Data Analysis
Didier Murillo
2026-02-03
library(dplyr)
library(readxl)
library(janitor)
library(ggplot2)
library(emmeans)
library(lme4)
library(lmerTest)Load data
data_long = readxl::read_xlsx("data_wide.xlsx") |>
janitor::clean_names()Prepare data
eda = data_long |>
mutate(
year = factor(year),
variety = factor(variety),
spacing = factor(spacing),
trt_vs = interaction(variety, spacing, sep = " | ")
)Boxplot by treatment for tuber_num_total_marketable
ggplot(
eda |> filter(!is.na(tuber_num_total_marketable)),
aes(x = trt_vs, y = tuber_num_total_marketable, fill = variety)
) +
geom_boxplot(outlier.alpha = 0.6) +
geom_jitter(width = 0.12, alpha = 0.6, size = 1.8) +
facet_wrap(~year, scales = "free_y") +
labs(
title = "Marketable tuber number by treatment",
x = "Treatment (Variety | Spacing)",
y = "tuber_num_total_marketable",
fill = "Variety"
) +
theme_minimal() +
theme(axis.text.x = element_text(angle = 30, hjust = 1))
Boxplot by treatment for cwt_per_a_total_marketable
ggplot(
eda |> filter(!is.na(cwt_per_a_total_marketable)),
aes(x = trt_vs, y = cwt_per_a_total_marketable, fill = variety)
) +
geom_boxplot(outlier.alpha = 0.6) +
geom_jitter(width = 0.12, alpha = 0.6, size = 1.8) +
facet_wrap(~year, scales = "free_y") +
labs(
title = "Marketable yield (cwt/acre) by treatment",
x = "Treatment (Variety | Spacing)",
y = "cwt_per_a_total_marketable",
fill = "Variety"
) +
theme_minimal() +
theme(axis.text.x = element_text(angle = 30, hjust = 1))
Interaction plot (tuber number marketable)
plot_num = data_long |>
filter(!is.na(tuber_num_total_marketable)) |>
group_by(year, variety, spacing) |>
summarise(
mean = mean(tuber_num_total_marketable),
se = sd(tuber_num_total_marketable) / sqrt(n()),
n = n(),
.groups = "drop"
) |>
arrange(year, variety, spacing)
ggplot(plot_num, aes(x = as.numeric(as.character(spacing)), y = mean, color = variety, group = variety)) +
geom_line(linewidth = 1) +
geom_point(size = 2) +
geom_errorbar(aes(ymin = mean - se, ymax = mean + se), width = 0.2, alpha = 0.35) +
facet_wrap(~year) +
labs(
title = "Variety × Spacing by Year",
subtitle = "Response: tuber_num_total_marketable (mean ± SE)",
x = "Spacing",
y = "Marketable tuber number",
color = "Variety"
) +
theme_minimal()
Interaction plot (marketable yield weight)
plot_cwt = data_long |>
filter(!is.na(cwt_per_a_total_marketable)) |>
group_by(year, variety, spacing) |>
summarise(
mean = mean(cwt_per_a_total_marketable),
se = sd(cwt_per_a_total_marketable) / sqrt(n()),
n = n(),
.groups = "drop"
) |>
arrange(year, variety, spacing)
ggplot(plot_cwt, aes(x = as.numeric(as.character(spacing)), y = mean, color = variety, group = variety)) +
geom_line(linewidth = 1) +
geom_point(size = 2) +
geom_errorbar(aes(ymin = mean - se, ymax = mean + se), width = 0.2, alpha = 0.35) +
facet_wrap(~year) +
labs(
title = "Variety × Spacing by Year",
subtitle = "Response: cwt_per_a_total_marketable (mean ± SE)",
x = "Spacing",
y = "Marketable yield (cwt/acre)",
color = "Variety"
) +
theme_minimal()
Models
Model for cwt_per_a_total_marketable 2025 data
dat = data_long |>
filter(!is.na(cwt_per_a_total_marketable )) |>
mutate(
year = factor(year),
rep = factor(rep),
variety = factor(variety),
spacing = factor(spacing) # treat as categorical levels for RCBD
)
dat_2025 = dat |>
filter(year == "2025")
m_2025 = lmer(
cwt_per_a_total_marketable ~ variety * spacing + (1 | rep),
data = dat_2025,
REML = TRUE
)
anova(m_2025) # fixed effects testsType III Analysis of Variance Table with Satterthwaite's method
Sum Sq Mean Sq NumDF DenDF F value Pr(>F)
variety 6203.0 6203.0 1 14.134 3.0175 0.1041
spacing 8619.1 4309.6 2 14.130 2.0965 0.1595
variety:spacing 3836.7 1918.3 2 14.130 0.9332 0.4162summary(m_2025) # estimatesLinear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: cwt_per_a_total_marketable ~ variety * spacing + (1 | rep)
Data: dat_2025
REML criterion at convergence: 190.5
Scaled residuals:
Min 1Q Median 3Q Max
-1.53534 -0.70236 -0.05073 0.64071 1.67185
Random effects:
Groups Name Variance Std.Dev.
