• Analysis steps based on set up from Bio415 Statistical Modeling for Biology
• ELISA Validation dataset like would be used for validation.
• Null Hypothesis There will be no statistical difference in the optical density between the different lots or runs
• Alternative Hypothesis There will be a statistical difference
• Significance Level alpha = 0.05
• From NIST Education & Training: Data Sets https://www.itl.nist.gov/div898/education/datasets.htm#anova

# Read in the data
elisa_data <- read_csv(
  "ELISA_Validation_Test_Data.csv", show_col_types = FALSE)

# Lot & Run read in as numerical values, 
# Convert the two columns into factor data
elisa_data$Lot <- as.factor(elisa_data$Lot)
elisa_data$Run <- as.factor(elisa_data$Run)

Evaluate the 5 different lots optical density results to see if there is a statistical difference between them.

tapply() used to graph a stripchart

• p-value = 0.2931 is greater than significance level 0.05
• Therefore: Equal Variance seen across the lots

# Fit a linear model.
lot_model <- lm(Density ~ Lot, data = elisa_data)
# Residuals from the fitted linear model.  
elisa_data$res <- residuals(lot_model)
# Levene's Test for equal variance
leveneTest(res ~ Lot, data = elisa_data)

Levene's Test for Homogeneity of Variance (center = median)
      Df F value Pr(>F)
group  4  1.2622 0.2931
      70               

# Normal Probability plot
qqnorm(elisa_data$res, pch = 1, col = "steelblue", frame = FALSE)
qqline(elisa_data$res, col = "darkred", lwd = 2)

• p-value = 0.0001839 is less than significance level 0.05
• Therefore: Normal Distribution is not seen in the data

  # Shapiro-Wilk test for normal distribution
shapiro.test(elisa_data$res)

    Shapiro-Wilk normality test

data:  elisa_data$res
W = 0.92148, p-value = 0.0001839

Data does show a slight skew. While ANOVA can except data that is slightly off of normal distribution, a transformation to more normal would be better to prevent false positives.

The data is not normally distributed but does show equal variance. Based on the skew, a sqrt transformation attempted to correct distribution

# Transform Data with sqrt transformation
elisa_data$Density_sqrt <- sqrt(elisa_data$Density)
# Fit a linear model on transformed data
lot_model_sqrt<- lm(Density_sqrt ~ Lot, data = elisa_data)
# Residuals from the fitted linear model of transformed data
elisa_data$res_sqrt <- residuals(lot_model_sqrt)

• p-value = 0.2575 is greater than significance level 0.05
• Therefore: Equal Variance still seen across the lots

# Levene's Test for equal variance for transformed data
leveneTest(res_sqrt ~ Lot, data = elisa_data)

Levene's Test for Homogeneity of Variance (center = median)
      Df F value Pr(>F)
group  4  1.3577 0.2575
      70               

• p-value = 0.0003208 is less than significance level 0.05
• Therefore: Normal Distribution is not seen in the transformed data.

# Shapiro-Wilk test for normal distribution of Transformed data
shapiro.test(elisa_data$res_sqrt)

    Shapiro-Wilk normality test

data:  elisa_data$res_sqrt
W = 0.92665, p-value = 0.0003208

_ Sqrt
- Log10
- Log2
- Turkey
- Box Cox Lambda 2
- Box Cox Lambda 6

• p-value = 0.001793 is less than significance level 0.05.
• Therefor it can be concluded that there is significant difference between the lots

# Kruskal-Wallis test 
kruskal.test(Density ~ Lot, data = elisa_data)

    Kruskal-Wallis rank sum test

data:  Density by Lot
Kruskal-Wallis chi-squared = 17.168, df = 4, p-value = 0.001793

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = x)

$Lot
           diff         lwr          upr     p adj
2-1  0.10620000 -0.06300149  0.275401486 0.4063682
3-1  0.06073333 -0.10846815  0.229934820 0.8521828
4-1  0.03560000 -0.13360149  0.204801486 0.9762779
5-1 -0.11686667 -0.28606815  0.052334820 0.3093379
3-2 -0.04546667 -0.21466815  0.123734820 0.9430949
4-2 -0.07060000 -0.23980149  0.098601486 0.7691944
5-2 -0.22306667 -0.39226815 -0.053865180 0.0038869
4-3 -0.02513333 -0.19433482  0.144068153 0.9935914
5-3 -0.17760000 -0.34680149 -0.008398514 0.0349390
5-4 -0.15246667 -0.32166815  0.016734820 0.0970768

