2.1. First stage bootstrap DEA
Economic production theory provides the basis for farms’ allocative efficiency analysis performed in this study. Production theory provides an analytical framework for making assumptions about farmers’ behavior when their environment changes. The central point of the theory is the production function. It describes the technical relationship that links the maximum physical quantity of products (\({\varvec{y}}_{\varvec{t}}\)) that a farm can produce at time (t) from the given physical quantities of inputs (\({\varvec{X}}_{\varvec{i}\varvec{t},}\)) given a production technology. Thus, the mathematical representation of the production function is as follows:
$${\text{y}}_{\text{t}}=\text{f}({\text{x}}_{1\text{t},}{\text{x}}_{2\text{t},}{\text{x}}_{3\text{t},}\dots {\text{x}}_{\text{n}\text{t}})$$
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where\(f\) represents the relationship between the inputs used and the outputs produced. The basic premise of production theory is that maximizing profits motivates the producer to choose the various crops to produce as well as the types of inputs to use and the quantities needed. For the same reason, the farmer also identifies the combination of different types of inputs that will minimize the cost of producing a given quantity of a product (Battese 1992). Regarding this assumption, economists have been very concerned with measuring the production efficiency of a farm. DEA is a well-known technique introduced in Charnes et al. (1978) for evaluating the relative efficiency of a set of decision-making units (DMUs) that use multiple inputs to produce multiple outputs. Using Data Envelopment Analysis, three parameters have often been calculated (Farrell 1957). DEA is used in this research to measure Allocative efficiency (AE). Also refers to as price efficiency, allocative efficiency, under an input-based approach reflects the ability of a farm to respond to economic signals or to use inputs in optimal proportions, given output and input prices. The allocative efficiency can be derived from the deconstruction of cost efficiency using technical efficiency (Sharma and Leung 2003). Technical efficiency with input orientation relates to the degree to which a farm uses the minimum feasible amount of each input to produce a given level of output. It reflects the ability of a farm to use production technology in practice. Under the variable returns to scale (VRS) DEA model for production technology, consider a farm with a vector of inputs \(x\) and outputs \(y\); \(X\) and \(Y\) are the input and output matrices of the sample, respectively. The input-based technical efficiency under VRS technology, \({TE}_{i}\), of the \({i}^{th}\)farm is defined as:
$${TE}_{i}=\widehat{{\theta }_{i}}=min\left\{{\theta }_{i}|{y}_{i}\le Y\lambda ; {\theta }_{i}{x}_{i}\ge X\lambda ;I\lambda =1;\lambda \ge 0\right\}$$
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Where \(\lambda\) is vector of constant (weights) that defines the linear combination of the peers \(I\lambda =1\) is the convexity constraint for assuming variable return to scale.
Next, cost efficiency relates to the degree to which a farm uses the minimum feasible cost to produce a given level of output, assuming unchanged production technology and input prices. It measures the abilities of farm managers to choose appropriate technology and an optimal mix of input. For the \({i}^{th}\)farm of \(N\) farms, the DEA framework under VRS technology to calculate the minimum feasible cost to produce a given level of output and input prices is as follows:
$${w}_{i}^{{\prime }}\stackrel{-}{{x}_{i}}=min\left\{{w}_{i}^{{\prime }}{x}_{i}|{y}_{i}\le Y\lambda ; {x}_{i}\ge X\lambda ;I\lambda =1;\lambda \ge 0\right\}$$
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Where \(w\) is a vector of input prices for \({i}^{th}\) farm, and \(\stackrel{-}{{x}_{i}}\) is a vector of cost minimizing inputs levels for the \({i}^{th}\) farm calculated by the linear programming. Cost efficiency for the \({i}^{th}\) farm, \({CE}_{i}\), is how close the farm achieved minimum cots by adjusting inputs levels for a given level of outputs. It is expressed as the ratio of the minimum cost (\({w}_{i}^{{\prime }}\stackrel{-}{{x}_{i}}\)) to observed cost (\({w}_{i}^{{\prime }}{x}_{i}\)):
$${CE}_{i}={w}_{i}^{{\prime }}\stackrel{-}{{x}_{i}}/{w}_{i}^{{\prime }}{x}_{i}$$
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As mentioned earlier, the measure for allocative efficiency is provided as the ratio of cost or economic efficiency to technical efficiency. The Allocative efficiency, \({AE}_{i}\), for the \({i}^{th}\) farm is calculated as follows:
$${AE}_{i}={CE}_{i}/{TE}_{i}$$
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The widespread use of DEA in production contexts mainly comes from its appealing nonparametric nature that avoids the need of functional specifications of the production technology. A major drawback of the conventional DEA models is the assumption that the included inputs and outputs are measured without noise, do not forward any axioms on the distributional structure of deviations from a best practice frontier, the estimation of efficiency is sample-specific, yielded estimates are positively exaggerated and offer no discussion of uncertainty surrounding the estimates due to sampling variations (Simar and Wilson 1995; Olesen and Petersen 2015; Long et al. 2020; Long 2021). Indeed, the whole surrounding economic system has a stochastic nature. As a consequence, ignoring data uncertainty might lead to inconsistent and