Palmer Amaranth Herbicide Efficacy: A Comprehensive Statistical Analysis
Simulated Greenhouse Experiment — RCBD with Repeated Measures & Dose-Response
Emmanouil Tsakas
2026-03-28
Introduction
This report presents a comprehensive statistical analysis of a simulated greenhouse experiment evaluating herbicide treatments against Palmer Amaranth (Amaranthus palmeri), one of the most problematic herbicide-resistant weed species in agriculture.
Study Design:
- Design: Randomized Complete Block Design (RCBD) with repeated measures
- Blocks: 4 (accounting for spatial variation in the greenhouse)
- Treatments: 6 herbicide regimes (including untreated control)
- Time points: 3 phenotyping assessments (10, 14, 18 Days After Spray)
- Replicates: 3 pots per block × treatment combination
- Response variables: Plant height, leaf area, fresh weight, true leaf count, mortality
Treatments tested:
| Code | Description |
|---|---|
| Untreated | No herbicide (control) |
| Herb_A_Low | Herbicide A at low rate (50 g/ha) |
| Herb_A_Mid | Herbicide A at medium rate (100 g/ha) |
| Herb_A_High | Herbicide A at high rate (200 g/ha) |
| Herb_AB_Low | Herbicide A + B combination, low rate |
| Herb_AB_High | Herbicide A + B combination, high rate |
1 Mock Data Generation
All data in this report is simulated to mimic a realistic greenhouse experiment. The simulation encodes biologically plausible treatment effects, growth trajectories, and correlated responses.
## Observations: 216 | Treatments: 6 | Blocks: 4 | Time points: 32 Data Quality & Exploration
2.1 Missing Data Assessment
Figure 2.1: Missing data pattern across the dataset.
The dataset has a small proportion of missing values in plant_height_cm and leaf_area_cm2, consistent with occasional measurement failures in greenhouse experiments. Complete-case analysis is used below, as missingness is minimal and random.
2.2 Summary Table (Table 1)
| Variable |
Treatment Group
|
p-value2 | ||||||
|---|---|---|---|---|---|---|---|---|
| Overall N = 2161 |
Untreated N = 361 |
Herb_A_Low N = 361 |
Herb_A_Mid N = 361 |
Herb_A_High N = 361 |
Herb_AB_Low N = 361 |
Herb_AB_High N = 361 |
||
| Plant Height (cm) | 18.9 (8.8) | 24.2 (10.2) | 23.3 (10.3) | 20.4 (8.5) | 17.2 (6.4) | 16.3 (6.0) | 12.3 (3.8) | <0.001 |
| Unknown | 6 | 0 | 1 | 4 | 1 | 0 | 0 | |
| Leaf Area (cm²) | 35.3 (23.9) | 47.4 (29.9) | 45.8 (28.5) | 37.8 (21.5) | 31.4 (18.2) | 30.0 (16.3) | 19.5 (12.6) | <0.001 |
| Unknown | 9 | 1 | 2 | 0 | 1 | 4 | 1 | |
| Fresh Weight (g) | 4.8 (3.7) | 6.4 (4.6) | 5.8 (4.2) | 5.5 (4.0) | 4.6 (3.1) | 3.9 (2.7) | 2.9 (1.8) | 0.001 |
| True Leaves (n) | 6.5 (2.9) | 7.3 (3.4) | 7.6 (3.6) | 7.0 (2.8) | 6.4 (2.7) | 5.8 (2.2) | 4.9 (1.9) | 0.006 |
| Mortality (%) | 15% | 2.8% | 5.6% | 0% | 11% | 33% | 39% | <0.001 |
| 1 Mean (SD); % | ||||||||
| 2 Kruskal-Wallis rank sum test; Pearson’s Chi-squared test | ||||||||
2.3 Raincloud Plots
Raincloud plots combine a half-violin (distribution shape), jittered raw data points, and a boxplot — providing a complete view of the data that standard boxplots cannot.
Figure 2.2: Distribution of plant height by treatment and time point.