Is there a difference in the runs that is possibly at fault of the lot issue? Run the same analysis set up - but using “Run” in place of “Lot”

tapply() used to graph a stripchart

• p-value = 0.332 is greater than significance level 0.05
• Therefore: Equal Variance still seen across the runs

# Fit a linear model.
run_model <- lm(Density ~ Run, data = elisa_data)
# Residuals from the fitted linear model.  
elisa_data$res2 <- residuals(run_model)
# Levene's Test for equal variance
leveneTest(res2 ~ Run, data = elisa_data)

Levene's Test for Homogeneity of Variance (center = median)
      Df F value Pr(>F)
group  4   1.169  0.332
      70               

# Normal Probability plot for transformed data
qqnorm(elisa_data$res2, pch = 1, col = "steelblue", frame = FALSE)
qqline(elisa_data$res2, col = "darkred", lwd = 2)

• p-value = 0.005266 is less than significance level 0.05
• Therefore: Normal Distribution is not seen in the data.

  # Shapiro-Wilk test for normal distribution
shapiro.test(elisa_data$res2)

    Shapiro-Wilk normality test

data:  elisa_data$res2
W = 0.95043, p-value = 0.005266

• p-value = 0.1015 is greater than significance level 0.05.
• Therefor it can be concluded that there is not a significant difference between the runs.

# Kruskal-Wallis test 
kruskal.test(Density ~ Run, data = elisa_data)

    Kruskal-Wallis rank sum test

data:  Density by Run
Kruskal-Wallis chi-squared = 7.7415, df = 4, p-value = 0.1015

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = x)

$Run
           diff         lwr          upr     p adj
2-1 -0.18000000 -0.35443685 -0.005563149 0.0397894
3-1 -0.08320000 -0.25763685  0.091236851 0.6701621
4-1 -0.02393333 -0.19837018  0.150503517 0.9952772
5-1 -0.03453333 -0.20897018  0.139903517 0.9810486
3-2  0.09680000 -0.07763685  0.271236851 0.5316030
4-2  0.15606667 -0.01837018  0.330503517 0.1010917
5-2  0.14546667 -0.02897018  0.319903517 0.1461686
4-3  0.05926667 -0.11517018  0.233703517 0.8755777
5-3  0.04866667 -0.12577018  0.223103517 0.9352236
5-4 -0.01060000 -0.18503685  0.163836851 0.9998074

The Box plot of the runs shows the outlier in run one and the elongated range of data in run two in comparison to the other runs.

Statistics <- grobTree(textGrob(
  "Kruskal-Wallis: chi-squared = 17.168, df = 4, p-value = 0.001793",
  x=0.01,  y=0.985, hjust=0,
  gp=gpar(col="black", fontsize=10, fontface="italic")))

#Violin plots of Optical Density by Lot
Lots_Plot <- ggplot(data = elisa_data, 
        aes(x = Lot, y = Density, fill = Lot)) + 
  geom_violin(trim=FALSE) + theme_classic() +
  geom_boxplot(width=0.1, color = "black", fill = "darkred")+
  scale_fill_manual(values=c("steelblue", "darkblue", 
                    "#56B4E9", "lightblue", "mediumblue"))+
  theme(legend.position="none")+
  labs(title = "Violin Plot of Lot Optical Densities") +
  annotation_custom(Statistics)

Statistics2 <- grobTree( textGrob(
 "Kruskal-Wallis: chi-squared = 7.7415, df = 4, p-value = 0.1015", 
           x=0.01, y=0.985, hjust=0,
  gp=gpar(col="black", fontsize=10, fontface="italic")))

#Violin plots of Optical Density by Run
Runs_Plot <- ggplot(data = elisa_data, 
                    aes(x = Run, y = Density, fill = Run)) + 
  geom_violin(trim=FALSE) + theme_classic() +
  geom_boxplot(width=0.1, color = "black", fill = "darkred")+
  scale_fill_manual(values=c("chartreuse4", "darkgreen", 
         "darkolivegreen", "darkseagreen4", "forestgreen"))+
  theme(legend.position="none")+
  labs(title = "Violin Plot of Run Optical Densities") +
  annotation_custom(Statistics2)