Profile Likelihood Intervals for Predicted Risk Contrasts

Risk Contrasts as Profile Targets

A regression model can predict a risk for a person with one set of covariates and a different risk for a person with another. The difference between those risks is often the quantity of scientific interest. It is not, in general, a regression coefficient: it depends on both the fitted model and the covariate patterns at which predictions are compared.

For example, in logistic regression the coefficient of a binary treatment is a log odds ratio. The corresponding risk difference also depends on the baseline covariates. Profile likelihood handles this by treating the risk difference itself as the target, while re-estimating the remaining model parameters at each fixed value of that target.

Definition: Let \(\boldsymbol{x}_a\) and \(\boldsymbol{x}_b\) be two specified covariate vectors, and let \(\mu(\boldsymbol{x};\boldsymbol{\beta})\) denote the model-predicted risk at covariates \(\boldsymbol{x}\). The predicted risk contrast from pattern \(a\) to pattern \(b\) is \(\Delta(\boldsymbol{\beta})=\mu(\boldsymbol{x}_b;\boldsymbol{\beta})-\mu(\boldsymbol{x}_a;\boldsymbol{\beta})\). For a logistic model, \(\mu(\boldsymbol{x};\boldsymbol{\beta})=\operatorname{expit}(\boldsymbol{x}^{T}\boldsymbol{\beta})\), where \(\operatorname{expit}(u)=1/(1+\exp(-u))\). A profile log-likelihood for a proposed contrast \(d\) is \(\ell_p(d)=\sup_{\boldsymbol{\beta}:\Delta(\boldsymbol{\beta})=d}\ell(\boldsymbol{\beta})\).

If the unrestricted maximum-likelihood estimate exists, write \(\hat{\boldsymbol{\beta}}\) for an estimator attaining it. A likelihood-ratio profile confidence set with cutoff \(c\) is

\mathcal{C}=\left\{d:2\left[\ell(\hat{\boldsymbol{\beta}})-\ell_p(d)\right]\le c\right\}.

For a regular, identifiable, scalar contrast whose true value is in the interior of its possible range, the usual large-sample cutoff is a quantile of a chi-squared distribution with one degree of freedom. A cutoff calibrated by a suitable bootstrap can be used instead. The optimization defining \(\ell_p(d)\) is the central operation: at each \(d\), it finds the best-fitting model that has exactly that risk difference.

Two Independent Binomial Risks

A simple setting makes the constraint explicit. Suppose independent groups \(0\) and \(1\) have binomial counts \(Y_j\sim\operatorname{Binomial}(n_j,p_j)\), for \(j=0,1\). Their risk difference is \(p_1-p_0\). At a proposed value \(d\), set \(p_0=p\) and \(p_1=p+d\). The feasible values of \(p\) satisfy \(0<p<1\) and \(0<p+d<1\). Ignoring binomial coefficients, which do not depend on the risks, the constrained log-likelihood is

\ell_d(p)=y_0\log p+(n_0-y_0)\log(1-p) +y_1\log(p+d)+(n_1-y_1)\log(1-p-d).

At an interior constrained maximum, differentiation gives the equation below. A solution must also maximize the constrained log-likelihood over its feasible domain; a score root alone does not establish that it is the maximum.

\ell_d'(p)=\frac{y_0}{p}-\frac{n_0-y_0}{1-p} +\frac{y_1}{p+d}-\frac{n_1-y_1}{1-p-d}=0.

Result: In the two-group binomial model, profiling a proposed risk difference \(d\) reduces the constrained optimization to a one-dimensional maximization over feasible \(p\). If the constrained maximum is interior, its risk \(p_0\) satisfies the displayed score equation, and \(p_1=p_0+d\).

Worked Example: A Constrained Fit at a Risk Difference of 0.2

Suppose group 0 has 4 events among 20 observations and group 1 has 12 events among 20 observations. The unrestricted maximum-likelihood estimates are \(\hat p_0=4/20=0.2\) and \(\hat p_1=12/20=0.6\), so the fitted risk difference is \(0.4\). Find the constrained fit for a proposed difference \(d=0.2\).

Write \(p_0=p\) and \(p_1=p+0.2\). The score equation becomes

\frac{4}{p}-\frac{16}{1-p}+\frac{12}{p+0.2}-\frac{8}{0.8-p}=0.