Figure 2.3: Distribution of leaf area by treatment and time point.
2.4 Correlation Matrix
Figure 2.4: Pearson correlation matrix of all continuous response variables.
Response variables are positively correlated, consistent with overall plant vigor — larger plants are taller, have more leaves, and weigh more. This multicollinearity motivates the multivariate analysis in Section 4.
3 Treatment Efficacy — Univariate
3.1 Mixed ANOVA — Plant Height
Time is modeled as a within-subject factor (repeated measures on the same pots), while block and condition are between-subject factors.
ANOVA Table
Effect | df | MSE | F | ges | p.value |
|---|---|---|---|---|---|
block | 3, 42 | 8.25 | 11.95 *** | .271 | <.001 |
condition | 5, 42 | 8.25 | 85.27 *** | .815 | <.001 |
block:condition | 15, 42 | 8.25 | 1.12 | .149 | .366 |
time_f | 1.59, 66.82 | 6.75 | 841.62 *** | .919 | <.001 |
block:time_f | 4.77, 66.82 | 6.75 | 2.15 + | .080 | .073 |
condition:time_f | 7.95, 66.82 | 6.75 | 22.49 *** | .602 | <.001 |
block:condition:time_f | 23.86, 66.82 | 6.75 | 1.29 | .206 | .208 |
Interaction Plots
Figure 3.1: Estimated marginal means: condition × time interaction.
3.2 Dunnett’s Test — Treatment vs. Control
Dunnett’s test is the appropriate many-to-one comparison when comparing multiple treatments against a single control. It is more powerful than Bonferroni for this specific design.
contrast | time_f | estimate | SE | df | t.ratio | p.value | significance |
|---|---|---|---|---|---|---|---|
Herb_A_Low - Untreated | t10 | -0.552 | 0.506 | 42.000 | -1.093 | 0.683 | ns |
Herb_A_Mid - Untreated | t10 | -1.417 | 0.548 | 42.000 | -2.584 | 0.056 | ns |
Herb_A_High - Untreated | t10 | -1.976 | 0.506 | 42.000 | -3.908 | 0.002 | ** |
Herb_AB_Low - Untreated | t10 | -2.674 | 0.490 | 42.000 | -5.453 | 0.000 | *** |
Herb_AB_High - Untreated | t10 | -4.436 | 0.490 | 42.000 | -9.045 | 0.000 | *** |
Herb_A_Low - Untreated | t14 | -1.134 | 0.969 | 42.000 | -1.170 | 0.635 | ns |
Herb_A_Mid - Untreated | t14 | -4.839 | 1.051 | 42.000 | -4.602 | 0.000 | *** |
Herb_A_High - Untreated | t14 | -6.044 | 0.969 | 42.000 | -6.235 | 0.000 | *** |
Herb_AB_Low - Untreated | t14 | -7.559 | 0.940 | 42.000 | -8.039 | 0.000 | *** |
Herb_AB_High - Untreated | t14 | -10.690 | 0.940 | 42.000 | -11.368 | 0.000 | *** |
Herb_A_Low - Untreated | t18 | -1.510 | 1.472 | 42.000 | -1.026 | 0.723 | ns |
Herb_A_Mid - Untreated | t18 | -5.665 | 1.597 | 42.000 | -3.548 | 0.004 | ** |
Herb_A_High - Untreated | t18 | -12.126 | 1.472 | 42.000 | -8.238 | 0.000 | *** |
Herb_AB_Low - Untreated | t18 | -13.448 | 1.428 | 42.000 | -9.417 | 0.000 | *** |
Herb_AB_High - Untreated | t18 | -20.690 | 1.428 | 42.000 | -14.488 | 0.000 | *** |
Abbott’s Efficacy
Percent reduction relative to untreated control — the industry standard metric in weed science:
\[\text{Efficacy (\%)} = \frac{\bar{Y}_{\text{control}} - \bar{Y}_{\text{treated}}}{\bar{Y}_{\text{control}}} \times 100\]
Figure 3.2: Herbicide efficacy (percent reduction from control) by treatment and time.
