In analyzing DTMV adoption decisions, we need to address whether a potential adopter is informed about its existence and has physical access to seed and at a price that is affordable. Once the DTMVs are released, information about their existence is disseminated through multiple channels that include (i) onfarm trials, (ii) demonstration plots controlled by agricultural extension agents, (iii) field days for farmers, (iv) agricultural shows to which farmers are invited and farmertofarmer exchange of information occurs, and (v) varietal promotion. The seed is usually produced by private seed companies and can be distributed by government, public sector agencies, cooperatives, and the private sector—agrodealers or, as is often the case, by a combination of all of these.
As the DTMVs are new and the target population is not universally exposed to them, observed sample adoption rates do not consistently represent the true population adoption parameters, even when based on a randomly selected sample. The reason is that researchers and extension workers have a tendency to target progressive farmers first, while farmers selfselect into exposure (Diagne 2006). To account for selection bias, Diagne and Demont (2007) use the counterfactual average treatment effect (ATE) framework, which allows for both nonparametric and parametric methods to derive consistent estimates. The ATE parameter measures the effect, or impact, of a “treatment” on a person randomly selected in the population (Rubin, 1974; Wooldridge 2002). But as expressed by Donstop et al. (2013), apart from a lack of awareness, there are is another constraint, which is the lack of access to seed. The farmer can be aware of DTMV but cannot become an adopter if (s)he does not have access to them. DTMV awareness and seed access are, therefore, both necessary conditions for adoption. Donstop et al. (2013) also show that while it is possible to observe farmers can be aware of improved varieties without getting access to their seed, it is not possible to observe the seed access status among farmers that are not aware of the existence of DTMVs (Fig. 1). By extension, the farmers can be aware of DTMVs without having access to their seed at an affordable price, but we do not know the status in terms of accessibility to affordable DTMV seed among farmers that are unaware of the existence of DTMVs and among those that have no physical access to seed. As in the case of Donstop et al. (2013), in this paper, we use the term “access” to imply physical availability of the seed in the farmer’s environment and not the acquisition availability (affordability). Our extension in this study is that we also explore how the acquisition (price) affordability of DTMV seed affects adoption rates.
To obtain the access and affordability variables, we collected information on all possible reasons for not adopting the DTMVs through individual interviews among households that were aware of DTMVs but did not adopt them (Fig. 1). At the first stage, all farmers were asked whether they knew specific DTMVs. At the second stage, for those who reported having knowledge (denoted by w) of DTMVs (w = 1), the following specific question was asked: “Did you grow any of the DTMVs in the 2015 planting season?” When a farmer responded that (s)he did not grow any DTMV, (s)he was asked to provide reasons for not growing them. A wide range of responses were recorded; however, of interest were responses related to seed accessibility and seed affordability. We denote s to stand for the (physical) access to seed status of a farmer, with s = 1 for farmers who had access to seed and s = 0 for farmers who had no access to seed. For farmers who did not know about DTMV (that is w = 0), they were not asked about seed access. As expressed by Donstop et al. (2013), this implies that we do not have information on seed access status of the farmers who were not aware of DTMVs. Indeed, some of the farmers who are not aware of DTMVs may actually have access to DTMV seed even though they are not aware of its existence. As expressed by Donstop et al. (2013), this could be the case, for example, when the variety is present in the village, but the farmer is not aware of the variety. We denote p to stand for the seed acquisition affordability status of a farmer, with p = 1 for farmers that had access to seed at an affordable price and p = 0 for farmers who had no access to affordable seed. For farmers who did not know about DTMV (that is w = 0), they were not asked about seed affordability. As in the case of seed availability, this implies that we do not have information on access to the “affordable” seed status of the farmers who were not aware of DTMVs. Indeed, some of the farmers who are not aware of DTMVs may actually be able to afford DTMV seed.
Based on the earlier explanation, the physical seed access status variable is either 0 or 1 and it is only observed among individuals that are aware of DTMVs. Hence, the awareness and the physical access–unrestricted potential adoption rate is always greater than or equal to the awareness–unrestricted one. Similarly, the awareness–, physical access–, and acquisition affordability–unrestricted potential adoption rate is always greater than or equal to awareness– and physical access–unrestricted potential adoption rate.
In what follows, we extend the ATE adoption framework proposed by Diagne and Demont (2007) to estimate three types of potential adoption rates; (i) the awareness–unrestricted; (ii) the awareness–access–unrestricted; and (iii) the awareness–access–affordability–unrestricted DTMV potential adoption rates and the associated adoption gaps in Uganda, as well as the determinants of DTMV awareness, access , affordability, and adoption.
We adopted the potential outcome framework of Rubin (1974), in which every farmer in the population has theoretically eight potential adoption outcomes:

