Data

data <- read.table("data_pcr_fgf9.txt", header = T, sep = "\t", dec = ".", fill = T)
data <- na.omit(data)
data$genotype <- factor(data$genotype, levels = c("WT", "Mut"))
data$Time <- factor(data$Time, 
                    levels = c("iPS", "M1_36h00", "M2_06h00", "M2_12h00", "M2_24h00", "M2_36h00", "M2_48h00", "M3_24h00", "M3_48h00"))
data %>%
  kbl() %>%
  kable_paper("hover", full_width = F) %>%
  kable_styling(bootstrap_options = c("striped", "hover")) %>%
  scroll_box(height = "300px")
Sample_Reference genotype dCt_NA Experiment Time
1 21-121 WT 12.46154 iPS09 M1_36h00
2 21-122 WT 12.54687 iPS09 M1_36h00
3 21-123 WT 11.82234 iPS09 M1_36h00
4 21-124 WT 13.02666 iPS09 M1_36h00
5 21-125 WT 11.69485 iPS09 M1_36h00
6 21-126 Mut 11.17416 iPS09 M1_36h00
7 21-127 Mut 11.65472 iPS09 M1_36h00
8 21-128 Mut 12.31313 iPS09 M1_36h00
9 21-129 Mut 11.03377 iPS09 M1_36h00
10 21-130 Mut 10.35606 iPS09 M1_36h00
11 22-59 WT 12.87990 iPS12 M1_36h00
12 22-60 WT 12.49654 iPS12 M1_36h00
13 22-61 WT 12.58122 iPS12 M1_36h00
14 22-62 WT 12.51467 iPS12 M1_36h00
15 22-63 WT 12.85105 iPS12 M1_36h00
16 22-64 WT 12.36693 iPS12 M1_36h00
17 22-65 Mut 11.37795 iPS12 M1_36h00
18 22-66 Mut 11.40796 iPS12 M1_36h00
20 22-68 Mut 11.42081 iPS12 M1_36h00
22 22-70 Mut 11.36159 iPS12 M1_36h00
23 23-181 WT 13.15268 iPS26a M1_36h00
24 23-182 WT 12.81992 iPS26a M1_36h00
25 23-183 WT 13.18746 iPS26a M1_36h00
26 23-184 WT 13.04699 iPS26a M1_36h00
27 23-185 WT 12.88933 iPS26a M1_36h00
28 23-186 WT 13.16146 iPS26a M1_36h00
29 23-187 WT 12.80593 iPS26b M1_36h00
30 23-188 WT 13.15962 iPS26b M1_36h00
31 23-189 WT 13.10211 iPS26b M1_36h00
32 23-190 WT 13.18347 iPS26b M1_36h00
33 23-191 WT 13.26135 iPS26b M1_36h00
34 23-192 WT 13.01085 iPS26b M1_36h00
35 23-193 Mut 12.14293 iPS26a M1_36h00
36 23-194 Mut 12.27821 iPS26a M1_36h00
37 23-195 Mut 12.46465 iPS26a M1_36h00
38 23-196 Mut 12.46030 iPS26a M1_36h00
39 23-197 Mut 12.55558 iPS26a M1_36h00
40 23-198 Mut 12.13226 iPS26a M1_36h00
41 23-199 Mut 12.91510 iPS26b M1_36h00
42 23-200 Mut 12.79076 iPS26b M1_36h00
43 23-201 Mut 12.69690 iPS26b M1_36h00
44 23-202 Mut 13.05962 iPS26b M1_36h00
45 23-203 Mut 12.93654 iPS26b M1_36h00
46 23-204 Mut 12.63284 iPS26b M1_36h00
47 22-71 WT 13.56473 iPS12 M2_06h00
48 22-72 WT 13.50299 iPS12 M2_06h00
50 22-74 WT 13.55734 iPS12 M2_06h00
51 22-75 WT 13.38498 iPS12 M2_06h00
53 22-77 Mut 11.97065 iPS12 M2_06h00
54 22-78 Mut 11.81595 iPS12 M2_06h00
55 22-79 Mut 11.84649 iPS12 M2_06h00
56 22-80 Mut 11.93076 iPS12 M2_06h00
57 22-81 Mut 12.26734 iPS12 M2_06h00
58 22-82 Mut 12.24013 iPS12 M2_06h00
59 23-205 WT 12.43467 iPS26a M2_06h00
60 23-206 WT 12.25262 iPS26a M2_06h00
61 23-207 WT 12.54939 iPS26a M2_06h00
62 23-208 WT 12.26983 iPS26b M2_06h00
63 23-209 WT 12.11422 iPS26b M2_06h00
64 23-210 WT 12.23498 iPS26b M2_06h00
65 23-211 Mut 11.98416 iPS26a M2_06h00