3.3 Effect Sizes
Modern statistical practice emphasizes effect sizes alongside p-values. We report partial eta-squared for ANOVA effects and Cohen’s d for pairwise comparisons.
Partial Eta-Squared (ANOVA)
Parameter | Eta2_partial | CI | CI_low | CI_high |
|---|---|---|---|---|
block | 0.461 | 0.950 | 0.253 | 1.000 |
condition | 0.910 | 0.950 | 0.865 | 1.000 |
block:condition | 0.286 | 0.950 | 0.000 | 1.000 |
time_f | 0.952 | 0.950 | 0.938 | 1.000 |
block:time_f | 0.133 | 0.950 | 0.000 | 1.000 |
condition:time_f | 0.728 | 0.950 | 0.635 | 1.000 |
block:condition:time_f | 0.315 | 0.950 | 0.000 | 1.000 |
Cohen’s d — Forest Plot
Figure 3.3: Standardized mean differences (Cohen’s d) for each treatment vs. control at the final time point.
3.4 GLMM — Count Data (True Leaves)
The number of true leaves is a count variable. Standard ANOVA assumes normality, which is inappropriate for counts — especially at low values. We fit a negative binomial GLMM to properly model the data-generating process.
## Family: nbinom2 ( log )
## Formula: number_true_leaves ~ condition * time_f + (1 | block/pot_id)
## Data: dat
##
## AIC BIC logLik -2*log(L) df.resid
## NA NA NA NA 195
##
## Random effects:
##
## Conditional model:
## Groups Name Variance Std.Dev.
## pot_id:block (Intercept) 4.823e-10 2.196e-05
## block (Intercept) 2.236e-03 4.729e-02
## Number of obs: 216, groups: pot_id:block, 72; block, 4
##
## Dispersion parameter for nbinom2 family (): 3.18e+08
##
## Conditional model:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 1.385e+00 1.463e-01 9.470 < 2e-16 ***
## conditionHerb_A_Low -6.454e-02 2.075e-01 -0.311 0.7558
## conditionHerb_A_Mid 1.679e-06 2.041e-01 0.000 1.0000
## conditionHerb_A_High -6.454e-02 2.075e-01 -0.311 0.7558
## conditionHerb_AB_Low -1.335e-01 2.113e-01 -0.632 0.5274
## conditionHerb_AB_High -2.877e-01 2.205e-01 -1.305 0.1920
## time_ft14 4.595e-01 1.844e-01 2.492 0.0127 *
## time_ft18 1.070e+00 1.673e-01 6.400 1.56e-10 ***
## conditionHerb_A_Low:time_ft14 1.997e-01 2.602e-01 0.768 0.4428
## conditionHerb_A_Mid:time_ft14 2.597e-02 2.601e-01 0.100 0.9205
## conditionHerb_A_High:time_ft14 2.426e-02 2.644e-01 0.092 0.9269
## conditionHerb_AB_Low:time_ft14 7.489e-03 2.697e-01 0.028 0.9778
## conditionHerb_AB_High:time_ft14 1.001e-01 2.786e-01 0.359 0.7195
## conditionHerb_A_Low:time_ft18 7.165e-02 2.394e-01 0.299 0.7647
## conditionHerb_A_Mid:time_ft18 -1.133e-01 2.383e-01 -0.475 0.6344
## conditionHerb_A_High:time_ft18 -1.586e-01 2.432e-01 -0.652 0.5142
## conditionHerb_AB_Low:time_ft18 -2.231e-01 2.490e-01 -0.896 0.3701
## conditionHerb_AB_High:time_ft18 -2.845e-01 2.616e-01 -1.088 0.2767
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1GLMM Diagnostics (DHARMa)
DHARMa uses simulation-based residuals to assess model fit for non-normal distributions — more appropriate than standard residual plots for GLMMs.
Figure 3.4: Simulation-based residual diagnostics for the negative binomial GLMM.
