Data

data <- read.table("data_pcr_sox9.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 21-85 WT 8.218367 iPS07 M1_36h00
2 21-86 WT 7.438973 iPS07 M1_36h00
3 21-92 Mut 7.678646 iPS07 M1_36h00
4 21-93 Mut 8.323477 iPS07 M1_36h00
5 21-94 Mut 7.918076 iPS07 M1_36h00
6 21-95 Mut 7.324906 iPS07 M1_36h00
7 21-96 Mut 7.471019 iPS07 M1_36h00
8 21-121 WT 7.155493 iPS09 M1_36h00
9 21-122 WT 6.790472 iPS09 M1_36h00
10 21-123 WT 7.016722 iPS09 M1_36h00
11 21-124 WT 6.890930 iPS09 M1_36h00
12 21-125 WT 7.129077 iPS09 M1_36h00
13 21-126 Mut 6.508825 iPS09 M1_36h00
14 21-127 Mut 6.393469 iPS09 M1_36h00
15 21-128 Mut 6.223399 iPS09 M1_36h00
16 21-129 Mut 6.215586 iPS09 M1_36h00
17 21-130 Mut 6.137569 iPS09 M1_36h00
18 22-59 WT 7.216948 iPS12 M1_36h00
19 22-60 WT 7.410435 iPS12 M1_36h00
20 22-61 WT 7.238029 iPS12 M1_36h00
21 22-62 WT 7.030680 iPS12 M1_36h00
22 22-63 WT 7.309076 iPS12 M1_36h00
24 22-65 Mut 6.146671 iPS12 M1_36h00
25 22-66 Mut 6.165782 iPS12 M1_36h00
26 22-67 Mut 6.178772 iPS12 M1_36h00
27 22-68 Mut 6.225652 iPS12 M1_36h00
29 22-70 Mut 6.277790 iPS12 M1_36h00
30 23-181 WT 7.805886 iPS26a M1_36h00
31 23-182 WT 8.208788 iPS26a M1_36h00
32 23-183 WT 8.270644 iPS26a M1_36h00
33 23-184 WT 7.906137 iPS26a M1_36h00
34 23-185 WT 8.309672 iPS26a M1_36h00
35 23-186 WT 8.331521 iPS26a M1_36h00
36 23-187 WT 8.659797 iPS26b M1_36h00
37 23-188 WT 9.364963 iPS26b M1_36h00
38 23-189 WT 8.528527 iPS26b M1_36h00
39 23-190 WT 8.782401 iPS26b M1_36h00
40 23-191 WT 9.599367 iPS26b M1_36h00
41 23-192 WT 8.303328 iPS26b M1_36h00
42 23-193 Mut 8.651905 iPS26a M1_36h00
43 23-194 Mut 8.376658 iPS26a M1_36h00
44 23-195 Mut 8.386757 iPS26a M1_36h00
45 23-196 Mut 8.517802 iPS26a M1_36h00
46 23-197 Mut 8.688960 iPS26a M1_36h00
47 23-198 Mut 8.492869 iPS26a M1_36h00
48 23-199 Mut 7.945904 iPS26b M1_36h00
49 23-200 Mut 8.245824 iPS26b M1_36h00
50 23-201 Mut 8.143124 iPS26b M1_36h00
51 23-202 Mut 8.167578 iPS26b M1_36h00
52 23-203 Mut 8.187634 iPS26b M1_36h00
53 23-204 Mut 8.617719 iPS26b M1_36h00
54 22-71 WT 8.647383 iPS12 M2_06h00
55 22-72 WT 9.412343 iPS12 M2_06h00
