### Two way analysis of variance using R

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### Introduction

Two way analysis of variance can be used when you have one measurement variable and two nominal variables, and each value of one nominal variable is found in combination with each value of the other nominal variable. It tests three null hypotheses.

H0: All means of genotypes are equal

H1: At least two means of genotypes are unequal

The means of observations grouped by the other factor are the same.

H0: All means of gender are equal

H1: At least two means of gender are unequal

There is no interaction between the two factors. The interaction test tells you whether the effects of one factor depend on the other factor.

H0: There is no interaction between genotype and gender

H1: There is significant interaction

When the interaction term is significant, the usual advice is that you should not test the effects of the individual factors.

### Repeated measures

One experimental design that people analyze with a two-way ANOVA is repeated measures. In this design the observation has been made on the same individual more than once. This usually involves measurements taken at different time points or at different places. Repeated measures experiments are often done without replication, although they could be done with replication.

### Randomized blocks

Another experimental design that is analyzed by a two-way ANOVA is randomized blocks. This often occurs in agriculture, where you may want to test different treatments on small plots within larger blocks of land. Because the larger blocks may differ in some way that may affect the measurement variable, the data are analyzed with a two-way ANOVA, with the block as one of the nominal variables. Each treatment is applied to one or more plot within the larger block, and the positions of the treatments are assigned at random. Here, the example used shows the enzyme activity of mannose-6-phosphate isomerase and MPI genotypes in the amphipod crustacean. Amphipods were separated by gender to know whether the gender also affected enzyme activity.

### Import data

I often recommend to first clear all the objects or values in global environment using rm(list = ls(all = TRUE)) before importing the data set. You can also clear the plots using graphics.off() and clear everything in console using shell() function.

rm(list = ls(all = TRUE))
graphics.off()
shell("cls")

Now let’s import the data set using read.csv() function. I have already saved the data file as CSV (comma delimited file) in the working directory. The file argument specify the file name with extension CSV. In header argument you can set a logical value that will indicate whether the data file contains first variable names as first row. In my data set the file contains the variable names in the first row, so I shall use TRUE for this argument. The head() function will print the first six rows of the data set.

#   genotype gender activity
# 1       FF Female     3.39
# 2       FF Female     3.34
# 3       FF Female     3.59
# 4       FO Female     3.58
# 5       FO Female     4.12
# 6       FO Female     4.72

### Format variables

The str() or structure function will tell us the format for each column in the data frame. It gives information about rows and columns. It also gives information whether the variables are being read as integer, factor or number.

In some cases, we may have a variable coded as character or integer that we would like R to recognize as a factor variable. For this purpose as.factor() function is used. In our data file the genotype and gender are being read as character in R instead of being recognized as factor variables. To change it to factor assign the function of as.factor() where x argument represent the factor component of the data set (data$genotype). This will change the structure of the variable genotype and R will recognize it as factor. Similarly, assign the same function for second variable by replacing genotype with gender in previous command. Now using again the str() function will show the second variable is being read as factor. str(data) # 'data.frame': 18 obs. of 3 variables: #$ genotype: Factor w/ 3 levels "FF","FO","OO": 1 1 1 2 2 2 3 3 3 1 ...
#  $gender : Factor w/ 2 levels "Female","Male": 1 1 1 1 1 1 1 1 1 2 ... #$ activity: num  3.39 3.34 3.59 3.58 4.12 4.72 5.06 4.05 4.09 2.2 ...
data$genotype <- as.factor(x = data$genotype)
data$gender <- as.factor(x = data$gender)
str(data)
# 'data.frame': 18 obs. of  3 variables:
#  $genotype: Factor w/ 3 levels "FF","FO","OO": 1 1 1 2 2 2 3 3 3 1 ... #$ gender  : Factor w/ 2 levels "Female","Male": 1 1 1 1 1 1 1 1 1 2 ...
#  $activity: num 3.39 3.34 3.59 3.58 4.12 4.72 5.06 4.05 4.09 2.2 ... ### Fit two way analysis of variance model First attach the data set to the R search path using attach() function. This means that the database is searched by R when evaluating a variable, so objects in the database can be accessed by simply giving their names. This will help us to reduce the argument data$genotypes to genotypes only.

We can access the functions of two way analysis of variance using R stats package. Load this package using the library() function before fitting the two way analysis of variance model. The two way analysis of variance model can be fitted in two ways as per the objectives or to get the information a researcher is most interested in. If you want to see the main effects without interaction then fit analysis of variance model using aov() function.

