Why the Table Needs More Than a Threshold
A tipping point is easy to quote and easy to misread. A statement such as “the conclusion tipped at 1.4” does not tell readers what changed, how the outcome was measured, whether the conclusion depended on an estimate or an uncertainty interval, or how finely the sensitivity parameter was searched. A useful table keeps those pieces together: sensitivity settings, treatment-effect estimates, uncertainty intervals, decision status, and the precision supported by the calculation.
The table reports results of a specified sensitivity analysis; it does not establish that any sensitivity setting is true or plausible. Its purpose is to let a reader see what the analysis concludes across the settings considered and exactly how the reported threshold was obtained.
Specify the Rule Before Formatting the Results
A table can only be interpreted if the analysis choices behind it are explicit. State the treatment-effect contrast and its units, the outcome scale and direction, the sensitivity parameter and its units, and what a change in that parameter means. Also state the inferential procedure, the confidence level of the interval, and the decision rule. For example, a rule might classify evidence as “supports a positive effect” when the lower endpoint of a two-sided 95% confidence interval is greater than zero. That is not the same rule as classifying by whether the point estimate is positive.
Keep the rule stable across rows unless a row-specific rule is necessary and clearly explained. If the analysis uses the confidence-interval conclusion, do not label a row “no effect”: an interval that includes zero does not demonstrate that the effect is zero. Prefer precise labels such as “supports positive effect” and “does not support positive effect under this rule.” A table may use shorter labels, but define them immediately.
The estimate and interval must correspond to the same sensitivity setting and analysis. They should also be reported on the same effect scale throughout the table. If a transformation or a different scale is needed for a particular analysis, identify it rather than placing unlike estimates in one column without warning.
Worked Examples
Worked Example: A One-Parameter Table With an Interval-Based Rule
Consider a hypothetical randomized study with a continuous outcome measured in points, where positive treatment-minus-control differences favor treatment. Let \(\delta\) be a sensitivity parameter in outcome points, with larger values representing progressively less favorable assumptions for the treatment group. Suppose the stipulated sensitivity analysis gives an estimate \(\hat{\theta}(\delta)=1.20-0.30\delta\) points and a standard error of 0.40 points at every setting. Use a two-sided 95% interval with normal critical value 1.96, and classify a result as supporting a positive effect only if the lower endpoint is greater than zero.
At each setting, the interval is \(\hat{\theta}(\delta)\pm1.96(0.40)\), so its half-width is \(0.784\) points. The lower endpoint is therefore \(1.20-0.30\delta-0.784\). The threshold solves
The positive-effect conclusion applies below this value and does not apply at or above it, because the rule requires the lower endpoint to be strictly greater than zero. The arithmetic for selected rows is shown in the table. Values are hypothetical; the estimate and standard error are stipulated components of this example.
| \(\delta\) (points) | Estimate (points) | 95% confidence interval (points) | Status |
|---|---|---|---|
| 0.0 | 1.20 | [0.416, 1.984] | Supports positive effect |
| 1.0 | 0.90 | [0.116, 1.684] | Supports positive effect |
| 1.3 | 0.81 | [0.026, 1.594] | Supports positive effect |
| 1.4 | 0.78 | [-0.004, 1.564] | Does not support positive effect |
For example, at \(\delta=1.3\), the estimate is \(1.20-0.30(1.3)=0.81\) points and the interval is \(0.81\pm0.784=[0.026,1.594]\) points. The status changes between the evaluated settings 1.3 and 1.4. The formula supports an interpolated threshold of 1.3867 under the stipulated linear estimate and constant standard error. A clear report would say both things: the grid brackets the change between 1.3 and 1.4, while the model-based calculation gives approximately 1.387 points. It would not present 1.3867 as if the data had directly identified the threshold to four decimal places.
