The impact of tobacco prices on smoking onset in Vietnam: duration analyses of retrospective data

Abstract

The benefits of preventing smoking onset are well known, and even just delaying smoking onset conveys benefits. Tobacco control policies are of critical importance to low-income countries with high smoking rates such as Vietnam where smoking prevalence is greater than 55 % in young men between the ages of 25 and 45. Using a survey of teens and young adults, I conducted duration analyses to explore the impact of tobacco price on smoking onset. The results suggest that tobacco prices in Vietnam have a statistically significant and fairly substantial effect on the onset of smoking. Increases in average tobacco prices, measured by an index of tobacco prices and by the prices of two popular brands, are found to delay smoking onset. Of particular interest is the finding that Vietnamese youth are more sensitive to changes in prices of a popular international brand that has had favourable tax treatment since the late 1990s.

Appendix: The split population duration model

Following the notation of Schmidt and Witte [79, 80], let F be an unobservable variable indicating whether an individual i would or would not eventually start smoking (i.e. fail). Formally:

$$ P({\text{eventually}}\;{\text{fail}}) = P(F = 1) = \delta $$

$$ P({\text{never}}\;{\text{fail}}) = P(F = 0) = 1 - \delta $$

Let g(t\F = 1) and G(t\F = 1) be the conditional density of survival times and its corresponding cumulative distribution function for individuals who eventually fail. Let T be the length of the follow-up period (i.e. T _i indicates censoring time). Let R be an observable indicator so that R_i = 1 if there is failure by time T and R_i = 0 if not. For the starters (R _i  = 1), the unconditional density is:

$$ P(R = 1) = P(F = 1)P(t < T\backslash F = 1)g(t\backslash t < T,F = 1) $$

$$ = P(F = 1)g(t/F = 1) $$

$$ = \delta g(t/F = 1) $$

For the nonstarters (R _i  = 0), the unconditional density is:

$$ P(R = 0) = P(F = 0) + P(F = 1)P(t > T/F = 1) $$

$$ = (1 - \delta ) + \delta G(t/F = 1) $$

Combining the two unconditional densities yields the following likelihood function:

$$ L = \prod\limits_{i = 1}^{N} {\delta_{i} g} (t\left| {F = 1} \right.)^{{R_{i} }} (1 - \delta_{i} + \delta_{i} G(t\left| {F = 1} \right.))^{{1 - R_{i} }} $$

The contribution of individual i to the log-likelihood function is:

$$ {\text{In }}L = R_{i} \left( {{\text{In}}\delta_{i} + {\text{lng}}(t\left| {F = 1} \right.)} \right) + (1 - R_{i} ){\text{In}}\left( {1 - \delta_{i} + \delta_{i} G(t\left| {F = 1)} \right.} \right) $$

The probability δ _i is typically modelled as a logit or a probit but can also be modelled as a complementary log–log (cloglog). Unlike probit or logit for which the response curve is symmetric about δ _i  = 0.5, the cloglog model has a response curve that is asymmetric [23]. Formally [23]:

$$ {\text{logit:}}\,\delta_{i} = \frac{{e^{{a'z_{i} }} }}{{1 + e^{{^{{a'z_{i} }} }} }} $$

$$ {\text{probit:}}\,\delta_{i} = \Upphi (a'z_{i} ) $$

$$ {\text{cloglog:}}\,\delta = 1 - e^{{\left( { - e^{{a'z_{i} }} } \right)}} $$

where z _i is a vector of time invariant covariates, $\Upphi$ is the cumulative density function for the standard normal distribution, and α is a parameter vector. When δ_i = 1 for all individuals, the split population duration model reduces to a standard duration model.

Both theory and data indicate that the hazard rate first increases and then decreases, which rules out exponential and Weibull duration models. Log-logistic and lognormal models are typically used. Formally, the probability density function g(t/F = 1) and the survival function G(t/F = 1) for the log-logistic distributions are [23]:

$$ g(t\left| {F = 1} \right.) = \lambda P(\lambda t)^{\rho - 1} /\left( {1 + (\lambda t)^{\rho } } \right)^{2} $$

$$ G(t\left| {F = 1} \right.) = 1/1 + (\lambda t)^{\rho } $$

where ρ is a shape parameter, $\lambda = e^{ - \beta 'X}$, X_i is a vector of time invariant and time variant covariates, and β is a parameter vector.