Assumption for inferential regression

[Mrgarrels]

I was reading about regressional inference and one of the assumptions stated was:

The standard deviation of the responses about the population line is the same for all values of the explanatory variable.

I understand this as:

If X has a certain value the mean value of Y for that X value has a certain standard deviation. That standard deviation of Y is equal for all X values.

I have two questions about that:

1. Do i interpret the first statement correctly?

2. If so, should the standard deviation be exactly the same or is it allowed to differ slightly?

I am sorry if this is a stupid question but i couldnt find it on the internet.

Thank you in advance

 

[ondansetron]

I was reading about regressional inference and one of the assumptions stated was:

The standard deviation of the responses about the population line is the same for all values of the explanatory variable.

I understand this as:

If X has a certain value the mean value of Y for that X value has a certain standard deviation. That standard deviation of Y is equal for all X values.

I have two questions about that:

1. Do i interpret the first statement correctly?

2. If so, should the standard deviation be exactly the same or is it allowed to differ slightly?

I am sorry if this is a stupid question but i couldnt find it on the internet.

Thank you in advance

The assumption is typically phrased in terms of the error variance, but the implication is there for standard deviation.


1. Your interpretation is correct. Remember, though, that this means for every possible combination of the values of the independent variables (male 55 years old, woman 55 years old, male 56 years old, female 56 years old, and so on, assuming only gender and age were in the model). As you can tell, that gets more tricky to think about as we add more independent variables. This can be checked (informally, but quite effectively) by creating a scatter plot of the regression residuals (as an estimate of errors, on the Y-axis) against the respective, predicted values of Y (X-axis). This allows us to look at the assumption in two dimensions (using predicted Y-values is a way to reduce the dimensions that arise from having more than one independent variable). You should expect to see no discernible pattern in the residuals as the predicted y-values change (implicitly, as the combinations of x-values change). Some classic violation patterns are cones/triangles/football/bullet patterns in the residuals.


2. In theory, it should be the same, but in practice you'll never have it identical due to sampling variation. In general, you examine the assumption and hope you don't see any evidence that it's violated (i.e. too different).

 

[Mrgarrels]

Thank you for your reply ondansetron that is a handy trick indeed. Is there a rule of thumb (like a proportional value) about how different "too different" is?

 

[ondansetron]

Thank you for your reply ondansetron that is a handy trick indeed. Is there a rule of thumb (like a proportional value) about how different "too different" is?

Not that I've ever used (but that doesn't mean there is one). If you feel uncomfortable just looking at the graph (and most people are at first, differentiating "insufficient data" from "outliers" from "unequal variance" can be tricky), you can look into formal tests for heteroscedasticity (fancy word for nonconstant variance). Breusch-Pagan test for heteroscedasticity and White test for heteroscedasticity are options you may want to consider. You're essentially fitting a model with the squared residuals as Y and the independent variables (and or transformations in the White test) as X variables. There are a couple ways to conduct the test (I'd recommend reading up on these methods if you're interested in trying any of them). If it's significant it indicates that the X-variables are significant predictors of the squared residuals (Y), meaning that there is evidence of nonconstant variance. An alternative form of the White test uses predicted y-values and squared predicted y-values as independent variables for predicting squared residuals of the original regression. (Again, predicted Y and predicted Y-squared are used as a way to reduce dimensions and conserve degrees of freedom.)

I haven't used any of these more than a handful of times, so I can't speak too authoritatively on them. Hence, I'm a fan of the scatter plot method (you also don't run the risk of making a Type I or Type II error if you're not doing a formal test). If you're unsure from the visual method, you can also take the corrective action (heteroscedasticity consistent standard errors or transforming Y, for example), and compare the results of the two methods (i.e. are the conclusions still similar? be careful if you've transformed Y, though, because interpretations might change).

Lastly, anything you do should be reported in your research including why you did it and what the outcome was (if this is for research).