Why scale conversion matters
Clinical experts often express treatment effects in the units they use to discuss patient outcomes: points on a symptom scale, minutes of recovery time, or a reduction in a measured score. A trial model may use a different unit, or may express the effect relative to a standard deviation. A prior must describe the same clinical belief after conversion; changing the number without changing its uncertainty, direction, or reference scale can produce a materially different prior.
The safest approach is to define the treatment contrast and the scale transformation before writing down the model-scale prior. In particular, make explicit which arm is subtracted from which, whether larger outcomes are better or worse, and what quantity is used to standardize a raw effect.
Transform the whole belief, not just its center
Suppose a prior belief about an effect in clinical units is represented by a random variable \(X\), with mean \(m\) and variance \(v\). If the model uses the affine conversion \(Y=a+bX\), where \(a\) and \(b\) are fixed and \(b\ne 0\), then the transformed mean and variance are
The variance uses \(b^2\), not \(b\). Thus, if \(X\sim N(m,v)\), then \(Y\sim N(a+bm,b^2v)\). The standard deviation is multiplied by \(|b|\). A negative \(b\) reverses the direction of the effect, but does not make its uncertainty negative.
This result follows directly by writing \(X=m+\sqrt{v}Z\), where \(Z\sim N(0,1)\). Then
A linear transformation of a normal random variable is normal; its center is \(a+bm\), and the squared coefficient of \(Z\) gives variance \(b^2v\). More generally, any interval \([L,U]\) transforms to \([\min(a+bL,a+bU),\max(a+bL,a+bU)]\). When \(b<0\), the transformed endpoints switch order.
For a treatment contrast, adding the same constant to all scores does not change the difference: \((\bar{Y}_T+c)-(\bar{Y}_C+c)=\bar{Y}_T-\bar{Y}_C\). Multiplying scores by a conversion factor does change the contrast and its uncertainty by that factor. This distinction is useful when a scale has been shifted to a new zero point or converted to different units.
Worked Example: Converting a Symptom Score to a Ten-Point Scale
Suppose a hypothetical symptom score ranges from 0 to 100, with higher scores indicating worse symptoms. The treatment effect is defined as treatment mean minus control mean, so a negative value favors treatment. A clinician’s prior on this difference is \(\theta_{100}\sim N(-8,6^2)\) score points. The model records the same score on a 0-to-10 scale, obtained by dividing every score by 10. Convert the prior.
Because both arm scores are divided by 10, their difference is also divided by 10: \[ \theta_{10}=0.1\theta_{100}. \] Therefore, \[ E(\theta_{10})=0.1(-8)=-0.8\text{ points},\qquad \operatorname{SD}(\theta_{10})=0.1(6)=0.6\text{ points}. \] The variance is \(0.1^2(6^2)=0.36\) squared points, so \[ \theta_{10}\sim N(-0.8,0.36). \]
For a check on the uncertainty conversion, the central 90% interval under the original normal prior is approximately \(-8\pm1.645(6)\), or \([-17.87,1.87]\) points. Dividing both endpoints by 10 gives \([-1.787,0.187]\) points on the ten-point scale. The converted interval remains a 90% interval under the transformed prior. Its negative values still represent benefit under the stated direction convention.
Direction is a scale choice too
Clinical language and model notation do not always point in the same direction. A clinician might describe “improvement” as a positive number, while a model defines the effect as treatment minus control on a scale where lower scores are better. These are opposite contrasts. Convert the sign before using a belief in the model; otherwise, the prior can favor the wrong treatment effect.
Worked Example: Converting Positive Improvement to a Model Contrast
In a hypothetical fatigue trial, the original score ranges from 0 to 40, with higher scores indicating worse fatigue. A clinician describes improvement as control mean minus treatment mean and assigns it the prior \(B\sim N(6,3^2)\) points. The trial model instead defines \(\theta_{40}\) as treatment mean minus control mean. The model then reports scores on a 0-to-100 scale, obtained by multiplying the original scores by \(2.5\). Find the model-scale prior.
First reverse the contrast: \[ \theta_{40}=-B. \] Thus, \(E(\theta_{40})=-6\) points and \(\operatorname{Var}(\theta_{40})=3^2=9\) squared points. Now convert units: \[ \theta_{100}=2.5\theta_{40}. \] It follows that \[ E(\theta_{100})=2.5(-6)=-15\text{ points},\qquad \operatorname{Var}(\theta_{100})=2.5^2(9)=56.25\text{ squared points}. \] Equivalently, the standard deviation is \(2.5(3)=7.5\) points, giving \(\theta_{100}\sim N(-15,56.25)\). A negative model contrast favors treatment because higher fatigue scores are worse. The positive six-point improvement belief and the negative model contrast describe the same direction of clinical benefit.
Standardizing a raw effect
When the model uses a standardized mean difference, define the denominator before converting. A common form is \(\delta=\theta/\sigma_{\mathrm{ref}}\), where \(\theta\) is the raw treatment-minus-control mean difference and \(\sigma_{\mathrm{ref}}>0\) is a specified reference standard deviation. If that reference SD is treated as fixed, standardization is a linear rescaling: a prior \(\theta\sim N(m,v)\) implies \(\delta\sim N(m/\sigma_{\mathrm{ref}},v/\sigma_{\mathrm{ref}}^2)\). The result is dimensionless.
The reference SD is not interchangeable with any available SD. It might be a prespecified baseline SD or another agreed reference quantity. The SD of change scores, an arm-specific SD, or an SD estimated from a different population may produce a different standardized effect. State the choice and its units so that the standardized prior can be interpreted and reproduced.
