# Flawed multiple linear regression in academia? Heteroscedasticity's effect on p-value?

I believe I have found a paper in academia that has used a flawed multiple linear regression. I have downloaded the data set and replicated their regression results. I have done some diagnostics and have found this to my surprise:

There clearly is heteroscedasticity in the model, right? Hence, this violates the assumption of MLR that there is homoscedasticity.

Thus far I have found that heteroscedasticity has an effect on p-value, i.e. that it makes p-values for independent variables' association with dependent variable smaller. Thus, with heterscedasticity, the MLR model can show significant relationships between IVs and DVs, when in reality the significance is absent.

Is my understanding correct? Any useful resources on what heteroscedasticity entails for the MLR model's results?

Appreciate it.

• What was the dependent variable? The dependent variable seems to have an upper bound which is visible as a line in the first plot. May 21, 2021 at 14:57
• It was a continous variable (it is a ratio, so it takes values from 0 to 1). It measures the share of total loan's funds disbursed. May 21, 2021 at 14:59

This isn't heteroscedasticity you are looking at, but truncation.

You can see this very clearly in the first plot: No combination of the fitted + residual exceeds a certain number, causing this sudden imaginary diagonal line, past which no observations exist. In the scale-location plot, this strange shape reveals that the data are truncated at $$1$$.

It is easy to simulate some truncated data and show that the diagnostic plots indeed display this diagonal cutoff, as well as the strange V-shape in the scale-location plot:

set.seed(1234)
n      <- 1000
beta_0 <- 1.5
beta_1 <- 0.5
x      <- rnorm(n)
y      <- beta_0 + beta_1 * x + rnorm(n, 0, 0.5)
y      <- pmin(y, 1)
plot(lm(y ~ x))


The real question isn't what to conclude from these diagnostic plots, but rather what these data are. If you include a reference to the paper you read, we could see why the data are bounded, and whether that renders their conclusions invalid or not.

Edit: In the comments you explained these are ratios. That gives you the actual answer to whether their approach is flawed (it probably is). Rather than an ordinary linear model, the authors should probably have used e.g. logistic regression using the original values that made up these ratios.

• Yes, but those numbers are calculated from a ratio of one divided by another, right? If those original numbers are not available, you can use beta regression for number bounded between $0$ and $1$. May 21, 2021 at 15:07
• You can use logistic regression even if the numbers are unknown (the ratios). This is sometimes called fractional outcome regression (e.g. in Stata). Logistic regression models the conditional log-odds for Y=1 with no requirement that the outcome is binary or anything like that. You'd probably want to use robust standard errors though. If there are any $0$s or $1$s in the data, beta regression can't be used. May 21, 2021 at 15:12
• Here is a good introduction if you're interested. May 21, 2021 at 15:18
• @KenLee OLS assumes conditional normality, and while it may work well for approximately normal errors, a number bounded between zero and one like this is nowhere near approximate normality. May 21, 2021 at 15:20
• A histogram is much harder to judge than a QQ-plot. In the QQ-plot you can clearly see problematic deviation form normality (strong deviation from the straight line). The diagnostic plots for logistic regression need not look better, because you don't assume normality in logistic regression. It is a better model from a theoretical standpoint. May 22, 2021 at 8:58