12.2 – One way ANOVA

Introduction
Three important terms you’ll need for one-way ANOVA
Comments on experimental design
ANOVA sources of variation
Mean squares
Worked example with R
Questions
Quiz
Data set
Chapter 12 contents

Introduction

In ANalysis Of VAriance, or ANOVA for short, we likewise have tools to test the null hypothesis of no difference between between categorical independent variables — often called factors when there’s just a few levels to keep track of — and a single, dependent response variable. But now, the response variable is quantitative, not qualitative like the $\chi^2$ tests.

Analysis of variance, ANOVA, is such an important statistical test in biology that we will take the time to “build it” from scratch. We begin by reminding you of where you’ve been with the material.

We already saw an example of this null hypothesis. When there’s only one factor (but with two or more levels), we call the analysis of means and “one-way ANOVA.” In the independent sample t-test, we tested whether two groups had the same mean. We made the connection between the confidence interval of the difference between the sample means and whether or not it includes zero (ie, no difference between the means). In ANOVA, we extend this idea to a test of whether two or more groups have the same mean. In fact, if you perform an ANOVA on only two groups, you will get the exact same answer as the independent two-sample t-test, although they use different distributions of critical values (t for the t-test, F for the ANOVA — a nice little fact for you square the t-test statistic, you’ll get the F-test statistic: $t^{2} = F$ ).

Let’s say we have an experiment where we’ve looked at the effect of different three different calorie levels on weight change in middle-aged men.

I’ve created a simulated dataset which we will use in our ANOVA discussions. The data set is available at the end of this page (scroll down or click here).

We might graph the mean weight changes ( $\pm$ SEM); Below are two possible outcomes of our experiment (Fig 1).

Figure 1. Hypothetical results, box plots.

Figure 1. Hypothetical results of an experiment, box plots. Left, no difference among groups; Right, large differences among groups.

As statisticians and scientists, we would first calculate an overall or grand mean for the entire sample of observations; we know this as the sample mean whose symbol is $\bar{X}$ . But this overall mean is made up of the average of the sample means. If the null hypothesis is true, then the all of the sample means all estimate the overall mean. Put another way, being a member of a treatment group doesn’t matter, ie, there is no systematic effect, all differences among subjects are due to random chance.

$\begin{align*} H_{0} : \bar{X} = \bar{X}_{A} = \bar{X}_{B} = \bar{X}_{C} \end{align*}$

The hypotheses among are three groups or treatment levels then are:

$\begin{align*} H_{0} : \bar{X}_{A} = \bar{X}_{B} = \bar{X}_{C}&& \end{align*}$

Null that there are no differences among the group means. And the alternative hypotheses include any (or all) of the following possibilities

$\begin{align*} H_{0} : \bar{X}_{A} \ne \bar{X}_{B} = \bar{X}_{C} \end{align*}$

or maybe

$\begin{align*} H_{0} : \bar{X}_{A} = \bar{X}_{B} \ne \bar{X}_{C} \end{align*}$

or… have we covered all possible alternate outcomes among three groups?

In either case, we could use one-way ANOVA to test for “statistically significant differences.”

Three important terms you’ll need for one-way ANOVA

FACTOR: We have one factor of interest. For example, a factor of interest might be

Diet fed to hypertensive subjects (men and women)
Distribution of coral reef sea cucumber species in archipelagos
Antibiotic drug therapy for adolescents with Acne vulgaris (click see Webster 2002 for review).

💡 Mike’s Tips

Character vs factor. In R programming, while it is true that a treatment variable can be a character vector like Gender*, such vectors are handled by R as string. In contrast, by claiming Gender as a factor, R handles the variable as a nominal data type. Hence, ANOVA and other GLM functions expect any treatment variable to be declared as a factor:

Gender <- c("Gender non-conforming", "Men", "Non-binary", "Transgender", "Women", "Declined"); Gender
[1] "Gender non-conforming" "Men" "Non-binary" "Transgender" 
[5] "Women" "Declined"

# it’s a string vector, not a factor

is.factor(Gender)
[1] FALSE

Change string vector to a factor.

Gender <- as.factor(Gender); is.factor(Gender) [1] TRUE

* For discussions of ethics of language inclusivity and whether or not such information should be gathered, see Henrickson et al 2020.

LEVELS: We can have multiple levels (2 or more) within the single factor. Some examples of levels for the Factors listed:

Three diets (DASH), diet rich in fruits & vegetables, control diet)
Five archipelagos (Hawaiian Islands, Line Islands, Marshal Islands, Bonin Islands, and Ryukyu Islands)
Five antibiotics (ciprofloxacin, cotrimoxazole, erythromycin, doxycycline, minocycline).