56 22-73 WT 9.096348 iPS12 M2_06h00
57 22-74 WT 9.970729 iPS12 M2_06h00
58 22-75 WT 9.938560 iPS12 M2_06h00
61 22-78 Mut 8.184610 iPS12 M2_06h00
62 22-79 Mut 9.098640 iPS12 M2_06h00
63 22-80 Mut 9.536485 iPS12 M2_06h00
64 22-81 Mut 9.921793 iPS12 M2_06h00
65 22-82 Mut 9.317197 iPS12 M2_06h00
66 23-205 WT 8.907813 iPS26a M2_06h00
67 23-206 WT 9.083361 iPS26a M2_06h00
68 23-207 WT 9.431648 iPS26a M2_06h00
69 23-208 WT 9.592413 iPS26b M2_06h00
70 23-209 WT 10.033072 iPS26b M2_06h00
71 23-210 WT 9.555004 iPS26b M2_06h00
72 23-211 Mut 9.213553 iPS26a M2_06h00
73 23-212 Mut 9.090539 iPS26a M2_06h00
74 23-213 Mut 9.233725 iPS26a M2_06h00
75 23-214 Mut 8.873874 iPS26b M2_06h00
76 23-215 Mut 9.282277 iPS26b M2_06h00
77 23-216 Mut 9.747164 iPS26b M2_06h00
79 22-84 WT 9.020415 iPS12 M2_12h00
82 22-87 WT 9.084134 iPS12 M2_12h00
83 22-88 WT 8.918715 iPS12 M2_12h00
84 22-89 Mut 8.507957 iPS12 M2_12h00
85 22-90 Mut 8.034786 iPS12 M2_12h00
86 22-91 Mut 10.790611 iPS12 M2_12h00
87 22-92 Mut 8.389786 iPS12 M2_12h00
88 22-93 Mut 9.721986 iPS12 M2_12h00
89 22-94 Mut 10.103291 iPS12 M2_12h00
90 23-217 WT 8.711112 iPS26a M2_12h00
91 23-218 WT 9.002685 iPS26a M2_12h00
92 23-219 WT 9.338509 iPS26a M2_12h00
93 23-220 WT 9.601033 iPS26b M2_12h00
94 23-221 WT 10.156795 iPS26b M2_12h00
95 23-222 WT 9.660587 iPS26b M2_12h00
96 23-223 Mut 8.859579 iPS26a M2_12h00
97 23-224 Mut 8.848585 iPS26a M2_12h00
98 23-225 Mut 8.798873 iPS26a M2_12h00
99 23-226 Mut 8.992127 iPS26b M2_12h00
100 23-227 Mut 9.084956 iPS26b M2_12h00
101 23-228 Mut 9.934845 iPS26b M2_12h00
102 21-131 WT 7.072322 iPS09 M2_24h00
103 21-132 WT 6.148624 iPS09 M2_24h00
104 21-133 WT 6.861230 iPS09 M2_24h00
105 21-134 WT 5.888372 iPS09 M2_24h00
106 21-135 WT 5.811749 iPS09 M2_24h00
107 21-136 Mut 5.876976 iPS09 M2_24h00
108 21-137 Mut 6.042642 iPS09 M2_24h00
109 21-138 Mut 5.872281 iPS09 M2_24h00
110 21-139 Mut 5.887489 iPS09 M2_24h00
111 21-140 Mut 6.076356 iPS09 M2_24h00
112 22-95 WT 6.351077 iPS12 M2_24h00
113 22-96 WT 7.320212 iPS12 M2_24h00
114 22-97 WT 6.419391 iPS12 M2_24h00
115 22-98 WT 6.707798 iPS12 M2_24h00