Solving this equation over \(0<p<0.8\) gives \(p\) approximately \(0.29038\). Thus the constrained fitted risks are approximately \(0.29038\) and \(0.49038\). For a check on the numerical solution, substituting \(p=0.2904\) into the left side gives approximately \(-0.002576\), close to zero; the root is slightly smaller.

The unrestricted log-likelihood minus the constrained log-likelihood, with the constrained risks rounded to \(0.2904\) and \(0.4904\), is approximately

4\log\left(\frac{0.2}{0.2904}\right) +16\log\left(\frac{0.8}{0.7096}\right) +12\log\left(\frac{0.6}{0.4904}\right) +8\log\left(\frac{0.4}{0.5096}\right) \approx 0.9100.

Consequently, the likelihood-ratio statistic for \(d=0.2\) is approximately \(2(0.9100)=1.8200\). This is the profile evidence at that candidate contrast; comparing it with the chosen cutoff determines whether \(0.2\) belongs to the confidence set. The fitted risks are not obtained by simply moving both unrestricted risks by the same amount. They are re-estimated jointly under the constraint.

Why the Covariate Patterns Matter

In a regression model, changing either covariate vector changes the target function \(\Delta(\boldsymbol{\beta})\). It therefore changes the constraint imposed during profiling, even when the data and likelihood remain the same. This is consequential because the constrained fit must accommodate all observations, not just the two predictions named by the contrast.

A direct comparison of pairs in a saturated categorical-risk model illustrates the point. Each covariate pattern has its own risk, and the unrestricted maximum-likelihood estimate for that risk is its observed proportion. To test a zero contrast between two patterns, the constraint sets their risks equal. The joint constrained maximum is their pooled observed proportion.

Result: For two independent binomial groups with a common risk under the constraint \(p_a=p_b\), the constrained maximum-likelihood estimate is \(\tilde p=(y_a+y_b)/(n_a+n_b)\), provided this proportion is interior. To see this, their constrained log-likelihood, apart from constants, is \((y_a+y_b)\log p+(n_a+n_b-y_a-y_b)\log(1-p)\). Its derivative is \((y_a+y_b)/p-(n_a+n_b-y_a-y_b)/(1-p)\). Setting this to zero gives \(p=\tilde p\); the second derivative is negative for \(0<p<1\), so this is the maximum. \(\square\)

Worked Example: Changing the Pair Changes the Profile Evidence

Suppose three covariate patterns have event counts \(4/20\), \(12/20\), and \(8/20\), labelled A, B, and C. In this saturated model, compare the profile likelihood at zero contrast for A versus B with that for A versus C.

For A versus B, the constrained common risk is \((4+12)/(20+20)=0.4\). The unrestricted risks are \(0.2\) and \(0.6\). The log-likelihood loss under the constraint is

4\log(0.2/0.4)+16\log(0.8/0.6) +12\log(0.6/0.4)+8\log(0.4/0.6) \approx 3.4522.

The likelihood-ratio statistic is therefore approximately \(2(3.4522)=6.9044\).

For A versus C, the constrained common risk is \((4+8)/40=0.3\). The corresponding loss and statistic are

4\log(0.2/0.3)+16\log(0.8/0.7) +8\log(0.4/0.3)+12\log(0.6/0.7) \approx 0.9663,\qquad 2(0.9663)\approx 1.9326.

For a cutoff of \(3.841\), zero is excluded for the A-versus-B contrast but included for A versus C. The data and fitting method have not changed; the pair of patterns defining the target has. In a nonsaturated regression model, the effect of changing patterns can be even more involved because the constrained fit must adjust shared regression coefficients to satisfy the new prediction constraint.

Risk Differences in Logistic Regression

For logistic regression, the risk difference is a nonlinear function of the regression parameters. Its gradient shows explicitly how the chosen patterns affect the local profile geometry. With \(\boldsymbol{x}_a\) and \(\boldsymbol{x}_b\) fixed, define \(\mu_j=\operatorname{expit}(\boldsymbol{x}_j^T\boldsymbol{\beta})\). Differentiation gives

\nabla_{\boldsymbol{\beta}}\Delta(\boldsymbol{\beta}) =\mu_b(1-\mu_b)\boldsymbol{x}_b -\mu_a(1-\mu_a)\boldsymbol{x}_a.