(i)
An outcome with awareness and access to seed at an affordable price, say y_{111} (that is, y_{111} is the outcome when w = 1, and s = 1 and p = 1 )

(ii)
An outcome when is aware and has access to seed but when seed is sold at a price farmers cannot afford, say y_{110} (that is, y_{110} is the outcome when w = 1, s = 1 and p = 0 )

(iii)
An outcome with awareness, with affordable seed price, but farmers do not have access to seed, say y_{101} (that is, y_{101} is the outcome when w = 1, s = 0, and p = 1 )

(iv)
An outcome with awareness of DTMV, but one does not have access to seed and the seed price is not affordable, say y_{100} (that is, y_{100} is the outcome when w = 1, s = 0, and p = 0 )

(v)
An outcome without awareness of DTMV, but having access to seed and at a price that is affordable, say y_{011} (that is, y_{011} is the outcome when w = 0, s = 1, and p = 1 )

(vi)
An outcome without awareness of DTMV and with access to seed but the seed price is not affordable, say y_{010} (that is, y_{010}is the outcome when w = 0, s = 1, and p = 0 )

(vii)
An outcome without awareness of DTMV and with no access to seed but the seed price is affordable say y_{001} (that is, y_{001}is the outcome when w = 0, s = 0, and p = 1 )