116 22-99 WT 6.634364 iPS12 M2_24h00
118 22-101 Mut 8.235849 iPS12 M2_24h00
119 22-102 Mut 7.831369 iPS12 M2_24h00
120 22-103 Mut 7.470043 iPS12 M2_24h00
121 22-104 Mut 8.119498 iPS12 M2_24h00
122 22-105 Mut 7.521480 iPS12 M2_24h00
124 23-229 WT 7.802969 iPS26a M2_36h00
125 23-230 WT 8.205312 iPS26a M2_36h00
126 23-231 WT 7.656192 iPS26a M2_36h00
127 23-232 WT 8.402710 iPS26b M2_36h00
128 23-233 WT 8.795423 iPS26b M2_36h00
129 23-234 WT 8.378504 iPS26b M2_36h00
130 23-235 Mut 6.582401 iPS26a M2_36h00
131 23-236 Mut 6.747534 iPS26a M2_36h00
132 23-237 Mut 6.531096 iPS26a M2_36h00
133 23-238 Mut 7.753417 iPS26b M2_36h00
134 23-239 Mut 7.723957 iPS26b M2_36h00
135 23-240 Mut 8.200265 iPS26b M2_36h00
137 22-108 WT 6.848177 iPS12 M2_48h00
138 22-109 WT 7.004958 iPS12 M2_48h00
139 22-110 WT 6.707018 iPS12 M2_48h00
140 22-111 WT 6.935081 iPS12 M2_48h00
142 22-113 Mut 6.154825 iPS12 M2_48h00
143 22-114 Mut 6.564262 iPS12 M2_48h00
144 22-115 Mut 7.006395 iPS12 M2_48h00
145 22-116 Mut 6.189247 iPS12 M2_48h00
146 22-117 Mut 6.806225 iPS12 M2_48h00
147 22-118 Mut 7.087976 iPS12 M2_48h00
148 23-241 WT 6.685244 iPS26a M2_48h00
149 23-242 WT 7.268945 iPS26a M2_48h00
150 23-243 WT 7.125162 iPS26a M2_48h00
151 23-244 WT 7.656195 iPS26b M2_48h00
152 23-245 WT 8.237021 iPS26b M2_48h00
153 23-246 WT 7.509246 iPS26b M2_48h00
154 23-247 Mut 7.002440 iPS26a M2_48h00
155 23-248 Mut 6.398525 iPS26a M2_48h00
156 23-249 Mut 6.832315 iPS26a M2_48h00
157 23-250 Mut 6.877812 iPS26b M2_48h00
158 23-251 Mut 7.117477 iPS26b M2_48h00
159 23-252 Mut 6.949720 iPS26b M2_48h00
160 23-253 WT 6.013914 iPS26a M3_24h00
161 23-254 WT 6.249470 iPS26a M3_24h00
162 23-255 WT 6.339054 iPS26a M3_24h00
163 23-256 WT 6.901734 iPS26b M3_24h00
164 23-257 WT 7.495945 iPS26b M3_24h00
165 23-258 WT 6.809646 iPS26b M3_24h00
166 23-259 Mut 5.969940 iPS26a M3_24h00
167 23-260 Mut 6.134913 iPS26a M3_24h00
168 23-261 Mut 5.966927 iPS26a M3_24h00
169 23-262 Mut 5.637016 iPS26b M3_24h00
170 23-263 Mut 5.838424 iPS26b M3_24h00
171 23-264 Mut 6.327909 iPS26b M3_24h00
172 21-146 WT 5.258954 iPS09 M3_48h00
