Assumption 1: Groups of data represent random samples from their respective populations.

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1 Tutorial 6: Comparing Two Groups Assumptions The following methods for comparing two groups are based on several assumptions. The type of test you use will vary based on whether these assumptions are met or not. Assumption 1: Groups of data represent random samples from their respective populations. Assumption 2: Numerical data are normally distributed (follow a Gaussian curve) in each population. Assumption 3: The standard deviations of the sample groups are equal. You may also see this referred to as the assumption of homogeneity of variance. If Assumption 1 is violated, then you should probably not even be performing statistical analyses on your data, so let s move on to violations of the other two assumptions If both Assumption 2 and 3 are met, proceed with a Two-sample T-test. If Assumption 2 is met, but Assumption 3 is violated, proceed with Welch s approximate T-test. If Assumption 2 is violated, but Assumption 3 is met, proceed with Mann-Whitney U-test. If both Assumption 2 and 3 are violated, your situation is far less than ideal. Your best option is to try transforming your data to hopefully meet the assumption of normality, which would allow you to perform Welch s approximate T-test. If that doesn t work, you may be forced to perform a Mann-Whitney U-test; however, the results should be interpreted with caution because the power of this analysis is greatly reduced when variances are unequal. Normality Testing Before doing any statistical tests, you should first test the normality of your data to see whether it follows a Gaussian distribution (normal bell curve). If data are normal, you should proceed with parametric analyses. If data are non-normal, you should either attempt transformations to make data normal or proceed with non-parametric alternatives. The Shapiro-Wilk normality test, shapiro.test(), is the most commonly used normality test. The output will give a P-value, which if less than 0.05 indicates that data are non-normal. shapiro.test(dataset$measurement) D'Agostino-Pearson Omnibus K2 normality test, dagotest(), is used for data a sample size of at least 20. It requires the fbasics package. library(fbasics) dagotest(measurement) 1

2 Variance Testing The variance of two normally distributed samples can be compared using an F test. The group variable in the second example must only have two groups (levels) for this method to work. var.test(sample1, sample2) var.test(dataset$measurement~dataset$group) Bartlett test for homogeneity of variances is for use on normally data, but can handle unequal sample sizes. bartlett.test(dataset$measurement~dataset$group) Levene test used on non-normal data (Modified Levene s Equal Variance). Requires the car package. library(car) levenetest(dataset$measurement, dataset$group) Data Transformations If data are non-normally distributed, it is common to transform data in an attempt to make them normal. Every attempt at transformation should be followed by a normality test, and you should also graph your data to see what it looks like. Logarithmic transformation is used for right-skewed data with no negative values. If there are any zero values in your dataset, add one to all values. Natural Log You can assign your transformed data to a name ( logdata in this case). If the dataset contains zeros, add 1 to all values (accomplished by adding +1 ) log(dataset$measurement) logdata=log(dataset$measurement) log(dataset$measurement+1) Base 10 Log log10(dataset$measurement) 2

3 Square-root transformation is used on count data following the Poisson distribution. sqrt(dataset$measurement) Arc-sine transformation is used on proportion (percentage) data. asin(dataset$measurement/100) asin(sqrt(dataset$measurement/100)) Note: There are many more options for transformation not listed here. These are just some of the more common ones. Adding a Column to an Existing Data frame After performing a transformation or other calculation, it is useful to append the new data to your original dataset (dataframe format). Such calculations produce a vector of the same length as the number of rows in the data frame can be added to the data frame for the duration of your R session. This does not affect the original data file, but if you wish to save those changes, you can export the newer data frame for later use (covered at the end of this tutorial). This example takes an existing column of measurements in the dataset, multiplies each value by 5 and places them in a new column newcolumn at the end of the data frame. dataset$newcolumn=dataset$measurement*5 This takes an existing column of measurements in the dataset, natural log transforms each data point and places them in a new column logdata at the end of the data frame. dataset$logdata=log(dataset$measurement) Two-sample T-test Used to test whether the means of two groups are equal. For manipulative studies, a t-test compares one control group and one treatment group. For observational studies, a t-test compares two groups that differ in some categorical way (location, habitat, community type, etc.). The default is a Welch two-sample t-test that does not assume equal variance. There is an argument, var.equal=, that is by default set to FALSE. t.test(sample1, sample2, var.equal=false) 3