This function requires formula as its argument to represent what output you wants to get from it. For main effects model use the response variable (activity in this case) separated by genotype and gender. In this model formula the response variable is separated from factor variables using tilde operator ~. The factor variables are combined using the plus sign. To print this model use anova() function for this object.

Now let’s fit the same model including the interaction effect. Use the same command as previously used to fit main effects by combining the variable factors with an additional argument genotype:gender representing the interaction term.

attach(data)
library(stats)

# Main effect
aov.res1 <- aov(activity ~
genotype +
gender)
# Interaction effect
aov.res2 <- aov(activity ~
genotype +
gender +
genotype:gender)
# OR
aov.res2 <- aov(activity ~
genotype*gender)
anova(aov.res1)
# Analysis of Variance Table
#
# Response: activity
#           Df Sum Sq Mean Sq F value   Pr(>F)
# genotype   2 6.2525 3.12627 10.4111 0.001698 **
# gender     1 1.3833 1.38334  4.6068 0.049855 *
# Residuals 14 4.2039 0.30028
# ---
# Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(aov.res2)
# Analysis of Variance Table
#
# Response: activity
#                 Df Sum Sq Mean Sq F value   Pr(>F)
# genotype         2 6.2525 3.12627 10.9923 0.001938 **
# gender           1 1.3833 1.38334  4.8640 0.047669 *
# genotype:gender  2 0.7911 0.39554  1.3908 0.286269
# Residuals       12 3.4129 0.28441
# ---
# Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Enzyme activity is significantly affected by the both factor variables individually. However, this effect was non-significant for interaction term.

### Compute Tukey Honest Significant differences

I shall use TukeyHSD() to compute the significant differences among the mean values. This test create a set of confidence intervals on the differences between the means of the levels of a factor with the specified family-wise probability of coverage. The intervals are based on the Studentized range statistic, Tukey’s ‘Honest Significant Difference’ method.

This test requires certain arguments. The x argument specify the fitted model object (aov.res1 in this case), usually an aov() fit. The argument ordered specify a logical value indicating if the levels of the factor should be ordered according to increasing average in the sample before taking differences. If ordered is set to TRUE then the calculated differences in the means will all be positive. The significant differences will be those for which the lower end point is positive.

The argument which specify a character vector listing terms in the fitted model for which the intervals should be calculated. Default value specify all the terms. You can also specify the variable factor name for this argument. For interaction use ”genotype:gender” as the value for which argument. For confidence level use conf.level argument to set its value. Default value for this argument is 0.95 representing the 5% level of significance.

# Mean comparison test for genotypes
TukeyHSD(x = aov.res1,
ordered = FALSE,
which = "genotype",
conf.level = 0.95)
#   Tukey multiple comparisons of means
#     95% family-wise confidence level
#
# Fit: aov(formula = activity ~ genotype + gender)
#
# $genotype # diff lwr upr p adj # FO-FF 0.7866667 -0.04137865 1.614712 0.0635754 # OO-FF 1.4416667 0.61362135 2.269712 0.0012185 # OO-FO 0.6550000 -0.17304532 1.483045 0.1322481 # Mean comparison test for gender TukeyHSD(x = aov.res1, ordered = FALSE, which = "gender", conf.level = 0.95) # Tukey multiple comparisons of means # 95% family-wise confidence level # # Fit: aov(formula = activity ~ genotype + gender) # #$gender
#                   diff       lwr           upr    p adj
# Male-Female -0.5544444 -1.108486 -0.0004029292 0.049855
# Mean comparison test for interaction (genotype:gender)
TukeyHSD(x = aov.res2,
ordered = FALSE,
which = "genotype:gender",
conf.level = 0.95)
#   Tukey multiple comparisons of means
#     95% family-wise confidence level
#
# Fit: aov(formula = activity ~ genotype * gender)
#
# \$genotype:gender
#                           diff        lwr        upr     p adj
# FO:Female-FF:Female  0.7000000 -0.7625920  2.1625920 0.6090747
# OO:Female-FF:Female  0.9600000 -0.5025920  2.4225920 0.3028696
# FF:Male-FF:Female   -0.9333333 -2.3959253  0.5292586 0.3288700
# FO:Male-FF:Female   -0.0600000 -1.5225920  1.4025920 0.9999909
# OO:Male-FF:Female    0.9900000 -0.4725920  2.4525920 0.2754196
# OO:Female-FO:Female  0.2600000 -1.2025920  1.7225920 0.9892495
# FF:Male-FO:Female   -1.6333333 -3.0959253 -0.1707414 0.0258232
# FO:Male-FO:Female   -0.7600000 -2.2225920  0.7025920 0.5305000
# OO:Male-FO:Female    0.2900000 -1.1725920  1.7525920 0.9825802
# FF:Male-OO:Female   -1.8933333 -3.3559253 -0.4307414 0.0094477
# FO:Male-OO:Female   -1.0200000 -2.4825920  0.4425920 0.2498774
# OO:Male-OO:Female    0.0300000 -1.4325920  1.4925920 0.9999997
# FO:Male-FF:Male      0.8733333 -0.5892586  2.3359253 0.3927173
# OO:Male-FF:Male      1.9233333  0.4607414  3.3859253 0.0084220
# OO:Male-FO:Male      1.0500000 -0.4125920  2.5125920 0.2262177

