Why the Outcome Scale Matters
A difference-in-differences (DiD) contrast compares the change in an outcome for a treated group with the change for a comparison group. But “change” depends on the scale of the outcome. A change of 10 dollars is additive; a change from 100 to 110 dollars is a 10 percent increase. Both descriptions may be useful, but they are not interchangeable. Choosing the scale is therefore part of defining the causal question, not merely a technical choice made when fitting a model.
On the original scale, a DiD estimates an additive effect when parallel trends holds for the untreated outcome on that scale. On a logarithmic scale, it instead estimates an effect in mean log outcomes, which can be expressed as a ratio contrast. A log(1 + outcome) transformation accommodates zero values, but its contrast concerns 1 plus the outcome, not the outcome itself. These interpretations require different care when reporting results.
Define the Contrast Before Interpreting It
Let \(G=1\) denote the treated group and \(G=0\) the comparison group. Let \(t=0\) denote the pre-intervention period and \(t=1\) the post-intervention period. For a chosen transformation \(g\), define the observed group-period mean on that scale as \(m^g_{a,t}=E[g(Y)\mid G=a,t]\), where \(a\) is 0 or 1. The population DiD contrast on that scale is
For the additive outcome scale, \(g(y)=y\). For a log outcome, \(g(y)=\log(y)\), which requires positive outcomes. For an outcome that can be zero, a common alternative is \(g(y)=\log(1+y)\). Each transformation defines a different contrast.
If the intervention is absent before the post period, the observed pre-period outcome for the treated group equals its untreated pre-period potential outcome. If the comparison group remains untreated, its observed outcomes are untreated outcomes in both periods. Under consistency, no anticipation, no interference between units, and parallel trends on the chosen scale, the DiD contrast identifies the treated group’s average post-period effect on that scale:
For \(g(y)=y\), this is the average additive treatment effect among treated units in the post period. For \(g(y)=\log(y)\), it is an average effect in log outcomes, not an average effect in outcome units. The assumption is about mean changes on the chosen scale; it does not require the groups to have equal baseline levels.
Worked Example: An Additive Effect in Outcome Units
Worked Example: An Additive Effect in Outcome Units
Suppose a hypothetical program changes monthly earnings. The following are group means, in dollars per month, for treated and comparison workers:
| Group | Before | After |
|---|---|---|
| Treated | $1,800 | $2,050 |
| Comparison | $2,000 | $2,100 |
The additive-scale DiD is
The treated group’s mean earnings rose by $250 per month, while the comparison group’s rose by $100 per month. The DiD is therefore $150 per month. Its causal interpretation requires that, without the program, the treated group’s mean earnings would have increased by the same $100 as the comparison group’s. Under that additive parallel-trends assumption and the other DiD conditions, the estimate is an average increase of $150 per month for treated workers in the post period.
Why Transforming the Outcome Changes the Question
A nonlinear transformation generally does not preserve mean differences. In particular, the mean of log outcomes is not the log of the arithmetic mean outcome:
Thus a log-scale DiD must be calculated from group-period means of individual log outcomes, or from a model that estimates the corresponding mean-log contrast. Taking logs of the four arithmetic cell means instead answers a different question when outcomes vary within cells.
For positive outcomes, a log-scale contrast can also be written as a log ratio-of-ratios:
Exponentiating gives a ratio-of-ratios. If the log-scale parallel-trends assumption holds, the exponentiated causal contrast is the ratio of geometric means of the treated group’s post-period potential outcomes with and without treatment. Subtracting 1 expresses that ratio as a proportional difference in geometric means. It is not generally the percentage change in arithmetic means, nor the average individual percentage effect.
Worked Example: The Same Data on Additive and Log Scales
Suppose, hypothetically, every unit in each group-period cell has the same positive monthly earnings, measured in dollars. The cell values are $80 before and $100 after for the treated group, and $100 before and $110 after for the comparison group. Because each cell is constant, its arithmetic mean and geometric mean equal the stated value.
On the additive scale, the contrast is
On the log scale, the contrast is
Exponentiating gives \(\exp(0.1278)\approx1.136\), or a ratio-of-ratios about 13.6 percent above 1. For a causal interpretation on this scale, the treated group’s untreated mean log earnings must have followed the same change as the comparison group’s mean log earnings. The comparison group’s earnings ratio is \(110/100=1.10\); that trend would predict untreated post-period earnings of \(80(1.10)=88\) for the treated group. The observed $100 is about 13.6 percent above $88.
The additive contrast of $10 has a different counterfactual implication: additive parallel trends would predict untreated post-period earnings of \(80+(110-100)=90\), making the additive effect $10. The two contrasts answer different questions and invoke different parallel-trends assumptions. The observed cell means alone do not establish which assumption is more credible.
Using Log(1 + Outcome) When Zeros Occur
If some outcomes are zero, \(\log(Y)\) is undefined for those observations. The transformation \(\log(1+Y)\) is defined for nonnegative outcomes, including zero. Its DiD is still a contrast in group-period means, now of the individual values \(\log(1+Y)\). Exponentiating that contrast gives a ratio of geometric means of \(1+Y\); it does not give a ratio of geometric means of \(Y\) or a direct percentage effect on \(Y\).