Incidence Rate Ratios
For count models, exponentiated coefficients give incidence rate ratios (IRR) — the multiplicative change in expected leaf count relative to the reference level.
term | IRR | conf.low | conf.high | p.value |
|---|---|---|---|---|
(Intercept) | 3.996 | 3.000 | 5.322 | 0.000 |
conditionHerb_A_Low | 0.938 | 0.624 | 1.408 | 0.756 |
conditionHerb_A_Mid | 1.000 | 0.670 | 1.492 | 1.000 |
conditionHerb_A_High | 0.938 | 0.624 | 1.408 | 0.756 |
conditionHerb_AB_Low | 0.875 | 0.578 | 1.324 | 0.527 |
conditionHerb_AB_High | 0.750 | 0.487 | 1.155 | 0.192 |
time_ft14 | 1.583 | 1.103 | 2.273 | 0.013 |
time_ft18 | 2.917 | 2.101 | 4.048 | 0.000 |
conditionHerb_A_Low:time_ft14 | 1.221 | 0.733 | 2.033 | 0.443 |
conditionHerb_A_Mid:time_ft14 | 1.026 | 0.616 | 1.709 | 0.920 |
conditionHerb_A_High:time_ft14 | 1.025 | 0.610 | 1.720 | 0.927 |
conditionHerb_AB_Low:time_ft14 | 1.008 | 0.594 | 1.709 | 0.978 |
conditionHerb_AB_High:time_ft14 | 1.105 | 0.640 | 1.908 | 0.719 |
conditionHerb_A_Low:time_ft18 | 1.074 | 0.672 | 1.717 | 0.765 |
conditionHerb_A_Mid:time_ft18 | 0.893 | 0.560 | 1.425 | 0.634 |
conditionHerb_A_High:time_ft18 | 0.853 | 0.530 | 1.374 | 0.514 |
conditionHerb_AB_Low:time_ft18 | 0.800 | 0.491 | 1.303 | 0.370 |
conditionHerb_AB_High:time_ft18 | 0.752 | 0.451 | 1.256 | 0.277 |
3.5 Model Diagnostics — Mixed ANOVA
Figure 3.5: Comprehensive model diagnostic panel for the mixed ANOVA.
4 Multivariate Analysis
Individual ANOVAs for each response variable ignore the correlation structure between outcomes. A multivariate approach tests whether treatments affect the overall plant phenotype.
4.1 MANOVA
## Df Pillai approx F num Df den Df Pr(>F)
## condition 5 0.87768 10.119 20 720 < 2.2e-16 ***
## time_f 2 1.01342 45.711 8 356 < 2.2e-16 ***
## block 3 0.37402 6.374 12 537 1.258e-10 ***
## condition:time_f 10 0.81070 4.575 40 720 < 2.2e-16 ***
## Residuals 180
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 14.2 PCA Biplot
Principal Component Analysis reveals how treatments separate in the multivariate response space and which variables drive that separation.
Figure 4.1: PCA biplot of plant responses, colored by treatment. Ellipses represent 95% confidence regions.
Figure 4.2: Variance explained by each principal component.
4.3 PERMANOVA — Robustness Check
PERMANOVA is a non-parametric alternative that makes no distributional assumptions. It tests whether group centroids differ in multivariate space.
Df | SumOfSqs | R2 | F | Pr(>F) |
|---|---|---|---|---|
20.000 | 728.563 | 0.911 | 91.788 | 0.001 |
180.000 | 71.437 | 0.089 | ||
200.000 | 800.000 | 1.000 |
5 Dose-Response Analysis
For dose-response modeling, we use the final time point (18 DAS) and fresh weight as the primary endpoint — the standard efficacy metric in herbicide trials.
5.1 Four-Parameter Log-Logistic Model
The 4-parameter log-logistic (LL.4) model is the standard in dose-response analysis:
\[f(x) = c + \frac{d - c}{1 + \exp(b(\log(x) - \log(e)))}\]
where \(c\) = lower limit, \(d\) = upper limit, \(b\) = slope at \(e\), and \(e\) = ED50.