rep (Intercept) 1264 35.56
Residual 2056 45.34
Number of obs: 23, groups: rep, 4
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 493.8911 32.0490 12.6090 15.410 1.49e-09 ***
varietyLakeview 0.6773 34.9997 14.2482 0.019 0.985
spacing10 62.8100 34.9997 14.2482 1.795 0.094 .
spacing12 3.0674 34.9997 14.2482 0.088 0.931
varietyLakeview:spacing10 -64.8197 47.4637 14.1652 -1.366 0.193
varietyLakeview:spacing12 -36.7080 47.4637 14.1652 -0.773 0.452
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) vrtyLk spcn10 spcn12 vrL:10
varietyLkvw -0.634
spacing10 -0.634 0.580
spacing12 -0.634 0.580 0.580
vrtyLkvw:10 0.467 -0.737 -0.737 -0.428
vrtyLkvw:12 0.467 -0.737 -0.428 -0.737 0.544# Get the estimated marginal means ----------------------------------------
emm = emmeans(m_2025, ~ variety * spacing)
emm variety spacing emmean SE df lower.CL upper.CL
Caribou 8 494 32.2 12.5 424 564
Lakeview 8 495 28.8 10.1 430 559
Caribou 10 557 28.8 10.1 493 621
Lakeview 10 493 28.8 10.1 428 557
Caribou 12 497 28.8 10.1 433 561
Lakeview 12 461 28.8 10.1 397 525
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 pairs(emm) # all pairwise contrast estimate SE df t.ratio p.value
Caribou spacing8 - Lakeview spacing8 -0.677 35.2 14.2 -0.019 1.0000
Caribou spacing8 - Caribou spacing10 -62.810 35.2 14.2 -1.785 0.5038
Caribou spacing8 - Lakeview spacing10 1.332 35.2 14.2 0.038 1.0000
Caribou spacing8 - Caribou spacing12 -3.067 35.2 14.2 -0.087 1.0000
Caribou spacing8 - Lakeview spacing12 32.963 35.2 14.2 0.937 0.9301
Lakeview spacing8 - Caribou spacing10 -62.133 32.1 14.0 -1.938 0.4207
Lakeview spacing8 - Lakeview spacing10 2.010 32.1 14.0 0.063 1.0000
Lakeview spacing8 - Caribou spacing12 -2.390 32.1 14.0 -0.075 1.0000
Lakeview spacing8 - Lakeview spacing12 33.641 32.1 14.0 1.049 0.8930
Caribou spacing10 - Lakeview spacing10 64.142 32.1 14.0 2.001 0.3886
Caribou spacing10 - Caribou spacing12 59.743 32.1 14.0 1.863 0.4606
Caribou spacing10 - Lakeview spacing12 95.773 32.1 14.0 2.987 0.0839
Lakeview spacing10 - Caribou spacing12 -4.400 32.1 14.0 -0.137 1.0000
Lakeview spacing10 - Lakeview spacing12 31.631 32.1 14.0 0.987 0.9148
Caribou spacing12 - Lakeview spacing12 36.031 32.1 14.0 1.124 0.8636
Degrees-of-freedom method: kenward-roger
P value adjustment: tukey method for comparing a family of 6 estimates pairs(emm, by = "variety") # spacing comparisons within varietyvariety = Caribou:
contrast estimate SE df t.ratio p.value
spacing8 - spacing10 -62.81 35.2 14.2 -1.785 0.2098