RESPONSE: There is one outcome or measurement variable. This variable must be quantitative (ie, on the ratio or interval scale). Continuing our examples then

Reduction in systolic pressure
Numbers of individual sea cucumbers in a plot
number of microcomedo.

Note 1

A comedo is a clogged pore in the skin; a microcomedo refers to the small plug. Yes, I had to look that up, too.

The response variable can be just about anything we can measure, but because ANOVA is a parametric test, the response variable must be normally distributed!

Comments on experimental design

As we discuss ANOVA, keep in mind we are talking about analyzing results from an experiment. Understanding statistics, and in particular ANOVA, informs how to plan an experiment. The basic experimental design is called a completely randomized experimental design, or CDR, where treatments are assigned to experimental units at random.

In this experimental design, subjects (experimental units) must be randomly assigned to each of these levels of the factor. That is, each individual should have had the same probability of being found in any one of the levels of the factor. The design is complete because randomization is conducted for all levels, all factors.

Thinking about how you would describe an experiment with three levels of some treatment, we would have

Table 1. Summary statistics of three levels for some ratio scale response variable.

Level 1

$\bar_{X}_{1}$

sample mean₁

$s_{1}$

sample standard deviation₁

Level 2

$\bar_{X}_{2}$

sample mean₂

$s_{2}$

sample standard deviation₂

Level 3

$\bar_{X}_{3}$

sample mean₃

$s_{3}$

sample standard deviation₃

Note 2

Table 1 is the basis for creating the box plots in Figure 1.

ANOVA sources of variation

ANOVA works by partitioning total variability about the means (the grand mean, the group means). We will discuss the multiple samples and how the ANOVA works in terms of the sources of variation. There are two “sources” of variation that can occur:

Within Group Variation
Among Groups Variation

So let’s look first at the variability within groups, also called the Within Group Variation.

Consider and experiment to see if DASH diet reduces systolic blood pressure in USA middle-aged men and women with hypertension (Moore et al 2001). After eight weeks we have

We get the corrected sum of squares, SS, for within group

$\begin{align*} within-groups \ SS = \sum_{i=1}^{k}\left [ \sum_{j=1}^{n_{i}}\left ( X_{ij}-\bar{X}_{i} \right )^2 \right ] \end{align*}$

and the degrees of freedom, DF, for within groups

$\begin{align*} within-groups \ DF = \sum_{i=1}^{k}\left ( n_{i}-1 \right ) = N - k \end{align*}$

where i is the identity of the groups, $X_{ij}$ is the individual observations within group i, $\bar{X}_{i}$ is the group i mean, n_i is the sample size within each group, N is the total sample size of all subjects for all groups, and k is the number of groups.

Importantly, this value is also referred to as “error sums of squares” in ANOVA. It’s importance is as follows — In our example, the within group variability would be zero if and only if all subjects within the same diet lost exactly the same amount of weight. This is hardly ever the case of course in a real experiment. Because there are almost always some other unknown factors or measurement error that affect the response variable, there will be some unknown variation among individuals who received the same treatment (within the same group). Thus, the error variance will generally be larger than zero.

The first point to consider: your ANOVA will never result in statistical differences among the groups if the error variance is greater than the second type of variability, the variability between groups.

The second type of variability in ANOVA is that due to the groups or treatments. Individuals given a calorie-restricted diet will lose some weight; individuals allowed to eat a calorie-rich diet likely will gain weight, therefore there will be variability (a difference) due to the treatment. So we can calculate the variability among groups. We get the corrected sum of squares for among groups

$\begin{align*} among-groups \ SS = \sum_{i=1}^{k}n_{i}\left ( X_{i}-\bar{X}\right )^2 \end{align*}$

and the degrees of freedom for among groups

$\begin{align*} among-groups \ DF = k - 1 \end{align*}$

where i is the identity of the groups, $\bar{X}$ is the grand mean as defined in Measures of Central Tendency (Chapter 3.1), $\bar{X}_{i}$ is the group i mean, n_i is the sample size within each group, N is the total sample size of all subjects for all groups, and k is the number of groups.

The sums of squares here is simply subtracting the mean of each population from the overall mean.

If the Factor is not important in explaining the variation among individuals then all the population means will be similar and the sums of squares among populations would be small.
If the Factor is important in explaining some of the variation among the individuals then all the population means will NOT be the same and the sums of squares among populations would be large.