misleading results (Beraldi and Bruni 2020). This consideration becomes even more relevant when DEA is used as a predicting performance evaluation technique (Beraldi and Bruni 2012). These factors make conventional DEA difficult to use in drawing statistical inferences, and broad policy implications (Long et al. 2020). Acknowledging the need of explicitly assess risk and uncertainty in the DEA approach, a considerable effort has been devoted to extending the original DEA models (Olesen and Petersen 2015). One direction of DEA extension allows the model to be able to handle estimated deviations from frontier practice as random deviations within a statistical (or econometric) framework. A statistical framework requires an axiomatic approach to a statistical model including a specification of a sampling procedure sometimes denoted a Data Generating Process (DGP). A given set of data is regarded as just a sample from a large population and, and that the sample distribution of the efficiency scores should be estimated or approximated, e.g., using a bootstrapping procedure. This yields consistent estimators and inference in the form of, e.g., confidence intervals have high priority (Olesen and Petersen 2015). An example is the single bootstrap procedure proposed by Simar and Wilson (1995) which relies on the simple idea of repeatedly simulating the data generation process and applying the original estimator to each simulated sample so that resampled estimates mimic the sampling distribution of the original estimator. The empirical distribution of the resampled estimates can thus be used to make statistical inferences (Long 2021). Later on Simar and Wilson (2007) developed Algorithm 1 and Algorithm 2 based on bootstrapping techniques, to resolve not only the problem of biased estimated but also the correlation of error term after regression of efficiency estimates. Output oriented with estimates ranged from 1, the algorithm 1 has been modified by Badunenko and Tauchmann (2019), allowing for an analysis of input-oriented efficiency with efficiency scores ranging from 0 to 1.
2.2. Risk attitude impacts pathways
Many important economic decisions involve a level of risk originating from uncertainty due to: (i) the inability to precisely control or measure the causal factors of events; (ii) limited capacity to process information; (iii) the cost of information (Edwards 1954; Chavas 2004; Cayatte 2009). Several theories have emerged in the literature to explain farm household production choices. The recognition that farmers’ behavior is typically influenced by several natural hazards, market, and social uncertainties in developing countries has shifted the way of understanding their production decisions (Chavas 2004; Mendola 2007; Berkhout et al. 2010). These conditions pose risks to farmers’ production and make farmers very cautious in their decision-making. Uncertainty and household attitude toward risk (eg. risk aversion) were therefore acknowledged to play a central role in farmers’ production decisions. Risk aversion has direct consequences on the optimum resource-use level (Torkamani and Hardaker 1996; La rovere 1997; Fraval 2000; Berkhout et al. 2010). Consequently, the profit-maximizing peasant theory and the neoclassical agricultural household models which are the first two production choices theories were subject to severe criticisms. These reactions have entailed the inclusion of risk attitude within farmers' production choices based on the expected utility approach, which is one way of conceptualizing peasant risk-aversion theory (Mendola 2007). The expected utility theory originates from Bernoulli's postulate stating that the consequences of an event have a moral value or a utility greater than the monetary value for the economic agent in his decision-making (Olivero 2019). Based on precise axioms (comparability, transitivity, continuity, independence), was introduced two centuries later in economic theory by Von Neumann and Morgenstern, and dominated the decision-making modeling in risky situations (Chavas 2004; Concina 2014; Hardaker et al. 2015). Abreha (2007) and Van Der Waal and Versluis (2017) showed that decision-making, whether it is the choice of the crop to produce or the quantity to produce, narrows down to the choice between different risky options given the decision maker's risk attitude. More explicitly, farmers make choices from available risky alternatives, based on their risk preferences about outcomes and their beliefs about the probability of their occurrence. Both farmer behavior and its revealed risk attitude (e.g. risk aversion) are reflected in its utility function (Mendola 2007). As a matter of fact, farmers choose the mix of inputs to produce a given output before the harvest and without knowing the true state of nature. Productive factors are thus allocated according to farmers’ beliefs on risk and risk aversion (Wiens 1976). Consequently, the yield and returns that a crop farmer obtains will always be risky for several reasons, such as weather risks, pest and disease risk, and market risks. The obtained post-harvest output is hardly the same as the anticipated output at planting (Lien et al. 2023). In such situation, a risk-averse farmer prefers to avoid the worst possible outcome by taking a safer position, keeping in mind the possibility of incurring a significant loss. This results in suboptimal economic decisions with respect to input allocation. In that sense, farmers are likely to have different optimal input and output allocations when risk aversion is taken into account (Nalunga et al. 2019).
