example, to obtain the response value for a new observation corresponding to 47 units of and 31 units of , the value is calculated using:

A friendly introduction to linear regression (using Python)

Forecasting with regression requires the Excel add-in called "," and linear programming requires the Excel add-in called"." How you check to see if these are activated on yourcomputer, and how to activate them if they are not active, varies withExcel version. Here are instructions for the most common versions. IfExcel will not let you activate Data Analysis and Solver, you must usea different computer.


Start Excel, then click Tools and look for Data Analysis and forSolver. If both are there, press Esc (escape) and continue withthe respective assignment. Otherwise click Tools, Add-Ins, andcheck the boxes for Analysis ToolPak and for Solver, then clickOK. Click Tools again, and both tools should be there.


Start Excel 2007 and click the Data tab at the top. Look to seeif Data Analysis and Solver show in the Analysis section at the farright. If both are there, continue with the respectiveassignment. Otherwise,
-click the “Office Button” at top left
-click the Excel Options button near the bottom of the resulting window
-click the Add-ins button on the left of the next screen
-near the bottom at Manage Excel Add-ins, click Go
-check the boxes for Analysis ToolPak and Solver Add-in if they are notalready checked, then click OK
-click the Data tab as above and verify that the add-ins show.


Start Excel 2010 and click the Data tab at the top. Look to seeif Data Analysis and Solver show in the Analysis section at the farright. If both are there, continue with the respectiveassignment. Otherwise,
-click the File tab at top left
-click the Options button near the bottom of the left side
-click the Add-ins button near the bottom left of the next screen
-near the bottom at Manage Excel Add-ins, click Go
-check the boxes for Analysis ToolPak and Solver Add-in if they are notalready checked, then click OK
-click the Data tab as above and verify that the add-ins show.

Correlation and Linear Regression | R-bloggers

This interpretation of the regression line says start with the mean of Y, and slide up or down the regression line times the deviation of X. For example, look back at Figure 2. Look for the deviation of X from the mean. Note the similarity to ANOVA, where you have a grand mean and each factor in the model is in terms of deviations from that mean.