Finally, we can identify the total variation in the entire experiment. We have the total sum of squares.

$\begin{align*} Total \ SS = among-groups \ SS + within-groups \ SS \end{align*}$

Thus, the insight of ANOVA is that variation in the dataset may be attributed to a portion explained by differences among the groups and differences among individual observations within each group. The inference comes from recognizing that if the among group effect is greater than the within group effect, then there will be a difference due to the treatment levels.

Mean squares

To decide whether the variation associated with the among group differences are greater than the within group variation, we calculate ratios of the sums of squares. These are called Mean Squares or MS for short. The ratio of the Mean Squares is called F, the test statistic for ANOVA.

For the one-way ANOVA we will have two Mean Squares and one F, tested with degrees of freedom for both the numerator $MS_{groups}$ , and denominator $MS_{error}$ ).

The Mean Square for (among) groups is

$\begin{align*} MS_{group} = \frac{among-groups \ SS}{among-groups \ DF} \end{align*}$

The Mean Square for error is

$\begin{align*} MS_{error} = \frac{within-groups \ SS}{within-groups \ DF} \end{align*}$

And finally, the value for F, the test statistic for ANOVA, is

$\begin{align*} F = \frac{MS_{group}}{MS_{error}} \end{align*}$

Worked example with R

A factor with three levels, A, B, and C

group <- c("A", "A", "A", "B", "B", "B", "C", "C", "C")

and their responses, simulated

response <- c(10.8, 11.8, 12.8, 6.5, 7, 8, 3.8, 2.8, 3)

We create a data frame

all <- data.frame(group, response)

Of course, you could place the data into a worksheet and then import the worksheet into R. Regardless, we now have our dataset.

Now, call the ANOVA function, aov, and assign the results to an object (eg, Model.1)

Model.1 <- aov(response ~ group, data=all)

Now, visualize the ANOVA table

summary(Model.1)

and the output from R, the ANOVA table, is shown below

Table 2. Output from aov() command, the ANOVA table, for the “Difference” outcome variable.

           Df   Sum Sq    Mean Sq   F value       Pr(>F) 
group       2   111.16      55.58     89.49    0.0000341 ***
Residuals   6     3.73       0.62
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Let’s take the ANOVA table one row at a time

The first row has subject headers defining the columns.
The second row of the table “groups” contains statistics due to group, and provides the comparisons among groups.
The third row, “Residuals” is the error, or the differences within groups.

Moving across the columns, then, for each row, we have in turn,

the degrees of freedom (there were 3 groups, therefore 2 DF for group),
the Sums of Squares, the Mean Squares,
the value of F, and finally,
the P-value.

R provides a helpful guide on the last line of the ANOVA summary table, the “Signif[icance] codes,” which highlights the magnitude of the P-value.

Note 3

On the use of “error” and “residuals.” Error refers to the difference between an observed value and the true population value. Residual is the difference between an observed value and a value predicted by a model. In ANOVA, the “error variance” or “residual variance” is the variation within groups unexplained by the tested factor. It is what remains after the effects of the model are accounted for.

What to report? ANOVA problems can be much more complicated than the simple one-way ANOVA introduced here. For complex ANOVA problems, report the ANOVA table itself! But for the one-way ANOVA it would be sufficient to report the test statistic, the degrees of freedom, and the p-value, as we have in previous chapters (eg, t-test, chi-square, etc.). Thus, we would report:

F = 89.49, df = 2 and 6, p = 0.0000341

where F = 89.49 is the test statistic, df = 2 (degrees of freedom for the among group mean square) and 6 (degrees of freedom for the within group mean square), and p = 0.0000341 is the p-value.

In Rcmdr, the appropriate command for the one-way ANOVA is simply

Rcmdr: Statistics → Means

Screenshot Rcmdr select one-way ANOVA

Figure 2. Screenshot Rcmdr select one-way ANOVA.

which brings up a simple dialog. R Commander anticipates factor (Groups) and Response variable. Optional, choose Pairwise comparisons of means for post-hoc test (Tukey’s) and, if you do not want to assume equal variances (see Chapter 13), select Welch F-test.

Screenshot Rcmdr select one-way ANOVA options

Figure 3. Screenshot Rcmdr select one-way ANOVA options.

