Is it possible to check that this assumption is satisfied

This means that the variability in the response is changing as the predicted value increases. This is a problem, in part, because the observations with larger errors will have more pull or influence on the fitted model. An unusual pattern might also be caused by an outlier. Outliers can have a big influence on the fit of the regression line. In this example, we have one obvious outlier. Many of the residuals with lower predicted values are positive these are above the center line of zero , whereas many of the residuals for higher predicted values are negative.

The one extreme outlier is essentially tilting the regression line. As a result, the model will not predict well for many of the observations. In addition to the residual versus predicted plot, there are other residual plots we can use to check regression assumptions. A histogram of residuals and a normal probability plot of residuals can be used to evaluate whether our residuals are approximately normally distributed.

Note that we check the residuals for normality. Our response and predictor variables do not need to be normally distributed in order to fit a linear regression model.

If the data are time series data, collected sequentially over time, a plot of the residuals over time can be used to determine whether the independence assumption has been met. How do we address these issues? We can use different strategies depending on the nature of the problem. For example, we might build a more complex model, such as a polynomial model, to address curvature. RegressIt is an excellent tool for interactive presentations, online teaching of regression, and development of videos of examples of regression modeling.

It includes extensive built-in documentation and pop-up teaching notes as well as some novel features to support systematic grading and auditing of student work on a large scale.

There is a separate logistic regression version with highly interactive tables and charts that runs on PC's. RegressIt also now includes a two-way interface with R that allows you to run linear and logistic regression models in R without writing any code whatsoever.

If you have been using Excel's own Data Analysis add-in for regression Analysis Toolpak , this is the time to stop. It has not changed since it was first introduced in , and it was a poor design even then. It's a toy a clumsy one at that , not a tool for serious work. Visit this page for a discussion: What's wrong with Excel's Analysis Toolpak for regression.

Four assumptions of regression. Testing for linear and additivity of predictive relationships. Testing for independence lack of correlation of errors. Testing for homoscedasticity constant variance of errors.

Testing for normality of the error distribution. There are four principal assumptions which justify the use of linear regression models for purposes of inference or prediction:. If any of these assumptions is violated i. More details of these assumptions, and the justification for them or not in particular cases, is given on the introduction to regression page. Ideally your statistical software will automatically provide charts and statistics that test whether these assumptions are satisfied for any given model.

RegressIt does provide such output and in graphic detail. The normal quantile plots from those models are also shown at the bottom of this page. These are important considerations in any form of statistical modeling, and they should be given due attention, although they do not refer to properties of the linear regression equation per se. Return to top of page.

Violations of linearity or additivity are extremely serious: if you fit a linear model to data which are nonlinearly or nonadditively related, your predictions are likely to be seriously in error, especially when you extrapolate beyond the range of the sample data. How to diagnose : nonlinearity is usually most evident in a plot of observed versus predicted values or a plot of residuals versus predicted values , which are a part of standard regression output.

The points should be symmetrically distributed around a diagonal line in the former plot or around horizontal line in the latter plot, with a roughly constant variance. The residual-versus-predicted-plot is better than the observed-versus-predicted plot for this purpose, because it eliminates the visual distraction of a sloping pattern.

Look carefully for evidence of a "bowed" pattern, indicating that the model makes systematic errors whenever it is making unusually large or small predictions.

In multiple regression models, nonlinearity or nonadditivity may also be revealed by systematic patterns in plots of the residuals versus individual independent variables. For example, if the data are strictly positive, the log transformation is an option. The logarithm base does not matter--all log functions are same up to linear scaling--although the natural log is usually preferred because small changes in the natural log are equivalent to percentage changes.

See these notes for more details. If a log transformation is applied to the dependent variable only, this is equivalent to assuming that it grows or decays exponentially as a function of the independent variables.

If a log transformation is applied to both the dependent variable and the independent variables, this is equivalent to assuming that the effects of the independent variables are multiplicative rather than additive in their original units. This means that, on the margin, a small percentage change in one of the independent variables induces a proportional percentage change in the expected value of the dependent variable, other things being equal.

Models of this kind are commonly used in modeling price-demand relationships, as illustrated on the beer sales example on this web site.

Another possibility to consider is adding another regressor that is a nonlinear function of one of the other variables. Higher-order terms of this kind cubic, etc. This sort of "polynomial curve fitting" can be a nice way to draw a smooth curve through a wavy pattern of points in fact, it is a trend-line option on scatterplots on Excel , but it is usually a terrible way to extrapolate outside the range of the sample data.

Finally, it may be that you have overlooked some entirely different independent variable that explains or corrects for the nonlinear pattern or interactions among variables that you are seeing in your residual plots. In that case the shape of the pattern, together with economic or physical reasoning, may suggest some likely suspects. For example, if the strength of the linear relationship between Y and X 1 depends on the level of some other variable X 2 , this could perhaps be addressed by creating a new independent variable that is the product of X 1 and X 2.

In the case of time series data, if the trend in Y is believed to have changed at a particular point in time, then the addition of a piecewise linear trend variable one whose string of values looks like 0, 0, …, 0, 1, 2, 3, … could be used to fit the kink in the data.

Such a variable can be considered as the product of a trend variable and a dummy variable. The null hypothesis is the default assumption that no relationship exists between two different measured phenomena.

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I Accept Show Purposes. Your Money. Personal Finance. Your Practice. Popular Courses. Economy Economics. Table of Contents Expand. The T-Test. T-Test Assumptions. Key Takeaways A t-test is a statistic method used to determine if there is a significant difference between the means of two groups based on a sample of data.

The test relies on a set of assumptions for it to be interpreted properly and with validity.