spacing8 - spacing12 -3.07 35.2 14.2 -0.087 0.9958
spacing10 - spacing12 59.74 32.1 14.0 1.863 0.1858
variety = Lakeview:
contrast estimate SE df t.ratio p.value
spacing8 - spacing10 2.01 32.1 14.0 0.063 0.9978
spacing8 - spacing12 33.64 32.1 14.0 1.049 0.5594
spacing10 - spacing12 31.63 32.1 14.0 0.987 0.5969
Degrees-of-freedom method: kenward-roger
P value adjustment: tukey method for comparing a family of 3 estimates pairs(emm, by = "spacing") # variety comparisons within spacingspacing = 8:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview -0.677 35.2 14.2 -0.019 0.9849
spacing = 10:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview 64.142 32.1 14.0 2.001 0.0652
spacing = 12:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview 36.031 32.1 14.0 1.124 0.2800
Degrees-of-freedom method: kenward-roger Conclusions on 2025
For 2025, the RCBD mixed model (cwt_per_a_total_marketable ~ variety * spacing + (1|rep)) found no statistically significant effects of variety (p = 0.104), spacing (p = 0.160), or variety×spacing interaction (p = 0.416) on marketable yield. Estimated marginal means were broadly similar across treatment combinations. Tukey-adjusted pairwise tests did not detect significant differences within varieties or within spacings. A possible Caribou advantage at spacing 10 appeared as a nominal trend in simple-effects comparisons (difference ≈ 64.1 cwt/ac, p = 0.065), but this was not significant under broader family-wise adjustment. Overall, 2025 evidence supports comparable marketable yield across tested variety-spacing combinations, with limited power due to small/unbalanced data and a need for confirmation using additional years.
Model for cwt_per_a_total_marketable 2024 data
dat_2024 = dat |>
filter(year == "2024")
m_2024 = lmer(
cwt_per_a_total_marketable ~ variety * spacing + (1 | rep),
data = dat_2024,
REML = TRUE
)
anova(m_2024)Type III Analysis of Variance Table with Satterthwaite's method
Sum Sq Mean Sq NumDF DenDF F value Pr(>F)
variety 23110.0 23110.0 1 18 4.6356 0.04512 *
spacing 12585.9 6293.0 2 18 1.2623 0.30689
variety:spacing 2390.2 1195.1 2 18 0.2397 0.78932
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(m_2024)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: cwt_per_a_total_marketable ~ variety * spacing + (1 | rep)
Data: dat_2024
REML criterion at convergence: 212.7
Scaled residuals:
Min 1Q Median 3Q Max
-1.99544 -0.57408 0.06128 0.47099 1.66761
Random effects:
Groups Name Variance Std.Dev.