Questions

1. Review the example ANOVA Table and confirm the following

How many levels of the treatment were there?
How many sampling units were there?
Confirm the calculation of $MS_{group}$ and $MS_{error}$ using the formulas contained in the text.
Confirm the calculation of F using the formula contained in the text.
The degrees of freedom for the F statistic in this example were 2 and 6 (F_2,6). Assuming a two-tailed test with Type I error rate of 5%, What is the critical value of the F distribution (see Appendix 20.4)

2. Repeat the one-way ANOVA using the simulated data, but this time, calculate the ANOVA problem for the “No.difference” response variable.

3. Leaf lengths from three strains of Arabidopsis thaliana plants grown in common garden are shown in Fig 2. Data are provided for you in the following R script.

arabid <- c("wt","wt","wt","AS1","AS1","AS1","AS2","AS2","AS2")
leaf <- c(4.909,5.736,5.108,6.956,5.809,6.888,4.768,4.209,4.065)
leaves <- data.frame(arabid,leaf)

Write out a null and alternative hypotheses
Conduct a test of the null hypothesis by one-way ANOVA
Report the value of the test statistic, the degrees of freedom, and the P-value
Do you accept or reject the null hypothesis? Explain your choice.

Box plot of length of leaves from three strains of Arabidopsis thaliana

Figure 4. Box plot of lengths of leaves of a 10-day old plant from on of three strains of Arabidopsis thaliana.

4. Return to your answer to question 7 from Chapter 12.1 and review your answer and modify as appropriate to correct your language to that presented here about factors and levels.

5. Conduct the one-way ANOVA test on the Comet assay data presented question 7 from Chapter 12.1. Data are copied for your convenience to end of this page. Obtain the ANOVA table and report the value of the test statistics, degrees of freedom, and p-value.
a. Based on the ANOVA results, do you accept or reject the null hypothesis? Explain your choice.

Quiz Chapter 12.2

One-way ANOVA

Data used in this page

Difference, no difference

Table 12.2 Difference or No Difference

Group	No.difference	Difference
A	12.04822	11.336161
A	12.67584	13.476142
A	12.99568	12.96121
A	12.01745	11.746712
A	12.34854	11.275492
B	12.17643	7.167262
B	12.77201	5.136788
B	12.07137	6.820242
B	12.94258	5.318743
B	12.0767	7.153992
C	12.58212	3.344218
C	12.69263	3.792337
C	12.60226	2.444438
C	12.02534	2.576014
C	12.6042	4.575672

Simulated data

Comet assay, antioxidant properties of tea

Data presented in Chapter 12.1

Chapter 12 contents

5.2 – Experimental units, Sampling units

Introduction
Experimental units
Experimental and sampling units often, but not always the same
Replication: Groups and individuals as sampling units
Conclusion
Questions
Quiz
Chapter 5 contents

Introduction.

Sampling units refer to the measured items, the focus of data collection; samples are selected from populations. Often, sampling units are the same thing as individuals. For example, if we are interested in the knowing whether men are more obese than women in Hawai’i, we would select individuals from the population; we would measure individuals. Thus, the sampling unit would be individuals, and the measurement unit would be percent body fat recorded for each individual. The data set would be the collection of all body fat measures for all individuals in the study, and we would then make inferences (draw conclusions) about the differences, if any, between adult males and females for body fat.

But sampling units can also refer to something more restrictive than the individual. For example, we may be interested in how stable, or consistent, is a person’s body fat over time. If we take a body fat measure once per year over a decade on the same group of adults, then the sampling unit refers to each observation of body fat recorded (once per year, ten times for an individual), and the population we are most likely to be referring to is the collection of all such readings (ten is arbitrary — we could have potentially measured the same individual thousands of times).

In some cases the researcher may wish to compare groups of individuals and not the individuals themselves. For example, a 2001 study sought to see if family structure influenced the metabolic (glycemic) control of children with diabetes (Thompson et al 2001). The researchers compared how well metabolic control was achieved in children of single parents and two-parent families. Thus, the sampling units would be families and not the individual children.

Experimental units.

Experimental units refers to the level at which treatments are independently applied in a study. Often, but not always, treatments are applied directly to individuals and therefore the sampling units and experimental units in these cases would be the same.

Question 1: What is the sampling unit in the following cell experiment?

