Ultimately, we want to use one sample to estimate both the farm-adoption share and the agricultural-area adoption share. We simulate the complete sampling process, from selecting districts and villages to selecting farms, while varying the sampling choices at all three stages.
Count-based probability-proportional-to-size (PPS) sampling uses farm counts as the measure of size. Area-based PPS uses net sown area.
Both methods produce small errors. Across seven scenarios with 30 percent farm adoption, no farm-size or observed village predictor, and limited within-village clustering, root mean squared error (RMSE) for estimates pooled across the six study states remains below 3 percentage points.
Relative to area-based village PPS, count-based village PPS lowers farm-adoption RMSE by 0.14 to 0.22 percentage points.
Relative to count-based village PPS, area-based village PPS lowers agricultural-area adoption RMSE by 0.60 to 0.78 percentage points.
These two measures, the farm-adoption share and the agricultural-area adoption share, answer different policy questions. A design that represents farm households efficiently may not represent agricultural area equally well.
The numerator counts farm households that use an innovation. Each farm household contributes once, regardless of farm size.
Here, \(A_i\) equals one when farm \(i\) adopts, and \(N\) is the number of farms.
The numerator measures land under an innovation. An adopting farm may use the innovation on only part of its agricultural area.
Here, \(a_i\) is farm \(i\)'s agricultural area, and \(u_i\) is the share of that area under the innovation.
The farm-adoption and agricultural-area adoption equations define population shares. We estimate both shares from the sample using inverse-probability survey weights, as explained in Section 4.
Compound here means that each simulated design combines one method for selecting districts, one method for selecting villages, and one farm-size cutoff within villages.
A measure of size determines how PPS inclusion probabilities vary across units. We use the same district measure of size throughout and compare two village measures of size.
| Sampling feature | What we hold fixed or compare |
|---|---|
| District measure of size | Use average relevant-crop area from the Directorate of Economics and Statistics (DES) for 2022-23 through 2024-25 in every district design |
| District selection method | Select 50 distinct districts directly using the inclusion probabilities implied by relevant-crop area, or order the district frame by one of six candidate variables before systematic PPS |
| Village measure of size | Use either farm counts or net sown area to determine village inclusion probabilities |
| Village selection method | Select 10 distinct villages directly using village inclusion probabilities, or order the village frame by one candidate variable at a time before systematic PPS |
| Farm-size cutoff | Compare state-specific p70, p80, and p90 thresholds while assigning 5 of 15 interviews above the cutoff |
Direct PPS selects distinct units using unit inclusion probabilities without first ordering a sampling frame. Systematic PPS orders units, chooses one random start, and selects at regular intervals along the cumulative measure of size.
Relevant-crop area excludes clearly irrelevant crops such as sugarcane. Each replication selects 50 districts and 10 villages per district. A selected village contributes 15 farms when its frame contains at least 15 farms. Smaller villages contribute every available farm. Under area-based PPS with a p80 cutoff, about 1 to 8 percent of selected villages across states contain fewer than 15 farms.
At a 30 percent farm-adoption rate, we compare 13 adoption scenarios, 7 district selection methods, and 19 combinations of village measure of size, village selection method, and farm-size cutoff. Additional simulations repeat the top five sampling designs at 5 and 60 percent farm adoption.
Evaluating a candidate ordering variable requires an assumption about how adoption relates to that variable. Each scenario states that assumption explicitly.
We add a shared unobserved village component to every farm's adoption propensity. The intraclass correlation coefficient (ICC) measures the share of unobserved variation attributable to the common village component. The latent ICC is the underlying simulation ICC in the continuous adoption variable, before we convert it into a yes-or-no adoption outcome.
| Scenario group | Relationship with adoption | Within-village clustering |
|---|---|---|
| No observed predictor 1 scenario |
No observed district, village, or farm-size characteristic predicts adoption | Latent ICC 0.03 |
| One district predictor 6 scenarios |
One candidate district characteristic predicts adoption. Farm size and observed village characteristics do not | Latent ICC 0.03 |
| District, village, and farm-size predictors 6 scenarios |
One candidate characteristic predicts adoption at both district and village levels. Adoption also rises with farm size | Latent ICC 0.15 |
Six candidate characteristics are irrigation, cultivator share, agricultural area per cultivator, literacy, agricultural market access, and longitude. For an adoption probability \(p\), log odds equal \(\log[p/(1-p)]\). The log-odds scale keeps predicted probabilities between zero and one.