The pathways of the impact of risk attitudes on resource allocation have been presented in the theory of change (Fig. 1).
2.3. Modelling the heterogeneous effect of risk attitude
The theoretical underpinning of this study combines the production theory and control function approach. Through economic theory, the researcher provides an analytical framework for analyzing farmers’ efficiency, behavior, and decision-making under production and market constraints. Based on the structural econometric analysis, the latter enables assessing the impact of farmers' risk attitudes on agricultural efficiency using the potential outcomes framework. The focus is on evaluating the causal effect of farmers' risk attitudes on their agricultural efficiency. To describe the "potential outcomes framework" in this study, and following Ju and Lee (2017), Wooldridge (2015), and Lewbel (2007), we assume that farmers' risk attitude is the "treatment" (\(D\)) and their allocative efficiency is the "outcome" (\(Y\)). For each farm, we define a score of allocative efficiency associated with (\({Y}_{d}\)) and without \(\left({Y}_{0}\right)\) a given level of risk attitude. The results (\({Y}_{d}\)) can be discrete, continuous, or mixed (discrete-continuous) random variables (Heckman et al. 2006). We assume that each farmer seeks to maximize their level of allocative efficiency. The set of mutually exclusive risk attitude levels studied is \(d=0, \cdots ,\text{R}\), where \(d=0\) is the reference representing an average person who is not risk-averse.
Given that the treatment is ordered, we have:
$$\varvec{D} \equiv (1-{D}_{0}){D}_{r}$$
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The observed allocative efficiency’s scores can be written as follows:
$${Y}_{i}={D}_{0i}{Y}_{0i}+\sum _{0}^{R}{\delta }_{di}{Y}_{di}$$
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Where \({\delta }_{di}\equiv 1[{D}_{ri} =d]\)
In the present study, three potential allocative efficiency are to be considered: allocative efficiency when the farmer is "slightly risk-aversive" for \(D=1\), allocative efficiency when the farmer is "highly risk-aversive" for \(D=2\), and allocative efficiency when the farmer is "extremely risk-aversive" for\(D=3\). Accordingly, let \({X}_{i}\) be the observed characteristics of the different farms and farmers, \({\beta }_{d}\), the parameters to be estimated, and \({U}_{d}\), the stochastic term that captures the unobserved characteristics such that \(E\left[{U}_{d}|{X}_{i}\right]=0\). Potential allocative efficiency scores are modeled as follows (Ju and Lee 2017):
$${Y}_{0}={X}_{0}{\beta }_{0}+\sum _{0}^{R}{\beta }_{0}{\delta }_{0}+{U}_{0}$$
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\({ Y}_{d}={X}_{i}{\beta }_{d}+\sum _{0}^{R}{\beta }_{d}{\delta }_{di}+{U}_{d}\) (9) When the farmer becomes aware of his risk attitude and has a priori information on the allocative efficiency score linked to it, he makes anticipated choices based on this information. Although the expected levels of efficiency resulting from each level of risk attitude are not observed, the link between the "potential outcome framework" and the structural econometrics literature can be established by assuming that they can be inferred by a latent variable \({D}^{*}\). The latter can be thought of as the net gain from having a given risk attitude \(\left(E\left({\pi }_{d}-{\pi }_{0}\right)\right)\). It is expressed as follows:
$${D}_{i}^{*}=w{x}_{i}+{\epsilon }_{d}$$