66 23-212 Mut 12.09958 iPS26a M2_06h00
67 23-213 Mut 11.70651 iPS26a M2_06h00
68 23-214 Mut 12.17826 iPS26b M2_06h00
69 23-215 Mut 12.16790 iPS26b M2_06h00
70 23-216 Mut 12.05350 iPS26b M2_06h00
71 22-83 WT 13.68977 iPS12 M2_12h00
72 22-84 WT 12.88334 iPS12 M2_12h00
73 22-85 WT 12.89130 iPS12 M2_12h00
74 22-86 WT 13.41430 iPS12 M2_12h00
75 22-87 WT 12.91940 iPS12 M2_12h00
76 22-88 WT 13.12297 iPS12 M2_12h00
77 22-89 Mut 11.22897 iPS12 M2_12h00
78 22-90 Mut 11.13885 iPS12 M2_12h00
79 22-91 Mut 11.79417 iPS12 M2_12h00
80 22-92 Mut 11.09804 iPS12 M2_12h00
81 22-93 Mut 11.55017 iPS12 M2_12h00
82 22-94 Mut 11.25895 iPS12 M2_12h00
83 23-217 WT 12.01773 iPS26a M2_12h00
84 23-218 WT 11.98648 iPS26a M2_12h00
85 23-219 WT 12.69373 iPS26a M2_12h00
86 23-220 WT 11.80496 iPS26b M2_12h00
87 23-221 WT 11.77526 iPS26b M2_12h00
88 23-222 WT 11.91763 iPS26b M2_12h00
89 23-223 Mut 12.18104 iPS26a M2_12h00
90 23-224 Mut 12.18077 iPS26a M2_12h00
91 23-225 Mut 12.10585 iPS26a M2_12h00
92 23-226 Mut 11.81561 iPS26b M2_12h00
93 23-227 Mut 12.16261 iPS26b M2_12h00
94 23-228 Mut 11.96268 iPS26b M2_12h00
95 21-131 WT 12.98639 iPS09 M2_24h00
96 21-132 WT 12.27557 iPS09 M2_24h00
97 21-133 WT 13.07213 iPS09 M2_24h00
98 21-134 WT 12.03032 iPS09 M2_24h00
99 21-135 WT 12.27773 iPS09 M2_24h00
100 21-136 Mut 10.17652 iPS09 M2_24h00
101 21-137 Mut 11.27970 iPS09 M2_24h00
102 21-138 Mut 10.56130 iPS09 M2_24h00
103 21-139 Mut 10.53302 iPS09 M2_24h00
104 21-140 Mut 11.17189 iPS09 M2_24h00
105 21-131 WT 12.98639 iPS09 M2_24h00
106 21-132 WT 12.27557 iPS09 M2_24h00
107 21-133 WT 13.07213 iPS09 M2_24h00
108 21-134 WT 12.03032 iPS09 M2_24h00
109 21-135 WT 12.27773 iPS09 M2_24h00
110 21-136 Mut 10.17652 iPS09 M2_24h00
111 21-137 Mut 11.27970 iPS09 M2_24h00
112 21-138 Mut 10.56130 iPS09 M2_24h00
113 21-139 Mut 10.53302 iPS09 M2_24h00
114 21-140 Mut 11.17189 iPS09 M2_24h00
115 23-229 WT 13.59186 iPS26a M2_36h00
116 23-230 WT 13.68539 iPS26a M2_36h00
117 23-231 WT 13.77563 iPS26a M2_36h00
118 23-232 WT 13.06587 iPS26b M2_36h00
119 23-233 WT 13.13479 iPS26b M2_36h00
120 23-234 WT 13.49237 iPS26b M2_36h00
121 23-235 Mut 14.13716 iPS26a M2_36h00
122 23-236 Mut 13.90191 iPS26a M2_36h00
123 23-237 Mut 13.81031 iPS26a M2_36h00
124 23-238 Mut 13.01396 iPS26b M2_36h00
125 23-239 Mut 13.23651 iPS26b M2_36h00
126 23-240 Mut 13.49843 iPS26b M2_36h00
127 22-107 WT 12.54905 iPS12 M2_48h00
128 22-108 WT 13.24822 iPS12 M2_48h00
129 22-109 WT 12.85570 iPS12 M2_48h00
130 22-110 WT 12.58749 iPS12 M2_48h00
131 22-111 WT 13.29151 iPS12 M2_48h00
133 22-113 Mut 11.57384 iPS12 M2_48h00
134 22-114 Mut 11.96141 iPS12 M2_48h00
135 22-115 Mut 12.33813 iPS12 M2_48h00
136 22-116 Mut 11.64191 iPS12 M2_48h00
137 22-117 Mut 12.01675 iPS12 M2_48h00
138 22-118 Mut 12.00895 iPS12 M2_48h00
139 23-241 WT 14.18000 iPS26a M2_48h00
140 23-242 WT 14.49000 iPS26a M2_48h00