173 21-147 WT 5.410088 iPS09 M3_48h00
174 21-148 WT 5.203708 iPS09 M3_48h00
175 21-149 WT 5.762150 iPS09 M3_48h00
176 21-150 WT 6.472869 iPS09 M3_48h00
177 21-151 Mut 5.540165 iPS09 M3_48h00
178 21-152 Mut 5.188900 iPS09 M3_48h00
179 21-153 Mut 5.219813 iPS09 M3_48h00
180 21-154 Mut 4.884440 iPS09 M3_48h00
182 23-265 WT 5.363178 iPS26a M3_48h00
183 23-266 WT 5.996024 iPS26a M3_48h00
184 23-267 WT 5.936177 iPS26a M3_48h00
185 23-268 WT 5.940215 iPS26b M3_48h00
186 23-269 WT 6.572287 iPS26b M3_48h00
187 23-270 WT 6.301134 iPS26b M3_48h00
188 23-271 Mut 6.074183 iPS26a M3_48h00
189 23-272 Mut 6.149558 iPS26a M3_48h00
190 23-273 Mut 6.485945 iPS26a M3_48h00
191 23-274 Mut 5.658109 iPS26b M3_48h00
192 23-275 Mut 5.861195 iPS26b M3_48h00
193 23-276 Mut 6.101198 iPS26b M3_48h00
194 21-107 WT 8.980294 iPS07 iPS
195 21-108 WT 12.253138 iPS07 iPS
196 21-109 WT 11.262143 iPS07 iPS
197 21-110 WT 5.931992 iPS07 iPS
198 21-111 WT 7.690653 iPS07 iPS
199 21-112 WT 6.797914 iPS07 iPS
200 21-113 WT 6.584005 iPS07 iPS
201 21-115 Mut 7.904360 iPS09 iPS
202 21-116 Mut 7.688884 iPS09 iPS
203 21-117 Mut 7.521963 iPS09 iPS
204 21-118 Mut 7.066838 iPS09 iPS
205 21-119 Mut 8.406699 iPS09 iPS
206 21-120 Mut 8.107497 iPS09 iPS
208 23-158 WT 7.879775 iPS19 iPS
209 23-159 WT 7.920730 iPS19 iPS
210 23-160 WT 7.795535 iPS19 iPS
211 23-161 WT 7.604310 iPS19 iPS
212 23-162 WT 7.651695 iPS19 iPS
213 23-169 WT 7.124371 iPS19 iPS
214 23-170 WT 8.392830 iPS19 iPS
215 23-171 WT 7.877830 iPS19 iPS
217 23-173 WT 7.917894 iPS19 iPS
218 23-174 WT 7.250314 iPS19 iPS
219 23-163 Mut 7.699100 iPS19 iPS
220 23-164 Mut 7.808285 iPS19 iPS
221 23-165 Mut 8.324655 iPS19 iPS
222 23-166 Mut 7.988800 iPS19 iPS
223 23-167 Mut 8.286595 iPS19 iPS
224 23-168 Mut 8.002540 iPS19 iPS
225 23-175 Mut 7.786711 iPS19 iPS
227 23-177 Mut 7.996272 iPS19 iPS
228 23-178 Mut 8.175131 iPS19 iPS
229 23-179 Mut 8.349876 iPS19 iPS
230 23-180 Mut 8.746550 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: 457.6
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.7315 -0.5306 -0.0415  0.3605  5.6969 