##
## Model fitted: Log-logistic (ED50 as parameter) (4 parms)
##
## Parameter estimates:
##
## Estimate Std. Error t-value p-value
## Slope:(Intercept) 0.93554 0.44733 2.0914 0.04023 *
## Lower:(Intercept) -7.52160 26.41050 -0.2848 0.77667
## Upper:(Intercept) 12.33283 0.48787 25.2790 < 2e-16 ***
## ED50:(Intercept) 1002.67992 2485.01249 0.4035 0.68785
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error:
##
## 1.714482 (68 degrees of freedom)Model Comparison
Model | AIC | LogLik | deltaAIC |
|---|---|---|---|
Log-Logistic (LL.4) | 287.84 | -138.92 | 0.00 |
Weibull Type 1 (W1.4) | 287.86 | -138.93 | 0.01 |
Weibull Type 2 (W2.4) | 287.95 | -138.97 | 0.11 |
5.2 Dose-Response Curve with Confidence Band
Figure 5.1: Fitted dose-response curve (LL.4) with 95% confidence band. Points are individual observations; the red dashed line marks the ED50.
5.3 Effective Doses with Confidence Intervals
ED Level | Dose (g/ha) | Std. Error | 95% CI Lower | 95% CI Upper |
|---|---|---|---|---|
e:1:10 | 95.76 | 138.29 | -180.20 | 371.72 |
e:1:25 | 309.86 | 603.93 | -895.26 | 1,514.99 |
e:1:50 | 1,002.68 | 2,485.01 | -3,956.09 | 5,961.45 |
e:1:75 | 3,244.56 | 9,796.47 | -16,303.98 | 22,793.11 |
e:1:90 | 10,499.05 | 37,446.70 | -64,224.68 | 85,222.78 |
5.4 Bootstrap Confidence Intervals for ED50
Delta-method CIs assume asymptotic normality. Bootstrap CIs provide a distribution-free alternative, particularly useful with small sample sizes.
## ED50 Bootstrap Estimate: 2106.26 g/ha## 95% Percentile CI: [ 196.57 , 7662.49 ] g/ha
Figure 5.2: Bootstrap distribution of ED50 estimates (1000 resamples). The red line marks the point estimate; blue dashed lines mark the 95% percentile CI.
Residual Diagnostics — Dose-Response Model
Figure 5.3: Residual diagnostics for the dose-response model.
## [1] 15 63Conclusions
Key Findings:
Treatment effects are significant — Both the mixed ANOVA (condition × time interaction) and MANOVA (Pillai’s trace) confirm that herbicide treatments significantly affect Palmer Amaranth growth.
Dose-response relationship is clear — The 4-parameter log-logistic model provides a good fit. The estimated ED50 indicates the dose required for 50% fresh weight reduction.
Combination treatments are most effective — The high-rate A+B combination shows the largest effect sizes (Cohen’s d) and highest Abbott’s efficacy across all time points.
Effects intensify over time — Treatment suppression grows from 10 to 18 DAS, as shown by the condition × time interaction.
Multivariate confirmation — PCA and PERMANOVA confirm that treatments separate plants in multivariate phenotype space, not just on individual endpoints.
