Cancer premiums and latency effects: A risk tradeoff approach for valuing reductions in fatal cancer risks

Abstract

Many studies estimate individuals’ values for avoiding fatality risks; however, most value-of-statistical-life studies focus on accident-related deaths. Consequently, little is known about preferences for avoiding other fatal risks, such as cancer. Cancer may engender strong feelings of dread, leading to a “cancer premium,” but cancer latency periods may have the opposite effect. Using a national survey, we elicit relative preferences for avoiding fatal cancer and auto-accident risks. We find strong preferences for avoiding cancer risks. With a 5-year latency, they are valued roughly three times greater than immediate accident risks, declining to 50% greater for a 25-year latency.

Appendices

Appendix A: Proofs of latency and survival probability effects on MER

1.1 The effect of cancer latency on MER

As shown in Eq. 3, the effect on MER of increasing latency (t) depends on how it affects the lifetime utility of the cancer profile. Expanding on Eq. 4, the effect on U(C(t), Y) of increasing latency by one period, from t to t + 1, can be written as

$$ \begin{array}{*{20}l} {{U{\left( {C{\left( {t + 1} \right)},Y} \right)} - U{\left( {C{\left( t \right)},Y} \right)}} \hfill} \\ {{ = {\left( {{\sum\limits_{i = 1}^{t - 1} {{\left( {s_{i} } \right)}} }{\left( {d_{i} } \right)}u^{h} {\left( {y_{i} } \right)}} \right)} + {\left( {s_{t} } \right)}{\left( {d_{t} } \right)}u^{h} {\left( {y_{t} } \right)} + {\left( {s_{{t + 1}} } \right)}{\left( {d_{{t + 1}} } \right)}u^{c} {\left( {y_{{t + 1}} } \right)}} \hfill} \\ {{ - {\left( {{\sum\limits_{i = 1}^{t - 1} {{\left( {s_{i} } \right)}} }{\left( {d_{i} } \right)}u^{h} {\left( {y_{i} } \right)}} \right)} - {\left( {s_{t} } \right)}{\left( {d_{t} } \right)}u^{c} {\left( {y_{t} } \right)}} \hfill} \\ {{ = {\left( {s_{t} } \right)}{\left( {d_{t} } \right)}{\left[ {u^{h} {\left( {y_{i} } \right)} - u^{c} {\left( {y_{t} } \right)}} \right]} + {\left( {s_{{t + 1}} } \right)}{\left( {d_{{t + 1}} } \right)}u^{c} {\left( {y_{{t + 1}} } \right)}} \hfill} \\ \end{array} $$

(11)

For simplicity, the duration of each time period, as indexed by i, is the same as the duration of cancer morbidity (i.e., between diagnosis and death). The first term in Eq. 11 will be positive as long as the utility of a period in normal health, u^h(y), is greater than with cancer, u^c(y). Even if the utility of a period (t or t + 1) with cancer is negative, the second term in this expression will also be less in absolute value terms than the first, as long as u^h(y) is positive (and y_t + 1 is not substantially less than y_t). Consequently, an increase in latency should increase U(C(t),Y) and decrease MER. This result is consistent with the intuition that extending cancer latency will reduce aversion to cancer risks.

1.2 The effect of survival probability on MER

In contrast to a change in cancer latency, an increase in survival probability, s_t, will affect MER through both the cancer and the normal health utility profiles. To examine the effect of increasing s_t on the MER, we must examine its effect on the ratio U(C,Y)/U(H,Y). To do this we first define the following expressions:

$$ \begin{array}{*{20}c} {A = {\sum\limits_{i = 0}^{t - 1} {{\left( {s_{i} } \right)}} }{\left( {d_{i} } \right)}u^{h} {\left( {y_{i} } \right)}} \\ {B = {\left( {d_{t} } \right)}u^{c} {\left( {y_{t} } \right)}} \\ {C = {\sum\limits_{j = t}^\infty {{\left( {s_{{jt}} } \right)}} }{\left( {d_{j} } \right)}u^{h} {\left( {y_{j} } \right)}} \\ {R = \frac{{U{\left( {C{\left( t \right)},Y} \right)}}} {{U{\left( {H,Y} \right)}}} = \frac{{A + s_{t} B}} {{A + s_{t} C}}} \\ \end{array} $$

(12)

Where

s_ij :

probability of surviving to period j conditional on surviving to period t.

Differentiating the lifetime utility ratio R with respect to s_t, we get:

$$ \frac{{\partial R}} {{\partial s_{t} }} = \frac{{A{\left( {B - C} \right)}}} {{{\left( {A + s_{t} C} \right)}^{2} }} < 0\quad {\text{if}}\;A > 0\;{\text{and}}\;B < C $$

(13)

These results imply that as long as the healthy state provides positive utility, and this utility is greater than the utility in the cancer state, then, for a given latency t, increasing the probability of survival to t, will decrease U(C(t),Y) and increase MER. In other words, for a given cancer latency, increasing the probability of survival for that period will increase aversion to cancer risks.

Appendix B: Survey descriptions of cancer morbidity

Depending on the type of cancer, respondents were presented with the following descriptions of symptoms due to cancer illness (the duration of cancer illness, which also varied across respondents, was illustrated using a time line):

• When liver cancer develops it causes symptoms that include discomfort in the upper abdomen on the right side, pain around the right shoulder blade, and yellowing of the skin (jaundice). Liver cancer also causes bloating, weight loss, fatigue, and general weakness. These symptoms usually become much worse as the illness progresses.

• When brain cancer develops it causes symptoms that include nausea, vomiting, headaches and seizures. Brain cancers can also lead to mental changes such as problems with memory, speech, and concentration, as well as to severe intellectual problems and confusion. These symptoms usually become much worse as the illness progresses.

• When stomach cancer develops it causes symptoms that include severe and constant stomach pain (especially after eating), nausea, vomiting, weight loss and general weakness. These symptoms usually become much worse as the illness progresses.

All respondents were presented with the following description of morbidity associated with cancer treatment:

• Treatment with radiation, chemotherapy, or surgery can often help to slow the progress of the disease, but they can also cause side effects such as pain, nausea, vomiting, diarrhea, and hair loss. They also weaken the immune system, which makes one more vulnerable to other illnesses like pneumonia. In cases of fatal [stomach/liver/brain] cancer, these treatments do not cure the disease. Even with treatment, most patients survive for only a few years after the first symptoms appear.

Appendix C: Sample selection model

Table 4 reports the results of a Heckman sample selection probit model applied to the main model reported in Table 3. The sample for the selection equation is the group of households in the KN panel who were invited to take the survey, and the sample for the main analysis is the subset of respondents who completed the survey, indicated a preference for Location A or B in the choice task, and did not contradict their own choice in a follow up question.⁹ The main probit equation is adapted from model (3) in Table 3—it excludes the demographic variables age, gender, race, education, and household size, which are jointly not significant. These five variables plus the income and urban variables are included in the sample selection equation because they are available for the full KN sample. Although age, race, education, household income, and household size are all statistically significant at a 0.10 level in the selection probit, the Wald test for independence of the two equations can not be rejected at a 0.10 level. This result indicates that the main probit equation does not suffer from sample selection bias.

Table 4 Heckman sample selection probit results