Worked Example: Standardizing a Prior for a Continuous Outcome
Suppose a hypothetical clinical outcome is measured in points, with higher values indicating better health. The treatment effect is treatment minus control, and the prior is \(\theta\sim N(3,5^2)\) points. For this model, the agreed fixed reference standard deviation is \(\sigma_{\mathrm{ref}}=12\) points. Convert the prior and a clinically meaningful threshold of 4 points.
The standardized effect is \(\delta=\theta/12\). Its prior mean and variance are \[ E(\delta)=\frac{3}{12}=0.25,\qquad \operatorname{Var}(\delta)=\frac{5^2}{12^2} =\frac{25}{144}=0.1736. \] Hence \(\delta\sim N(0.25,0.1736)\), with standard deviation \(5/12=0.4167\). The 4-point clinical threshold corresponds to \(4/12=0.3333\) standardized units.
For context, the central 95% interval under the raw normal prior is \(3\pm1.96(5)=[-6.8,12.8]\) points. Dividing by 12 gives approximately \([-0.5667,1.067]\) standardized units. The transformed interval includes effects in both directions, despite the positive prior mean. The mean alone does not summarize the full prior belief.
When simple rescaling is not enough
The formulas above require a fixed affine conversion. If the denominator in a standardized effect is uncertain, \(\theta/\sigma_{\mathrm{ref}}\) is a ratio of uncertain quantities, not a fixed multiple of \(\theta\). In general, its mean and variance are not \(E(\theta)/E(\sigma_{\mathrm{ref}})\) and \(\operatorname{Var}(\theta)/E(\sigma_{\mathrm{ref}})^2\). Dependence between numerator and denominator can matter as well. In that situation, the joint assumptions about both quantities must be specified and the ratio distribution derived or computed from that joint distribution; a fixed-denominator conversion is not justified.
Likewise, a nonlinear scale conversion does not generally preserve a normal distribution. For a monotone nonlinear function \(g\), quantile endpoints can be transformed as \(g(L)\) and \(g(U)\) (reversing their order if \(g\) decreases), but transforming only the mean and standard deviation using a local approximation may not preserve the tails or the interval probability. If the model parameter is on a nonlinear scale, define the transformation and check the resulting distribution rather than assuming it remains normal.
A practical diagnostic is to work one plausible value through the conversion in words and arithmetic. If a 5-point treatment benefit becomes a 5-point harm, check the contrast direction. If a change from a 0-to-100 scale to a 0-to-10 scale leaves the prior SD unchanged, check whether the uncertainty was converted. If a standardized effect has units of points, check the denominator. These checks often reveal errors before the prior is used.
Choosing the scale for communication and modeling
| Scale | Strength | Key requirement or limitation |
|---|---|---|
| Raw clinical units | Directly interpretable against the instrument and clinical thresholds. | Effects from different instruments may not be directly comparable. |
| Rescaled clinical units | Can make numbers easier to report without changing the underlying effect. | Both the center and uncertainty must use the same conversion factor. |
| Standardized mean difference | Expresses an effect relative to a reference SD and can aid comparison across scales. | Interpretation depends on the selected reference SD and can be less immediately clinical. |
These are alternative representations, not competing clinical claims. A raw-scale prior makes clinical thresholds easy to see; a standardized prior may suit a model or comparison that requires dimensionless effects. Whichever representation is used, preserve the same treatment contrast, clinical direction, and uncertainty. Retaining the original clinical-scale description alongside the converted specification makes the modeling choice easier to audit.
Common mistakes and limitations
- Changing the mean but not the uncertainty. If scores are divided by 10, the prior SD must also be divided by 10 and its variance by 100.
- Ignoring contrast direction. “Improvement” is not necessarily treatment minus control. Write both definitions and convert the sign explicitly.
- Leaving out the standardization denominator. A standardized effect is not fully defined until its reference SD is stated.
- Treating an uncertain denominator as fixed. Dividing by a point estimate may be a deliberate approximation, but it does not represent uncertainty in that denominator.
- Assuming every transformation preserves normality. Fixed affine transformations preserve normality; nonlinear transformations generally do not.
A scale conversion cannot repair an outcome definition that is ambiguous, an unsuitable reference SD, or an elicited belief that does not match the model’s treatment contrast. It only translates a clearly stated belief. Confirm the outcome, arm order, units, direction, and denominator before treating the converted parameters as equivalent.
Check Your Understanding
For each question, identify the conversion and check that its direction and uncertainty are preserved. Solutions follow the questions.
- A prior for treatment-minus-control is \(N(10,4^2)\) points. A new score is the old score multiplied by 0.5. Give the transformed mean and variance.
- On a scale where higher scores are worse, a clinician defines benefit as control minus treatment and believes it is \(N(2,3^2)\) points. What is the prior for treatment minus control?
- A raw effect prior is \(N(-4,6^2)\) points and the fixed reference SD is 8 points. Find the standardized mean and variance.
- Why is it generally incorrect to divide a raw-effect prior mean and SD by the mean of an uncertain reference SD and call the result the exact standardized prior?
Solutions. (1) The multiplier is \(0.5\), so the mean is \(0.5(10)=5\) points and the variance is \(0.5^2(16)=4\) squared points. The transformed prior is \(N(5,4)\). (2) The model contrast is the negative of benefit. Its mean is \(-2\) points and its variance remains \(3^2=9\) squared points, so the prior is \(N(-2,9)\). (3) Divide the mean by 8 and the variance by \(8^2\): the standardized mean is \(-4/8=-0.5\), and the variance is \(36/64=0.5625\). (4) The denominator is random, so the ratio’s distribution depends on its uncertainty and any dependence with the numerator. Dividing by its mean ignores those features and is not generally an exact transformation.