Reproducibility
Searchable Raw Data
Session Info
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## insight 1.4.6 2026-02-04 [1] CRAN (R 4.5.3)
## IRanges 2.44.0 2025-10-29 [1] https://bioc-release.r-universe.dev (R 4.5.2)
## iterators 1.0.14 2022-02-05 [1] CRAN (R 4.5.1)
## janitor * 2.2.1 2024-12-22 [1] CRAN (R 4.5.2)
## jquerylib 0.1.4 2021-04-26 [1] CRAN (R 4.5.1)
## jsonlite 2.0.0 2025-03-27 [1] CRAN (R 4.5.1)
## knitr 1.50 2025-03-16 [1] CRAN (R 4.5.1)
## labeling 0.4.3 2023-08-29 [1] CRAN (R 4.5.0)
## later 1.4.4 2025-08-27 [1] CRAN (R 4.5.1)
## lattice 0.22-7 2025-04-02 [2] CRAN (R 4.5.1)
## leaps 3.2 2024-06-10 [1] CRAN (R 4.5.3)
## lifecycle 1.0.4 2023-11-07 [1] CRAN (R 4.5.1)
## listenv 0.10.0 2025-11-02 [1] CRAN (R 4.5.2)
## litedown 0.7 2025-04-08 [1] CRAN (R 4.5.1)
## lme4 * 1.1-37 2025-03-26 [1] CRAN (R 4.5.1)
## lmerTest 3.2-1 2026-03-05 [1] CRAN (R 4.5.3)
## lubridate * 1.9.4 2024-12-08 [1] CRAN (R 4.5.1)
## magrittr 2.0.3 2022-03-30 [1] CRAN (R 4.5.1)
## markdown 2.0 2025-03-23 [1] CRAN (R 4.5.1)
## MASS * 7.3-65 2025-02-28 [2] CRAN (R 4.5.1)
## Matrix * 1.7-3 2025-03-11 [2] CRAN (R 4.5.1)
## matrixStats 1.5.0 2025-01-07 [1] CRAN (R 4.5.2)
## memoise 2.0.1 2021-11-26 [1] CRAN (R 4.5.1)
## mgcv 1.9-3 2025-04-04 [2] CRAN (R 4.5.1)
## mime 0.13 2025-03-17 [1] CRAN (R 4.5.0)
## minqa 1.2.8 2024-08-17 [1] CRAN (R 4.5.1)
## multcomp * 1.4-30 2026-03-09 [1] CRAN (R 4.5.3)
## multcompView 0.1-11 2026-02-16 [1] CRAN (R 4.5.3)
## mvtnorm * 1.3-3 2025-01-10 [1] CRAN (R 4.5.1)
## naniar * 1.1.0 2024-03-05 [1] CRAN (R 4.5.3)
## nlme 3.1-168 2025-03-31 [2] CRAN (R 4.5.1)
## nloptr 2.2.1 2025-03-17 [1] CRAN (R 4.5.1)
## numDeriv 2016.8-1.1 2019-06-06 [1] CRAN (R 4.5.0)
## officer 0.7.0 2025-09-03 [1] CRAN (R 4.5.1)
## openssl 2.3.3 2025-05-26 [1] CRAN (R 4.5.1)
## parallelly 1.46.1 2026-01-08 [1] CRAN (R 4.5.2)
## parameters 0.28.3 2025-11-25 [1] CRAN (R 4.5.3)
## patchwork * 1.3.2 2025-08-25 [1] CRAN (R 4.5.1)
## performance * 0.16.0 2026-02-04 [1] CRAN (R 4.5.3)
## permute * 0.9-10 2026-02-06 [1] CRAN (R 4.5.3)
## pillar 1.11.0 2025-07-04 [1] CRAN (R 4.5.1)
## pkgconfig 2.0.3 2019-09-22 [1] CRAN (R 4.5.1)
## plotrix 3.8-4 2023-11-10 [1] CRAN (R 4.5.0)
## plyr 1.8.9 2023-10-02 [1] CRAN (R 4.5.1)
## png 0.1-8 2022-11-29 [1] CRAN (R 4.5.0)
## promises 1.3.3 2025-05-29 [1] CRAN (R 4.5.1)
## purrr * 1.1.0 2025-07-10 [1] CRAN (R 4.5.1)
## qgam 2.0.0 2025-04-10 [1] CRAN (R 4.5.3)
## R6 2.6.1 2025-02-15 [1] CRAN (R 4.5.1)
