Checking unimodality using isotonic regression: an application to breast cancer mortality rates

Abstract

In some diseases it is well-known that a unimodal mortality pattern exists. A clear example in developed countries is breast cancer, where mortality increased sharply until the nineties and then decreased. This clear unimodal pattern is not necessarily applicable to all regions within a country. In this paper, we develop statistical tools to check if the unimodality pattern persists within regions using order restricted inference. Break points as well as confidence intervals are also provided. In addition, a new test for checking monotonicity against unimodality is derived allowing to discriminate between a simple increasing pattern and an up-then-down response pattern. A comparison with the widely used joinpoint regression technique under unimodality is provided. We show that the joinpoint technique could fail when the underlying function is not piecewise linear. Results will be illustrated using age-specific breast cancer mortality data from Spain in the period 1975–2005.

Appendix

The mathematical tools for understanding the proofs included in this appendix are not basic. For those readers who are not expert, we recommend first to read carefully the classical book by Robertson et al. (1988). New notation is also used to simplify the proofs. Let us define U as the union of cones $U_q$ such that $U=\cup _{q=t_1}^{t_n}U_{q}$, where $U_q=\{{\mathbf{r}}\in {\mathfrak{R}}^{n} | \, r_{t_1}\le \cdots \le r_{q} \ge r_{q+1}\ge \cdots \ge r_{t_n}\}$. Then, the testing problems (1) and (2) can be reformulated as follows

$$H_{1q}:{\mathbf{r}} \in U_q \quad vs. \quad H_{2}-H_{1q},$$

(10)

$$H_{0q}:{\mathbf{r}} \in M _{q} \quad vs. \quad H_{1q}- H_{0q},$$

(11)

where $M_{q}=M \cap U_{q}$. $R_{d}$ is referred as the set $D=d$. The estimators ${\widehat{\mathbf{r}}}^{1q}$ and ${\widehat{\mathbf{r}}}^{0q}$ are given by the weighted projection of ${\mathbf{v}}$ with weights $W=\text{ diag }(w_{t_1},\ldots ,w_{t_n})$ in the convex cones $U_{q}$ and $M_{q}$, defined as $P_{W}(\mathbf{v}|U_{q})$ and $P_{W}(\mathbf{v}|M_{q})$ respectively [see Robertson et al. (1988)]. In order to simplify notation the weight matrix is eliminated from the projection operator in what follows.

Lemma 1 shows that the cones defining the null and the alternative hypotheses in (11) verify a non oblique property. Then, Lemma 1 is used in the proof of Theorem 3.

Lemma 1

• (i) Let $L_{U}$ and $L_{M}$ be subsets verifying $P({\mathbf{v}}|U_q)=P({\mathbf{v}}|L_{U}), P(P({\mathbf{v}}|U_q)|M_q)=P({\mathbf{v}}|L_{M})$. Then, $L_{M} \subset L_{U}$

• (ii) The regions $M_q=M\cap U_q$ and $U_q$ are non oblique

  $$P(P({\mathbf{v}}|U_q)|M_q)=P({\mathbf{v}}|M_q ).$$

1.1 Proof of Lemma 1

• (i) $L_{U}$ and $L_{M}$ can be defined by a set of linear inequalities as follows: $\ L_{M}=\left\{ {\mathbf{v}}\in {\mathbb{R}}^{n} |v_{i}=v_{i+1}, i \in B \right\}$ and $\L _{U}=\left\{{\mathbf{v}}\in {\mathbb{R}}^{n} /v_{i}=v_{i+1}, i \in A \right\}$, where $B, A \subset \{1,\ldots ,n-1\}$. The results will follow by showing that $A \subset B$. Let $i \in A$, then $P(\mathbf{v}|U_q)_{i}=P(\mathbf{v}|U_q)_{i+1}$. Now, as the cone $M_q$ is an acute cone we have that $P(P({\mathbf{v}}|U_q)|M_q)_{i}=P(P({\mathbf{v}}|U_q)|M_q)_{i+1}$ [see Menéndez and Salvador (1991)] and the result follows.

