*2.4. Statistical Analysis*

Two statistical models were used to evaluate OS-based VRN in-field fertilizer N rate, lint yield, and net return (NR) relationships with farm field characteristics. The first is a general linear model for the fertilizer N managemen<sup>t</sup> mean di fferences. The sub-plot lint yields (YLD), fertilizer N rates (FNs), N e fficiency (FNEFF), and NRs (FNRs) that are summarized in Table 2 were used to construct the regressions' dependent variables. The dependent variables were created using paired sub-plot observations in each strip-plot to measure di fferences between VRN 1 and the FP (VRN 1-FP) and VRN 2 and the FP (VRN 2-FP). For example, field 1, replication 1, and sub-plot 1 for the VRN 1 treatment versus field 1, replication 1, and sub-plot 1 for the FP treatment. This procedure resulted in 1263 observations available for each of the regressions (Table 3). Fixed e ffects included in the mean di fference regressions are landscape, soil, and weather characteristics georeferenced to each sub-plot. To account for potential di fferences in landscape and soil characteristics between paired VRN and FP sub-plot observations within each replication, observations were omitted from the regressions if soil characteristics di ffered between the two sub-plots. For example, if soil texture di ffered across field 1, replication 1, and sub-plot 1 for the VRN treatment versus sub-plot 1 for the FP treatment, then the observation was omitted from the regressions; if not, the observation was retained for the estimation. The summary statistics for the landscape, soil, and weather variables used as fixed e ffects in the mean di fference regressions are also presented in Table 3.



a ΔYLD, difference in optical sensing-based variable rate nitrogen managemen<sup>t</sup> and farmer practice nitrogen managemen<sup>t</sup> lint yields (kg ha−1). b ΔFN, difference in optical sensing-based variable rate nitrogen managemen<sup>t</sup> and farmer practice nitrogen managemen<sup>t</sup> fertilizer nitrogen rates (kg ha−1). c ΔFNEFF, difference in optical sensing-based variable rate nitrogen managemen<sup>t</sup> and farmer practice nitrogen managemen<sup>t</sup> fertilizer nitrogen efficiency measured as lint yield divided by fertilizer nitrogen rate (index). d ΔFNR, difference in optical sensing-based variable rate nitrogen managemen<sup>t</sup> and farmer practice nitrogen managemen<sup>t</sup> net returns (USD ha−1). e Yprob, if optical sensing-based variable rate nitrogen managemen<sup>t</sup> lint yield is less than farmer practice nitrogen managemen<sup>t</sup> lint yield, then 1; else 0. f Nprob, if optical sensing-based variable rate nitrogen managemen<sup>t</sup> fertilizer nitrogen rate is less than farmer practice nitrogen managemen<sup>t</sup> fertilizer nitrogen, then 1; else 0. g NEFFprob, if optical sensing-based variable rate nitrogen managemen<sup>t</sup> nitrogen efficiency is less than farmer practice nitrogen managemen<sup>t</sup> nitrogen efficiency, then 1; else 0. h NRprob, if optical sensing-based variable rate nitrogen managemen<sup>t</sup> net return is less than farmer practice nitrogen managemen<sup>t</sup> net return, then 1; else 0. i Soil texture index, 1 = Clay, 2 = Silt, 3 = Loam, and 4 = Sand. Sand is the reference variable in the regressions. Sources: [30,31]. j Elevation, vertical distance above sea level (m). Source: [29]. k WHC, water holding capacity (volume fraction). Source: [30]. l SOM, soil organic matter (%). Source: [30]. m SEI, soil erosion index. n Depth, soil depth (cm) from the top of the soil to the base of the soil horizon. Source: [30] o GDD, growing degree days, 1 April through 1 October, base 15.6 degrees Celsius: Source: [33]. ν p, 0–1 variable indicating variable rate nitrogen treatment using normalized difference vegetative index and either digital yield maps (Mississippi and Tennessee), soil productivity zones (Louisiana), and soil zones (Missouri).