(viii)
An outcome without awareness of DTMV and without access to seed and when the seed price is not affordable say y_{000} (that is, y_{000}is the outcome when w = 0, s = 0, and p = 0 ).
Hence, the observed adoption outcome y can be expressed relative to the eight potential adoption outcomes as:
$$ y={wspy}_{111}+ ws\left(1p\right){y}_{110}+w\left(1s\right){py}_{101}+w\left(1s\right)\left(1p\right){y}_{100}+\left(1w\right){spy}_{011}+\left(1w\right)\ s\ \left(1p\right){y}_{010}+\left(1w\right)\left(1s\right){y}_{001}+\left(1w\right)\left(1s\right)\left(1p\right){y}_{000} $$
(1)
Since awareness, physical seed access, and seed price affordability are necessary conditions for adoption in that order, we have y_{101} = y_{100} = y_{001} = y_{010} = y_{001} = y_{011} = y_{000} = 0.
Hence, Eq. (1) is reduced to:
The potential outcome is always 0 when the farmer is not aware, and/or does not have access to seed and/or not have access at an affordable price. It follows that y_{111}, which is the potential outcome, is also the treatment effect of a given farmer when the farmer is aware and has physical seed access and seed access at an affordable price. The average treatment effect of awareness and physical access to seed at an affordable price is expressed as the expected value E(y_{111}).
If we consider awareness as a treatment, the awareness–unrestricted potential adoption outcome can be derived from Eq. (2) by setting w = 1 and expressed as follows:
$$ {y}_1^{\ast }={spy}_{111} $$
(3)
Similarly, by setting s = 1, the physical seed access–unrestricted potential adoption outcome\( {y}_1^{\ast \ast } \) is defined as:
$$ {y}_1^{\ast \ast }={wpy}_{111} $$
(4)
After setting p = 1, the seed acquisition affordability–unrestricted potential adoption outcome can also be expressed as:
$$ {y}_1^{\ast \ast \ast }={wsy}_{111} $$
(5)
Similarly, the awareness and physical seed access–unrestricted potential adoption outcome is by setting (w, s) = (1, 1) expressed as:
$$ {y}_{11}^{\ast }={py}_{111} $$
(6)
The awareness and acquisition affordability–unrestricted potential adoption outcome is by setting (w, p ) = (1, 1) expressed as:
$$ {y}_{11}^{\ast \ast }={sy}_{111} $$
(7)
The physical seed access and acquisition affordability–unrestricted potential adoption outcome is by setting (s, p) = (1, 1) expressed as:
$$ {y}_{11}^{\ast \ast \ast }={wy}_{111} $$
(8)
The average treatment effect (ATE) of awareness, physical seed access, and acquisition affordability as measured by the expected value E(y_{111}) is the potential adoption rate when the full population is aware of DTMVs and has physical access to the seed for DTMVs at a price affordable by the full population. This is different from the potential adoption rate when the full population is only aware of DTMVs \( E\left({y}_1^{\ast}\right) \), and it is also different from the potential adoption rate when the full population only has physical access to DTMV seed \( E\left({y}_1^{\ast \ast}\right) \). It is also different from the population potential adoption rate when the full population has access to seed at an affordable price (with some not necessarily being aware), which is measured by the parameter \( E\left({y}_1^{\ast \ast \ast}\right) \). Three more joint bivariate potential adoption rates (Eqs. 6–8) correspond to awareness and physical access to seed (\( {y}_{11}^{\ast}\Big), \)awareness and acquisition affordability of seed (\( {y}_{11}^{\ast \ast}\Big) \), and physical seed access and acquisition affordability of seed (\( {y}_{11}^{\ast \ast \ast}\Big). \)
To distinguish the seven population potential adoption rates, we call parameter E(y_{111}) the awareness–physical seed access at affordable prices unconstrained potential adoption rate (ATE_{wsp}), whereas \( E\left({y}_1^{\ast}\right) \), \( E\left({y}_1^{\ast \ast}\right) \) and \( E\left({y}_1^{\ast \ast \ast}\right) \) are called awareness unconstrained (ATE_{w}), access unconstrained (ATE_{s}), and affordability–unconstrained (ATE_{p}) population potential adoption rates, respectively. \( E\Big({y}_{11}^{\ast } \)), \( E\Big({y}_{11}^{\ast \ast } \)), and \( E\Big({y}_{11}^{\ast \ast \ast } \)) are called the joint bivariate potential adoption rates corresponding to awareness and physical seed access (ATE_{ws}), awareness and seed affordability (ATE_{wp}), and physical seed access and seed affordability ATE_{sp}), respectively.
Among the seven population potential adoption rates defined above, we restrict our empirical estimation to only three: ATE_{w}=E(\( {y}_1^{\ast}\Big),{ATE}_{ws} \)=E(\( {y}_{11}^{\ast}\Big), \) and ATE_{wsp}=E( y_{111}). The exclusion of the two marginal potential adoption rates (related to physical seed access (ATE_{s}) and acquisition affordability (ATE_{p})) from the empirical analysis is justified by the fact that the two variables (i.e., s and p) are observed only for the aware subsample (i.e., for w = 1) which makes it difficult to estimate them without further assumptions. The same is true for the excluded joint bivariate potential adoption rate related to physical seed access and acquisition affordability (\( {ATE}_{sp}=E\left({y}_{11}^{\ast \ast \ast}\right) \)). The exclusion of the joint bivariate potential adoption rate related to awareness and acquisition affordability (\( {ATE}_{wp}=E\left({y}_{11}^{\ast \ast \ast}\right)\Big) \) from the empirical analysis is justified by the fact that it measures the same quantity as the potential adoption rate under unrestricted joint awareness–physical access–acquisition affordability (ATE_{wsp} = E(y_{111})) since it is measured only for those with physical access to seed (s = 1). The choice of three potential adoption rates ATE_{w}=E(\( {y}_1^{\ast}\Big),{ATE}_{ws} \)=E(\( {y}_{11}^{\ast}\Big), \) and ATE_{wsp}=E( y_{111}) for the empirical analysis is justified by their policy relevance in two ways. First, understanding the marginal adoption changes resulting from awareness creation should inform policy on the level of investment required for improving the adoption of DTMVs through activities that enhance the awareness about DTMVs among the farming population. Second, understanding the marginal increase in adoption rates resulting from increased seed availability and affordability should be useful to seed suppliers in forecasting the potential demand for DTMV seed at given market prices and should also inform public policy regarding the magnitude of price support required to enhance farmer’s adoption of DTMVs.