• (ii) Let $L_{U}$ and $L_{M}$ be subspaces verifying $P({\mathbf{v}}|U_q)=P({\mathbf{v}}|L_{U}), P(P({\mathbf{v}}|U_q)|M_q)=P({\mathbf{v}}|L_{M})$. Then, from basic properties of projections onto subspaces and convex cones, we have that for a given ${\mathbf{v}}$

  $${\mathbf{v}}=P({\mathbf{v}}|U_q)+P({\mathbf{v}}|{U_q}^{p})=P({\mathbf{v}}|U_q|M_q)+P({\mathbf{v}}|U_q|{M_q}^{p})+P({\mathbf{v}}|{U_q}^{p}),$$

  from (i) we have that $L_{U}\subset L_{M}$, and then,

  $${\mathbf{v}}=p({\mathbf{v}}|L_{U})+P({\mathbf{v}}|L_{U}^{\bot })=P({\mathbf{v}}|L_{M})+P({\mathbf{v}}|L_{M}^{\bot }\cap L_{U})+P({\mathbf{v}}|L_{U}^{\bot }).$$

  Now, as $M_q\subset U_q$, and both are closed convex cones, then ${U_q}^{p}\subset {M_q}^{p}$ and

  $$P({\mathbf{v}}|U_q|{M_q}^{p})+P({\mathbf{v}}|{U_q}^{p})=P({\mathbf{v}}|L_{M}^{\bot }\cap L_{U})+P({\mathbf{v}}|L_{U}^{\bot })\epsilon M_q^{p}.$$

  We also have that

  $$P({\mathbf{v}}|U_q|M_q)=P({\mathbf{v}}|L_{M})\epsilon M_q,$$

  and

  $$<P({\mathbf{v}}|L_{M}),P({\mathbf{v}}|L_{M}^{\bot }\cap L_{U})+P({\mathbf{v}}|L_{U}^{\bot })>\le 0.$$

  Then, from statements above and basic properties of projections onto convex cones, we have that

  $$P({\mathbf{v}}|L_{M})=P({\mathbf{v}}|M_q).$$

  (12)

  Now, for ${\mathbf{v}} \in {\mathfrak{R}}^{n}$ and $L_U$ and $L_M$ subspaces as in (i), we have that

  $$P(P({\mathbf{v}}|U_q)|M_q)=P({\mathbf{v}}|L_{M})=P({\mathbf{v}}|M_q)$$

  and this last statement shows that $M_q$ and $U_q$ are non-oblique. $\square$

1.2 Proof of Theorem 3

• (i) From Lemma 1 (ii), Lemma 2.2. in Menéndez et al. (1992), and basic properties of polar cones, $T_{0q}$ is given by

  $$T_{0q}=\Vert P({\mathbf{v}}|U_q\cap {M_q}^{p})\Vert ^{2}=\Vert P({\mathbf{v}}|U_q\cap {M}^{p})\Vert ^{2}.$$

  (13)

  Let $R_{d}$ be given by $R_{d}=\{{\mathbf{v}} \in {\mathfrak{R}}^{n}|P({\mathbf{v}}|U_q\cap {M}^{p})=P({{v}}|L), dim L=d\}$. From Lemma 1 is easy to prove that $R_{d}=\{{\mathbf{v}} \in {\mathfrak{R}}^{n}|P({\mathbf{v}}|U_q)=P({\mathbf{v}}|L_{U}),P({\mathbf{v}}|M_q)=P({\mathbf{v}}|L_{M}),d=dim(L_{U})-dim(L_{M})\}$. Now Shapiro (1988) shows that, for ${\mathbf{r}}_0$, the conditional distribution of $T_{0q}$ given to the subsets $R_{d}$ is a chi-squared with d degrees of freedom and the result follows.

• (ii) From (i) and equality

  $$R_{0}=\{{\mathbf{v}} \in {\mathfrak{R}}^{n}|P({\mathbf{v}}|U_q)=P({\mathbf{v}}|M_q)\}=\{{\mathbf{v}} \in {\mathfrak{R}}^{n}/ T_{0q}({\mathbf{v}})=0\},$$

  we have that

  $$\begin{aligned} \Pi _q({\mathbf{r}}_0)&=\sum _{d}pr_{r_{0}}\left( T_{0q}\ge c(d)\right| {\mathbf{v}} \in R_{d})pr_{r_{0}}( {\mathbf{v}} \in R_{d})\\&=\sum _{d}pr\left( \chi ^2_{d}>c(d) \right) pr_{r_{0}}( {\mathbf{v}} \in R_{d}) \\&=\frac{\alpha }{1-pr_{r_{0}}(T_{0q}=0)} \sum _{d}pr_{r_{0}}( {\mathbf{v}} \in R_{d}) \\&=\frac{\alpha }{1-pr_{r_{0}}(T_{0q}=0)}(1-pr_{r_{0}}({\mathbf{v}} \in R_{0}))=\alpha \end{aligned}$$

  and the results follow.$\square$