# Help in understanding summation notation in published article

I am reading an article on cost-effectiveness analysis and would like some help in interpreting an equation given by the authors for the primary outcome of interest, "lifetime productivity costs." The article in question is Tilford et al.'s (2009) "Labor Market Productivity Costs for Caregivers of Children with Spina Bifida: A Population-Based Analysis," published in the journal Medical Decision Making (1). I would like to use the methods in this analysis to guide a similar analysis done for a different disease.

Background and context

As the title suggests, in this U.S.-based paper, the authors estimated the labor market productivity costs of caregivers of children with spina bifida (SB), a birth defect involving the spine and spinal cord that can cause physical and sometimes mental disabilities among affected individuals. SB is preventable through adequate intake of folic acid during pregnancy. SB causes lifelong disabilities and parents of individuals with SB may face the decision of dropping out of the labor market in order to provide care for their SB-affected child. One of the goals of the study was to highlight the value of prevention by quantifying the economic costs of the disease. The authors compared the productivity costs of caregivers of children with SB to those of caregivers of children from the general population. They found that caregivers of children with SB on average worked 9.2 hours less per week than caregivers of children without disabilities, which translates to a lifetime productivity cost of $133,755 in 2002 dollars. Methods The author used the following equation to estimate lifetime productivity costs (LPC): $$LPC=\sum_{t}\frac{w_t(s_{t}^{SB}H_{t}^{SB}-s_{t}^{o}H_{t}^{o})}{(1+r)^t}$$ Where: •$t$is "the age of the child defined over the relative age range in which the child's disability is likely to impact labor market outcomes," here the authors assume that the child would need assistance from birth to age 21 •$w_t$is the age-sex adjusted hourly wage that could be earned in the absence of caring for a child with disabilities •$s_t$is the child's survival probability •$H_t$is the average number of hours worked per year • the superscript$SB$refers to caregivers of children with SB • the superscript$o$refers to caregivers of children from the general population -and$r$is the discount rate (the authors used 3%) Questions 1. I interpreted$\sum_{t}(\cdot)$as "sum$(\cdot)$over [all valid] values of$t$," meaning, rather than summing over all$i$individuals in a dataset. Is this correct? 2. As a follow-up question to (1), the summation notation does not have an upper limit (i.e., it's not$\sum_{t=1}^4$), but the article stated that they assumed the child would need care to 21 years of age. If my interpretation of the summation as described is correct, does that mean the summation can be rewritten as:$LPC = \frac{w_0(s_{0}^{SB}H_{0}^{SB}-s_{0}^{o}H_{0}^{o})}{(1+0.03)^0} +...+ \frac{w_{21}(s_{21}^{SB}H_{21}^{SB}-s_{21}^{o}H_{21}^{o})}{(1+0.03)^{21}}$? 3. If (1) and (2) are correct, does this mean that I would need the average (or median, depending on the distribution in the data) of number of hours$H$at each age child age$t$in my dataset, since the equation again indexes the age$t$instead of an individual child/caregiver$i$? 4. Finally, would$s$be the age-specific survival probabilities, from age 0 to 21 for children with SB and children in the general population (e.g. from Kaplan-Meier life tables from previous studies)? References 1. Tilford JM, Grosse SD, Goodman AC, Li K (2009) Labor market productivity costs for caregivers of children with spina bifida: a population-based analysis. Med Decis Mak 29(1):23–32. doi:10.1177/0272989X08322014. URL: http://mdm.sagepub.com/content/29/1/23 ## 1 Answer You can write summation in detailed fashion$ \sum_{i=1}^n x_i $, less precisely$\sum_i x_i$, or even as$\sum x$. The more complicated is your formula, the more precise you need to be in your notation. For example,$\sum_i x_i$is pretty clear way of saying "sum all the$x_i$there are". There are also other ways of writing summation, e.g.$\sum_{i \in C} x_i$(sum for all$i$indexes in set$C$), or$\sum_{4 < i \le 16} x_i$(sum all$i$indexes in given range),$\sum_{i,j} x_{ij} = \sum_i \sum_j x_{ij}$(sum over all the$i$and$j$indexes) etc. Notation needs to be readable, you can use it flexibly as far as it serves this purpose. As about (3), this follows from your question since you defined$H_t$as "average number of hours worked per year". Question (4) cannot be answered from the information you've provided in the question -- if it's unclear, e-mail the authors of the paper. • Thanks, Tim. As (4) is concerned,$s$was described as the survival probability of the child, so I interpreted$s_t$as the age- (or age-band) specific probability of survival. My broader question is whether I was interpreting the summation correctly as summing over some aggregate measure by$t\$ (age), be it survival, wage, etc., rather than individual children Dec 14, 2016 at 15:33
• @MarquisdeCarabas you and the paper refer to it as "average number..." so I can't see how it could not be an aggregate measure.
– Tim
Dec 14, 2016 at 15:36