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. 2025 Aug 22;15:30946. doi: 10.1038/s41598-025-16804-w

Table 3.

Determinants of digital training participation: instrument Variables.

IV – 1 IV – 2 IV – 3 Interaction term (1*2*3)
Digital training participation 1.729*** (0.351) 3.042* (1.697) 1.636** (0.885) 2.395*** (0.716)
Cragg-Donald Wald F Statistic 27.65** 44.73*** 29.28*** 35.72***
Wu-Hausman F test 11.37*** 17.91*** 8.52** 13.71***
Anderson canonical correlation LM statistic 249.7** 218.6*** 164.4*** 281.4***
VIF 1.37 2.25 1.83 1.41
Breusch-Pagan test statistic

5.28

(P-value 0.071)

4.84

(P-value 0.063)

6.26

(P-value 0.151)

5.19

(P-value 0.107)

Hansen J statistic

4.35

(P-value 0.29)

3.29

(P-value 0.27)

3.94

(P-value 0.35)

2.46

(P-value 0.31)

N = 723, endogenous variable (digital training participation), three IVs (1) access to internet connection (ATC) (2) distance to training center, and (3) farm radius, and additional explanatory variables included: household head age, education of household head, access to market, livestock ownership, off-farm income, access to credit, skill development, tenancy status, farming experience, marital status. The Cragg-Donald Wald F Statistic measures the relevance of the instruments used (F-stat > 10 = ***p < 0.01). *p < 0.1; **p < 0.05; ***p < 0.01. A Variance Inflation Factor (VIF) greater than 10 indicates potential multicollinearity. The Hansen J test assesses over-identification, with a statistic greater than 0.15 suggesting that we fail to reject the null hypothesis, supporting the validity of the instruments. The Breusch-Pagan test measures heteroscedasticity, and a p-value greater than 0.05 indicates no evidence of heteroscedasticity.