A technician thaws a cryo tube containing about ½ million A549 cells (Foster et al 1998) and grows the cells in a T-75 culture flask (the 75 corresponds to 75 cm² growing area) in a CO₂ incubator at 37 °C. After the cells reach about 70% confluency, which may represent hundreds of thousands of cells, the technician aliquots 1000 cells into twelve wells of a plate for a total of 12000 cells. To three wells the technician adds a cytokine, to another three wells he adds a cytokine inhibitor, and to another three wells he adds both the cytokine and it’s inhibitor, and to the last three wells he adds DMSO, which was the solvent for both the cytokine and the inhibitor. He then returns the plate to the incubator and 24 hours later extracts all of the cells and performs a multiplex quantitative PCR to determine gene expression of several target genes known to be relevant to cell proliferation.

The described experiment would be an example of a routine, but not trivial procedure the technician would do in the course of working on the project in a molecular biology laboratory.

The choices for numbers provided in the description we may consider for the number of sampling units are:

1000 cells
12,000 cells
½ million cells
3 wells
12 wells
The target genes
None of the above

The correct answer is E, the 12 wells. In experimental design, the experimental/sampling unit is the smallest physical entity to which a specific treatment is independently applied and from which a measurement is taken. Treatments (cytokine, inhibitor, both, or DMSO) are applied to individual wells, and all cells within a single well are harvested and pooled together for a single qPCR measurement.

To find the sampling unit, work backward from the qPCR measurements to find the smallest entity that is independently treated and measured. The target genes are simply the response variables being measured, and individual cells are not independent because all 1,000 cells in a well share the same medium, receive the same treatment, and are pooled together during extraction. While all cells originate from a single thawed cryo tube, that tube represents the shared stock population rather than the sampling unit. Because the technician independently applies treatments to and harvests each of the 12 wells, each well acts as an independent sampling unit, making the correct answer E. 12 wells.

Experimental and sampling units often, but not always the same.

Question 2: What is the experimental unit in the described cell experiment?

1000 cells
12,000 cells
½ million cells
3 wells
12 wells
The target genes
None of the above

The correct answer is E, 12 wells. Noted above, the technician applied treatments to 12 wells. DMSO was added to all wells. There were three treatments, cytokine, inhibitor, and cytokine-inhibitor (Table 1).

In this experiment, the experimental unit and the sampling unit are both 1 well. They would differ under two main circumstances:

Subsampling Within a Well: If the technician applies treatments to the 12 wells but measures individual cells (eg, via single-cell RNA-seq) or splits each well’s extract into multiple qPCR reactions, the experimental unit is 1 well (12 total), while the sampling unit is 1 cell or 1 reaction.
Treating a Larger Group at Once: If the technician places four 3-well plates into four different incubators (eg, varying CO2 levels) and still harvests each well separately, the treatment is applied to the whole plate. Here, the experimental unit is 1 plate (4 total), while the sampling unit remains 1 well (12 total).

How This Connects to Pseudoreplication

When the experimental and sampling units differ, pseudoreplication occurs if you treat the smaller sampling units as independent experimental units in your statistical analysis (Hulbert 1984; Lazic 2010; Lazic et al 2018). For example, if we measured 100 individual cells from each of the 12 wells and ran a statistical test with n = 1,200 instead of accounting for the 12 wells, you would artificially inflate your sample size and risk false-positive results.

Table 1. Translate experiment description to a table to better visualize the design

Wells	Treatment
A1	Cytokine
A2	Cytokine
A3	Cytokine
A4	DMSO
B1	Inhibitor
B2	Inhibitor
B3	Inhibitor
B4	DMSO
C1	Cytokine+Inhibitor
C2	Cytokine+Inhibitor
C4	Cytokine+Inhibitor
C4	DMSO

💡 Mike’s Tips

Use package ggplate to create a simple graphic to visualize a culture plate.

table01.ch05.2 <- data.frame(
     well = c("A1", "A2", "A3", "B1", "B2", "B3", "C1", 
              "C2", "C3", "A4", "B4", "C4", "D4"),
     treatment = c(rep("Cytokine", 3), rep("Inhibitor", 3), 
              rep("Cytokine + Inhibitor", 3), rep("DMSO", 4))
     )
plate_plot(data = table01.ch05.2, 
     position = Well, 
     value = Treatment,
     plate_size = 12,
     plate_type = "round")

which results in this nice plate layout (Fig 1)

Plate layout for Table 1. Plot made with plate_plot() function from R package ggplate.

Figure 1. Plate layout for Table 1. Plot made with plate_plot() function from R package ggplate.

If observations were made, then the same function can display values. Add a column of values to the dataset (eg, “Values”), then add another value = line, eg, value = Values to the R code.

Replication: Groups and individuals as sampling units.

