# Visual Demonstration of Residual Confounding

To drive home the point that dichotomizing a continuous confounder creates problems of residual confounding in regression analyses, I'm wondering if there are any available visual demonstrations (not DAGS) for the average biomedical researcher (amongst whom I include myself).

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Dec 27, 2021 at 5:12
• Why not just plot prediction - true relationship for a simulated problem? Dec 27, 2021 at 5:17
• @Demetri Thank you for the suggestion. I'm wondering if it is possible to visually show the "mechanism" of residual confounding - that is, a demonstration that categorization weakens the associations of the confounder with both predictor and outcome, thereby leading to residual confounding. Dec 27, 2021 at 23:02
• @Yonghao Confounding is about bias, not just bias towards the null. My answer to your question shows 2 ways dichotmozation can damage inference: one at the level of the relationship between the predictor and the outcome, the other at the model level. Dec 28, 2021 at 0:07

## 1 Answer

Confounding is about bias. There are at least two ways I can think of in which dichotomization can bias a regression:

• Model Bias. Dichotomizing a continuous predictor transforms the conditional mean from a continuous (possibly smooth, depending on the functional form) function to a discontinuous constant function. It would be good to ask one's self if we truly believe the phenomenon to display this sort of behaviour (I tend to think most things in medicine do not behave like this). Hence, dichotomizing biases our estimates towards the set of functions which are discontinuous and constant. That's bad, as we will see.

• Bias in Estimates of Association. Regression Analysis, so far as it is from observational data and not part of some causal framework, is all about associations. The coefficients of the model $$\hat{\beta}_j$$ are the observed association between $$x_j$$ and the outcome. The estimated association can differ from the true association in expectation. This is the sort of standard statistical bias you might see in many senior undergrad or graduate level texts

$$\operatorname{Bias} = E(\hat{\beta_j} - \beta_j)$$

with the estimate being called unbiased if this expectation is 0.

In what follows, I demonstrate that dichomotization leads to both kinds of bias in a very simple model.

Let's simulate some data from the following data generating process

$$y \vert x \sim \mathcal{N}(\mu(x), 1)$$

$$\mu(x) = 2x + 1$$

Let's treat $$x$$ as if it were continuous, when in reality I will bucket it into bins of length 0.1 for means of exposition. Recall linear model fit via OLS has a few key properties, one being that the expectation of the residual should be 0 and independent of the predictor.

Let's generate data, fit a model, and get the residuals say 1000 times. Let's take the expectation of the residual across each $$x$$ value. The expectation should be 0 for each $$x$$ value. We can demonstrate that with some R code (which I will include at the end of the answer).

I've gone ahead and included approximate 95% confidence intervals as well. Not every residual CI covers 0, but this is to be expected (in fact, 5% should fail to do so), but this is largely in line with what we would expect. Let's do this again, but now we will dichotomize $$x$$ at the sample median (perhaps the most charitable point to split at so that both groups have approximately the same number of observations).

The effect of dichotomization is that the expectation for the residual is no longer 0 and is dependent on the covariate. This is residual confounding, as I understand it. In particular, this is the result of model bias. Our chosen class of functions (discontinuous constant functions) can not properly accommodate our data, leading to a relationship between the predictor and the residuals. This is the same sort of diagnostic one might perform to see if there are any non-linear relationships in the predictor (the difference being here that I have aggregated residuals over 1000 simulations, and in practice one usually has 1 dataset). After all, a prediction is an estimate, and it seems that many estimates in this model are biased (even within groups determined by the dichotomization).

Not only are the residuals not consistent with the modelling assumptions, the estimate of the observed association is also biased. If the assumptions of OLS are met, then the estimate of $$\beta$$ (the slope, or really any parameter in the model) should be unbiased. This means that, under repeated simulation, the expectation of the difference between estimate and truth should be 0. We can demonstrate that this is not the case when the data are dichotomized. In fact, for this problem, the estimated relationship is about 4 units higher (which makes sense after you think about the structure of the problem).

The difference is even more striking when you plot the difference between estimated and truth on the same axis.

Note that the bias can be fixed by correctly adjusting for the distance between mean of points which belong to each group (shown below) but this does not improve the model bias (the systematic tendency for the model to under/over estimate given a particular value of the predictor). Additionally, if you're going to correct the dichotmomized estimate in this way, you might as well just estimate the slope correctly. This is just getting the right answer via the wrong way and calling the entire approach valid.

What can we conclude from this answer? Somewhat tongue in cheek, we might conclude dichotmization bad. More sincerely, dichotmization needlessly biases our class of possible conditional expectations towards a class of functions which, at least in my opinion, is rarely observed in practice. Indeed, if I could opine slighly longer, dichotmization is not so much about statistical efficiency more than it is cognitive economy; people dichotmize because it is easier to think about 2 numbers than a continuum of them.