141 23-243 WT 14.61000 iPS26a M2_48h00
142 23-244 WT 13.65000 iPS26b M2_48h00
143 23-245 WT 14.22000 iPS26b M2_48h00
144 23-246 WT 13.72000 iPS26b M2_48h00
145 23-247 Mut 13.89000 iPS26a M2_48h00
146 23-248 Mut 13.55000 iPS26a M2_48h00
147 23-249 Mut 13.70000 iPS26a M2_48h00
148 23-250 Mut 14.13000 iPS26b M2_48h00
149 23-251 Mut 14.38000 iPS26b M2_48h00
150 23-252 Mut 14.16000 iPS26b M2_48h00
151 21-131 WT 12.98639 iPS09 M2_24h00
152 21-132 WT 12.27557 iPS09 M2_24h00
153 21-133 WT 13.07213 iPS09 M2_24h00
154 21-134 WT 12.03032 iPS09 M2_24h00
155 21-135 WT 12.27773 iPS09 M2_24h00
156 21-136 Mut 10.17652 iPS09 M2_24h00
157 21-137 Mut 11.27970 iPS09 M2_24h00
158 21-138 Mut 10.56130 iPS09 M2_24h00
159 21-139 Mut 10.53302 iPS09 M2_24h00
160 21-140 Mut 11.17189 iPS09 M2_24h00
161 23-253 WT 13.69454 iPS26a M3_24h00
162 23-254 WT 13.51696 iPS26a M3_24h00
163 23-255 WT 14.25398 iPS26a M3_24h00
164 23-256 WT 12.10233 iPS26b M3_24h00
165 23-257 WT 12.73347 iPS26b M3_24h00
166 23-258 WT 11.99639 iPS26b M3_24h00
167 23-259 Mut 11.79358 iPS26a M3_24h00
168 23-260 Mut 11.49572 iPS26a M3_24h00
169 23-261 Mut 11.67450 iPS26a M3_24h00
170 23-262 Mut 11.94252 iPS26b M3_24h00
171 23-263 Mut 12.14255 iPS26b M3_24h00
172 23-264 Mut 12.12307 iPS26b M3_24h00
173 21-146 WT 13.75476 iPS09 M3_48h00
174 21-147 WT 12.86544 iPS09 M3_48h00
175 21-148 WT 13.21333 iPS09 M3_48h00
176 21-149 WT 13.60240 iPS09 M3_48h00
177 21-150 WT 12.21891 iPS09 M3_48h00
179 21-152 Mut 12.00811 iPS09 M3_48h00
181 21-154 Mut 12.17217 iPS09 M3_48h00
182 21-155 Mut 12.36491 iPS09 M3_48h00
183 23-265 WT 12.52975 iPS26a M3_48h00
184 23-266 WT 13.02921 iPS26a M3_48h00
185 23-267 WT 13.43431 iPS26a M3_48h00
186 23-268 WT 10.79103 iPS26b M3_48h00
187 23-269 WT 10.88930 iPS26b M3_48h00
188 23-270 WT 10.73752 iPS26b M3_48h00
189 23-271 Mut 11.57271 iPS26a M3_48h00
190 23-272 Mut 11.34341 iPS26a M3_48h00
191 23-273 Mut 11.47600 iPS26a M3_48h00
192 23-274 Mut 10.90606 iPS26b M3_48h00
193 23-275 Mut 11.47358 iPS26b M3_48h00
194 23-276 Mut 11.50285 iPS26b M3_48h00
195 23-157 WT 14.62692 iPS19 iPS
196 23-158 WT 14.53885 iPS19 iPS
197 23-159 WT 14.30192 iPS19 iPS
198 23-160 WT 14.82520 iPS19 iPS
199 23-161 WT 14.59451 iPS19 iPS
200 23-162 WT 14.32377 iPS19 iPS
201 23-169 WT 13.96617 iPS19 iPS
202 23-170 WT 14.13391 iPS19 iPS
203 23-171 WT 13.90911 iPS19 iPS
204 23-172 WT 14.53716 iPS19 iPS
205 23-173 WT 14.19427 iPS19 iPS
206 23-174 WT 13.80354 iPS19 iPS
207 23-163 Mut 15.02069 iPS19 iPS
208 23-164 Mut 15.12220 iPS19 iPS
209 23-165 Mut 15.04068 iPS19 iPS
210 23-166 Mut 14.46016 iPS19 iPS
211 23-167 Mut 14.83635 iPS19 iPS
212 23-168 Mut 14.55928 iPS19 iPS
213 23-175 Mut 14.65537 iPS19 iPS
214 23-176 Mut 14.53501 iPS19 iPS
215 23-177 Mut 14.64444 iPS19 iPS
216 23-178 Mut 14.34309 iPS19 iPS
217 23-179 Mut 14.52701 iPS19 iPS
218 23-180 Mut 14.55937 iPS19 iPS

