Estimating variability in a response variable not accounted for by measured predictors using linear regression and R

Question

Background

I have a data set of patients who were operated on at two different hospitals, A and B. Lymph nodes were removed from each patient during the operation and counted, this is saved as LN_reviewed for each patient. I want to know how much variability there is in the lymph node number that is not accounted for by gender, the year of the operation, or the age of the patient when operated on. My assumption (hypothesis) is that any additional variability is likely due to the pathologist at the institution (this was not actually measured in my study).

My question

What is the best way to go about estimating the variability in lymph node number that is not accounted for by gender (a factor), year (a continuous variable), or age (a continuous variable)?

My initial attempt at answering this question

I built a linear regression model using the number of lymph nodes as the response variable and the gender, operation year, and operation age as predictors. I am not sure how to interpret the results to answer my specific question. Should I be looking at the R squared? If so, is there a way to get a confidence interval for it? Thanks to anyone who can help. If you think I am going about this the wrong way, please let me know.

Call:
lm(formula = LN_reviewed ~ Gender + Operation__year + Operation__age, data = sample_data)

Residuals:
    Min      1Q  Median      3Q     Max 
-49.436 -15.280  -0.495  13.450  61.564 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)     -1.190e+04  1.459e+03  -8.159 9.95e-14 ***
GenderMALE      -5.542e+00  4.685e+00  -1.183    0.239    
Operation__year  5.980e+00  7.296e-01   8.196 8.01e-14 ***
Operation__age  -2.524e-01  1.675e-01  -1.507    0.134    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 22.46 on 158 degrees of freedom
Multiple R-squared:  0.2999,    Adjusted R-squared:  0.2866 
F-statistic: 22.56 on 3 and 158 DF,  p-value: 3.268e-12

Answer 1

Yes, the unexplained variance in your model involves $R^2$: specifically, it is $1-R^2$. From Wikipedia:

The squared coefficient of multiple correlation can also be computed as the fraction of variance of the dependent variable that is explained by the independent variables, which in turn is 1 minus the unexplained fraction. The unexplained fraction can be computed as the sum of squared residuals—that is, the sum of the squares of the prediction errors—divided by the sum of the squared deviations of the values of the dependent variable from its expected value.

This page calculates confidence intervals for $R^2$. The formula it offers is as follows:

$$R^2\pm t_{\frac{1-\alpha}{2} ,n-k-1}SE_{R^2}$$ where $R^2$ is the squared multiple correlation, $α$ is the desired confidence interval percentage, $SE_{R^2}$ is the standard error for $R^2$, $t$ is a $t$-value, $k$ is the number of predictors in the model, and $n$ is the total sample size.

Standard error for an $R^2$ value (Olkin and Finn's approximation): $$SE_{R^2}\approx(\frac{4R^2(1-R^2)^2(n-k-1)^2}{(n^2-1)(3+n)})^\frac{1}{2}$$ where $R^2$ is the squared multiple correlation, $k$ is the number of predictors in the model, and $n$ is the total sample size.

One admonition about your hypothesis: you can't safely assume that all unexplained variance is due to one particular unmeasured factor. If, for instance, your data on age and year of operation are counted in natural numbers as years, you might say whatever fractional information that leaves out (months, days, and so on) is measurement error. If for no other reason than that, your predictors might produce some error in their predictions. Of course, you could probably also think of other unmeasured factors that might affect your dependent variable aside from the one you already had in mind (e.g., genetic variation). You'd need a very strong theoretical reason to assume that only one unmeasured source of error in your model's predictions is substantial, let alone that it is the one missing piece to the complete picture.

Furthermore, I don't see how you could even make a good guess at how much of your unexplained variance would be explained by adding a measurement of your hypothesized source to the model. Multiple regression coefficients can differ in strength quite a lot from bivariate correlations due to interactions, mediation, and suppression. I'm guessing that if you knew enough about those relationships within your model to inform a decent guesstimate, you probably wouldn't have much interest in this analysis at all. Hope this answer helps nonetheless!