Why Put Balance in a Plot?
A before-and-after balance table records standardized differences precisely, but a long table can make it difficult to see which covariates changed most, whether some remain imbalanced, or whether one adjustment stage improved balance while another did not. A Love plot presents those same summaries graphically: covariates appear on one axis, and their absolute standardized differences appear on the other. Each adjustment stage is represented by a distinct marker or series.
The plot is a design diagnostic. It helps compare measured covariate balance before and after weighting, matching, or another adjustment. It does not show treatment effects, establish that treatment assignment is unconfounded, or certify that a design is adequate. The earlier material on standardized mean differences supplies the underlying calculations; here the focus is on displaying and comparing those values faithfully.
A standard layout puts covariates on the vertical axis and absolute standardized differences on the horizontal axis. The horizontal axis begins at zero and extends far enough to show every point. Distinct markers, colours, or line types identify adjustment stages; a legend states which is which. Vertical reference lines, when included, are labelled with their numerical values and described as selected reference values. They should not be mistaken for confidence limits or automatic pass/fail boundaries.
Using absolute values makes magnitudes easy to compare: a point at \(0.30\) is farther from zero than a point at \(0.08\). The cost is that the plot no longer reveals direction. If the treated group had a larger mean before adjustment, the absolute-value point does not show that fact. A signed balance table or a separate signed plot can preserve direction when it matters.
Building a Love Plot Carefully
The plot is only as useful as its inputs and labels. First choose the covariates to display and identify every adjustment stage. For example, stages might be unadjusted, propensity-score weighted, and matched. Next calculate the signed standardized difference for each covariate at each stage, using the corresponding weighted or matched summaries. For a fair visual comparison, use a consistent denominator convention across stages. The detailed choice of denominator belongs to the calculation; it should not change silently from one plotted stage to another.
Then take absolute values, place one point for each covariate-stage pair, and label the stages. Keep the same covariate ordering across stages. Ordering covariates by their largest plotted absolute difference can make the most pronounced imbalances easy to find, but ordering alphabetically or by a prespecified substantive grouping is also reasonable. State the ordering rule if the graph could otherwise be confusing.
Name the covariates and adjustment stages to be compared, including the unadjusted stage if it is useful as a baseline.
Calculate each signed standardized difference using the appropriate covariate formula and a consistent denominator convention across stages.
Use the absolute value for the horizontal coordinate, one row per covariate, and a distinct visual marker for each stage.
Label axes, units or covariate definitions as needed, the stage legend, and any numerical reference lines. Check that the axis includes all points and that no covariate has been omitted.
Worked Example: Comparing Three Stages for Two Covariates
Suppose a hypothetical treatment study reports age and smoking status at three stages: unadjusted, weighted, and matched. For age, use the unadjusted control-group standard deviation of 10 years as the fixed denominator. For smoking, use the unadjusted control proportion \(p_0=0.25\), giving a fixed denominator \(\sqrt{p_0(1-p_0)}\). These choices are held constant across stages so that the displayed changes are not caused by changing denominators.
For age, the unadjusted treated and control means are 58 and 53 years. Thus the signed standardized difference is \((58-53)/10=0.50\), and the plotted absolute value is \(0.50\). After weighting, the means are 56.4 and 55.4 years, so \((56.4-55.4)/10=0.10\). After matching, the means are 55.8 and 55.2 years, so \((55.8-55.2)/10=0.06\). The three age coordinates are therefore \(0.50\), \(0.10\), and \(0.06\).
For smoking, the fixed denominator is \(\sqrt{0.25(1-0.25)}=\sqrt{0.1875}=0.4330\). The unadjusted treated proportion is \(0.40\), so the signed difference is \((0.40-0.25)/0.4330=0.3464\), with plotted absolute value \(0.3464\). The weighted proportions are \(0.32\) and \(0.30\), giving \((0.32-0.30)/0.4330=0.0462\). The matched proportions are \(0.35\) and \(0.33\), giving the same difference, \((0.35-0.33)/0.4330=0.0462\).
The plot has two rows, age and smoking, and three points on each row. Its legend identifies unadjusted, weighted, and matched points. Both covariates have smaller displayed magnitudes after either adjustment stage than before adjustment. The plot does not tell us whether those changes imply acceptable balance; that judgment requires a separate, explicit assessment.
Reading Stages and Reference Lines
A Love plot is especially helpful when it shows several stages at once. On each covariate row, compare the horizontal positions of the stage-specific points. Movement toward zero indicates a smaller absolute standardized difference for that covariate under the plotted standardization. A point that remains far to the right relative to other covariates signals a covariate to inspect further; it does not by itself explain why the imbalance persists.
Reference lines help the reader orient to the scale. For instance, a plot might show vertical lines at \(0\), \(0.10\), and \(0.20\), each labelled. The zero line marks exact equality of the standardized summaries. The other lines are numerical reference values selected for display; their presence does not make them universal standards. The next step in interpreting standardized-difference thresholds is to consider what such values mean in context. Here, the essential plotting rule is to mark lines clearly and avoid describing them as formal significance cutoffs.
Use an axis that begins at zero. Truncating the horizontal axis can exaggerate small differences between plotted points, while an axis that extends far beyond the largest value can compress relevant variation. If a covariate has a very large value, do not silently remove it or set an axis limit that clips it. Show it, or clearly disclose any alternative scale or panel used.
Worked Example: Keeping Direction and Magnitude Straight
In a hypothetical matched sample, a continuous covariate has a signed standardized difference of \(-0.24\), while a binary covariate has a signed standardized difference of \(0.12\). A Love plot uses absolute values, so the plotted coordinates are \(|-0.24|=0.24\) and \(|0.12|=0.12\). The first point is farther from zero, so its absolute standardized difference is larger.