Second, the observed relationship is biased (in this case away from the null, making the relationship seem more extreme than it actually is, but I could just as easily create an example where the bias is towards the null). This is a particularly pernicious form of bias, because it can go largely undetected without critical analysis of the model, unlike model bias. I will leave you to wonder about how this bias can effect care, should the regression analysis be used in some sort of evidence based investigation.

I should add that the simplicity of the model is not a limitation. These sorts of phenomenon can appear in multiple regressions, and in different ways, in GLMs (like logistic regression).



# Predictor buckted at bins of width 0.1
x = seq(-3.0, 3.0, 0.1)
N = length(x)

# Simulate the data 1000 times and return a residual for each predictor.
r = replicate(1000,{
y = 2*x + 1 + rnorm(N, 0, 1)
fit = lm(y~x)
resid(fit)
})

e = rowMeans(r)
s = apply(r, 1, function(x) sd(x)/sqrt(length(x)))
plot(x, e, pch=20, ylim = c(-0.15, 0.15), main='Expectation of Residuals Under No Dichotimization', ylab=expression(y-hat(y)))
arrows(x0=x, y0=e-2*s, x1=x, y1=e+2*s, code=3, angle=90, length=0, col="black", lwd=2)
abline(h=0, col='dark grey')

r2 = replicate(1000,{
y = 2*x + 1 + rnorm(N, 0, 1)
z = as.numeric(x>0)
fit = lm(y~z)
resid(fit)
})

e = rowMeans(r2)
s = apply(r2, 1, function(x) sd(x)/sqrt(length(x)))
plot(x, e, pch=20,, main='Expectation of Residuals Under Dichotomization', ylab=expression(y-hat(y)))
arrows(x0=x, y0=e-2*s, x1=x, y1=e+2*s, code=3, angle=90, length=0, col="black", lwd=2)
abline(h=0, col='dark grey')

no_dichot = replicate(1000, {
x = seq(-3.0, 3.0, 0.1)
N = length(x)
true_slope=2
y =true_slope*x + 1 + rnorm(N, 0, 1)
fit = lm(y~x)
estimated_slope = coef(fit)[2]
estimated_slope -true_slope

})

dichot = replicate(1000, {
x = seq(-3.0, 3.0, 0.1)
N = length(x)
true_slope=2
z = as.numeric(x>0)
y =true_slope*x + 1 + rnorm(N, 0, 1)
fit = lm(y~z)
estimated_slope = coef(fit)[2]
estimated_slope -true_slope

})

hist(no_dichot,
col=rgb(1, 0, 0, 0.5),
breaks = seq(-2, 6, 0.05),
xlab=expression(hat(beta)-beta),
main='Difference between estimated relationship and true relationship (1000 simulations)',
xlim = c(-2, 6))

hist(dichot,
col=rgb(0, 0, 1, 0.5),
breaks = seq(-2, 6, 0.05),
add=T)

legend("topright", c("Continuous", "Dichotomize"), col=c(rgb(1, 0, 0, 0.5), rgb(0, 0, 1, 0.5)), lwd=10)


• Thank you for your eloquent exposition. Prior to reading your answer, I do not know that residual confounding occurs when "the expectation for the residual is no longer 0 and is dependent on the covariate." In my more traditional way of thinking, I am, however, seeking a visual demonstration of how controlling for a dichotomized confounder (say, Z) distorts the X-Y association. Dec 28, 2021 at 4:46
• @Yonghao You're going to need to elaborate on what you mean by "visual demonstration". I've provided you a visualization which depicts the statistical bias resulting from dichotomization. What are you looking for exactly? Additionally, it is not that "residual confounding occurs when "the expectation for the residual is no longer 0 ", it is that there exists a relationship between the residuals and the predictor that the model can not accommodate for. Residual confounding could exist when there exists a non-linear relationship between the predictor & outcome, but we use a linear effect. Dec 28, 2021 at 4:49
• my sincere apologies for the vagueness. Papers that have extolled the virtues of using all information in the variable have reported (and contrasted) the crude beta of the (unadjusted) X-Y association, the beta when adjusting for a dichotomized confounder, and the beta when adjusting for the (original) continuous confounder. Whilst convincing, I thought a visual demonstration on this beta distortion (that is, the X-Y slope changes) and more importantly, why it occurs would be a very useful supplement. Dec 28, 2021 at 8:09
• @Yonghao The visual demonstration of the bias comes from plotting the predictions of the model across values of the predictor. As I have mentioned, dichotomizing will yield something that looks like a step function. The bias is evident as we would never, even in the asymptotic limit, fit a model which looks like the true effect unless there is a null effect of the variable. Dec 28, 2021 at 15:59
• @Yonghao The statistical bias comes from the fact we are estimating the difference in group means rather than the slope. You can see this in my example. the difference in group means is 2*mean(x[x>=0]) - 2*mean(x[x<0]) == 6.1 rather than the slope of 2. This is the quantity the regression is estimating (without correcting for the distance between these points), hence the bias of approximately 4 I show in my plot. Dec 28, 2021 at 16:01