So far we've been making the stable unit treatment value assumption due to Rubin. This assumption is two fold. First, there are not multiple versions of the treatment, and second, the values of a unit potential outcomes depend only on the treatment assignment administered to that unit, not the assignments of other units. The first part could be violated if the treatment, for example, a surgical procedure is performed by different surgeons, as a unit's outcome could depend on the surgeon. Ultimately, a judgment that there are no multiple treatments is subjective, and in many instances where this assumption is made as an example just given it may be violated to some extent. The second part of the stable unit treatment value assumption is typically referred to as interference. Now, no interference is reasonable for many applications. For example, a unit's outcome under surgery will not depend on whether or not some other unit is assigned or not assigned to surgery, but this assumption is often violated in social contexts. As examples, Hong and Raudenbush consider the case of a student's reading scores. They assume a student's scores are affected by the assignment of the other students in his or her class, but not by the assignment of students and other classes. Sobel considers the moving to opportunity study, in which families in housing projects were randomly assigned to either receive a housing voucher to move, or voucher plus assistance with the requirement that the move must be to a low poverty area, or to a control group receiving no assistance. The participants in the study within a housing project knew one another. Sobel points out that whether or not a unit moves to a low poverty area could depend not only on net subjects assignment, but the assignment of other subjects in the project. For example, family A may move with the family receives a voucher plus assistance and family B which is friendly with family A also receives a voucher plus assistance. But if family B is assigned to the control group, family A might not move. Hudgen and Halloran, who are hereafter referred to as HH, consider the case of infectious diseases. Suppose that an individual is unvaccinated, contrast that with the case in which all individuals with whom he is in contact are vaccinated or unvaccinated. It is easy to imagine that in the former case, he does not contract the disease and in the latter case, he does contract the disease. In all these cases, an individual's assignments stays the same, but the assignment of other units varies. Consequently, the individual gets a different result. This has been called a spillover effect. It has also been called an indirect effect in contracts to a comparison of a unit's outcome when that unit receives treatment and does not receive treatment. But since we've been using indirect effect in another way, I will hereafter refer to such effect as spillovers. Broadly speaking then there are two types of effects. One, direct by which we mean the effect on the unit when that unit is exposed to the treatment or not, set effect now possibly dependant on the assignments of other units, as in the MTO example above, where A moves to a low poverty area, B is assigned a voucher plus assistance, and does not move if not assigned a voucher plus assistance. Two, spill over in which a units assignment is the same but the assignment of other units vary, also as in the MTO example where A receives a voucher plus assistance, and A moves if B receives a voucher plus assistance, and does not move if B does not receive a voucher plus assistance. Before formalizing these ideas in the general case, we introduce them in the simplest possible setting, a population of two-person dyads for example, a husband and a wife. Each of whom can be exposed or not to an exercise program, after which each member's cardiovascular fitness is measured. Now it's natural to think in this context that the husband's outcome or the wife's outcome may depend not only on whether or not he or she was assigned to treatment, but the partners assignment as well through spillover. However, we will assume that there is no interference amongst different cross different dyads. Suppose a simple random sample of size n has been taken from a population P of dyads. The dyads are labeled j through J, capital J, and the units i equals 1, i equals 2 for wife. One for husband. Let Z_1j equals 1 if the husband in dyad J receives treatment and 0 otherwise, and define Z_2J analogously for the wife. Now, for all pairs j and j star, we assume that assignments in dyad j star do not affect outcomes in dyad j. We can then represent the potential outcomes for dyad J, as Y sub ij. Z_1, Z_2, where i is an element of one or two. i equals one for the husband's outcomes, and Z is zero or one depending on whether or not you treat or not treat so Z can be 0,0. Nobody gets treated in the dyad, that can be 1,1. Everybody gets treated in the dyad et cetera. Now, we denote the observed outcome y j. Both at a randomized experiment has been conducted with assignment probabilities, probability Z_1j, little z_1, Z_2J equals little z_2 et cetera. Okay. So, we're going to define the total effect for unit i and dyad j as yj_11 minus yj_00. So, that's the case where both parties in the dyad get treated, and the case where neither gets treated. Now, we're going to decompose the total effect as the sum of the direct effect and the spillover effect. The total effect may be decomposed as the sum of the direct effect, and the spillover effect respectively. So, you see this decomposition, the first one compares both traded to the case where the husband is not treated and the wife is treated, and then it's plus the case where the husband is not treated and the wife is treated versus no one is treated in the dyad. The second case is a different contrast. In the second case, the husband is treated and the wife is not versus the case where neither is treated, and it's plus the case where both are treated minus the case where the husband is treated. Now, in general, these two decompositions will give different values for the constituent effects. So, to fix ideas, let y equal 1, that's the husband. So, we're looking at husband outcomes, and consider the first of the two decompositions. The total effect of treating both husband and wife versus treating neither, is the sum of the direct effect of treatment for the husband when the wife is also treated, and the spillover effect from treating the wife when the husband is not treated, since spill over onto the husband because i is one. In the second, the total effect is the direct effect of treatment for the husband when the wife is untreated, plus the spill over onto the husband from treating the wife when he is treated. Then, now if you want to consider the wife, just interchange the words husband and wife above. Make sure you understand it. That's not a bad thing to do. Okay. So, now if we average over the husbands in the population or the wives, we'll get average potential outcomes, and I put the dot to say that we're averaging over the dyads j. Okay. So, if we average of yi is one, and we average over j, we're averaging over all the husbands. As we're assuming this is a randomized study, yi dot bar. So, that's the sample average amongst the husbands, in the group where both are treated that consistent estimate of the expected value for the husbands under treatment of both husband and wife. Similar remarks will apply to the average values for the other potential outcomes. Thus, the average direct and spill-over effects can be consistently estimated using the sample averages. All right. We now proceed to the more general case, above, to illustrate we use superpopulation inference. As much of the literature on interferences evolved using design-based inference, following the treatments in Sobel and HH, we use this mode of inference to introduce the general case.