rep (Intercept) 0 0.00
Residual 4985 70.61
Number of obs: 24, groups: rep, 4
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 471.227 35.304 18.000 13.348 8.93e-11 ***
varietyLakeview 84.684 49.927 18.000 1.696 0.107
spacing10 36.254 49.927 18.000 0.726 0.477
spacing12 -5.196 49.927 18.000 -0.104 0.918
varietyLakeview:spacing10 -19.315 70.607 18.000 -0.274 0.788
varietyLakeview:spacing12 -48.553 70.607 18.000 -0.688 0.500
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) vrtyLk spcn10 spcn12 vrL:10
varietyLkvw -0.707
spacing10 -0.707 0.500
spacing12 -0.707 0.500 0.500
vrtyLkvw:10 0.500 -0.707 -0.707 -0.354
vrtyLkvw:12 0.500 -0.707 -0.354 -0.707 0.500
optimizer (nloptwrap) convergence code: 0 (OK)
boundary (singular) fit: see help('isSingular')emm_2024 = emmeans(m_2024, ~ variety * spacing)
emm_2024 variety spacing emmean SE df lower.CL upper.CL
Caribou 8 471 35.3 18 397 545
Lakeview 8 556 35.3 18 482 630
Caribou 10 507 35.3 18 433 582
Lakeview 10 573 35.3 18 499 647
Caribou 12 466 35.3 18 392 540
Lakeview 12 502 35.3 18 428 576
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 pairs(emm_2024) contrast estimate SE df t.ratio p.value
Caribou spacing8 - Lakeview spacing8 -84.68 49.9 15 -1.696 0.5540
Caribou spacing8 - Caribou spacing10 -36.25 49.9 15 -0.726 0.9755
Caribou spacing8 - Lakeview spacing10 -101.62 49.9 15 -2.035 0.3686
Caribou spacing8 - Caribou spacing12 5.20 49.9 15 0.104 1.0000
Caribou spacing8 - Lakeview spacing12 -30.94 49.9 15 -0.620 0.9878
Lakeview spacing8 - Caribou spacing10 48.43 49.9 15 0.970 0.9205
Lakeview spacing8 - Lakeview spacing10 -16.94 49.9 15 -0.339 0.9993
Lakeview spacing8 - Caribou spacing12 89.88 49.9 15 1.800 0.4939
Lakeview spacing8 - Lakeview spacing12 53.75 49.9 15 1.077 0.8832
Caribou spacing10 - Lakeview spacing10 -65.37 49.9 15 -1.309 0.7758
Caribou spacing10 - Caribou spacing12 41.45 49.9 15 0.830 0.9571
Caribou spacing10 - Lakeview spacing12 5.32 49.9 15 0.107 1.0000
Lakeview spacing10 - Caribou spacing12 106.82 49.9 15 2.140 0.3196
Lakeview spacing10 - Lakeview spacing12 70.69 49.9 15 1.416 0.7177
Caribou spacing12 - Lakeview spacing12 -36.13 49.9 15 -0.724 0.9759
Degrees-of-freedom method: kenward-roger
P value adjustment: tukey method for comparing a family of 6 estimates pairs(emm_2024, by = "variety") variety = Caribou:
contrast estimate SE df t.ratio p.value
spacing8 - spacing10 -36.3 49.9 15 -0.726 0.7521
spacing8 - spacing12 5.2 49.9 15 0.104 0.9940
spacing10 - spacing12 41.4 49.9 15 0.830 0.6906
variety = Lakeview:
contrast estimate SE df t.ratio p.value
spacing8 - spacing10 -16.9 49.9 15 -0.339 0.9388
spacing8 - spacing12 53.7 49.9 15 1.077 0.5424
spacing10 - spacing12 70.7 49.9 15 1.416 0.3581
Degrees-of-freedom method: kenward-roger
P value adjustment: tukey method for comparing a family of 3 estimates pairs(emm_2024, by = "spacing") spacing = 8:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview -84.7 49.9 15 -1.696 0.1105
spacing = 10:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview -65.4 49.9 15 -1.309 0.2101
spacing = 12:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview -36.1 49.9 15 -0.724 0.4804
Degrees-of-freedom method: kenward-roger Conclusions on 2024
For 2024, the RCBD mixed model (cwt_per_a_total_marketable ~ variety * spacing + (1|rep)) found a statistically significant main effect of variety (p = 0.04512), but no significant effects of spacing (p = 0.30689) or variety×spacing interaction (p = 0.78932) on marketable yield. Estimated marginal means showed Lakeview higher than Caribou across all spacings (8: 556 vs 471; 10: 573 vs 507; 12: 502 vs 466 cwt/ac). Tukey-adjusted pairwise tests did not detect significant differences among the six treatment combinations, nor significant variety differences within individual spacings (spacing 8: p = 0.1105; spacing 10: p = 0.2101; spacing 12: p = 0.4804), and spacing contrasts within each variety were also non-significant. The model also had a singular (boundary) fit with block variance estimated at zero (SD_rep = 0.00; Var_rep = 0), while residual variation was substantial (SD_residual = 70.61; Var_residual = 4985), so blocking contributed little in 2024 and inference should be interpreted with that in mind. Overall, 2024 evidence suggests an average yield advantage for Lakeview across spacings, without evidence of spacing-dependent effects or interaction.