The general linear model for the fertilizer N managemen<sup>t</sup> mean differences was:

$$
\Delta Y\_{ijklt} = \mu + X\_{lt}\theta + \nu + q\_{\bar{f}} + q\_{\bar{k}(\bar{f})} + \varepsilon\_{ijklt} \tag{2}
$$

where *i* = 1 (VRN 1 − FP), 2 (VRN 2 − FP); *j* = 1, ... , 21 farm field locations; *k* = 1, 2, and 3 replications on fields; *l* = 1, ... , 8 to 10 replication sub-plots within each strip-plot; *t* = 2011, 2012, 2013, and 2014; Δ*Yijklt* = *YVRNi* − *YFP* is defined as the mean difference in the response variable *Y* (lint yields (ΔYLD, kg ha−1), fertilizer N rates (ΔFN, kg ha−1), YLD/FN (ΔFNEFF, index), and NR (ΔFNR, USD ha−1)) for VRN 1 or VRN 2 compared to the FP; μ is the conditional mean; *X* includes sub-plot measurements on soil texture (clay, silt, loam, and sand), elevation above sea level (m), soil water-holding capacity (volume fraction), soil organic matter (%), soil depth (cm), soil erosion index, and seasonal growing degree days (degrees Celsius); β is a vector of the estimated average landscape, soil, and weather effects on Δ*Y*; and ν is a 0–1 variable indicating the VRN 2 treatment. The parameters ϕ*j* and ϕ*k*(*j*) are the farm field location random effects and the nested random effects from replications in farm field locations, with ϕ*j* ∼ *N*0, <sup>σ</sup>2ϕ*j* and ϕ*k*(*j*) ∼ *N* 0, <sup>σ</sup>2ϕ*k*(*j*). The model error is *eijkt* ∼ *N*0, σ2*e* [42].

The models using Equation (2) were estimated using the MIXED model procedure and restricted maximum likelihood in SAS 9.2 [43]. The sand soil texture 0–1 variable was dropped to estimate regressions and was included as the reference variable in the intercept term. The mean difference models were evaluated for multicollinearity using variance inflation factors (VIF) estimated using the REG model procedure in SAS 9.2 [43]. VIF exceeding 10 may indicate that multicollinearity is increasing the size of the parameters' standard errors [44]. Models estimated using Equation (2) tested the null hypotheses that mean yields, fertilizer N rates, NRs, and N efficiency were not different between VRN and FP, holding landscape, soil, and weather factors constant.

The second statistical model is estimated as a mixed logistic regression:

$$\Pr\left(VRN\_{\text{ijklt}} > FP\_{\text{ijklt}} \middle| X\_{\text{lt}}\right) = \text{Logistic}\left(\mu + X\_{\text{lt}}\beta + \upsilon + q\_{\text{j}} + q\_{k(\text{j})} + \varepsilon\_{\text{ijklt}}\right) \tag{3}$$

where Pr(*VRNijklt* > *FPijklt Xlt*) is the probability that the response variable (lint yields (YLD, kg ha−1), fertilizer N rates (FN, kg ha−1), YLD/N (NEFF, index), and NRs (FNR, USD ha−1)) for VRN falls above or below the FP value. The sub-plot data summarized in Table 2 were used to construct the logit regressions' dependent variables and are presented in Table 3. The binary dependent variables using the paired sub-plot observations in each strip-plot were calculated as:

$$\text{If } \text{YLD}^{VRN\_l} - \text{YLD}^{FP} < 0, \text{ then } \text{YLD}prob = 1; \text{ else, } \text{YLD}prob = 0; \tag{4}$$

$$\text{If } FN^{VRN\_i} - FN^{FP} > 0, \text{ then } FNprob = 1; \text{ else, } FNprob = 0; \tag{5}$$

$$\text{If } FNEF^{VEN\_i} - FNEF^{V'} < 0, \text{ then } FNEFprob = 1; \text{ else, } FNEFprob = 0, \text{ and} \tag{6}$$

$$\text{If } FNR^{VRN\_i} - FNR^{FP} < 0, \text{ then } FNRprob = 1; \text{ else, } FNRprob = 0. \tag{7}$$

Equations (4)–(7) were estimated for each binary dependent variable with the same set of fixed effects summarized in Table 3 and the same random effects used for the mean difference regressions described above. The logit models were estimated using the GLIMIX model procedure and restricted maximum likelihood in SAS 9.2 [43]. Multicollinearity was also evaluated in the logit regressions with the same procedures used for the mean difference regressions [44]. The odds ratios calculated using the estimated coefficients β of these logistic regressions are used to test the hypotheses comparing FP and OS-based VRN. Each covariate's impact on the odds VRN < FP is exp(β). In percent terms, the change in the log odds probability that VRN lint yields, N rates, NEFF, or NRs exceeded those of the FP is 100 × [exp(β) − 1]. The null hypotheses for Equations (4)–(7) was that the N managemen<sup>t</sup> regime does not affect the probability that yields, N rates, N efficiency, and NRs differ for VRN versus the FP, holding soil, landscape, and weather variables constant.
