I think that there is an ordering of importance of assumptions about the residuals - most important - no outliers, no dependence in time or space, homoscedasticity, normality ; with the latter being the least important. I too favour a graphical approach or a procedure that is robust/resistant to non fulfillment rather than a test. To these are added : not too much collinearity, linearity, and the real difficult one - no important omitted variables.

An easy book for you is J. Pallant's SPSS survival manual. She refers to Tabachnik and Fidell: Using multivariate statistics often but she is clearer. Mahalanobis and Cook's is used for identification of outliers. The critical values depend on the number of independent variables IV: 2 IV = 13,82; 3 IV= 16,27; 4 IV= 18.47; 5 IV= 20,52; 6 IV= 22,46 and 7 IV = 24,32

You can find the residuals or the difference between actual and predicted Values or MSE mean square error.

mahalanobis

Hello Angel Tabullo. It might be good to start with some discussion of what the assumptions actually are. We should also acknowledge that exactly which assumptions are required depend on what one wants to do. Here is a set of slides in which I've summarized some good info on all of this as presented (mostly) in Jeff Wooldridge's well-known econometrics textbook.

• https://sites.google.com/a/lakeheadu.ca/bweaver/Home/statistics/files/OLS_regression_assumptions_Wooldridge.pdf

One further suggestion I would make is that graphical methods for assessing how well the assumptions are met are generally preferable to statistical tests of the assumptions. Statistical tests of the assumptions of some other procedure often have the problem that they are under-powered when samples are small and become over-powered as samples become larger. HTH.

An excellent reference is the book by Albert Madansky: Prescriptions for Working Statisticians.

Or you could just Google it with your question as the query:

check linear regression assumptions?

Best, David Booth

Thank you so much for your suggestions, I'll make sure to get the literature!

There are some plots for checking the linear regression i.e., Q-Q plots for normality and residual plots

These plots helps to verify the assumptions of linear regression

I think that there is an ordering of importance of assumptions about the residuals - most important - no outliers, no dependence in time or space, homoscedasticity, normality ; with the latter being the least important. I too favour a graphical approach or a procedure that is robust/resistant to non fulfillment rather than a test. To these are added : not too much collinearity, linearity, and the real difficult one - no important omitted variables.

Angel Tabullo This may be of help. I found it easy to follow.

https://statistics.laerd.com/spss-tutorials/multiple-regression-using-spss-statistics.php

to check normality assumption of linear regression, you can use plots like Histogram, Q Q plot and tests like kurtosis and skewness.

Carry out the normality test, goodness of fit test.. For further study Introduction to econometrics with examples in Reviews by M.O Adenomon and V.A Micheal

Also keep in mind, Angel, that different regressions may have different requirements; so, should you change "linear" into something else, you might end up with a rather complete different story...

Angel -

The classical things to look for do often turn out to be less important than an intro textbook might indicate. Normality of estimated residuals is not such a big deal, and many people think that the data should be "normally" distributed which is not true at all. Data for establishment surveys are typically highly skewed. Further, heteroscedasticity is arguably the norm, especially for highly skewed data, or anytime that the population member sizes vary greatly. (For regressions of form y = y* + e, which is most useful for predictions for finite populations, please see

https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression.)

I think it is better to use a graphical analysis than Cook's distance, as you can get a better idea of the "big picture" (overall situation) with graphics, particularly scatterplots. You can look at individual variable pairs and more.

Consider a graphical residual analysis with predicted y values on the x-axis, and the estimated residuals on the y-axis. You can find information on the internet regarding interpretation. Note that this works for rather complex regressions, as well as the simplest. For (simple) one regressor (predictor, independent variable) linear regression, with a zero intercept, predicted y is bx, and because relative size is what matters in regression, we can use x instead of bx, which may make a more familiar graph for many people.

Such "graphical residual analyses" are to study model fit. A "cross-validation" is used to study whether one might have overfit to the particular sample used.

You might find something of interest in https://www.researchgate.net/publication/313925533_Comparing_Model_Performances_Graphically.

Best wishes - Jim

Allaa Ali, it seems that you find a lot of questions/answers useful.

• https://www.researchgate.net/profile/Alaa_Ali3/answers

Unfortunately, people who are subscribed to the questions or discussions where you post that useless message end up wasting their time when they are notified of a new post.

Bruce, et.al. -

Regarding confidence intervals for regression coefficients, and prediction intervals for prediction errors:

In your notes (Bruce) from Wooldridge and Vittinghoff, et.al., about 'normality' for confidence intervals for OLS regression coefficients, it reminds me of having wondered before about 'normality' for prediction intervals. I've done prediction intervals for weighted least squares (WLS) regression, specifically the classical ratio estimator, where near the origin for a ratio estimator the intervals would go lopsided if you don't allow negative numbers, but beyond that, there is the assumption of normality that is often used in confidence and prediction intervals.

For any multiple linear regression, you can write

y = y* + e,

as shown in https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression. The y* is any predicted y, with any number of predictors, where OLS is just a special case of WLS.

For large samples, the regression coefficients are approximately normally distributed, regardless of the distribution of the estimated residuals. But I have considered before, "What about the distribution of the prediction error?"

The estimated variance of the prediction error is made up of a model part, with a component for each predictor, and a part from the residual. For WLS, for that 'residual component,' there is a factor that is the inverse of the regression weight, and a factor which is sigma-squared for the random factors of the estimated residuals. (For OLS, for the estimated variance of the prediction error, the component associated with the estimated residuals is just sigma-squared for the estimated residuals, which results from equal regression weights.)

At any rate, it seems to me that since the component/part of the estimated variance of the prediction error which comes just from the estimated residuals is the part that Scott Fortman-Roe (http://scott.fortmann-roe.com/docs/BiasVariance.html) calls the "irreducible error," sigma-squared, then this component of the estimated variance of the prediction error, which is sometimes a large part of it, will keep the large sample effect on the distribution of the regression coefficients from being as helpful for prediction intervals. That is, the prediction intervals will not be asymptotically normally distributed because of this irreducible error.

Have you, Bruce, or anyone else, seen anything on this. I think I wrote a ResearchGate question on it but didn't get anywhere.

This seems like an important 'corollary' to the part in the notes that Bruce attached which said that for large n, "Normality of the errors becomes less important...." - I think that though that applies to regression coefficients, it is only partially applicable to prediction errors.

Has anyone seen anything on this? Any thoughts?

Thanks - Jim