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where \({D}^{*}\)is the status of risk attitude and \({x}_{i}\equiv (X, Z)\) is a vector k×1 of regressors that affect risk attitude by excluding a constant term, \(w\) is a conformal vector of parameters and \(\epsilon\) is the error term indicating heterogeneity not observed for processing and can be a function of \({U}_{i}\). The individual effect of risk attitude on farm allocative efficiency is the difference in farm allocative efficiency with the actual individual level of risk attitude and without this level of risk attitude is given by \({\varDelta }_{d}= {Y}_{\text{d}} - {Y}_{0}\) (Angrist et al. 1996; Heckman et al. 1997). However, at any given time, a farmer can only have one level of risk attitude and none of the other 3 levels can be observed. The farm's allocative efficiency score corresponding to another level of isk attitude is unknown. Therefore, the effects of treatment at the individual level cannot be directly estimated (Heckman et al. 2006). Thus, the population-level endpoint, including the mean treatment effect (ATE), is mainly evaluated. It estimates, for a farmer selected from the population, his average gain from a level \(d\) of risk attitude displayed compared to level \(d=0\). It is expressed as follows:
$$ATE\left(d,0 \right)= E\left[AE,{D}_{d}= 1 \right]-E\left[AE,{D}_{0} =0\right]$$
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The average effect of the treatment on the treated (ATE1 or ATT) also estimated. We observe the real allocative efficiency score of the farm \({Y}_{i}\), written in the form of a switching regression model as:
$${Y}_{i} = {Y}_{0} + {D}_{d}\left({Y}_{d} - {Y}_{0} \right)$$
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Using conventional regression notation, the realized allocative efficiency ( \({Y}_{i} )\)of the farm can be written as follows:
\({Y}_{i} ={\delta }+ X\gamma + {D}_{d}\left[X\lambda + {\eta }_{d}\right]+{\vartheta }_{d}\) (13) With 𝛾 = \({\beta }_{0}\), 𝜆 = \({\beta }_{d}\) − \({\beta }_{0}\), \({\eta }_{d}\) = \({U}_{d}-U\), and \({\vartheta }_{d}={U}_{i}\)
The coefficient of the treatment \({D}_{d }\)in the Eq. (13) is the individual treatment effect \({\varDelta }_{d}\)on the individual allocative efficiency of the holding associated with the risk attitude. It consists of two elements. The first is the conditional mean treatment effect at \(X = x\), \(ATE\left(X\right) = X\lambda\). It estimates the average gain of having a risk attitude \(d\) different from non-risk aversion (baseline) for a randomly selected farmer with characteristics \(X\). The second component, \({\eta }_{d}={U}_{\text{d}}-{U}_{0}\) is called the idiosyncratic gain for a particular farmer with a given risk-risk attitude (Heckman et al. 1997). It is the unobserved heterogeneity, which implies that the effect of having an attitude to risk \(d\) can vary from one farmer to another even after controlling for observable heterogeneity using the \(X\) covariates.
Streams of literature on allocative efficiency reveal that observable characteristics such as farmer characteristics (age, gender, education level, farming experience, access to extension services, access to credit, and improved management), which determine levels of allocative efficiency, also affect their attitude to risk (Wik et al. 2004; Okoruwa et al. 2014; Ullah et al. 2015; Van Winsen et al. 2016; Akhtar et al. 2017; Oluwatusin 2017; Ahmad et al. 2020; Sapkota 2021). To avoid an erroneous conclusion of a causal relationship between risk attitude and allocative efficiency, these common factors must be controlled (Adegbola 2010). In addition, the attitude towards risk is an intention, an orientation of the decision maker towards risk-taking which is inherent to him (Rohrmann 2008). This means that it is a behavioral variable that is not observable and whose assignment process to an individual is fuzzy. This effect makes it difficult to identify the impact of risk attitude. Furthermore, certain unobservable factors, such as entrepreneurial skills, management skills, social networks, and transaction costs, can influence both risk attitude and levels of allocative efficiency. Because of these unobservable factors, the impact may be exaggerated even if risk attitude had no direct effect. From this literature, the present study adopts a heterogeneous response model.