## 
## Random effects:
##  Groups     Name        Variance Std.Dev.
##  Experiment (Intercept) 0.2681   0.5178  
##  Residual               0.4495   0.6704  
## Number of obs: 215, groups:  Experiment, 6
## 
## Fixed effects:
##                           Estimate Std. Error        df t value Pr(>|t|)    
## (Intercept)                8.15495    0.29967  14.25120  27.213 1.09e-13 ***
## genotypeMut                0.33169    0.24767 196.60108   1.339 0.182038    
## TimeM1_36h00              -0.36183    0.28461 162.37753  -1.271 0.205427    
## TimeM2_06h00               1.05839    0.33577 169.58220   3.152 0.001917 ** 
## TimeM2_12h00               0.84969    0.34879 174.26223   2.436 0.015853 *  
## TimeM2_24h00              -1.23628    0.33557 179.83523  -3.684 0.000303 ***
## TimeM2_36h00              -0.38937    0.38564 183.42225  -1.010 0.313984    
## TimeM2_48h00              -1.19595    0.34160 171.79215  -3.501 0.000591 ***
## TimeM3_24h00              -1.96126    0.38564 183.42225  -5.086 9.00e-07 ***
## TimeM3_48h00              -2.22632    0.32728 176.86558  -6.803 1.53e-10 ***
## genotypeMut:TimeM1_36h00  -0.75834    0.31751 196.52502  -2.388 0.017869 *  
## genotypeMut:TimeM2_06h00  -0.52886    0.37824 194.61164  -1.398 0.163642    
## genotypeMut:TimeM2_12h00  -0.35212    0.38611 194.53838  -0.912 0.362920    
## genotypeMut:TimeM2_24h00   0.04019    0.38889 194.49677   0.103 0.917793    
## genotypeMut:TimeM2_36h00  -1.28210    0.45953 193.88784  -2.790 0.005797 ** 
## genotypeMut:TimeM2_48h00  -0.72982    0.37926 194.59818  -1.924 0.055773 .  
## genotypeMut:TimeM3_24h00  -0.98747    0.45953 193.88784  -2.149 0.032886 *  
## genotypeMut:TimeM3_48h00  -0.51707    0.38445 194.59300  -1.345 0.180205    
## ---
## 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        4.4017  1    0.03590 *  
## Time          461.9116  8    < 2e-16 ***
## genotype:Time  15.4779  8    0.05049 .  
## ---
## 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  -0.3317 0.249 197  -1.330  0.1851
## 
## Time = M1_36h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut   0.4267 0.190 193   2.250  0.0256
## 
## Time = M2_06h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut   0.1972 0.286 192   0.690  0.4912
## 
## Time = M2_12h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut   0.0204 0.296 192   0.069  0.9451
## 
## Time = M2_24h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut  -0.3719 0.300 192  -1.240  0.2164
## 
## Time = M2_36h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut   0.9504 0.387 192   2.455  0.0150
## 
## Time = M2_48h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut   0.3981 0.287 192   1.385  0.1675
## 
## Time = M3_24h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut   0.6558 0.387 192   1.694  0.0919
## 
## Time = M3_48h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut   0.1854 0.293 192   0.633  0.5278
## 
## 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  M2_48h00  M3_24h00 
## 0.3245508 0.1149889 0.5937511 0.9451419 0.3245508 0.1149889 0.3245508 0.2755613 
##  M3_48h00 
## 0.5937511

From these p-values, we can conclude that the genotype does not significantly impact the expression level of SOX9 at any timepoint.

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

emmip_output<-emmip(mod, genotype ~ Time)
emmip_dataframe<-emmip_output$data
emmip_dataframe
##  genotype Time     yvar    SE    df tvar xvar    
##  WT       iPS      8.15 0.305 14.11 WT   iPS     
##  Mut      iPS      8.49 0.297 13.72 Mut  iPS     
##  WT       M1_36h00 7.79 0.258  8.88 WT   M1_36h00
##  Mut      M1_36h00 7.37 0.251  8.14 Mut  M1_36h00
##  WT       M2_06h00 9.21 0.306 16.56 WT   M2_06h00
##  Mut      M2_06h00 9.02 0.306 16.56 Mut  M2_06h00
##  WT       M2_12h00 9.00 0.320 19.74 WT   M2_12h00
##  Mut      M2_12h00 8.98 0.300 15.46 Mut  M2_12h00
##  WT       M2_24h00 6.92 0.312 18.30 WT   M2_24h00
##  Mut      M2_24h00 7.29 0.312 18.30 Mut  M2_24h00
##  WT       M2_36h00 7.77 0.360 30.48 WT   M2_36h00
##  Mut      M2_36h00 6.82 0.360 30.48 Mut  M2_36h00
##  WT       M2_48h00 6.96 0.312 17.94 WT   M2_48h00
##  Mut      M2_48h00 6.56 0.300 15.46 Mut  M2_48h00
##  WT       M3_24h00 6.19 0.360 30.48 WT   M3_24h00
##  Mut      M3_24h00 5.54 0.360 30.48 Mut  M3_24h00
##  WT       M3_48h00 5.93 0.302 16.25 WT   M3_48h00
##  Mut      M3_48h00 5.74 0.308 17.62 Mut  M3_48h00
## 
## Degrees-of-freedom method: kenward-roger
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 not affect SOX9 expression at any timepoint.

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