### Compute mean and standard error

Before visualizing main and interaction effects first compute mean values and standard error. Load the library dplyr that states the grammar of data manipulation. Connect data set to group it by first genotypes and then summarise it to get mean values and standard error using group_by() and summarise(), respectively. For connection we shall use %>% pipe operator. In function summarise() set functions for the objects avg_A and se to compute mean and standard error values. Print function will display the results for this object.

Similarly, get mean values and standard error for second variable factor using the same command except connecting the data set to group_by() gender using %>% pipe operator. Print function for this object will display the results. Again use the same command for interaction term. Here you will connect the data set to group_by() genotype and gender using the %>% pipe operator. Print function for this object will display the results. Attach these three objects to access its components by simply giving their names while visualizing plots.

library(dplyr)
# For main effects
MeanSE_A = data %>%
group_by(genotype) %>%
summarise(avg_A = mean(activity),
se = sd(activity)/sqrt(length(activity)))

MeanSE_B = data %>%
group_by(gender) %>%
summarise(avg_B = mean(activity),
se = sd(activity)/sqrt(length(activity)))
print(MeanSE_A)
# # A tibble: 3 x 3
#   genotype avg_A    se
#   <fct>    <dbl> <dbl>
# 1 FF        2.97 0.223
# 2 FO        3.76 0.240
# 3 OO        4.42 0.281
print(MeanSE_B)
# # A tibble: 2 x 3
#   gender avg_B    se
#   <fct>  <dbl> <dbl>
# 1 Female  3.99 0.198
# 2 Male    3.44 0.326
# For interaction term
MeanSE_AB = data %>%
group_by(genotype, gender) %>%
summarise(avg_AB = mean(activity),
se = sd(activity)/sqrt(length(activity)))
print(MeanSE_AB)
# # A tibble: 6 x 4
# # Groups:   genotype [3]
#   genotype gender avg_AB     se
#   <fct>    <fct>   <dbl>  <dbl>
# 1 FF       Female   3.44 0.0764
# 2 FF       Male     2.51 0.157
# 3 FO       Female   4.14 0.329
# 4 FO       Male     3.38 0.186
# 5 OO       Female   4.40 0.330
# 6 OO       Male     4.43 0.535
attach(MeanSE_A)
attach(MeanSE_B)
attach(MeanSE_AB)

### Visualize effects

#### Plot genotype

Let’s start to display the main and interaction effects using grammar of graphics ggplot() function. This will require to load the package ggplot2. Create an object p1 using the grammar of graphics function. This function requires data set (MeanSE_A) and some aesthetic mappings to use for the plot. In aesthetic mappings function the argument x specify the variable genotype and argument y specify mean values (avg_A). Print function will print a blank plot showing x-axis and y-axis labels.

Now let’s add layers to this plot object. For plotting bars use geom_bar() function. In stat argument set the value as identity to leave the data as is. In color argument specify the color for borders of the bars. The position argument specify the Position adjustment, either as a string, or the result of a call to a position adjustment function. The argument width specify the width of the bars. Printing this layer with plot object will display bars on the plot.

Let’s add the second layer to show error bars on the bars of the plot. For this use geom_errorbar() function. In aesthetic mappings you can specify the maximum ymax and minimum ymin values to show the heights of the error bars. Use the same position argument as used in previous command so that the error bars could place in the center of the bars. You can adjust width of the error bars by setting the value for width argument.

Let’s change the main title, X and Y axis labels of the plot. For this use labs() function. In title you can specify the main title of the plot. For X and Y argument specify the labels to be used for X and Y axis.

library(ggplot2)
# Create plot object
p1 = ggplot(MeanSE_A, aes(x = genotype,
y = avg_A))
# Adding layers to plot object
# Plotting bars
plotA = p1 +
geom_bar(stat = "identity",
color = "black",
position = position_dodge(width=0.9),
width = 0.8)