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<h1 class="title toc-ignore">Randomized, Basic Factorial
Experiments</h1>
</div>
<hr />
<p>The experimental designs in this section have 2 key characteristics
in common:</p>
<ul>
<li>Conditions are assigned to subjects (a.k.a. experimental units)
completely at random</li>
<li>The way error is accounted for in all these designs and the way the
requirements are checked can be approached the same way</li>
</ul>
<p>To assign something <strong>completely at random</strong> means that
each experimental unit has a known and equal chance of being selected
for a particular treatment and that no other considerations are taken
into account when making treatment assignments. Usually, for balance,
the same number of units are assigned to each treatment. For example, if
I have 4 treatment and 16 units, I may use a computer to randomly
shuffle the units into treatment groups.</p>
<p>Factorial experiments involve two or more factors that are crossed.
<strong>Factorial crossing</strong> means that each combination of
factor levels is considered as a treatment in the study. (A study with
just one factor is not technically a factorial design, but we will lump
it in with our discussion of factorial experiments her because of the
completely random treatment assignment).</p>
<p>Contrast a factorial design with the one-at-a-time approach. In a
one-at-a-time approach, if I had two factors I wanted to study I would
run two separate experiments to evaluate the effect of each factor on
the response one-at-time. Factorial designs have a couple of major
advantages over one-factor-at-a-time studies.</p>
<ol style="list-style-type: decimal">
<li>They are a more efficient use of our time and material: I can get
information about both of my factors from just one experimental
unit</li>
<li>Perhaps most importantly, factorial designs allow the researcher to
estimate interaction effects. Or in other words, we can observe how one
factor’s effect on the response variable changes for different levels of
the other factor.</li>
</ol>
<p>EXAMPLE of BF2???</p>
<p>In summary, “factorial” refers to how you determine which treatments
will be included in the study, and “completely randomized” refers to how
treatments are assigned to subjects.</p>
<hr />
<div id="bf1" class="section level2 tabset tabset-fade tabset-pills">
<h2 class="tabset tabset-fade tabset-pills">BF[1]</h2>
<p>A study with just one factor</p>
<div id="overview" class="section level3">
<h3>Overview</h3>
<p>In a basic one-way factorial design (BF[1]), only one factor is
purposefully varied. Each level of the factor is considered a treatment.
Each experimental unit is randomly assigned to exactly one
treatment.</p>
<div id="factor-structure" class="section level4">
<h4>Factor structure</h4>
<p>The factor structure for the model resulting from a completely
randomized, one factor with design is:</p>
<p><img src="images/BF1_factor_structure_with_equation.PNG" /></p>
<p>The above diagram illustrates a factor with 4 levels, 6 replications
at each level.</p>
</div>
<div id="hypotheses-and-model" class="section level4">
<h4>Hypotheses and model</h4>
<p>A more detailed description of the model for an ANOVA with just one
factor:</p>
<p><span class="math display">\[
y_{ij} = \mu + \alpha_i + \epsilon_{ij}
\]</span></p>
<p><span class="math inline">\(y_{ij}\)</span>: the <span
class="math inline">\(j^{th}\)</span> observation from treatment <span
class="math inline">\(i\)</span></p>
<p><span class="math inline">\(\mu\)</span>: the grand mean of the
dataset. Also referred to as an overall mean or benchmark.</p>
<p><span class="math inline">\(\alpha_i\)</span>: effect of treatment
<span class="math inline">\(i\)</span></p>
<p><span class="math inline">\(\epsilon_{ik}\)</span>: the error term,
or residual term of the model. There are <em>j</em> replicates for each
treatment. It represents the distance from an observation to its
treatment mean (or predicted value).</p>
<p>The null and alternative hypotheses can be expressed as:</p>
<p><span class="math display">\[
H_o: \alpha_1 = \alpha_2 = ... = 0
\]</span> <span class="math display">\[
H_a: \alpha_i \neq 0 \quad \text{for at least one }\ \alpha_i
\]</span></p>
</div>
<div id="assumptions" class="section level4">
<h4>Assumptions</h4>
<p>A one-way ANOVA model may be used to analyze data from a BF[1] design
if the following requirements are satisfied:</p>
<ol style="list-style-type: decimal">
<li>Each experimental unit is randomly assigned to only 1 treatment (or
factor level)</li>
<li>The error term of the model (<span
class="math inline">\(\epsilon_{ik}\)</span>) is normally distributed.