The plot cannot show that the continuous covariate’s signed difference was negative and the binary covariate’s was positive. A reader who needs to know which group had the higher mean or proportion must consult the signed values. The plot should not be described as showing that treatment and control are “above” or “below” one another: its horizontal positions represent magnitudes only.
If reference lines are drawn at \(0.10\) and \(0.20\), the first point lies to the right of both and the second lies to the right of the first line but to the left of the second. Those locations are descriptive comparisons with the drawn values, not evidence from a hypothesis test and not, by themselves, a decision about whether the design is adequate.
Multiple Categories and Clear Labelling
A categorical covariate with more than two levels may require more than one plotted value, depending on how its standardized differences were defined. A graph should not compress several level-specific differences into a single unexplained point. Either display each defined contrast or level with labels that identify it, or state a prespecified summary and how it was calculated. The plot must correspond to the quantities in the balance table.
Likewise, define abbreviations and use readable covariate names. If a table uses “BMI” and “baseline HbA1c,” the plot should not use unexplained codes such as “C3” and “C7.” If many covariates make the plot crowded, divide it into clearly named panels or use a larger figure rather than dropping variables without explanation. A plot intended to assess balance should include the covariates relevant to the design, not merely those with attractive results.
Worked Example: Auditing a Three-Stage Plot
A hypothetical balance table gives the following absolute standardized differences for three covariates:
| Covariate | Unadjusted | Weighted | Matched |
|---|---|---|---|
| Age | 0.42 | 0.11 | 0.08 |
| Baseline score | 0.18 | 0.23 | 0.14 |
| Smoking | 0.31 | 0.09 | 0.13 |
The correct plot has three points per covariate row. For age, the points are at \(0.42\), \(0.11\), and \(0.08\). For baseline score, they are at \(0.18\), \(0.23\), and \(0.14\); the weighted value is larger than the unadjusted value. For smoking, they are at \(0.31\), \(0.09\), and \(0.13\). A horizontal axis from \(0\) to \(0.45\) includes all nine values without clipping, and the legend identifies all three stages. Reference lines at \(0\), \(0.10\), and \(0.20\) are visibly labelled as reference values.
This example shows why multiple stages should be plotted together: weighting reduces the displayed magnitude for age and smoking but increases it for baseline score, while matching yields a different pattern. A statement that “adjustment improved balance” would conceal these covariate-specific differences. The plot supports a focused description of the pattern; it does not determine which stage should be selected without considering the design and the full set of diagnostics.
Diagnostics, Alternatives, and Limitations
Several checks can reveal a poorly constructed or misleading Love plot:
- Missing or mismatched stages. Verify that every stage in the legend has values for the same covariate set, or clearly explain any unavailable values.
- Inconsistent denominators. Recheck the calculations if a plotted change may reflect a changed standardization denominator rather than changed group summaries.
- Clipped points or misleading axis limits. Confirm that every point is visible and that the horizontal axis starts at zero.
- Unexplained categorical summaries. Make clear what each point represents when a variable has several levels.
- Confusing markers or reference lines. Use a legible legend and label reference values; do not make them look like confidence intervals.
A balance table is the main alternative for presenting the same standardized differences. It is more precise for looking up individual numbers and may be easier to audit, while the plot makes patterns across covariates and stages easier to scan. They complement one another: a plot can summarize the pattern, and a table can supply exact values. Neither replaces examination of other features of the covariate distributions. In particular, similar standardized differences do not establish that distributional shapes or relationships among covariates are similar.
A Love plot also has limits that follow from what it displays. It summarizes selected covariates, selected contrasts, and selected stages using standardized differences. It does not diagnose unmeasured confounding, establish causal identification, show uncertainty in the displayed values, or demonstrate that all relevant aspects of balance have been achieved. Treat it as a transparent visual summary of a defined set of calculations, not as a complete evaluation of an observational design.
Common Mistake
Wrong reasoning: “Every point is close to a reference line, so the study is balanced and the treatment effect is unbiased.” This confuses a descriptive display with a universal decision rule and a causal guarantee. Correction: report which covariates and stages are shown, how the standardized differences were calculated, and what the plot displays. Assess the pattern using a justified criterion and the broader design context; balance on plotted measured covariates does not establish balance on unmeasured variables or guarantee an unbiased effect estimate.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A covariate has signed standardized differences \(0.28\) before adjustment and \(-0.09\) after adjustment. What are the Love plot coordinates, and what information is lost?
- Why should the same denominator convention generally be used across stages in a before-and-after Love plot?
- A three-stage plot shows age values of \(0.35\), \(0.12\), and \(0.07\), and smoking values of \(0.08\), \(0.22\), and \(0.10\). Which stage has the smallest age magnitude? Does one stage have the smallest value for both covariates?
- What does a reference line at \(0.10\) establish, and what does it not establish?
- A categorical covariate has three levels, but the plot shows one point labelled only with the variable name. What should be checked?
Solutions. (1) The coordinates are \(0.28\) and \(0.09\). Taking absolute values removes the direction of the signed differences, so the plot does not show which group had the larger mean or proportion. (2) A fixed denominator makes stage-to-stage changes more directly comparable; otherwise a change in the plotted value could reflect a changed denominator as well as changed group summaries. (3) The third stage has the smallest age magnitude, \(0.07\). The first stage has the smallest smoking magnitude, \(0.08\), so no single stage is smallest for both covariates. (4) It marks the numerical location \(0.10\) on the displayed scale. It does not by itself establish a universal adequacy rule, statistical significance, or unbiased causal estimates. (5) Check how the categorical standardized differences were defined and whether the single point is an explicitly described summary; otherwise display and label the relevant level-specific values.