Combined model
m_all = lmer(
cwt_per_a_total_marketable ~ year * variety * spacing + (1 | year:rep),
data = dat,
REML = TRUE
)
anova(m_all)Type III Analysis of Variance Table with Satterthwaite's method
Sum Sq Mean Sq NumDF DenDF F value Pr(>F)
year 1134.7 1134.7 1 6.1202 0.3125 0.59601
variety 2522.3 2522.3 1 29.2974 0.6946 0.41135
spacing 20828.9 10414.4 2 29.2916 2.8680 0.07286 .
year:variety 26111.7 26111.7 1 29.2974 7.1907 0.01191 *
year:spacing 456.9 228.4 2 29.2916 0.0629 0.93915
variety:spacing 4682.9 2341.5 2 29.2916 0.6448 0.53206
year:variety:spacing 1858.1 929.1 2 29.2916 0.2558 0.77597
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(m_all)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: cwt_per_a_total_marketable ~ year * variety * spacing + (1 |
year:rep)
Data: dat
REML criterion at convergence: 406.4
Scaled residuals:
Min 1Q Median 3Q Max
-2.02626 -0.51610 0.04661 0.45532 1.77339
Random effects:
Groups Name Variance Std.Dev.
year:rep (Intercept) 562.1 23.71
Residual 3631.3 60.26
Number of obs: 47, groups: year:rep, 8
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 471.227 32.378 32.261 14.554 9.93e-16
year2025 21.062 49.252 33.315 0.428 0.6717
varietyLakeview 84.684 42.611 29.139 1.987 0.0563
spacing10 36.254 42.611 29.139 0.851 0.4018
spacing12 -5.196 42.611 29.139 -0.122 0.9038
year2025:varietyLakeview -82.405 62.931 29.586 -1.309 0.2005
year2025:spacing10 28.158 62.931 29.586 0.447 0.6578
year2025:spacing12 9.865 62.931 29.586 0.157 0.8765
varietyLakeview:spacing10 -19.315 60.260 29.139 -0.321 0.7509
varietyLakeview:spacing12 -48.553 60.260 29.139 -0.806 0.4269
year2025:varietyLakeview:spacing10 -47.107 87.130 29.373 -0.541 0.5928
year2025:varietyLakeview:spacing12 10.243 87.130 29.373 0.118 0.9072
(Intercept) ***
year2025
varietyLakeview .
spacing10
spacing12
year2025:varietyLakeview
year2025:spacing10
year2025:spacing12
varietyLakeview:spacing10
varietyLakeview:spacing12
year2025:varietyLakeview:spacing10
year2025:varietyLakeview:spacing12
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) yr2025 vrtyLk spcn10 spcn12 yr2025:L y2025:10 y2025:12
year2025 -0.657
varietyLkvw -0.658 0.433
spacing10 -0.658 0.433 0.500
spacing12 -0.658 0.433 0.500 0.500
yr2025:vrtL 0.446 -0.692 -0.677 -0.339 -0.339
yr2025:sp10 0.446 -0.692 -0.339 -0.677 -0.339 0.542
yr2025:sp12 0.446 -0.692 -0.339 -0.339 -0.677 0.542 0.542
vrtyLkvw:10 0.465 -0.306 -0.707 -0.707 -0.354 0.479 0.479 0.239
vrtyLkvw:12 0.465 -0.306 -0.707 -0.354 -0.707 0.479 0.239 0.479
yr2025:L:10 -0.322 0.500 0.489 0.489 0.245 -0.722 -0.722 -0.391
yr2025:L:12 -0.322 0.500 0.489 0.245 0.489 -0.722 -0.391 -0.722
vrL:10 vrL:12 y2025:L:10
year2025
varietyLkvw
spacing10
spacing12
yr2025:vrtL
yr2025:sp10
yr2025:sp12
vrtyLkvw:10
vrtyLkvw:12 0.500
yr2025:L:10 -0.692 -0.346
yr2025:L:12 -0.346 -0.692 0.522 # Year-specific treatment means
emm_by_year = emmeans(m_all, ~ variety * spacing | year)
emm_by_yearyear = 2024:
variety spacing emmean SE df lower.CL upper.CL
Caribou 8 471 32.4 32.2 405 537
Lakeview 8 556 32.4 32.2 490 622
Caribou 10 507 32.4 32.2 442 573
Lakeview 10 573 32.4 32.2 507 639
Caribou 12 466 32.4 32.2 400 532
