Which coefficient for which data, what regression coefficients actually mean, and the causal language that quietly gets theses into trouble.
Correlation measures how strongly two variables move together. Regression goes further and models an outcome from one or more predictors, giving you the size and direction of each relationship while holding the others constant. Neither establishes cause. The word “predictor” in regression describes a variable’s position in an equation, not its role in the world — and the gap between those two readings is where most regression chapters lose marks.
The choice follows from the data type and the shape of the relationship.
| Coefficient | Use with | Detects |
|---|---|---|
| Pearson’s r | Two continuous variables, roughly linear relationship | Linear association only |
| Spearman’s rho | Ordinal data, or continuous data that is skewed or has outliers | Monotonic association — consistently rising or falling, not necessarily straight |
| Kendall’s tau | Ordinal data, particularly small samples or many tied ranks | Monotonic association; often preferred over rho with many ties |
| Point-biserial | One continuous, one genuinely binary variable | Difference in means, expressed as a correlation |
Two points that repeatedly cause trouble.
Plot the data before trusting any coefficient. Pearson’s r measures straight-line association only. A strong curved relationship — performance rising with pressure and then falling — can return an r near zero, and reporting “no relationship” would be flatly wrong. A scatterplot takes seconds and prevents this.
Interpreting the size of r is field-specific. Conventional labels for small, medium and large correlations circulate widely, but what counts as a strong relationship differs enormously between disciplines — a correlation that would be unremarkable in psychometrics may be substantial in field research on human behaviour. Where your field has its own benchmarks, use and cite those rather than applying generic labels.
Squaring the coefficient gives the proportion of variance the two variables share, which is often more informative than r itself: a correlation of 0.30 sounds moderate until you note it corresponds to around nine per cent of shared variance.
Multiple regression estimates the relationship between each predictor and the outcome while holding the other predictors in the model constant. That final clause is the whole point, and it is what makes regression more useful than a series of correlations — and also what makes it easy to misread.
Unstandardised coefficients (B) are in the original units: a one-unit rise in the predictor is associated with a B-unit change in the outcome. These are what you report when the units are meaningful — years, pounds, points on a validated scale.
Standardised coefficients (beta) are expressed in standard deviations, which allows rough comparison of predictors measured on different scales within the same model. They should not be compared across different models or different samples, because the standard deviations differ.
R squared is the proportion of variance in the outcome explained by the model. Adjusted R squared penalises for the number of predictors, because adding predictors can only ever increase R squared, including useless ones. With several predictors, report the adjusted figure.
A caution on R squared: it measures explained variance, not importance or correctness. A model can explain a great deal and still be misspecified, and a low R squared is normal and acceptable in fields studying complex human behaviour. Treating it as a grade for your model is a misreading.
When the outcome is binary — passed or failed, readmitted or not — linear regression is inappropriate, because it can predict values below zero and above one. Logistic regression models the log-odds of the outcome instead.
The raw coefficients are in log-odds and are not directly interpretable. Exponentiating them gives odds ratios, which is what you report. An odds ratio above one means the odds of the outcome rise with the predictor; below one, they fall; and one means no association.
Two things to get right, because both are commonly wrong in theses.
An odds ratio is not a relative risk. They are close when the outcome is rare, and they diverge substantially when it is common. Reporting an odds ratio of 2.0 as “twice as likely” is a genuine error whenever the outcome is frequent, and it is the kind an examiner in health or social science will pick up.
Confidence intervals matter more than the p-value here. An interval crossing 1.0 indicates no reliable effect, and the width of the interval tells the reader how precisely the odds ratio has been estimated. Report them as standard.
Note also that the various pseudo R squared measures reported by software for logistic models are not equivalent to R squared in linear regression, are not interpretable as proportion of variance explained, and should be labelled by name rather than presented as a general measure of fit.
Predictors should be chosen from theory and prior evidence, decided before you look at the results. This sounds obvious and is routinely not what happens.
Stepwise selection — letting software add or remove predictors based on statistical criteria — is widely criticised in the methodological literature, and it is worth knowing why before using it. It capitalises on chance features of your particular sample, produces results that are unstable across samples, biases the resulting coefficients and standard errors, and substitutes an algorithm for the theoretical reasoning your thesis is supposed to demonstrate. Examiners familiar with the literature will ask about it, so if a supervisor recommends it, be ready to justify the choice or use a theory-driven model instead.