To describe the heterogeneous response pattern, in more formal terms, consider the allocative efficiency realized \({Y}_{i}\)in the Eq. (14). Next, the coefficient \({\eta }_{d}\)is replaced by its mean \(\stackrel{-}{\eta }=E\left[{\eta }_{d}\right]\)and its standard deviation \(\stackrel{\sim}{\eta }={\eta }_{d}-E\left[{\eta }_{d}\right]\). This transformation results in the following constant-coefficient model:
$${Y}_{i} = {\delta }+ X\gamma +{D}_{d}\stackrel{\sim}{X}\lambda +{D}_{d}{\stackrel{-}{\eta }}_{d}+{e}_{d}$$
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Where \(\stackrel{\sim}{X}= X-\stackrel{-}{X}\) denotes deprecated explanatory variables. By deprecating the \(X\) covariates before interacting them with \({D}_{d}\), we ensure that \(\stackrel{-}{{\eta }_{d}}\)is the average treatment effect at the means of \(X\). With this linear specification, it is also the unconditional ATE. \({e}_{d}={D}_{d}\stackrel{\sim}{{\eta }_{d}}+{\vartheta }_{d}\)is a composite error term comprising: the interaction term \(\stackrel{\sim}{{\eta }_{d}}\) with \({D}_{d}\)deviation from ATE, and the error term \({\vartheta }_{d}\). Assume that the two unobservable components of the error term \({e}_{d}\) are linearly related to the error term \({\epsilon }_{d}\)of the Eq. (10). Two important sub-cases of the heterogeneous response model can be derived. First, we hypothesize that \({D}_{d}\)is statistically independent of \(\stackrel{\sim}{\eta }\)given \(X\) and therefore uncorrelated as would occur when treatment is randomized between farmers. But \({D}_{d}\)is only related to \({\vartheta }_{d}\). This means that farmers who are more likely to display a given risk attitude due to unobserved characteristics differ in their pre-treatment characteristics from the ones who are less likely to display that risk attitude. We conclude that, after controlling for \({X}_{i}\)observables, farmers' risk attitude is not based on the idiosyncratic gains associated with it (Cornelissen et al. 2016). Second, we assume that the error term \({e}_{d}\) in Eq. (14) linearly depends on the unobservable \({\epsilon }_{d}\)in Eq. (10). The treatment effect \({\varDelta }_{d}\) varies even after accounting for \({X}_{i}\). This case arises when farmers make a positive sorting in risk attitude levels based on partial or full knowledge of their idiosyncratic gains from having a particular risk attitude. As a result, \(\stackrel{\sim}{\eta }\) and \({D}_{d}\)are positively correlated, or even conditional to \(X\), resulting in \(\left[{D}_{d}\stackrel{\sim}{{\eta }_{d}}\left|{D}_{d}\right.=1\right]>E\left[{D}_{d}\stackrel{\sim}{{\eta }_{d}}\left|{D}_{d}=0\right.\right]\). In addition, farmers with the same X's are influenced by their knowledge of their idiosyncratic earnings which differs from selection bias. Eq. (14) is referred to as a 'correlated random coefficient' or 'heterogeneous treatment effect' or 'essential heterogeneity' model. Thus, \(E\left({D}_{d}\stackrel{\sim}{{\eta }_{d}}+{\vartheta }_{d} |{X}_{i}=x, {Z}_{i}=z\right)\ne 0\). This is therefore the model used in this study to assess the heterogeneous effect of the risk attitude of plantain farmers on their allocative efficiency. Several econometric models have been developed to address the interaction of treatment effect with unobserved heterogeneity (Basu et al. 2007; Cornelissen et al. 2016).