Lakeview 12 502 32.4 32.2 436 568
year = 2025:
variety spacing emmean SE df lower.CL upper.CL
Caribou 8 492 37.4 33.9 416 568
Lakeview 8 495 32.4 32.2 429 561
Caribou 10 557 32.4 32.2 491 623
Lakeview 10 493 32.4 32.2 427 558
Caribou 12 497 32.4 32.2 431 563
Lakeview 12 461 32.4 32.2 395 527
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 # Pairwise within each year (all 6 combos per year)
pairs(emm_by_year, adjust = "tukey")year = 2024:
contrast estimate SE df t.ratio p.value
Caribou spacing8 - Lakeview spacing8 -84.68 42.6 29.0 -1.987 0.3734
Caribou spacing8 - Caribou spacing10 -36.25 42.6 29.0 -0.851 0.9550
Caribou spacing8 - Lakeview spacing10 -101.62 42.6 29.0 -2.385 0.1947
Caribou spacing8 - Caribou spacing12 5.20 42.6 29.0 0.122 1.0000
Caribou spacing8 - Lakeview spacing12 -30.94 42.6 29.0 -0.726 0.9771
Lakeview spacing8 - Caribou spacing10 48.43 42.6 29.0 1.137 0.8620
Lakeview spacing8 - Lakeview spacing10 -16.94 42.6 29.0 -0.398 0.9986
Lakeview spacing8 - Caribou spacing12 89.88 42.6 29.0 2.109 0.3106
Lakeview spacing8 - Lakeview spacing12 53.75 42.6 29.0 1.261 0.8029
Caribou spacing10 - Lakeview spacing10 -65.37 42.6 29.0 -1.534 0.6460
Caribou spacing10 - Caribou spacing12 41.45 42.6 29.0 0.973 0.9229
Caribou spacing10 - Lakeview spacing12 5.32 42.6 29.0 0.125 1.0000
Lakeview spacing10 - Caribou spacing12 106.82 42.6 29.0 2.507 0.1552
Lakeview spacing10 - Lakeview spacing12 70.69 42.6 29.0 1.659 0.5683
Caribou spacing12 - Lakeview spacing12 -36.13 42.6 29.0 -0.848 0.9556
year = 2025:
contrast estimate SE df t.ratio p.value
Caribou spacing8 - Lakeview spacing8 -2.28 46.6 29.8 -0.049 1.0000
Caribou spacing8 - Caribou spacing10 -64.41 46.6 29.8 -1.383 0.7363
Caribou spacing8 - Lakeview spacing10 -0.27 46.6 29.8 -0.006 1.0000
Caribou spacing8 - Caribou spacing12 -4.67 46.6 29.8 -0.100 1.0000
Caribou spacing8 - Lakeview spacing12 31.36 46.6 29.8 0.674 0.9836
Lakeview spacing8 - Caribou spacing10 -62.13 42.6 29.0 -1.458 0.6923
Lakeview spacing8 - Lakeview spacing10 2.01 42.6 29.0 0.047 1.0000
Lakeview spacing8 - Caribou spacing12 -2.39 42.6 29.0 -0.056 1.0000
Lakeview spacing8 - Lakeview spacing12 33.64 42.6 29.0 0.789 0.9671
Caribou spacing10 - Lakeview spacing10 64.14 42.6 29.0 1.505 0.6637
Caribou spacing10 - Caribou spacing12 59.74 42.6 29.0 1.402 0.7255
Caribou spacing10 - Lakeview spacing12 95.77 42.6 29.0 2.248 0.2478
Lakeview spacing10 - Caribou spacing12 -4.40 42.6 29.0 -0.103 1.0000
Lakeview spacing10 - Lakeview spacing12 31.63 42.6 29.0 0.742 0.9748
Caribou spacing12 - Lakeview spacing12 36.03 42.6 29.0 0.846 0.9561
Degrees-of-freedom method: kenward-roger
P value adjustment: tukey method for comparing a family of 6 estimates # Spacing comparisons within variety, by year
pairs(emm_by_year, by = c("year","variety"), adjust = "tukey")year = 2024, variety = Caribou:
contrast estimate SE df t.ratio p.value
spacing8 - spacing10 -36.25 42.6 29.0 -0.851 0.6750
spacing8 - spacing12 5.20 42.6 29.0 0.122 0.9918
spacing10 - spacing12 41.45 42.6 29.0 0.973 0.5996
year = 2025, variety = Caribou:
contrast estimate SE df t.ratio p.value
spacing8 - spacing10 -64.41 46.6 29.8 -1.383 0.3622
spacing8 - spacing12 -4.67 46.6 29.8 -0.100 0.9945
