Analysis of prostacyclin experiment

Author

Lieven Clement

1 Background

Researchers study the effect of arachidonic acid on prostacyclin level in blood plasma. They use 3 different doses of arachidonic acid: low (10 units), medium (25 units) and high dose (50 units) and measure the prostacycline levels in blood plasma using an elisa fluorescence measurement. Treatments are randomly assigned to rats and every rat originates from a different litter and is grown in a separate cage.

Code

library("tidyverse")

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2 Experimental design

There is one explanatory variable: the factor dose
There are three treatments: low, medium and high dose of arachidonic acid
The effect of the treatments is assessed on the response variable prostacyclin level in blood plasma
The rats are the experimental and observational units
The experiment is a completely randomized design (CRD)
Researchers are interested to assess if there is an effect of the arachidonic acid dose on the prostacyclin level and when there is an effect, they want to asssess for which dose levels significantly different effects occur.

3 Data

The data can be found in a table of Voorbeeld 5.23 in the bachelor cours of statistics.

Code

prostacyclin <- data.frame(
    prostac=c(19.2, 10.8, 33.6, 11.9, 15.9, 33.3, 81.1, 36.7, 58.0, 60.8, 50.6, 69.4, 30.8, 27.6, 13.2, 38.8, 37.0, 38.3, 65.2, 66.4, 63.2, 49.9, 89.5, 60.5, 102.9, 57.1, 76.7, 70.5, 66.4, 76.3, 83.1, 61.7, 101.5, 86.2, 115.9, 102.1),
    dose=factor(c(rep(10,12),rep(25,12),rep(50,12))))

4 Analysis

One-way anova to assess if there is an effect of the treatment. Tukey post-hoc test to assess pairwise differences between the treatment.

4.1 One-way anova

4.1.1 Data exploration

Code

prostacyclin |>
  ggplot(aes(x=dose, y=prostac)) +
  geom_boxplot(outlier.shape = NA) +
  geom_point(position = "jitter")

Code

prostacyclin |>
  ggplot(aes(sample=prostac)) +
  geom_qq() +
  geom_qq_line() +
  facet_grid(cols = vars(dose))

The boxplots show that the data are homoscedastic
The qq-plot show no severe deviations from normality

4.1.2 Anova

H0: On average there is no effect of the arachidonic acid treatment on the prostacyclin level in blood serum of rats
H1: The average prostacyclin level in blood serum of rats is different for at least two arachidonic dose levels

Code

library(car)

Loading required package: carData


Attaching package: 'car'

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Code

library(multcomp)

Loading required package: mvtnorm

Loading required package: survival

Loading required package: TH.data

Loading required package: MASS


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Code

lm1 <- lm(prostac~dose,data=prostacyclin)
Anova(lm1,type=3)

There is an extremely significant effect of arachidonic acid on the average prostacyclin blood concentration in rats (p<0.001).

4.1.3 Post-hoc

Code

tukey <- glht(lm1,linfct=mcp(dose="Tukey"))
summary(tukey)


     Simultaneous Tests for General Linear Hypotheses

Multiple Comparisons of Means: Tukey Contrasts


Fit: lm(formula = prostac ~ dose, data = prostacyclin)

Linear Hypotheses:
             Estimate Std. Error t value Pr(>|t|)    
25 - 10 == 0    8.258      8.698   0.949 0.613379    
50 - 10 == 0   43.258      8.698   4.974  < 1e-04 ***
50 - 25 == 0   35.000      8.698   4.024 0.000858 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Adjusted p values reported -- single-step method)

Code

confint(tukey)


     Simultaneous Confidence Intervals

Multiple Comparisons of Means: Tukey Contrasts


Fit: lm(formula = prostac ~ dose, data = prostacyclin)

Quantile = 2.4532
95% family-wise confidence level
 

Linear Hypotheses:
             Estimate lwr      upr     
25 - 10 == 0   8.2583 -13.0789  29.5956
50 - 10 == 0  43.2583  21.9211  64.5956
50 - 25 == 0  35.0000  13.6628  56.3372

The average prostacyclin concentration is higher in the high dose group than in the low and moderate dose group (both p-values are smaller than p<0.001). The average concentration in the high dose group is 43.3 ng/ml (95% CI [21.9, 64.6]) and 35 ng/ml (95% CI [13.7, 56.3]) higher than in the low and moderate dose group, respectively (both p<0.001). The difference in average prostacyclin concentration between the moderate and low dose group is not significant (p=0.61, 95% CI on average difference [-13.1, 29.6] ng/ml).

5 Conclusion

There is an extremely significant effect of arachidonic acid on the average prostacyclin blood concentration in rats (p<0.001). The average concentration in the high dose group is 43.3 ng/ml (95% CI [21.9, 64.6]) and 35 ng/ml (95% CI [13.7, 56.3]) higher than in the low and moderate dose group, respectively (both p<0.001). The difference in average prostacyclin concentration between the moderate and low dose group is not significant (p=0.61, 95% CI on average difference [-13.1, 29.6] ng/ml).