Simple Linear Regression and What R2 Means

The line almost everyone has already drawn

Open a results table, right-click a scatterplot in Excel, choose “Add Trendline,” tick “Display R-squared value on chart,” and a line and a number appear. Almost nobody drafting a manuscript figure thinks of that click as running a statistical test. It is one. The line Excel fits is the ordinary least-squares regression line — the same line, the same slope, the same R² that SPSS's Linear Regression procedure returns when the predictor and outcome are the same two columns.¹ The corpus of methods advice a resident is likely to have read covers multiple linear regression, with several predictors, and logistic regression, with a binary outcome. The single-predictor, continuous-outcome case underneath both of them — the trendline everyone has already drawn without naming it — rarely gets its own explanation of what the two numbers on the chart actually claim.

What the slope and its confidence interval actually claim

The regression coefficient, or slope, is the estimated average change in the outcome variable for a one-unit increase in the predictor.¹ That definition is doing more work than it looks like: the number is only interpretable in the units the predictor was measured in. A regression of body weight on height returns a slope of roughly 1.16 kg per centimetre; measure the same relationship in metres instead and the slope becomes roughly 115.9 kg per metre — the same underlying relationship, a very different-looking number, and a result that means nothing to a reader unless the units are stated alongside it.⁴

Whether that slope is distinguishable from a flat line (no relationship at all) is tested with a t statistic, and the 95% confidence interval around the slope is built from the same standard error: slope ± (critical t value × standard error).² If that interval excludes zero, the data are inconsistent with no linear relationship at conventional confidence — the same conclusion the t test and its P value give, reported as a range instead of a threshold. The interval says nothing about how much of the outcome the predictor explains; that is a separate question, answered by a separate number.

R² is not a grade, and it is not accuracy

R², the coefficient of determination, is the proportion of the total variability in the outcome that the regression line accounts for — formally, the explained sum of squares divided by the total sum of squares.² An R² of 0.17 means the predictor accounts for 17% of the variation in the outcome across the sample; it does not mean the model is 17% accurate, and it is not a percentage of patients whose outcome the line “got right.” A model can have an entirely valid, statistically significant slope and still leave the large majority of the outcome's variability unexplained, because most clinical and physiological outcomes are driven by more than one variable at once.⁵

What counts as a meaningful R² also depends on the field, not on a fixed universal cutoff: values that would be unremarkable in physical-science measurement are often unrealistic in clinical medicine, where an R² is doing the work of standing in for genetics, comorbidity, technique, and everything else that also drives the outcome.⁵ The number to report is R² itself, in the same sentence as the slope and its confidence interval — not a paraphrase of R² as how “good” or how “predictive” the model is.

A worked example

The following is illustrative, constructed for demonstration and not drawn from a real dataset. Suppose a single-centre series of 64 primary total hip arthroplasties records operative time (minutes) and estimated intraoperative blood loss (mL) for each case. A simple linear regression of blood loss on operative time returns a slope of 4.8 mL per minute (95% CI, 2.1-7.5), t₆₂ = 3.56, P < .001, and R² = 0.17. Written up in AMA format: “Operative time was positively associated with intraoperative blood loss (b = 4.8 mL/min; 95% CI, 2.1-7.5; R² = 0.17; P < .001).” Both things are true about that one line at once: the association is unlikely to be chance (the CI clears zero comfortably), and operative time on its own explains less than a fifth of why blood loss varied from patient to patient. A discussion section that reports the slope and CI but drops R² overstates how complete the explanation is; one that reports R² as “the model was 17% accurate” has converted a variance statistic into an accuracy statistic it was never built to be.

How to run it — SPSS vs StatsPlease

In IBM SPSS Statistics (documentation current to version 32), the path is Analyze ▸ Regression ▸ Linear….³ The outcome variable goes in the “Dependent” box and the single predictor goes in “Independent(s).” Under Statistics, “Estimates” is ticked by default and returns the coefficient, standard error, standardised beta, t, and significance; ticking “Confidence intervals” adds the range for each coefficient to the same table. Two output tables carry the numbers that matter: “Model Summary,” where R and R² live, and “Coefficients,” where the slope (unstandardised B), its confidence interval, t, and P value live. The most common misread is reading the standardised Beta in the Coefficients table as if it were the slope in the predictor's real units — it isn't, and it won't match the number you'd report in a Methods sentence.

In StatsPlease, the process collapses into upload and read. Upload the dataset (or the relevant columns). StatsPlease's deterministic engine identifies the appropriate test from the variable type and study design, checks the relevant assumptions, computes the result using fixed, non-LLM algorithms, and drafts the Methods/Results sentence in AMA format — the same number a reader would get running the test by hand in SPSS.

References

1. Kim HY. Statistical notes for clinical researchers: simple linear regression 1 – basic concepts. Restor Dent Endod. 2018;43(2):e21.
2. Kim HY. Statistical notes for clinical researchers: simple linear regression 2 – evaluation of regression line. Restor Dent Endod. 2018;43(3):e34.
3. Kim HY. Statistical notes for clinical researchers: simple linear regression 3 – residual analysis. Restor Dent Endod. 2019;44(1):e11.
4. Schneider A, Hommel G, Blettner M. Linear regression analysis: part 14 of a series on evaluation of scientific publications. Dtsch Arztebl Int. 2010;107(44):776-782.
5. Gupta A, Stead TS, Ganti L. Determining a meaningful R-squared value in clinical medicine. Acad Med Surg. 2024;1. doi:10.62186/001c.125154.