Testing Equality Of Category-Specific Slopes

What Does “Equal Slopes” Mean?

A predictor can affect the log odds of several response categories by different amounts. Asking whether its category-specific slopes are equal is a question about contrasts between coefficients. It is not automatically a question about whether the predictor has no effect. The distinction matters: several categories can share the same nonzero slope.

Use a baseline-category multinomial logit with response categories \(1,\ldots,J\), taking category \(J\) as the reference. For a scalar predictor \(x\), write the model as

Here \(\beta_j\) is the slope for the log odds of category \(j\) versus the reference category. For notation in comparisons involving the reference, set \(\beta_J=0\). This is a convenient convention for writing contrasts; it does not add a separately estimated parameter.

For example, equality between the slopes for categories 1 and 2 is \(H_0:\beta_1-\beta_2=0\). Equality of slopes for three categories can be expressed with two independent contrasts, such as \(\beta_1-\beta_3=0\) and \(\beta_2-\beta_3=0\). These two restrictions imply that all three slopes are equal. Adding the contrast \(\beta_1-\beta_2=0\) would be redundant.

Calculating the Uncertainty of a Contrast

The uncertainty of a difference depends on the covariance between its two coefficient estimates. If \(\hat\beta_j\) and \(\hat\beta_k\) are estimated slopes, then

This follows by expanding the variance of a linear combination. It applies whether the estimates are positively or negatively correlated. The covariance must come from the fitted model's covariance matrix; treating the estimates as independent without justification can give the wrong standard error.

To verify the result, write \(a^\mathsf{T}\hat\theta=\sum_s a_s\hat\theta_s\). By bilinearity of covariance,

Replacing the unknown covariance matrix by its estimate gives the stated estimated variance. For the difference, the two diagonal terms are the individual variances and the two off-diagonal terms combine to \(-2\widehat{\operatorname{Cov}}(\hat\beta_j,\hat\beta_k)\). \(\square\)

The Wald Test for Several Equality Restrictions

Suppose the parameter vector is \(\theta\), with estimator \(\hat\theta\), and consider \(q\) independent restrictions \(R\theta=r\). The Wald test compares the estimated restriction error \(R\hat\theta-r\) with its estimated covariance matrix. In the usual large-sample setting, the statistic has an approximate chi-square distribution with \(q\) degrees of freedom under the null.

For a multinomial logit fitted by maximum likelihood, these conditions are the standard large-sample conditions: in particular, the likelihood is sufficiently smooth near the true parameter, the information is nonsingular, and the observations satisfy the sampling assumptions supporting the stated asymptotic normality. The restriction covariance must be nonsingular; the full-row-rank condition and positive-definite \(V\) ensure this here.

To prove the result, let \(A=RVR^\mathsf{T}\), which is positive definite because \(R\) has full row rank. Under the null, \(R\theta_0-r=0\), so

Also, consistency of the covariance estimator implies \(nR\widehat{\operatorname{Var}}(\hat\theta)R^\mathsf{T}\xrightarrow{p}A\). Consequently, the Wald statistic can be written as

By the continuous mapping theorem and Slutsky's theorem, this converges in distribution to \(Z^\mathsf{T}A^{-1}Z\), where \(Z\sim N(0,A)\). The vector \(A^{-1/2}Z\) has distribution \(N(0,I_q)\), so \(Z^\mathsf{T}A^{-1}Z\) is the sum of squares of \(q\) independent standard normal variables. That sum has distribution \(\chi^2_q\), proving the result. \(\square\)

Equality Is Not the Same as No Predictor Effect

The no-effect hypothesis for predictor \(x\), in the baseline-category model, is \(\beta_1=\cdots=\beta_{J-1}=0\). It tests whether changing \(x\) leaves every modeled category-versus-reference log odds unchanged. In contrast, an equality test among selected nonreference categories tests whether their slopes match, regardless of their shared value. A predictor can therefore have a substantial effect while passing an equality test.

If the reference category is included in an equality claim, the convention \(\beta_J=0\) matters. For example, equality of slopes for categories 1 and \(J\) is \(\beta_1=0\). Requiring all categories to have equal slopes, including the reference, is equivalent to requiring every nonreference slope to be zero. But equality only among categories 1 and 2 does not impose that requirement.

Common Pitfall and Practical Interpretation

Wrong reasoning: “The test of equal slopes is not significant, so the predictor has no effect.” This confuses a difference, \(\beta_j-\beta_k\), with the coefficients themselves. A failure to reject equality does not establish that the common slope is zero, and it does not establish exact equality.

Correction: Write the null as an explicit contrast before choosing the test. For equality among selected categories, use independent differences and calculate their covariance from the full fitted covariance matrix. For no predictor effect, set all relevant nonreference slopes to zero and test those restrictions instead. The number of degrees of freedom is the rank of the independent restriction matrix, not the number of pairwise comparisons one can list.

These are statements about log-odds slopes. Equal slopes for two nonreference logits mean that the log odds of either category versus the baseline change at the same rate per unit increase in \(x\); they do not imply that the corresponding category probabilities change equally. Probability changes depend on the full set of category logits and on the covariate values.