Hello Zachary,

I think that bootstrapped (error estimates) anova or regression would likely be simpler and preferable to trying to find an acceptable transformation of the DV. Both methods have the same underlying mathematical models, so yes, "robust" methods are applicable to either.

For the other questions:

1. Independence of residuals hinges more on how the data were collected than on any other feature. Unfortunately, data collection scheme isn't manifest from the scores (whether observed, fitted, or residual scores are under consideration). While regression programs often incorporate (now somewhat dated) checks such as Durbin-Watson (1st order autocorrelation), that's only one way in which dependence can be present.

2. If you're asking whether categorical variables be used as IVs in linear models, then, yes, of course they can. If the variable has more than two levels, you have to re-express these as (k - 1) dummy variates, where k is the number of levels for the categorical variable. That's why the df in anova for a factor (typically a categorical variable) is k - 1.

Good luck with your work.

If you are only worried about points at the extreme, why are you suggesting least square procedures, use least absolute values or an M-estimator. However, as David Morse says, probably just bootstrapping is simplest.

David Morse

Thanks for the answer

The question " How do I test for a linear relationship between (1) IV and residuals and (2) IV and DV - when the IV are all categorical? " - this really meant ~ How do you address the linearity assumption of multiple linear regression when the independent variable is categorical? I see you can make plots when the DV and IV are continuous, but this isn't logical when the IV is categorical - you will just get parallel lines for each category.

Separately do you have any good resources on bootstrapping and the Durbin-Watson in STATA? Which of the anova or linear regression is more appropriate?

@Zachary, define what you mean by linear between two points. All relations will be linear in that usual sense.

Daniel Wright

Thanks for all your answers.

I dont see where I said "linear between two points"

Separate question

Can you do bootstrapping and robust estimates at the same time? I just performed it in stata but I want to know if it is appropriate.

Daniel Wright

I think I understand now, you are saying that " linear between two points " - you mean that by default a categorical variable is linear because you are connecting the point at 0 and 1 for the dichotomous variable?

Yes on the linear, because you are connecting just two points, and that defines the line. On bootstrapping, that is just a procedure you do separately from the statistical model. You sample from the data with replacement, run whatever procedure and write down the statistic of interest, repeat this a few thousand times, and then treat the distribution of the observed statistic as its distribution. One approach is then to treat the middle 95% of these values as the 95% CI (though there are CI procedures that tend to perform a little better). It doesn't work for every statistic, but is works for a lot. Here is what I have on the name in my teaching materials:

\begin{bbox}\textbf{Etymology of Bootstrap} \\

Pulling oneself up by their bootstrap--that loop on back of boots used to help pull them on--is physically impossible. If you pull hard you will fall flat on your face. It is ``a physical impossibility to lift yourself up by a bootstrap'' Rep. Alexandria Ocasio-Cortez (not surprisingly some on social media disagreed with AOC, tried this, and fell flat). It is a phrase used to suggest doing something that is impossible. The phrase is sometimes attributed to the fictional character Baron Munchausen, who pulled himself and his horse out of a swamp by pulling on the back of his hair, and equally impossible feat (Munchausen is based on the real world character M\"unchhausen, and is where name Munchausen syndrome comes from). \citet[p.~25]{Efron1979} said other names were considered: Swiss army knife, meat ax, swan-dive, jack-rabbit, and his favorite: shotgun. Tukey claimed shotgun was appropriate since it ``can blow the head off any problem if the statistician can stand the resulting mess'' \citep[as cited in][p.~25]{Efron1979}.

\end{bbox}

Zachary -

You know that the dependent variable can have any distribution, right? (The same goes for any continuous independent variable, if you had any.) It is just somewhat desirable that the estimated residuals be fairly close to 'normally' distributed. (I assume you were only manipulating the dependent variable to manipulate the estimated residuals.)

Regarding heteroscedasticity, there are examples that may help here:

https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity

Though none of those examples are for categorical independent variables, once you have the coefficient of heteroscedasticity, gamma, you then have the regression weight,

w = z^(-2 gamma),

where for z you could use your preliminary (OLS) predicted y. But this only helps you if your software will let you enter the regression weights. If so, then to do what is done in the above paper, you can use the following tool:

https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx

Cheers - Jim

Thanks for all the help

Final questions:

Robust standard errors vs Bootstrapping: Which is better for adjusting for non-normality of residuals?

Robust standard errors vs Bootstrapping: Which is better for adjusting for heterodiscacity?

I am using stata estat hettast in Stata but it isnt working with robust standard errors and bootstrapping. Is there a way to use this test while using those adjustments or is there an alternative?