Plots

We first examine the distribution of dCt_NA across the different timepoints and colored by experiments.

ggplot(data = data, aes(x = genotype, y = dCt_NA)) +
         geom_boxplot(outlier.shape = NA) +
         geom_jitter(aes(colour = Experiment), size = 0.8) +
         theme_classic() + facet_grid(. ~ Time)
Figure 1: dCt (raw data) as a function of the genotype at each time point

Figure 1: dCt (raw data) as a function of the genotype at each time point

We next look at the distribution of dCt_NA across the different experiments and colored by timepoints.

ggplot(data = data, aes(x = genotype, y = dCt_NA, colour = Time)) +
         geom_boxplot(outlier.shape = NA) +
         geom_jitter(position=position_jitterdodge(jitter.width = 0.1), size = 0.8) +
         theme_classic() + facet_grid(. ~ Experiment)
Figure 2: dCt (raw data) as a function of the genotype in each experiment

Figure 2: dCt (raw data) as a function of the genotype in each experiment

Analysis

We use a mixed-effect model, to analyse the dependent variable dCt_NA with respect to:

  • fixed effects (i.e., the genotype and Time variables)

  • random effects (i.e., the Experiment variable)

We include an interaction term between genotypeand Timepoint as we are interested in the effect of genotypeat each Timepoint and the effect of genotypeseems not to be homogeneous at each timepoint.

We obtain the following:

mod = lmer(dCt_NA ~ genotype*Time + (1 | Experiment), data = data)

