Lecture notes. Written from the material of GTPB Psls20, licensed CC BY 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.
22. Breast Cancer Dataset¶
What this covers¶
A worked case study in simple linear regression: is the expression of the gene S100A8 associated with expression of ESR1 in breast cancer tumours? It assumes the reader already has simple linear regression — the model \(Y_i \mid X_i \sim N(\beta_0+\beta_1X_i,\sigma^2)\), the t-test and confidence interval on the slope, and the residual diagnostics for checking the model's assumptions — and shows how those pieces are actually used on a real, noisy, biological dataset: why the data get transformed before fitting, how the assumptions are checked rather than assumed, and how a regression coefficient on a transformed scale gets turned back into a statement about fold changes.
The dataset and the question¶
The data are a subset of a published study of 32 breast cancer patients, all with an estrogen-receptor-positive (ER+) tumour treated with the hormone therapy tamoxifen. For each patient the data record the tumour's histological grade (1 or 3), lymph node status (affected and removed, or not), tumour size, and the expression of two genes measured in the tumour biopsy by microarray: ESR1 and S100A8.
The two genes matter for different reasons:
- ESR1 encodes the estrogen receptor and is active in around 75% of breast cancer tumours. A tumour that expresses ER is one that responds to hormone therapy, because tamoxifen works by interacting with the estrogen receptor and modulating the genes it controls.
- S100A8 is a member of the S100 protein family, which is frequently dysregulated in cancer. S100A8 expression is thought to suppress the local immune response and create an inflammatory environment that promotes tumour growth.
Both genes are biologically implicated in the tumour, but by different mechanisms. That raises a question that is purely statistical once it is posed: is the expression of S100A8 associated with the expression of ESR1 in these tumours, and how strong is that association?
Looking at the data before modelling¶
Before fitting anything, the data are plotted. With several variables recorded, a scatterplot
matrix (a grid of pairwise scatterplots, ggpairs) gives a first look at all the pairwise
relationships at once. Attention then narrows to the pair of interest: S100A8 expression against
ESR1 expression.
Plotted on the raw scale, the relationship is visibly there but not linear — it curves. A straight line fitted through the points does not track the trend that a smooth curve (a loess fit) picks out. This matters because Pearson correlation and simple linear regression are both built to detect linear association; a real but curved relationship can look weak to them even when it is strong.
Gene expression concentrations are typically right-skewed, and a standard remedy is a log transform. Taking \(\log_2\) of both ESR1 and S100A8 expression and re-plotting, the relationship becomes linear: the curve and the straight line now agree closely.
Pearson and Spearman correlation¶
Two correlation coefficients are computed, on both the raw and the log2-transformed data:
- Pearson correlation measures the strength of a linear association.
- Spearman correlation is Pearson correlation applied to the ranks of the data, so it measures the strength of any monotonic association, linear or not, and is unaffected by a monotonic transformation such as \(\log_2\).
Both correlations come out negative, consistent with the visual impression that higher ESR1 goes with lower S100A8. The instructive part is how they compare:
- On the raw scale, the Spearman correlation is much larger in absolute value than the Pearson correlation. Pearson correlation is penalised by the curvature: a real monotonic relationship that is not linear is not fully captured by a coefficient that only measures linear association.
- On the log2 scale, the Pearson correlation rises to match the Spearman correlation, because log-transforming has made the relationship linear. The Spearman correlation itself barely changes between the two scales — it cannot, since \(\log_2\) is a monotonic transformation and rank correlation depends only on the ordering of the values, not the scale they are measured on.
This is the diagnostic that justifies working on the log2 scale from here on: it is the scale on which the association is (close to) linear, so it is the scale on which a linear-regression model, and the Pearson correlation that goes with it, are appropriate tools.
The regression model¶
Having decided to work on the log2 scale, the association is modelled as simple linear regression:
where \(Y_i\) is the \(\log_2\)-transformed S100A8 expression of subject \(i\) and \(X_i\) is the \(\log_2\)-transformed ESR1 expression of subject \(i\).
Fitting this model (lm) requires its assumptions to hold, and they are not simply assumed —
they are checked:
- Independence. The subjects were sampled at random from the population, so their outcomes are treated as independent. This one is a property of the study design, not something read off a plot.
