Data

data <- read.table("data_pcr_wnt4.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")

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)

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)

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: 630
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -2.18630 -0.39623  0.00636  0.51528  3.11162 
## 
## Random effects:
##  Groups     Name        Variance Std.Dev.
##  Experiment (Intercept) 0.04406  0.2099  
##  Residual               1.43649  1.1985  
## Number of obs: 201, groups:  Experiment, 5
## 
## Fixed effects:
##                          Estimate Std. Error       df t value Pr(>|t|)    
## (Intercept)               11.8900     0.4047  10.6357  29.381 1.58e-11 ***
## genotypeMut                1.1529     0.4893 179.8338   2.356 0.019536 *  
## TimeM1_36h00               1.0835     0.4871  14.1355   2.224 0.042912 *  
## TimeM2_06h00              -0.8016     0.5549  22.0699  -1.445 0.162617    
## TimeM2_12h00               0.5927     0.5549  22.0699   1.068 0.296991    
## TimeM2_24h00              -3.0002     0.5692  23.1012  -5.271 2.36e-05 ***
## TimeM2_36h00              -2.1400     0.6479  36.5878  -3.303 0.002145 ** 
## TimeM2_48h00              -2.8073     0.5549  22.0699  -5.059 4.52e-05 ***
## TimeM3_24h00              -2.3711     0.6479  36.5878  -3.660 0.000791 ***
## TimeM3_48h00               0.1467     0.5547  22.1699   0.264 0.793846    
## genotypeMut:TimeM1_36h00  -1.0435     0.6060 179.8607  -1.722 0.086794 .  
## genotypeMut:TimeM2_06h00  -0.6196     0.7168 179.8898  -0.864 0.388485    
## genotypeMut:TimeM2_12h00  -0.8849     0.6998 179.8746  -1.265 0.207686    
## genotypeMut:TimeM2_24h00  -0.7588     0.7261 180.1573  -1.045 0.297369    
## genotypeMut:TimeM2_36h00  -1.9370     0.8475 179.8338  -2.286 0.023448 *  
## genotypeMut:TimeM2_48h00  -1.0599     0.7075 179.8338  -1.498 0.135884    
## genotypeMut:TimeM3_24h00  -0.9282     0.8475 179.8338  -1.095 0.274891    
## genotypeMut:TimeM3_48h00  -1.2082     0.7075 179.8338  -1.708 0.089421 .  
## ---
## 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)

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        2.3349  1     0.1265    
## Time          334.7602  8     <2e-16 ***
## genotype:Time   6.6091  8     0.5793    
## ---
## 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  -1.1529 0.489 180  -2.356  0.0195
## 
## Time = M1_36h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut  -0.1095 0.358 180  -0.306  0.7598
## 
## Time = M2_06h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut  -0.5333 0.524 180  -1.018  0.3100
## 
## Time = M2_12h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut  -0.2680 0.500 180  -0.535  0.5930
## 
## Time = M2_24h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut  -0.3941 0.537 181  -0.734  0.4639
## 
## Time = M2_36h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut   0.7841 0.692 180   1.133  0.2587
## 
## Time = M2_48h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut  -0.0930 0.511 180  -0.182  0.8557
## 
## Time = M3_24h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut  -0.2247 0.692 180  -0.325  0.7457
## 
## Time = M3_48h00:
##  contrast estimate    SE  df t.ratio p.value
##  WT - Mut   0.0553 0.511 180   0.108  0.9139
## 
## 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 
## 0.1758127 0.9139401 0.9139401 0.9139401 0.9139401 0.9139401 0.9139401 0.9139401 
##  M3_48h00 
## 0.9139401

From these p-values, we can conclude that the genotype does not significantly impact the expression level of WNT4 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      11.89 0.405  11.4 WT   iPS     
##  Mut      iPS      13.04 0.405  11.4 Mut  iPS     
##  WT       M1_36h00 12.97 0.271  33.6 WT   M1_36h00
##  Mut      M1_36h00 13.08 0.276  35.9 Mut  M1_36h00
##  WT       M2_06h00 11.09 0.383  78.8 WT   M2_06h00
##  Mut      M2_06h00 11.62 0.399  88.3 Mut  M2_06h00
##  WT       M2_12h00 12.48 0.383  78.8 WT   M2_12h00
##  Mut      M2_12h00 12.75 0.370  70.2 Mut  M2_12h00
##  WT       M2_24h00  8.89 0.408  77.9 WT   M2_24h00
##  Mut      M2_24h00  9.28 0.409  79.2 Mut  M2_24h00
##  WT       M2_36h00  9.75 0.512 125.2 WT   M2_36h00
##  Mut      M2_36h00  8.97 0.512 125.2 Mut  M2_36h00
##  WT       M2_48h00  9.08 0.383  78.8 WT   M2_48h00
##  Mut      M2_48h00  9.18 0.383  78.8 Mut  M2_48h00
##  WT       M3_24h00  9.52 0.512 125.2 WT   M3_24h00
##  Mut      M3_24h00  9.74 0.512 125.2 Mut  M3_24h00
##  WT       M3_48h00 12.04 0.383  80.0 WT   M3_48h00
##  Mut      M3_48h00 11.98 0.383  80.0 Mut  M3_48h00
## 
## Degrees-of-freedom method: kenward-roger

emmip(mod, genotype ~ 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))

Interpretation

Overall, the genotype does not affect WNT4 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