## ragg 1.5.0 2025-09-02 [1] CRAN (R 4.5.1)
## rbibutils 2.3 2024-10-04 [1] CRAN (R 4.5.1)
## RColorBrewer 1.1-3 2022-04-03 [1] CRAN (R 4.5.0)
## Rcpp 1.1.0 2025-07-02 [1] CRAN (R 4.5.1)
## Rdpack 2.6.4 2025-04-09 [1] CRAN (R 4.5.1)
## readr * 2.1.5 2024-01-10 [1] CRAN (R 4.5.1)
## reformulas 0.4.4 2026-02-02 [1] CRAN (R 4.5.3)
## reshape2 1.4.4 2020-04-09 [1] CRAN (R 4.5.1)
## rjson 0.2.23 2024-09-16 [1] CRAN (R 4.5.0)
## rlang 1.1.7 2026-01-09 [1] CRAN (R 4.5.3)
## rmarkdown 2.29 2024-11-04 [1] CRAN (R 4.5.1)
## rstatix 0.7.2 2023-02-01 [1] CRAN (R 4.5.1)
## S4Vectors 0.48.0 2025-10-29 [1] https://bioc-release.r-universe.dev (R 4.5.2)
## S7 0.2.0 2024-11-07 [1] CRAN (R 4.5.1)
## sandwich 3.1-1 2024-09-15 [1] CRAN (R 4.5.2)
## sass 0.4.10 2025-04-11 [1] CRAN (R 4.5.1)
## scales 1.4.0 2025-04-24 [1] CRAN (R 4.5.1)
## scatterplot3d 0.3-45 2026-02-23 [1] CRAN (R 4.5.2)
## see 0.13.0 2026-01-30 [1] CRAN (R 4.5.3)
## sessioninfo 1.2.3 2025-02-05 [1] CRAN (R 4.5.1)
## shape 1.4.6.1 2024-02-23 [1] CRAN (R 4.5.2)
## shiny 1.11.1 2025-07-03 [1] CRAN (R 4.5.1)
## snakecase 0.11.1 2023-08-27 [1] CRAN (R 4.5.1)
## stringi 1.8.7 2025-03-27 [1] CRAN (R 4.5.0)
## stringr * 1.5.1 2023-11-14 [1] CRAN (R 4.5.1)
## survival * 3.8-3 2024-12-17 [2] CRAN (R 4.5.1)
## systemfonts 1.3.2 2026-03-05 [1] CRAN (R 4.5.3)
## textshaping 1.0.3 2025-09-02 [1] CRAN (R 4.5.1)
## TH.data * 1.1-5 2025-11-17 [1] CRAN (R 4.5.3)
## tibble * 3.3.0 2025-06-08 [1] CRAN (R 4.5.1)
## tidyr * 1.3.1 2024-01-24 [1] CRAN (R 4.5.1)
## tidyselect 1.2.1 2024-03-11 [1] CRAN (R 4.5.1)
## tidyverse * 2.0.0 2023-02-22 [1] CRAN (R 4.5.1)
## timechange 0.3.0 2024-01-18 [1] CRAN (R 4.5.1)
## TMB 1.9.21 2026-03-23 [1] CRAN (R 4.5.3)
## tzdb 0.5.0 2025-03-15 [1] CRAN (R 4.5.1)
## uuid 1.2-1 2024-07-29 [1] CRAN (R 4.5.0)
## vctrs 0.6.5 2023-12-01 [1] CRAN (R 4.5.1)
## vegan * 2.7-3 2026-03-04 [1] CRAN (R 4.5.3)
## visdat 0.6.0 2023-02-02 [1] CRAN (R 4.5.3)
## withr 3.0.2 2024-10-28 [1] CRAN (R 4.5.1)
## xfun 0.52 2025-04-02 [1] CRAN (R 4.5.1)
## xml2 1.4.0 2025-08-20 [1] CRAN (R 4.5.1)
## xtable 1.8-4 2019-04-21 [1] CRAN (R 4.5.1)
## yaml 2.3.10 2024-07-26 [1] CRAN (R 4.5.0)
## zip 2.3.3 2025-05-13 [1] CRAN (R 4.5.1)
## zoo 1.8-14 2025-04-10 [1] CRAN (R 4.5.1)
##
## [1] C:/Users/emant/AppData/Local/R/win-library/4.5
## [2] C:/Program Files/R/R-4.5.1/library
## * ── Packages attached to the search path.
##
## ──────────────────────────────────────────────────────────────────────────────