Adding predictors has costs. Each one uses degrees of freedom, and models with many predictors relative to the sample size fit the noise in your data rather than the pattern — which is why they perform badly on any new sample. Rules of thumb about cases per predictor circulate widely and disagree with each other; a power analysis tied to your own model is the defensible alternative.
Hierarchical entry, where variables are added in theoretically motivated blocks, is often the strongest approach in doctoral work: it lets you show what a set of predictors adds over and above controls, which is usually the actual research question.
This is where otherwise competent regression chapters get damaged, and the fix costs nothing.
Regression on cross-sectional data establishes association, adjusted for the other variables in the model. It does not establish that changing a predictor would change the outcome. But the vocabulary pulls hard in the other direction: “predictor”, “effect”, “explained variance”, “impact” and “determinant” all sound causal, and it is easy to drift into them without intending a causal claim.
Defensible: higher workload was associated with lower reported wellbeing, controlling for tenure and role.
Not defensible on cross-sectional data: workload reduces wellbeing, or workload had a significant impact on wellbeing.
Controlling for variables does not solve this. It removes the influence of the variables you measured; it can do nothing about the ones you did not, and there is no statistical procedure that turns an observational design into an experimental one. Worth also remembering that a variable measured at the same time as the outcome could as easily be a consequence of it as a cause — the equation is indifferent to which side of it a variable sits on.
For a correlation: the coefficient, the sample size, the exact p-value, and a confidence interval where your style guide expects one. State which coefficient you used and why, if it was not Pearson.
For a regression: the model R squared and adjusted R squared, the overall model test, and for each predictor the unstandardised coefficient with its standard error, the standardised coefficient, the p-value and ideally a confidence interval. For logistic models, odds ratios with confidence intervals.
Then the parts most often left out, and most often asked about: which assumptions you checked and what you found, how missing data was handled, the sample size the model actually ran on, and whether the analysis was planned in advance or exploratory. A correlation matrix of all study variables is expected in most fields and is worth including as standard.
Correlation summarises how strongly two variables move together in a single symmetric coefficient — it does not distinguish which variable is which. Regression models an outcome from one or more predictors, gives the size and direction of each relationship while holding the others constant, and can be used to predict values. Neither establishes causation on observational data.
Use Spearman when your data are ordinal, when a continuous variable is markedly skewed, or when outliers are distorting the relationship — it works on ranks, so it is unaffected by extreme values and detects any consistently rising or falling relationship rather than only a straight-line one. Plot the data either way: a strong curved relationship can produce a Pearson coefficient near zero, which would lead you to report no relationship where a clear one exists.
The proportion of variance in the outcome that the model accounts for. Report adjusted R squared when you have several predictors, because unadjusted R squared can only rise as predictors are added, including useless ones. It is not a measure of whether the model is correct or important — a misspecified model can explain a lot, and a low value is entirely normal in fields studying complex human behaviour.
Not on cross-sectional observational data, regardless of how many variables you control for. Controlling removes the influence of variables you measured and can do nothing about those you did not, and a variable measured at the same time as the outcome could as easily be its consequence as its cause. Causal claims need random allocation, a credible natural experiment, or measurement over time with alternative explanations argued away. Keep the wording associational and the chapter is far harder to challenge.
It is best avoided, and if a supervisor recommends it you should be ready to defend the choice. Stepwise selection is widely criticised because it capitalises on chance features of your specific sample, produces unstable results across samples, biases coefficients and standard errors, and replaces the theoretical reasoning your thesis exists to demonstrate with an algorithm. Choose predictors from theory and prior evidence, and consider hierarchical entry in theoretically motivated blocks.
It is the multiplicative change in the odds of the outcome for a one-unit rise in the predictor: above one means the odds increase, below one they decrease, and one means no association. Report it with a confidence interval, since an interval crossing one indicates no reliable effect. Do not describe it as “twice as likely” unless the outcome is rare — odds ratios and relative risks diverge substantially when the outcome is common, and conflating them is a recognised error.
Most regression problems are visible in the diagnostics and the wording rather than the output itself. Send your model and variables, and a PhD in your field will go through it with you.
Leave an email or a WhatsApp number — whichever you prefer — and tick how we should reach you. We reach out within 30 minutes.