When a candidate characteristic predicts adoption, a one-standard-deviation increase in that characteristic raises log odds by 0.7. At a 30 percent starting probability, the increase raises predicted adoption to about 46 percent when other terms stay fixed.
We also add an unobserved district component to adoption propensity with a coefficient of 0.7. The unobserved component creates district variation that no candidate ordering can directly balance. We then calibrate farm-level probabilities so their average equals the simulated adoption rate for that district.
For adopters, we draw the share of agricultural area under the innovation from a beta distribution with mean 55 percent and concentration 6. A concentration parameter of 6 controls how widely farm-specific use shares vary around the mean. We set both values as simulation assumptions.
The simulation fixes one synthetic population, repeatedly draws the complete sample, and compares each estimate with the known population share.
The weight \(w_i\) is the inverse of farm \(i\)'s inclusion probability. Pooling the estimated numerators and denominators across all sampled districts matches how the survey will compute its estimates.
Lower RMSE is better. Every result below reports RMSE or an RMSE difference in percentage points. Horizontal bars show paired 95 percent Monte Carlo intervals.
The rule first removes clearly inferior designs, then compares the survivors under several weights on farm-adoption precision. The coefficient of variation (CV) divides RMSE by the true simulated adoption share.
The main comparison uses \(w=0.60\). Weights of 0.40 and 0.50 show whether a conclusion depends on placing slightly more or less weight on farm-adoption precision. No complete design meets every provisional CV target in every neutral condition, so the guardrails identify weak cells rather than select a design by themselves.
The full design varies district ordering together with village PPS, the farm-size cutoff, and village ordering.
Move left: the ordered district design has lower weighted RMSE
Move right: drawing districts without an observed ordering variable has lower weighted RMSE
Yellow marks the condition with no imposed observed relationship
With the stronger 0.7 relationship, the matched ordering lowers weighted RMSE for irrigation, cultivator share, area per cultivator, literacy, and longitude. The gain ranges from 0.17 to 0.42 percentage points. Market-access ordering raises weighted RMSE by 0.20 points even when market access predicts adoption.
Across all neutral conditions and all village-stage arms, every systematic district order either has a clear loss in some cells or falls more than 0.10 points behind the best order in some cells. The rule therefore keeps the direct unordered district PPS draw as the baseline choice.
A district gradient describes how strongly farm adoption varies with an observed district characteristic. A 0.35 cultivator-share gradient means that a one-standard-deviation increase in cultivator share raises adoption log odds by 0.35.
Starting from 30 percent adoption, the 0.35 gradient raises the predicted probability to about 38 percent, holding the unobserved component fixed.
The paired weighted-loss improvement is 0.16 to 0.17 percentage points across the three decision weights, and every paired interval excludes zero. District ordering is therefore a substantive assumption choice: cultivator-share ordering helps if that district gradient is credible, while the unordered draw is safer when no candidate relationship is imposed.
Count-based village PPS is more precise for the farm-adoption share. Area-based village PPS is more precise for the agricultural-area adoption share.
Dark teal represents count-based village PPS
Light teal represents area-based village PPS
Each row holds the district method, p90 cutoff, and 5 of 15 allocation fixed
Farm-adoption RMSE ranges from 1.99 to 2.74 points under count PPS and from 2.19 to 2.96 points under area PPS. Agricultural-area adoption RMSE ranges from 1.21 to 1.78 points under area PPS and from 1.86 to 2.55 points under count PPS. Both designs therefore keep six-state RMSE below 3 percentage points in the neutral conditions.
