Direct comparison
Random Intercepts vs. Random Slopes
Random intercepts give groups their own baseline; random slopes let an effect vary by group too. When each is justified, and the real cost of adding one.
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How do Random Intercept Only, Random Intercept + Random Slope compare side by side?
The table below compares Random Intercept Only, Random Intercept + Random Slope across 9 procurement-relevant dimensions, from what varies by group through typical software default.
Side-by-side comparison
| Dimension | Random Intercept Only | Random Intercept + Random Slope |
|---|---|---|
| What varies by group | Only the group's baseline (starting) level of the outcome. | The group's baseline AND the strength/direction of a chosen predictor's effect. |
| lme4 syntax (R) | (1 | group) | (1 + x | group) — correlated intercept/slope by default; (1 + x || group) forces them uncorrelated. |
| Parameters added per random effect | One variance component (intercept variance). | Three: intercept variance, slope variance, and the intercept–slope correlation (two if fit uncorrelated). |
| Minimum data to estimate reliably | Works with as few as 2 observations per group. | Needs enough within-group observations across enough distinct values of the predictor to estimate a group-specific trend — a handful of groups or one observation per group at a single predictor value cannot support it. |
| When it is the right call | Groups plausibly differ in overall level, but there is no reason to think a predictor's effect itself differs by group (e.g., patients differ in baseline severity, but the drug is assumed to work the same way for everyone). | The research question is partly ABOUT whether an effect varies by group, or the study design repeats the same predictor manipulation within each group (e.g., a within-subjects experiment, or a treatment effect that plausibly differs by clinic). |
| Effect on the fixed-effect standard error | Corrects standard errors for within-group correlation in the outcome level only. | Also corrects for the fact that the sample's average slope is itself only one draw from a distribution of group-level slopes — omitting a needed random slope is a well-documented source of anti-conservative (too-small) standard errors and inflated Type I error on that predictor's fixed effect. |
| Convergence / singular-fit risk | Low — a single variance component is usually well-identified even with modest data. | Higher — the extra variance and correlation parameters are a common source of "boundary (singular) fit" warnings when there is too little data per group to support them; the standard response is to simplify (drop the correlation term, or drop the random slope) rather than ignore the warning. |
| How to decide statistically | Baseline model. | Fit both models with ML (not REML) and compare via a likelihood-ratio test, or use a model-selection criterion (AIC/BIC) — Matuschek et al. (2017) showed that always including the theoretically maximal structure ("keep it maximal," per Barr, Levy, Scheepers & Tily 2013) can cost real power on typical datasets, and recommend a data-justified structure instead. |
| Typical software default | Often the starting point researchers reach for first, sometimes under-specifying a design that actually needs a random slope. | Not fit unless explicitly specified — no mainstream package adds a random slope automatically. |
Common questions
Common questions about Random Intercept Only vs Random Intercept + Random Slope
What is the actual difference between a random intercept and a random slope?
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A random intercept lets each group have its own baseline value for the outcome while assuming every predictor affects all groups identically. A random slope lets a specific predictor's effect itself differ from group to group — for example, letting a treatment's effect on symptom score vary from clinic to clinic, not just letting clinics differ in starting severity.
Do I need a random intercept before I can add a random slope?
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In standard practice, yes — a random slope for a predictor is specified in addition to a random intercept for the same grouping factor, not as a substitute for it. Fitting a random slope without a random intercept is possible in most software but is rarely the right model: it forces every group's regression line through the same starting point, which is usually not a realistic assumption.
How do I know if my design needs a random slope?
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Two questions: does the predictor vary WITHIN each group (a between-group-only predictor cannot have a meaningful random slope), and is there a substantive reason to expect its effect differs by group? If both are yes and you have enough within-group data to estimate it, a likelihood-ratio test (fit with ML) or AIC/BIC comparison against the random-intercept-only model is the standard way to decide.
Why did adding a random slope produce a singular-fit warning?
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A singular or boundary fit means the data can't support the extra variance and correlation parameters a random slope adds — typically because there are too few groups, too few observations per group across the predictor's range, or the true group-to-group variation in that effect is genuinely close to zero. The fix is to simplify the random-effects structure (drop the intercept-slope correlation term, or drop the random slope itself), not to publish the model with the warning unresolved.
Should I always fit the "maximal" random-effects structure recommended by Barr et al. (2013)?
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Not automatically. Barr, Levy, Scheepers & Tily (2013) argued for including every random slope the design justifies, to keep Type I error at its nominal rate. Matuschek et al. (2017) showed this can cost substantial statistical power on typical datasets and demonstrated that a data-justified (reduced, model-selected) structure often balances Type I error and power better than the maximal structure by default. Treat "maximal" as a starting point to test against simpler alternatives, not a rule to apply unconditionally.
Going deeper