This gradient is not simply the difference between the covariate vectors: each vector is weighted by the Bernoulli variance at its predicted risk. Thus the same coefficient uncertainty can translate into different uncertainty in the risk difference at different covariate values. The gradient is a local description, not a substitute for the constrained profile calculation.

Worked Example: Two Contrasts at Different Covariate Values

Consider a logistic model with a single quantitative covariate \(x\), an intercept, and fitted coefficients \(\hat\beta_0=-1\) and \(\hat\beta_1=0.8\). Compare the predicted risks at \(x=0,1,2\). The linear predictors are \(-1,-0.2,0.6\), so the risks are approximately \(0.26894,0.45017,0.64566\).

The contrast from \(x=0\) to \(x=1\) is \(0.45017-0.26894=0.18123\), while the contrast from \(x=0\) to \(x=2\) is \(0.64566-0.26894=0.37672\). Their constraints for a proposed difference \(d\), written in terms of arbitrary coefficients \((\beta_0,\beta_1)\), are respectively

\operatorname{expit}(\beta_0+\beta_1)-\operatorname{expit}(\beta_0)=d

and

\operatorname{expit}(\beta_0+2\beta_1)-\operatorname{expit}(\beta_0)=d.

At the fitted coefficients, the gradient for the first contrast is approximately \((0.05090,0.24752)^T\); for the second it is approximately \((0.03217,0.45757)^T\). For example, \(\operatorname{expit}(-0.2)[1-\operatorname{expit}(-0.2)]\approx0.24752\), while \(\operatorname{expit}(-1)[1-\operatorname{expit}(-1)]\approx0.19661\). Substituting these values into the gradient formula gives the stated components. The gradients differ in direction and size, so fixing the same proposed \(d\) produces different constrained surfaces in coefficient space and can yield differently shaped profile intervals.

Constructing and Interpreting the Confidence Set

For each feasible \(d\), maximize the log-likelihood subject to \(\Delta(\boldsymbol{\beta})=d\), then compare that maximum with the unrestricted maximum. The resulting set need not be symmetric around the maximum-likelihood contrast: logistic risks are nonlinear functions of coefficients, and the constrained nuisance-parameter estimates can move differently on the two sides. A confidence set can also be disconnected or extend to a boundary under relevant conditions, so interval endpoints should be found from the likelihood criterion rather than assumed to have a particular shape.

Specify the contrast.
State the covariate vectors, the direction of subtraction, and the model-predicted risk at each vector.

Find the unrestricted fit.
Compute the maximum log-likelihood and the fitted contrast. Verify that the model fit is well-defined for the data and parameterization.

Profile candidate values.
For each feasible \(d\), maximize the likelihood under \(\Delta(\boldsymbol{\beta})=d\), allowing all other parameters to re-estimate.

Apply the cutoff.
Retain precisely those \(d\) whose likelihood-ratio statistic is at most the specified cutoff; determine endpoints by solving the cutoff equation.

The risk-difference scale also imposes a natural range: because both risks lie between zero and one, \(-1\leq\Delta\leq1\). A profile computation must respect feasible parameter values at each proposed contrast. This bound does not guarantee that every value in the range is attainable under a particular regression model.

Common Pitfall

A tempting shortcut is to estimate a coefficient contrast or use the difference between two fitted risks and then attach a standard error while holding the other fitted coefficients fixed. That does not construct a profile likelihood interval. The profile fit at each candidate \(d\) re-estimates nuisance parameters subject to the risk-difference constraint; those adjustments are part of the likelihood evidence. Another error is to treat a logistic coefficient as a risk difference. A slope coefficient describes a change in log odds per covariate unit, whereas a predicted risk contrast depends on the two covariate patterns and on all coefficients determining their risks.

Key takeaway: A profile interval for a predicted risk contrast is defined by constrained likelihood maximization for that exact contrast and those exact covariate patterns. Changing the patterns changes the target, the constrained fit, and potentially the shape of the confidence set.

Check Your Understanding

Use the definitions and calculations above to answer the following questions.

In a logistic model, why is the coefficient of a binary treatment generally not equal to the predicted risk difference between treatment patterns?
For two independent binomial groups, write the feasible range of \(p_0\) when profiling a proposed risk difference \(d\).
In the three-pattern example, what constrained risk is used for the zero contrast between A and B, and how is it obtained?
Why can two risk contrasts based on different covariate pairs have different profile likelihood curves even when they use the same fitted model and data?
What must be re-estimated when profiling a candidate risk difference in a regression model?