Worked Example: Showing Two Sensitivity Parameters in One Table
Suppose a hypothetical analysis evaluates two sensitivity parameters, \(\delta_A\) and \(\delta_B\), each measured in outcome points. Positive treatment-minus-control estimates favor treatment. At every row the standard error is 0.30 points, so the two-sided 95% interval is the estimate plus or minus \(1.96(0.30)=0.588\) points. The stated decision rule again requires the lower endpoint to exceed zero.
For the setting \((\delta_A,\delta_B)=(0.5,0.5)\), suppose the estimate is 0.55 points. The interval is \(0.55\pm0.588=[-0.038,1.138]\) points, so the row does not support a positive effect under the specified rule. A compact table retains both parameter values rather than compressing them into an ambiguous label such as “scenario 3.”
| \(\delta_A\) (points) | \(\delta_B\) (points) | Estimate (points) | 95% confidence interval (points) | Status |
|---|---|---|---|---|
| 0.0 | 0.0 | 0.90 | [0.312, 1.488] | Supports positive effect |
| 0.5 | 0.0 | 0.72 | [0.132, 1.308] | Supports positive effect |
| 0.0 | 0.5 | 0.65 | [0.062, 1.238] | Supports positive effect |
| 0.5 | 0.5 | 0.55 | [-0.038, 1.138] | Does not support positive effect |
| 1.0 | 0.5 | 0.30 | [-0.288, 0.888] | Does not support positive effect |
The first three rows support a positive effect, while the last two do not. This table shows outcomes at the displayed combinations; it does not claim that every point between them has the same status. A two-parameter tipping boundary cannot generally be summarized by one threshold number. Pairing each sensitivity value with its estimate and interval makes the table interpretable, while the pattern across rows can be considered alongside a robustness-region summary.
Worked Example: Reporting Grid Resolution and an Interpolated Threshold
In another hypothetical one-parameter analysis, let the estimate be \(\hat{\theta}(\delta)=0.90-0.20\delta\) points and the standard error be 0.30 points. Use the same 95% interval rule: support for a positive effect requires the lower endpoint to exceed zero. The interval half-width is \(1.96(0.30)=0.588\) points. Suppose the analysis evaluates a grid in increments of 0.1 near the change in status.
At \(\delta=1.5\), the estimate is \(0.90-0.20(1.5)=0.60\), giving interval \(0.60\pm0.588=[0.012,1.188]\) points. At \(\delta=1.6\), the estimate is \(0.90-0.20(1.6)=0.58\), giving interval \(0.58\pm0.588=[-0.008,1.168]\) points. Thus the grid directly establishes a change between 1.5 and 1.6.
| \(\delta\) (points) | Estimate (points) | 95% confidence interval (points) | Status |
|---|---|---|---|
| 1.5 | 0.60 | [0.012, 1.188] | Supports positive effect |
| 1.6 | 0.58 | [-0.008, 1.168] | Does not support positive effect |
If the stipulated linear estimate and constant standard error are used to solve for the crossing, the calculation is
A careful report describes the grid bracket as \([1.5,1.6]\) and the interpolated value as approximately 1.56 under the specified formula. The second number is not more precise evidence from a finer search: it depends on the model between grid points and the assumed fixed standard error. If the analysis only evaluated the grid and did not justify interpolation, report the bracket, not an exact-looking value.
Deriving and Checking the Reported Threshold
The previous calculations illustrate a general justification. Suppose the decision is based on whether the lower endpoint of a two-sided interval exceeds zero. Write the estimate as \(\hat{\theta}(\delta)\), its standard error as \(SE(\delta)\), and the appropriate critical value as \(c\). The lower endpoint is \(\hat{\theta}(\delta)-cSE(\delta)\). A threshold at which the rule changes solves
If \(\hat{\theta}(\delta)=a-b\delta\), \(SE(\delta)=s\), and \(b>0\), solving this equation gives \(\delta^*=(a-cs)/b\). This expression is meaningful as a tipping threshold only if the starting conclusion has the stated status, the crossing lies in the sensitivity range being reported, and the status actually changes there. If the standard error varies with \(\delta\), the equation may require numerical solution; a table should then show the evaluated settings that bracket the crossing and describe any interpolation method.