The added 1 also depends on the outcome’s units. For counts, adding one count is well-defined, but for a continuous measure, changing units changes the numerical value of \(\log(1+Y)\). The scale and units therefore need to be reported explicitly. A transformation that is convenient computationally is not automatically the most meaningful causal scale.
Worked Example: Calculating a Log(1 + Outcome) DiD With Counts
Consider hypothetical nonnegative count outcomes, with two individuals in each group-period cell. These individual observations avoid treating a fractional count as though it were an individual count.
| Group | Before counts | After counts |
|---|---|---|
| Treated | 0, 2 | 1, 3 |
| Comparison | 0, 0 | 0, 2 |
For the treated group before the intervention, the mean transformed outcome is \((\log(1)+\log(3))/2=0.5493\). After the intervention it is \((\log(2)+\log(4))/2=1.0397\). For the comparison group, the corresponding means are \(0\) and \((\log(1)+\log(3))/2=0.5493\). Therefore,
Exponentiating gives \(\exp(-0.0589)\approx0.9428\). Under parallel trends on the log(1 + count) scale, the treated group’s post-period geometric mean of 1 plus the count is estimated to be about 5.72 percent lower than its counterfactual value without treatment. This is not a 5.72 percent decrease in the arithmetic mean count.
For comparison, the arithmetic mean counts are 1 and 2 for the treated group, and 0 and 1 for the comparison group. Their additive DiD is \((2-1)-(1-0)=0\) counts. The distinct results reflect distinct scales and assumptions. Also, the transformed treated-before mean is 0.5493, whereas \(\log(1+\text{mean count})=\log(2)\approx0.6931\). Substituting the cell’s mean count into the transformation would not reproduce the required mean of individual transformed outcomes.
A Short Justification of the Scale-Specific Result
The causal interpretation follows by adding and subtracting the treated group’s unobserved mean transformed outcome after the intervention, had treatment not occurred. Write \(m^g_{1,1}(0)=E[g(Y_1(0))\mid G=1]\) and \(m^g_{1,1}(1)=E[g(Y_1(1))\mid G=1]\). No anticipation makes the treated group’s observed pre-period mean equal to its untreated pre-period mean. The comparison group is untreated in both periods. Thus the observed DiD can be rearranged as
The second line separates the treated group’s average causal effect on the \(g\)-scale from the difference between the treated and comparison groups’ untreated trends on that scale. Parallel trends makes the latter difference zero, leaving the average treated effect. Changing \(g\) changes both that effect and the assumption needed to identify it.
Choosing and Reporting a Scale
A defensible scale is guided by the scientific question and by how outcome changes should be compared. An additive scale is natural when a fixed change in outcome units matters. A logarithmic scale may be useful when proportional changes are meaningful and outcomes are strictly positive. Log(1 + outcome) can retain zero observations, but its interpretation depends on the unit and remains a transformed-scale effect.
A saturated 2-by-2 regression of \(g(Y)\) on group, post period, and their interaction has an interaction coefficient equal to the DiD in \(g(Y)\). The coefficient’s interpretation follows the transformation: outcome units for \(g(Y)=Y\), log units for \(g(Y)=\log(Y)\), and log(1 + outcome) units for \(g(Y)=\log(1+Y)\). For a log outcome, exponentiating the interaction gives a ratio-of-ratios; for log(1 + outcome), it gives the corresponding ratio for 1 plus the outcome. Neither transformation silently converts a contrast into an additive effect in the original units.
When reporting a result, name the scale, state the units, identify the parallel-trends assumption that supports the causal claim, and describe the estimand in those terms. If the substantive question concerns original-unit effects but the analysis uses a transformed outcome, a transformed-scale estimate alone does not answer that question without further assumptions and a suitable method for translating scales.
Common Mistakes
- Assuming parallel trends is scale-free. Equal additive changes do not imply equal proportional changes. State parallel trends on the actual analysis scale.
- Taking the log of cell means and calling it a mean log. In general, \(\log(E[Y])\ne E[\log(Y)]\). Transform individual observations before averaging when the target is a mean transformed outcome.
- Calling a log interaction a percent change in the arithmetic mean. Exponentiation gives a ratio contrast in geometric means on the relevant scale, not generally an arithmetic-mean percentage effect.
- Treating log(1 + outcome) as a direct fix with no interpretive cost. It changes the estimand, and its numerical values depend on the units used for the outcome.
- Choosing a scale because its estimate is more favorable. The scale should reflect the scientific question and be selected for substantive reasons, not to obtain a preferred result.
Check Your Understanding
Use the scale-specific definitions and interpretations to answer the following questions.
- A study reports an additive DiD of 4.2 clinic visits per year. What is the causal interpretation under additive parallel trends?
- Why does a log-scale DiD generally differ from the log of an additive-scale DiD calculated from arithmetic cell means?
- A log-outcome interaction is 0.08. What ratio-of-ratios does it imply, approximately, and what does that number not establish by itself?
- Why must the individual observations be transformed before calculating a mean on the log(1 + outcome) scale?
- For a count outcome with zeros, what does exponentiating a log(1 + count) DiD describe, and why is it not automatically a percentage effect on the count itself?