4 Setting the var.equal= argument to TRUE treats variances as equal between groups and performs a normal two-sample t-test. To be proper, data should be checked for homogeneity of variance, var.test(), prior to performing any statistical test. t.test(sample1, sample2, var.equal=true) The above examples compare vectors of data. You can also compare columns in a dataset as long as they represent your sample groups. t.test(dataset$sample1, dataset$sample2, var.equal=t) If all of the sample data is together in one column (dataset$measurement) with a second column indicating the two groups (dataset$grouping), the tilde symbol (~) can be used to tell R to separate the measurements into the groups. The grouping variable must only have two levels (groups) for this to work. t.test(dataset$measurement~dataset$grouping, var.equal=t) Paired Two-sample T-test Used to test whether the mean difference between two groups equals a certain value. Common experimental designs: 1) Before and after same individuals/samples/plots measured before and after some treatment (before restoration vs. after restoration) or at different time points. 2) Adjacent plot design pairs of adjacent plots are used to control for localized environmental variability; one plot in each pair is manipulated Adding the paired=true argument performs a paired t-test. t.test(sample1, sample2, var.equal=true, paired=true) t.test(dataset$sample1, dataset$sample2, var.equal=true, paired=true) t.test(dataset$measurement~dataset$grouping, var.equal= TRUE, paired=true) 4

5 Two-Sample Wilcoxon Rank Sum (Mann-Whitney) Test This is a non-parametric equivalent for a two-sample t-test. This test is used when data are nonnormal (distribution of data do not follow normal curve). Note: This may be called a Mann-Whitney U-test in other statistical software packages. wilcox.test(sample1, sample2) wilcox.test(dataset$sample1, dataset$sample2) wilcox.test(dataset$measurement~dataset$grouping) Wilcoxon Signed Rank Test This is a non-parametric alternative to a paired t-test. wilcox.test(sample1, sample2, paired=true) wilcox.test(dataset$sample1, dataset$sample2, paired=t) wilcox.test(dataset$measurement~dataset$grouping, paired=true) Exporting Data from R Data frames, matrices, and other outputs generated in R can be exported to a.csv file to save and use later. If not using a working directory, a full pathname must be given for where the file is to be saved. In this example, dataset is the name of the dataset (data frame, matrix, etc.) you want to export. The file= argument tells R the location (pathname) for the location and name of the file you want to create. The pathname must be in quotes. write.csv(dataset, file="/users/johndoe/desktop/ dataset.csv") write.csv(dataset, file="c:/documents and Settings/ Owner/Desktop/dataset.csv") If you have set a working directory, you do not need provide the full pathname, only the name of the file you want to create is required. The resulting file will be saved in the folder you set as your working directory. setwd("/users/johndoe/desktop/") setwd("c:/documents and Settings/Owner/Desktop/") write.csv(dataset, dataset.csv") 5

6 Tutorial Code setwd("/users/johndoe/desktop/") example=read.csv("r_example_dataframe.csv") young=example[grep("young", example$age), "Richness"] young old=example[grep("old", example$age), "Richness"] old shapiro.test(young) shapiro.test(old) library(fbasics) dagotest(example$richness) var.test(young, old) var.test(example$richness~example$age) bartlett.test(example$richness~example$age) library(car) levenetest(example$richness, example$age) levenetest(example$richness~example$age) #natural log transformation log.young=log(young) log.young #base 10 log transformation log10.young=log10(young) log10.young #square root transformation sqrt.young=sqrt(young) sqrt.young 6

7 #arcsine transformation asin.young=asin(young/10) #divide Young richness values by max. value (10) to get proportion data asin.young #Two-sample (Student's) T-test t.test(example$richness~example$age, var.equal=t) t.test(example$richness~example$age, var.equal=f) #Paired T-test t.test(example$richness~example$age, paired=t) #Wilcoxon Rank-Sum Test (non-parametric) wilcox.test(example$richness~example$age) #Paired Wilcoxon (non-parametric) wilcox.test(example$richness~example$age, paired=t) #save.csv file without working directory set write.csv(example, file="/users/johndoe/desktop/r_example_dataframe_revised.csv") #With working directory set write.csv(example, file="r_example_dataframe_revised.csv") 7