- Linearity. Checked by plotting the residuals against the fitted values: if the model is missing curvature that is really in the data, the residuals will show a trend rather than being scattered evenly around zero.
- Equal variance (homoscedasticity). Checked from the same residuals-vs-fitted plot: the spread of the residuals should stay roughly constant across the range of fitted values, not fan out or narrow.
- Normally distributed residuals. Checked with a normal QQ-plot of the residuals: points should fall close to the diagonal, with no substantial systematic curve or heavy tail.
For this fit, the residuals are spread evenly around zero with no visible trend and roughly constant spread, and the QQ-plot shows no substantial deviation from a straight line. The assumptions hold well enough on the log2 scale to trust the inference that follows — which is exactly why the earlier step of finding the right scale mattered: the same diagnostics would have flagged a problem had the model been fit on the untransformed data.
Testing the association: the slope¶
The biological question — is there an association between the two genes' expression — translates into a statement about the slope of the fitted line. If \(\beta_1=0\), expression of ESR1 carries no linear information about expression of S100A8; if \(\beta_1\neq0\), it does. So the hypotheses are
The logic runs in the direction of falsification, not confirmation: data can never prove the alternative that an association exists, so instead the model tests whether the no-association null can be rejected. The two standard tools for this are the same ones used for any slope in simple linear regression:
- a t-test on \(\hat\beta_1\), read off the regression summary, and
- a confidence interval for \(\beta_1\), which adds information the test alone does not give: not just whether the association is detectable, but how large it plausibly is.
Both the test and the interval here point the same way: the association is extremely significant, and the interval is far enough from zero, and narrow enough, to say the association is not only statistically detectable but large enough to matter biologically.
Reading the coefficient as a fold change¶
The model was fit on the log2 scale, so \(\hat\beta_1\) is a slope in log2 units — not directly an interpretable biological quantity on its own. Exponentiating base 2 undoes the transform and turns the coefficient into a fold change: \(2^{\hat\beta_1}\) is the multiplicative change in (untransformed) S100A8 expression associated with a doubling of ESR1 expression, and applying the same transform to the endpoints of the confidence interval for \(\beta_1\) gives a confidence interval for that fold change.
This is the general move for interpreting a coefficient estimated on a log2 scale: additive on the log scale becomes multiplicative on the original scale, and the natural unit of \(X\) becomes "a doubling of \(X\)" rather than "one unit of \(X\)".
Conclusion¶
There is an extremely significant negative association between S100A8 expression and ESR1 expression. In fold-change terms: a patient whose ESR1 expression is twice that of another patient has, on average, a lower S100A8 expression by a factor read off from \(2^{\hat\beta_1}\), with the corresponding confidence interval giving the plausible range for that factor. The result is consistent with the biological picture that motivated the question: a tumour with an active estrogen receptor pathway (high ESR1, responsive to tamoxifen) tends to have less of the inflammation-promoting S100A8 signal.
Sources¶
This chapter is built entirely from the converted tutorial notes for this exercise; no slide deck or lecture transcript was supplied for it.
- Data, biological background, exploratory plots, log2 transform, and Pearson/Spearman
correlation:
docs/omics-statistics/gtpb/psls20/tutorialScripts/excercises/06_linearRegression/breastcancerExample/01-breast-cancer-dataset.md. - Regression model, assumption checking, hypothesis test on the slope, and back-transformation to
fold change:
docs/omics-statistics/gtpb/psls20/tutorialScripts/excercises/06_linearRegression/breastcancerExample/02-model.md. - The dataset is a subset of the study at https://doi.org/10.1093/jnci/djj052, cited in the notes but not otherwise described there.
- The notes give the R code that produces the scatterplots, residual and QQ diagnostic plots, and
the numerical output of
summary(lm),confint(lm), and the back-transformed coefficients, but the underlying.Rmdfile was not executed as part of this material, so no rendered plots or numeric estimates (coefficients, \(p\)-value, confidence interval bounds) were available to quote; the discussion above reports only the qualitative conclusions the notes themselves state.
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