summary(mod)
## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: dCt_NA ~ genotype * Time + (1 | Experiment)
##    Data: data
## 
## REML criterion at convergence: 394.6
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -2.81335 -0.54455 -0.00462  0.67134  2.33625 
## 
## Random effects:
##  Groups     Name        Variance Std.Dev.
##  Experiment (Intercept) 0.03876  0.1969  
##  Residual               0.35374  0.5948  
## Number of obs: 211, groups:  Experiment, 5
## 
## Fixed effects:
##                          Estimate Std. Error       df t value Pr(>|t|)    
## (Intercept)               14.3129     0.2612   6.0375  54.791 2.24e-09 ***
## genotypeMut                0.3790     0.2428 190.3308   1.561 0.120185    
## TimeM1_36h00              -1.5303     0.3055   7.2092  -5.009 0.001420 ** 
## TimeM2_06h00              -1.5131     0.3384  10.5791  -4.471 0.001041 ** 
## TimeM2_12h00              -1.6757     0.3301   9.5674  -5.076 0.000551 ***
## TimeM2_24h00              -1.7624     0.3345   8.9060  -5.269 0.000533 ***
## TimeM2_36h00              -0.9657     0.3730  15.3123  -2.589 0.020295 *  
## TimeM2_48h00              -0.7009     0.3339  10.0185  -2.099 0.062125 .  
## TimeM3_24h00              -1.3737     0.3730  15.3123  -3.683 0.002149 ** 
## TimeM3_48h00              -1.9026     0.3337  10.0256  -5.702 0.000196 ***
## genotypeMut:TimeM1_36h00  -1.1262     0.3021 190.3980  -3.728 0.000254 ***
## genotypeMut:TimeM2_06h00  -1.1129     0.3520 190.4137  -3.161 0.001827 ** 
## genotypeMut:TimeM2_12h00  -1.2656     0.3434 190.3308  -3.686 0.000297 ***
## genotypeMut:TimeM2_24h00  -2.1630     0.3258 190.3308  -6.640 3.20e-10 ***
## genotypeMut:TimeM2_36h00  -0.2370     0.4206 190.3308  -0.563 0.573792    
## genotypeMut:TimeM2_48h00  -1.0011     0.3473 190.3485  -2.882 0.004400 ** 
## genotypeMut:TimeM3_24h00  -1.5666     0.4206 190.3308  -3.725 0.000257 ***
## genotypeMut:TimeM3_48h00  -1.2090     0.3615 190.6436  -3.344 0.000994 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation matrix not shown by default, as p = 18 > 12.
## Use print(x, correlation=TRUE)  or
##     vcov(x)        if you need it
hist(residuals(mod), nclass = 50)
Figure 3: Histogram of the residuals from the linear mixed models

Figure 3: Histogram of the residuals from the linear mixed models

We can compute the marginal effects of the fixed effects and their interaction term.

Anova(mod)
## Analysis of Deviance Table (Type II Wald chisquare tests)
## 
## Response: dCt_NA
##                 Chisq Df Pr(>Chisq)    
## genotype       83.569  1  < 2.2e-16 ***
## Time          174.091  8  < 2.2e-16 ***
## genotype:Time  53.335  8   9.29e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We are interested in the genotype effect at each timepoint.

emm.all <- emmeans(mod,  ~ genotype | Time)
pairs(emm.all)
## Time = iPS:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut   -0.379 0.243 190  -1.561  0.1202
## 
## Time = M1_36h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut    0.747 0.180 190   4.157  <.0001
## 
## Time = M2_06h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut    0.734 0.255 190   2.878  0.0045
## 
## Time = M2_12h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut    0.887 0.243 190   3.651  0.0003
## 
## Time = M2_24h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut    1.784 0.217 190   8.214  <.0001
## 
## Time = M2_36h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut   -0.142 0.343 190  -0.414  0.6796
## 
## Time = M2_48h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut    0.622 0.248 190   2.505  0.0131
## 
## Time = M3_24h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut    1.188 0.343 190   3.459  0.0007
## 
## Time = M3_48h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut    0.830 0.268 191   3.096  0.0023
## 
## Degrees-of-freedom method: kenward-roger

We now adjust the p-values using the Benjamini-Hochberg to identify at which timepoints the dCT_NA are significantly different between WT and Mut.

p <- summary(pairs(emm.all))$p.value
adj.p <- p.adjust(p, method = "BH")
names(adj.p) <- levels(data$Time)
adj.p
##          iPS     M1_36h00     M2_06h00     M2_12h00     M2_24h00     M2_36h00 
## 1.352110e-01 2.194071e-04 6.683199e-03 1.011306e-03 2.900252e-13 6.795518e-01 
##     M2_48h00     M3_24h00     M3_48h00 
## 1.683259e-02 1.508062e-03 4.068326e-03

From these p-values, we can conclude that the genotype does not significantly impact the expression level of FGF9 at any timepoint.