2.4. The latent factor with instrumental variables approach
Based on the problems mentioned above, instrumental variables or the latent factor can be used. Due to the difficulty of finding good instruments, the likely weakness of the instruments, and the impossibility of testing the quality of the instrument other than by a justification based on the literature, latent factor models offer a good alternative. The latter has the potential to identify treatment effects even in the absence of instrumental variables and in cases of instrument weakness (Deb and Trivedi 2006a). Nevertheless, if the disadvantage of the instrumental variables’ method is the weakness of the instruments, the latent factor has functional form limitations. However, Banerjee and Basu (2021) argue that the latent factor model and the IV approach can be combined to circumvent these weaknesses and achieve better robustness. This combination significantly reduces the bias in estimating the causal effects of endogenous treatments. In this study, a three-factor latent model combined with instrumental variables (IV) was used to address the problem of bias in the estimates, often not taken into account in the empirical analysis, due to contamination or association of the selected instruments with outcomes independent of risk attitude (Carneiro et al. 2003; Aakvik et al. 2005; Deb and Trivedi 2006a; Mújica-Mota et al. 2020; Banerjee and Basu 2021). To construct the latent factor model, and following Aakvik et al. (2005), Carneiro et al. (2003), and Banerjee and Basu (2021), we assume the factor structure of \({\epsilon }_{d}\)and \({e}_{d}\)errors in the Eq. (10) and Eq. (14), written respectively as follows:
$${\epsilon }_{d}={{\delta }_{d}l}_{d}+{ϵ}_{d}$$
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$${e}_{d}={{\lambda }_{d}l}_{d}+{u}_{d}$$
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where \({l}_{d}\) are the latent (scalar) factors representing unobserved covariates (unobserved heterogeneity) such as farmer motivation, knowledge level, and innate managerial and technical abilities, \({ϵ}_{d}\) and \({u}_{d}\)are stochastic error terms; \({l}_{d},\) \({ϵ}_{d}\)and \({u}_{d}\)have a mean of zero, for each \(d\ne {d}^{{\prime }}\) they are mutually independent and are independent of exogenous variables in the selection and outcome equations. The parameters to be estimated \({\delta }_{d}\)and \({\lambda }_{d}\)are the factor loads for the selection and outcome equations, respectively.
A two-step approach was used to estimate the parameters. In the first step, the non-parametric bias-corrected DEA method was specified to estimate bias-corrected allocative efficiency scores using the single-bootstrap method (Badunenko and Tauchmann 2019; Long 2021). The model provides allocative efficiency scores between 0 and 1. Then, in the second step, the function of the predicted allocative efficiency scores and the risk attitude model are jointly estimated to analyze the effects of risk attitude. In this context, a model of correlated random coefficients for an ordered treatment and a discrete-continuous mixed outcome is specified (Carneiro et al. 2003). The farmer displays a risk attitude from a set of mutually exclusive and ordered \(d=\text{0,1},\text{2,3}\), implying a multiple-choice model. Considering the Eq. (10) and Eq. (15), the indirect utility associated with the nth level of risk attitude is:
$${D}_{i}^{\text{*}}=w{x}_{i}+\sum _{d=0}^{3}{{\delta }_{d}l}_{d}+{ϵ}_{d}$$
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Then, the probability of displaying a given the risk attitude can be inferred as follows (Wooldridge 2015; Ju and Lee 2017) :
$$\text{Pr}\left({D}_{i}|{x}_{i}, l\right)=g\left({x}_{i}{w}_{1}+\sum _{0}^{3}{{\delta }_{1}l}_{di},{x}_{i}{w}_{2}+\dots +\sum _{0}^{3}{{\delta }_{d}l}_{di}\right)$$
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In this study, we assume that g has an ordered probit structure defined as follows (Wooldridge 2015; Ju and Lee 2017):
$$Prob \left({D}_{i}=d|{x}_{i}\right)= {\Phi }\left({\mu }_{d}- w{x}_{i} \right)- {\Phi }\left({\mu }_{d-1}- w{x}_{i}\right), d=0, 1, 2, 3$$
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Thus, for a model with four levels of risk attitude, we have:
$${Pr }\left({D}_{i}=0|{x}_{i}\right)= \varPhi \left({\mu }_{0}- w{x}_{i} \right)$$
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$$Pr \left({D}_{i}=1|{x}_{i}\right)= \varPhi \left({\mu }_{1}- w{x}_{i} \right)-\varPhi \left({\mu }_{0}- w{x}_{i} \right)$$
$$Pr \left({D}_{i}=2|{x}_{i}\right)= \varPhi \left({\mu }_{2}- w{x}_{i} \right)-\varPhi \left({\mu }_{1}- w{x}_{i} \right)-\varPhi \left({\mu }_{0}- w{x}_{i} \right)$$
$$Pr \left({D}_{i}=3|{x}_{i}\right)= 1-\varPhi \left({\mu }_{2}- w{x}_{i} \right)-\varPhi \left({\mu }_{1}- w{x}_{i} \right)-\varPhi \left({\mu }_{0}- w{x}_{i} \right)$$