The correct identification of levels at which sampling independence occurs is crucial to successful interpretation of inferential statistics.

Note 2

Replication in Table 1: three cytokine, three cytokine-inhibitor, three with both.

Sampling error rate is evaluated at the level of the sampling units. Technical replication of sampling units allows one to evaluate errors of measurement (eg, instrument noise) (Blainey et al 2014) and other sources of technical sources of noise.

Note 3

Data noise refers to random or irrelevant information that corrupts the “true” signal in a dataset, introducing variability and potential errors. A nuisance variable is a specific type of noise that is fundamental to the data collection process but not of interest to the researcher, and its effect must be accounted for or removed via pre-processing steps. For approaches to control for noise, see Chapter 17.4 and Chapter 20.3 for more.

Replication of sampling units increases statistical power, the ability to correctly reject hypotheses. If the correct design reflects sampling units are groups and not individuals, then by counting the individuals as the independent sampling units would lead the researcher to think his design has more replication than it actually does. The consequence on the inferential statistics is that he will more likely reject a correct null hypothesis, in other words, the risk of elevated type I error occurs (Chapter 8 – Inferential statistics). This error, confusing individual and group sampling units, is called pseudoreplication (Lazic 2010).

Consider a simpler experimental design scenario depicted in Figure 2: Three different water treatments (eg, concentrations of synthetic progestins, Zeilinger et al. 2009) in bowls A, B, and C; three fish in bowl A, three fish in bowl B, and three fish in bowl C. The outcome variable might be a stress indicator, eg, plasma cortisol (Luz et al 2008).

Figure 2. Three aquariums, 3 fish. Image modified from https://www.pngrepo.com/svg/153528/aquarium

Figure 2. Three aquariums, three fish. Image modified from https://www.pngrepo.com/svg/153528/aquarium

Question 3: What were the experimental units for the fish in the bowl experiment (Fig 2)?

Three bowls
Nine fish
Three water treatments

The correct answer is A, 3 bowls. The treatments were allocated to the bowls, not to individual fish. The three fish in each bowl provides technical replication for the effects of bowl A, bowl B, and bowl C, but does not provide replication for the effects of the water treatments. Adding three bowls for each water treatment, each with three fish, would be the simplest correction of the design, but may not be available to the researcher because of space or cost limitations. The design would then include nine bowls and 27 fish. If resources are not available to simply scale up the design, then the researcher could repeat the study several times, taking care to control nuisance variables (Chapter 5.3). Alternatively, if the treatments were applied to the individual fish, then the experimental units become the individual fish and the bowls reduced to a blocking effect (Chapter 14.4), where differences may exist among the bowls, but they are no longer the level by which measurements are made.

Note 4

If pseudoreplication is present in a study, then this may be accounted for by specifying the error structure in a linear mixed model (eg, random effects, blocking effects, etc., see Chapter 14 and Chapter 18).

Question 4: What were the sampling units for the fish in the bowl experiment (Fig 2)?

Three bowls
Nine fish
Three water treatments

The correct answer is B, the individual fish. If instead of aqueous application of synthetic progestin, treatments were applied directly to each fish via injection, what would be the answers to Question 3 and Question 4?

Choices like these clearly involve additional compromises and assumptions about experimental design and inference about hypotheses.

Conclusion.

Sampling units, experimental units, and the concept of level at which units are independent within an experiment were introduced. Lack of independence yields the problem of pseudoreplication in an experiment, which will increase the chance that we will detect differences between our treatment groups, when no such difference exists!

Questions.

Figure 3. Three Miracle-Grow AeroGarden planters, each with nine seedlings of an Arabidopsis thaliana strain.

Figure 3. Three Miracle-Gro AeroGarden planters®, each with nine seedlings of an Arabidopsis thaliana strain.

Nine seeds each of three strains of Arabidopsis thaliana were germinated in three Miracle-Grow AeroGarden^® hydroponic planters (Fig 3). Each planter had nine or ten vials with sphagnum peat. All seeds from a strain were planted in the same apparatus, one seed per vial. With respect to this experimental design

1. Define experimental units and sampling units.

2. Provide design examples where the experimental units are the same as the sampling units.

3. With respect to the independent and dependent (outcome) variables, discuss impacts of nuisance variables.

Quiz Chapter 5.2

Experimental units, Sampling units

Chapter 5 contents

The basics explained
Experiments
Experimental and Sampling units
Replication, Bias, and Nuisance Variables
Clinical trials
Importance of randomization in experimental design
Sampling from Populations
References and suggested readings