We can plot the marginal means estimated by the mixed model for the genotypeas a function of Time.

emmip(mod, genotype ~ Time)
Figure 4: Mean dCt predicted by the linear model as a function of time

Figure 4: Mean dCt predicted by the linear model as a function of time

This plot is the same as the previous one but using the actual data. Note that this does not take into account the variability across experiments.

df <- aggregate(data[, 3], by = list(data$genotype, data$Time), mean)
ggplot(df, aes(x=Group.2, y = x, group = Group.1)) + geom_line(aes(color=Group.1)) + geom_point(aes(color=Group.1))
Figure 5: Mean dCt computed from the actual data (across experiments) as a function of time

Figure 5: Mean dCt computed from the actual data (across experiments) as a function of time

Interpretation

Overall, the genotype does affect FGF9 expression at every differentiation time point.

sessionInfo()
## R version 4.3.2 (2023-10-31)
## Platform: aarch64-apple-darwin20 (64-bit)
## Running under: macOS Sonoma 14.3.1
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
## LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## time zone: Europe/Paris
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] corrplot_0.92    car_3.1-2        carData_3.0-5    ggbeeswarm_0.7.2
##  [5] emmeans_1.9.0    lmerTest_3.1-3   lme4_1.1-35.1    Matrix_1.6-5    
##  [9] kableExtra_1.3.4 lubridate_1.9.3  forcats_1.0.0    stringr_1.5.1   
## [13] dplyr_1.1.4      purrr_1.0.2      readr_2.1.5      tidyr_1.3.0     
## [17] tibble_3.2.1     ggplot2_3.4.4    tidyverse_2.0.0 
## 
## loaded via a namespace (and not attached):
##  [1] tidyselect_1.2.0    viridisLite_0.4.2   farver_2.1.1       
##  [4] vipor_0.4.7         fastmap_1.1.1       TH.data_1.1-2      
##  [7] digest_0.6.34       estimability_1.4.1  timechange_0.3.0   
## [10] lifecycle_1.0.4     survival_3.5-7      magrittr_2.0.3     
## [13] compiler_4.3.2      rlang_1.1.3         sass_0.4.8         
## [16] tools_4.3.2         utf8_1.2.4          yaml_2.3.8         
## [19] knitr_1.45          labeling_0.4.3      xml2_1.3.6         
## [22] multcomp_1.4-25     abind_1.4-5         withr_3.0.0        
## [25] numDeriv_2016.8-1.1 grid_4.3.2          fansi_1.0.6        
## [28] xtable_1.8-4        colorspace_2.1-0    scales_1.3.0       
## [31] MASS_7.3-60.0.1     cli_3.6.2           mvtnorm_1.2-4      
## [34] rmarkdown_2.25      generics_0.1.3      rstudioapi_0.15.0  
## [37] httr_1.4.7          tzdb_0.4.0          minqa_1.2.6        
## [40] cachem_1.0.8        splines_4.3.2       parallel_4.3.2     
## [43] rvest_1.0.3         vctrs_0.6.5         boot_1.3-29        
## [46] webshot_0.5.5       sandwich_3.1-0      jsonlite_1.8.8     
## [49] hms_1.1.3           pbkrtest_0.5.2      beeswarm_0.4.0     
## [52] systemfonts_1.0.5   jquerylib_0.1.4     glue_1.7.0         
## [55] nloptr_2.0.3        codetools_0.2-19    stringi_1.8.3      
## [58] gtable_0.3.4        munsell_0.5.0       pillar_1.9.0       
## [61] htmltools_0.5.7     R6_2.5.1            evaluate_0.23      
## [64] lattice_0.22-5      highr_0.10          backports_1.4.1    
## [67] broom_1.0.5         bslib_0.6.1         Rcpp_1.0.12        
## [70] svglite_2.1.3       coda_0.19-4.1       nlme_3.1-164       
## [73] xfun_0.41           zoo_1.8-12          pkgconfig_2.0.3