Before finalizing a table, check that each interval follows from the stated procedure and that every status follows from the stated rule. Check the displayed precision against the underlying calculation. In particular, rounding an interval endpoint near zero can change its apparent status: retain enough digits to classify the row correctly, or show an unrounded status with a note that the displayed endpoint is rounded. Also verify that rows are sorted or grouped in a way that helps readers follow the change, and that any omitted grid points are not needed to support the stated bracket.
Choosing a Table Alongside Other Summaries
A table is strongest when readers need exact values at selected sensitivity settings: they can compare estimates, uncertainty, and status without estimating coordinates from a graphic. A figure can show a broad pattern or a boundary across many settings more efficiently, especially with two parameters. A concise narrative can then state the principal threshold and its interpretation. These are complementary forms, not competing definitions of the result: a narrative threshold alone cannot show the uncertainty intervals or the behavior at nearby settings, while a long table can obscure a broad pattern.
For multiple outcomes or analyses, make the relevant outcome or analysis identifiable in the table, either with a column or clearly separated panels. Keep units and decision rules visible when they differ. Do not combine rows with different outcome scales or interval-based criteria in a way that invites direct comparison without explanation. A table also cannot establish that the sensitivity values are scientifically plausible; that judgment requires a rationale stated in context.
Common Pitfall and Limitations
- Reporting only “the threshold was 1.4.” This hides the sensitivity parameter’s meaning, the decision rule, and whether 1.4 was a grid point or an interpolation. State the parameter and units, show nearby rows or the bracket, and identify the calculation behind any interpolated value.
- Calling an interval that includes zero “no effect.” Under a rule requiring an interval to exclude zero, the correct status is that the analysis does not support the specified positive-effect conclusion at that setting. The interval does not prove the effect is zero.
- Showing too few digits to reproduce the status. If a displayed lower limit rounds to 0.00, a reader may not be able to tell whether it is above or below zero. Give adequate precision or add an explicit note about classification using unrounded values.
- Assuming a table of selected rows establishes the full boundary. A few evaluated combinations show only those combinations. Do not claim a continuous region or an exact boundary unless the analysis supports that claim.
Tables are also conditional on the analysis choices they summarize. They do not account automatically for uncertainty in the estimated threshold, alternative model specifications, or uncertainty about which sensitivity settings are plausible. When those issues matter, report them as separate aspects of the sensitivity analysis rather than suggesting that decimal precision resolves them.
Check Your Understanding
Answer each question using the decision rule stated in the question. Brief solutions follow.
- A table uses the rule “supports a positive effect if the 95% interval’s lower endpoint is greater than zero.” At one setting the estimate is 0.4 points and the interval is [-0.1, 0.9] points. What status should the row receive, and what should it not be called?
- A status changes between evaluated sensitivity values 2.0 and 2.5. The analysis does not use interpolation. What threshold precision is directly supported?
- For \(\hat{\theta}(\delta)=1.0-0.2\delta\) points with fixed standard error 0.25 points, find the threshold under the rule that the two-sided 95% interval supports a positive effect only when its lower endpoint is above zero. Use \(c=1.96\).
- In a two-parameter table, what should be reported so a reader can identify the sensitivity setting for each estimate and interval?
Solutions. (1) “Does not support a positive effect under this rule,” because \(-0.1\) is not greater than zero. It should not be called “no effect.” (2) Report that the evaluated grid brackets the change between 2.0 and 2.5; without interpolation, the table does not support a more precise threshold. (3) The interval half-width is \(1.96(0.25)=0.49\) points. Solve \(1.0-0.2\delta-0.49=0\), giving \(\delta=0.51/0.2=2.55\). (4) Give both sensitivity-parameter values in separate, labeled columns on the same row as the estimate, interval, and status, with their units and the decision rule stated.