The major contribution of this paper is that this is the first attempt to estimate the joint average treatment effect of joint awareness, physical seed access, and acquisition affordability measured by the expected value E( y_{111}). This differs from the marginal adoption rate corresponding to awareness E(\( {y}_1^{\ast}\Big), \) defined in Diagne and Demont (2007) and also differs from the joint bivariate potential adoption rate corresponding to awareness and physical access E(\( {y}_{11}^{\ast}\Big) \) defined by Dontsop et al. (2013).
In this paper, the observed population adoption rate parameter (which is consistently estimated by the sample adoption rate computed from a random sample) is a measure of the population joint awareness–physical access–acquisition affordability and adoption rate which is the same as the population joint awareness, seed access, at affordable prices, and adoption rate as E(y) = E(wspy_{111}) and not a measure of the population joint awareness and adoption E(wy_{11}) rate as argued in Diagne and Demont (2007). Hence, in what follows, we use the notation JEAAA (joint awareness–access–affordability and adoption) for the observed population parameters (E(y)). It is also clear from the above that \( E(y)\le E\left({y}_1^{\ast}\right)=E\left(s{py}_{111}\right)\le E\left({y}_{111}\right) \) and \( E(y)\le E\left({y}_{11}^{\ast}\right)=E\left(p{y}_{111}\right)\le E\left({y}_{111}\right) \) (since w, s, and p are binary), meaning that the awareness–unconstrained and awareness–physical access–unconstrained, potential adoption rates are both greater than the observed actual adoption rate but always lower than the awareness–physical seed access–acquisition affordability unconstrained potential adoption rate.
We can then define three adoption gaps with one attributable to lack of seed access at affordable prices (Eq. 9), lack of physical seed access (Eq. 10), and lack of awareness (Eq. 11) as follows:
$$ {GAP}_{wsp}=E(y)E\left({y}_{111}\right)= JEAAA{ATE}_{wsp} $$
(9)
$$ {GAP}_{ws}=E(y)E\left({y}_{11}^{\ast}\right)= JEAA{ATE}_{ws} $$
(10)
$$ {GAP}_w=E(y)E\left({y}_1^{\ast}\right)= JEA{ATE}_w $$
(11)
where ATE_{wsp}is the average treatment effect parameter when joint awareness, physical seed access, and seed at affordable prices are the treatment variables. ATE_{ws} is the average treatment effect parameter when awareness and seed access, jointly, is the treatment variable and ATE_{w} is the average treatment effect parameter when awareness is the treatment variable.
According to the ATE framework, the awareness–unrestricted (ATE_{w}), the joint awareness–physical access–unrestricted (ATE_{ws}), and the joint awareness–physical access–affordability–unrestricted (ATE_{wsp}) potential adoption rates can be defined for various subpopulations by the values x in the support of some random variable X as the average treatment effects conditional on x, \( E\ \left({y}_1^{\ast }\ X=x\right) \), \( E\ \left({y}_{11}^{\ast}\kern0.5em X=x\right), \) and E (y_{111} X = x); E respectively (the conditional ATE parameters). It follows that the potential adoption rates in the subpopulation aware of DTMVs, in the subpopulation aware and with physical seed access, and in the subpopulation aware and with physical seed access at affordable prices, correspond to the average treatment effect on the treated (ATT) parameters and expressed as follows:
$$ {ATT}_w=E\Big({y}_1^{\ast}\leftw=1\right) $$
(12)
$$ {ATT}_{ws}=E\left({y}_{11}w=1,s=1\right) $$
(13)
$$ {ATT}_{wsp}=E\left({y}_{111}w=1,s=1,p=1\right) $$
(14)
The potential adoption rates in the untreated subpopulations are given by the respective ATE on the untreated (ATU) as follows:
$$ {ATU}_w=E\Big({y}_1^{\ast}\leftw=0\right) $$
(15)
$$ {ATU}_{ws}=E\left({y}_{11}^{\ast }w=0,s=0\right) $$
(16)
$$ {ATU}_{wsp}=E\left({y}_{111}w=0,s=0,p=0\right) $$
(17)
Furthermore, as in Diagne (2006, 2010) and Diagne and Demont (2007), we will define the awareness, awareness–physical seed access, and awareness–physical access–acquisition affordability population selection bias (PSB) parameters that measure the extent to which the three treatment status variables are not randomly distributed in the population, respectively, as:
$$ {PSB}_w={ATT}_w{ATE}_w=E\left({y}_1^{\ast }w=1\right)E\left({y}_1^{\ast}\right) $$
(18)
$$ {PSB}_{ws}={ATT}_{ws}{ATE}_{ws}=E\left({y}_{11}^{\ast }w=1,s={s}_1=1\right)E\left(\ {y}_{11}^{\ast}\right) $$
(19)
$$ {PSB}_{wsp}={ATT}_{wsp}{ATE}_{wsp}=E\left({y}_{111}w=1,s={s}_1=1,p={p}_1=1\right)E\left(\ {y}_{111}\right) $$
(20)
The empirical estimation involves the application of the ATE framework to provide consistent estimates of \( E\left({y}_1^{\ast}\right),\kern0.5em E\left({y}_{11}^{\ast}\right),\kern0.5em and\ E\left({y}_{111}\right). \) In fact, the parameters for \( {y}_1^{\ast } \) are identified and estimated exactly the same way as in Diagne and Demont (2007) using the w (awareness) variable while for the case of \( {y}_{11}^{\ast } \) and y_{111}, we use the ws and wsp variables, respectively. As shown in Fig. 1, all three variables are only observed for the farmers that are aware of DTMVs (that is, for farmers with w = 1) but the products ws and wsp are known for all farmers, as also shown above. It is assumed that the conditional independence assumption holds in all cases. As expressed in Donstop et al. (2013), it is assumed that the distributions of the treatment status variables w, ws, and wsp, are independent of the distribution of the potential outcomes \( {y}_1^{\ast } \), \( {y}_{11}^{\ast } \), and y_{111}, conditional on a vector of covariates x. That is, using the standard notation for conditional independence (A1): w ⊥ y_{1} ∣ x, w, s ⊥ y_{11} ∣ x, and w, s, p ⊥ y_{111} ∣ x. By the propriety of conditional independence, assumption (A1) also implies that \( w\perp {y}_1^{\ast}\mid x \) (Donstop et al. 2013). Therefore, we can use the same identification results and estimation procedures as in Diagne and Demont (2007) and Diagne (2012) to identify and estimate parameters related to the three treatments. The Additional file 1 elaborates on the parametric estimation of ATE.