Solved – Homogeneity of variance in regression

assumptionsheteroscedasticityregression

I have carried out simple linear regression and I am now checking the models meets the assumption of homogeneity of variance:

• am I correct in concluding that the Levenes tests which gave a p<0.05 indicates a violation of homogeneity of variance?

• some models contain 2 or 3 explanatory variables, but in some models, only 1 explanatory variable gave p<0.05 in the Levenes test. Therefore, should only those explanatory variables which gave p<0.05 be corrected?

• is transformation the best place to start when correcting for violation of homogeneity?

Best Answer

You don't "correct" your covariates; the heterogeneity is in the residuals.
If you're simply running Levene's test as a factorial ANOVA & finding heterogeneity for some of your explanatory variables, you probably have interactions with those variables that you're currently missing & need to add to your model.
Transformations of the response variable are a good place to start once you've accounted for possible omitted variables (such as interactions).

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Solved – Investigating robustness of logistic regression against violation of linearity of logit

The linearity assumption is so commonly violated in regression that it should be called a surprise rather than an assumption. Like other regression models, the logistic model is not robust to nonlinearity when you falsely assume linearity. Rather than detect nonlinearity using residuals or omnibus goodness of fit tests, it is better to use direct tests. For example, expand continuous predictors using regression splines and do a composite test of all the nonlinear terms. Better still don't test the terms and just expect nonlinearity. This approach is much better than trying different single-slope choices of transformations such as square root, log, etc., because statistical inference arise after such analyses will be incorrect because it does not have large enough numerator degrees of freedom.

Here's an example in R.

require(rms)
f <- lrm(y ~ rcs(age,4) + rcs(blood.pressure,5) + sex + rcs(height,4))
# Fits restricted cubic splines in 3 variables with default knots
# 4, 5, 4 knots = 2, 3, 2 nonlinear terms
Function(f)   # display algebraic form of fit
anova(f)      # obtain individual + combined linearity tests

Solved – Residual Diagnostics and Homogeneity of variances in linear mixed model

I think Question 1 and 2 are interconnected. First, the homogeneity of variance assumption comes from here, $\boldsymbol \epsilon \ \sim \ N(\mathbf{0, \sigma^2 I})$. But this assumption can be relaxed to more general variance structures, in which the homogeneity assumption is not necessary. That means it really depends on how the distribution of $\boldsymbol \epsilon$ is assumed.

Second, the conditional residuals are used to check the distribution of (thus any assumptions related to) $\boldsymbol \epsilon$, whereas the marginal residuals can be used to check the total variance structure.

Best Answer

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Solved – Investigating robustness of logistic regression against violation of linearity of logit

Solved – Residual Diagnostics and Homogeneity of variances in linear mixed model

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