Positive values mean area PPS has lower weighted RMSE
Neutral conditions isolate district relationships
Stress conditions also add farm-size and village relationships with stronger clustering
Positive values mean area PPS has lower weighted RMSE
The shaded band marks differences smaller than 0.05 percentage points
Circles are paired estimates and horizontal lines are 95 percent Monte Carlo intervals
At 5 percent adoption, the area-PPS advantage is 0.04 to 0.11 percentage points. Five of seven neutral conditions clear the 0.05-point decision rule with paired uncertainty. At 60 percent adoption, the advantage grows to 0.19 to 0.27 points and all seven conditions clear the rule.
The PPS choice is a real trade-off, but the absolute differences are small. Under 40/60, 50/50, or 60/40 weights, the compound evidence supports area-based village PPS.
The cutoff comparison fixes area-based village PPS and allocates 5 of 15 interviews above the state-specific threshold.
Negative values mean p90 has lower weighted RMSE
The shaded band marks differences smaller than 0.05 percentage points
Each row shows one neutral district relationship
At 30 percent adoption, p90 lowers weighted RMSE by 0.01 to 0.03 points across the seven neutral conditions. At 5 percent adoption, the differences range from a p80 advantage below 0.01 points to a p90 advantage of 0.02 points. At 60 percent adoption, p90 lowers weighted RMSE by 0.01 to 0.03 points. Every difference is too small to choose the cutoff on precision alone.
If a village has fewer than five farms above the cutoff, the design moves the remaining interviews below the cutoff. The sample still targets 15 interviews, but the planned larger-farm allocation becomes weaker.
Under area PPS, 86 to 99 percent of selected villages can supply five farms above p80. The range is 65 to 98 percent above p90. Punjab has the largest difference: 86 percent at p80 compared with 65 percent at p90.
Because the precision differences are small, p80's greater ability to supply five farms above the cutoff is the stronger argument for the operational choice.
In the 30 percent compound screen, agricultural area per cultivator has the lowest loss among the qualified area-PPS village orderings. The adoption-rate sensitivities compare this ordering with drawing villages directly from their PPS probabilities.
Negative values mean area-per-cultivator ordering has lower weighted RMSE
The shaded band marks differences smaller than 0.05 percentage points
Area PPS, p90, and 60 percent farm weight are fixed
At 30 percent adoption, area-per-cultivator ordering lowers point loss in six of seven neutral conditions. At 5 percent adoption, no comparison clears the 0.05-point rule with paired uncertainty. At 60 percent adoption, the irrigation condition clears the rule, while the other six conditions do not. The direction remains mixed across adoption rates and district relationships.
The compound evidence does not cleanly justify committing to one village ordering variable. Drawing villages directly from their PPS probabilities remains the simpler default.
The adoption-rate sensitivities compare one feature-assembled design, three direct feature comparisons, and one simple-random-sample benchmark.
All five designs draw districts directly from their PPS probabilities. The separate 0.35-gradient sensitivity compares cultivator-share district ordering with the direct district draw while holding the primary village design fixed.
| Design | Why it is included |
|---|---|
| Area PPS, p90, villages ordered by area per cultivator | Feature-assembled design from the 30 percent screen |
| Count PPS, p90, villages drawn without an observed order | Direct PPS comparison |
| Area PPS, p80, villages drawn without an observed order | Cutoff comparison |
| Area PPS, p90, villages drawn without an observed order | Village-order comparison |
| Simple random sample of 150 farms within each sampled district | Diagnostic precision benchmark |
The village-stage evidence therefore points to area-based PPS with a p80 cutoff, 5 of 15 interviews allocated above the cutoff, and villages drawn without sorting on an observed characteristic. The district-stage choice depends on whether the team considers a positive cultivator-share adoption gradient plausible enough to design around.