spacing10 - spacing12 59.74 42.6 29.0 1.402 0.3530
year = 2024, variety = Lakeview:
contrast estimate SE df t.ratio p.value
spacing8 - spacing10 -16.94 42.6 29.0 -0.398 0.9168
spacing8 - spacing12 53.75 42.6 29.0 1.261 0.4278
spacing10 - spacing12 70.69 42.6 29.0 1.659 0.2380
year = 2025, variety = Lakeview:
contrast estimate SE df t.ratio p.value
spacing8 - spacing10 2.01 42.6 29.0 0.047 0.9988
spacing8 - spacing12 33.64 42.6 29.0 0.789 0.7124
spacing10 - spacing12 31.63 42.6 29.0 0.742 0.7406
Degrees-of-freedom method: kenward-roger
P value adjustment: tukey method for comparing a family of 3 estimates # Variety comparisons within spacing, by year
pairs(emm_by_year, by = c("year","spacing"), adjust = "tukey")year = 2024, spacing = 8:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview -84.68 42.6 29.0 -1.987 0.0564
year = 2025, spacing = 8:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview -2.28 46.6 29.8 -0.049 0.9613
year = 2024, spacing = 10:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview -65.37 42.6 29.0 -1.534 0.1358
year = 2025, spacing = 10:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview 64.14 42.6 29.0 1.505 0.1431
year = 2024, spacing = 12:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview -36.13 42.6 29.0 -0.848 0.4034
year = 2025, spacing = 12:
contrast estimate SE df t.ratio p.value
Caribou - Lakeview 36.03 42.6 29.0 0.846 0.4047
Degrees-of-freedom method: kenward-roger Conclusion on combined model
For the combined model, the RCBD mixed model (cwt_per_a_total_marketable ~ year * variety * spacing + (1|year:rep)) found no statistically significant main effects of year (p = 0.59601) or variety (p = 0.41135), no significant year×spacing (p = 0.93915), variety×spacing (p = 0.53206), or year×variety×spacing (p = 0.77597) interactions, and only marginal evidence for spacing (p = 0.07286). The only statistically significant term was year×variety (p = 0.01191), indicating that the relative difference between Caribou and Lakeview changed by year. Consistent with that interaction, year-specific estimated marginal means showed Lakeview > Caribou in 2024 at all spacings (8: 556 vs 471; 10: 573 vs 507; 12: 502 vs 466 cwt/ac), while in 2025 the pattern shifted (8: 495 vs 492, Lakeview slightly higher; 10: 557 vs 493, Caribou higher; 12: 497 vs 461, Caribou higher). Tukey-adjusted pairwise tests within each year did not detect significant differences among the six variety×spacing combinations, and spacing contrasts within variety were non-significant in both years; variety contrasts within spacing were also non-significant after adjustment, with only a nominal trend in 2024 at spacing 8 (Caribou − Lakeview = -84.68 cwt/ac, p = 0.0564). Random-effect estimates indicated moderate block-within-year variability (SD_year:rep = 23.71) relative to residual variability (SD_residual = 60.26). Overall, combined-year evidence suggests no robust spacing or interaction pattern involving spacing, but clear year-dependence in variety performance; therefore, variety recommendations should be interpreted in a year-specific framework and confirmed with additional years.