Each subsection is collapsed by default. A link to a subsection opens it automatically.
The study frame excludes five West Bengal districts in the Eastern Himalayan Region: Alipurduar, Darjeeling, Jalpaiguri, Kalimpong, and Cooch Behar. Kolkata has no rural villages in the sampling frame. The compound experiment therefore uses 203 positive-support districts.
Purulia remains in the main frame. It is the only district in the West Bengal Eastern Plateau and Hills stratum and therefore enters the 50-district sample with certainty.
| Irrigation share | Irrigated area divided by irrigated plus unirrigated area |
| Cultivator share | Main and marginal cultivators divided by cultivators plus agricultural laborers |
| Agricultural area per cultivator | Net sown area divided by main and marginal cultivators |
| Literacy rate | Literate population divided by population age seven and above |
| Agricultural market access | Equal-weight mean of four village indicators: a market, weekly haat, agricultural marketing society, and agricultural credit society |
| Longitude | Longitude of the village centroid |
Village ordering uses the village-level value. District ordering aggregates the same information across villages in the district.
For district adoption, I subtract each state-by-zone stratum's unweighted mean and divide by its unweighted standard deviation. For village adoption, I apply the same calculation across villages within a district. A standardized value of one is therefore one standard deviation above the relevant mean.
We write district \(d\)'s target farm-adoption probability in stratum \(h\) under candidate \(k\) as \(q_{dhk}\). The intercept \(\alpha_{hk}\) sets the cultivator-weighted mean adoption rate in the stratum. The standardized candidate value is \(z_{dhk}\). The standardized unobserved district component is \(u_{dh}\). The main compound screen sets \(\beta_D\) to 0.7 when candidate \(k\) is active and to zero in the no-relationship condition.
We write farm \(i\)'s adoption probability in village \(v\) and district \(d\) as \(p_{ivd}\). The district intercept \(a_d\) makes the mean farm probability equal \(q_{dhk}\). Standardized log farm area is \(f_{ivd}\). The standardized active village candidate is \(z_{vd}\). The village component is \(b_{vd}\). Neutral conditions set \(\beta_F=\beta_V=0\) and use latent ICC 0.03. Stress conditions set \(\beta_F=\beta_V=0.7\) and use latent ICC 0.15.
The 2015-16 Agriculture Census reports the number and operated area of holdings in ten farm-size classes. For each state and percentile, I locate the class containing the requested cumulative share of holdings. Within that bounded class, I fit a truncated exponential distribution whose mean matches the published class mean. I then read the p70, p80, or p90 threshold from that fitted distribution.
| State | p70 (ha) | p80 (ha) | p90 (ha) |
|---|---|---|---|
| Bihar | 0.399 | 0.551 | 0.845 |
| Haryana | 2.130 | 3.154 | 4.978 |
| Odisha | 0.970 | 1.404 | 1.892 |
| Punjab | 4.107 | 5.090 | 7.446 |
| Uttar Pradesh | 0.692 | 0.992 | 1.631 |
| West Bengal | 0.795 | 0.963 | 1.677 |
Direct PPS selects the required number of distinct units from their inclusion probabilities without ordering the frame on an observed characteristic. The implementation uses fixed-size Sampford PPS without replacement.
Systematic PPS first orders units by one candidate characteristic. It then chooses one random start and selects units at regular intervals along the cumulative PPS size measure. District orders share the same random start within a stratum and replication. Village orders share the same random start within a district and replication.
Village farm counts, net sown area, irrigation, cultivator composition, literacy, market access, and location come from the Census village frame. State farm-size distributions come from the 2015-16 Agriculture Census. District PPS size measures come from average DES relevant-crop area for 2022-23 through 2024-25.
The figure script is explorations/sampling/python/plot_compound_dashboard.py. It reads the R=1,000 compound outputs at 30 percent adoption and the 5 percent, 60 percent, and 0.35-gradient sensitivity outputs. A manifest beside the figures records input and output hashes.