This assumption is met when the residuals are normally distributed (as
seen in a qq-plot).</li>
<li>The population variance of each group is equal. This is often called
the homogeneity of variance, or constant variance assumption. This is
considered met when each group of residuals in the residual
vs. fitted-values plot shows a similar vertical spread.</li>
</ol>
<hr />
</div>
</div>
<div id="design" class="section level3">
<h3>Design</h3>
<p>In a one factor design, one factor is purposefully varied and all
other factors are controlled in order to isolate the effect of just the
factor under study. Each level of the factor is considered a
treatment.</p>
<p>In a completely randomized design, each experimental unit is randomly
assigned to exactly 1 treatment. It is common to keep the number of
units assigned to each treatment the same to ensure balance. This can be
done by listing all the subjects, then listing the treatments, as seen
below:</p>
<table>
<thead>
<tr class="header">
<th align="center">Subject</th>
<th align="center">Treatment</th>
<th align="center">Order</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">Subject 1</td>
<td align="center">A</td>
<td align="center">1</td>
</tr>
<tr class="even">
<td align="center">Subject 2</td>
<td align="center">A</td>
<td align="center">2</td>
</tr>
<tr class="odd">
<td align="center">Subject 3</td>
<td align="center">B</td>
<td align="center">3</td>
</tr>
<tr class="even">
<td align="center">Subject 4</td>
<td align="center">B</td>
<td align="center">4</td>
</tr>
<tr class="odd">
<td align="center">Subject 5</td>
<td align="center">C</td>
<td align="center">5</td>
</tr>
<tr class="even">
<td align="center">Subject 6</td>
<td align="center">C</td>
<td align="center">6</td>
</tr>
</tbody>
</table>
<p>Then you randomly shuffle the treatment column. (You should also be
paying attention to the order in which subjects and treatments are being
experimented on as this could be a potential source of bias. In a BF[1],
you randomize the order also.) The result might look something like
this.</p>
<table>
<thead>
<tr class="header">
<th align="center">Subject</th>
<th align="center">Treatment</th>
<th align="center">Order</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">Subject 1</td>
<td align="center">A</td>
<td align="center">4</td>
</tr>
<tr class="even">
<td align="center">Subject 2</td>
<td align="center">B</td>
<td align="center">3</td>
</tr>
<tr class="odd">
<td align="center">Subject 3</td>
<td align="center">C</td>
<td align="center">6</td>
</tr>
<tr class="even">
<td align="center">Subject 4</td>
<td align="center">C</td>
<td align="center">5</td>
</tr>
<tr class="odd">
<td align="center">Subject 5</td>
<td align="center">A</td>
<td align="center">1</td>
</tr>
<tr class="even">
<td align="center">Subject 6</td>
<td align="center">B</td>
<td align="center">2</td>
</tr>
</tbody>
</table>
<p>You may notice in the above example that, even with randomization,
treatment C occurs in the last 2 observations. If we were truly
concerned about the order we could be more strategic and implement a
<em>blocked design</em> to prevent “unlucky” ordering and pairing.</p>
<p>Consider the following example. An experiment was done to asses
different modes of virtual training in how to launch a lifeboat. Sixteen
students in the maritime safety training institute were a part of the
study, and each of the students were assigned one of four possible
virtual training experiences. The four experiences included:
Lecture/Materials (Control) (1), Monitor/Keyboard (2), Head Monitor
Display/Joypad (3), and Head Monitor Display/Wearables (4). The response
variable was the student’s performance on a procedural knowledge
assessment (performance is defined as their level of improvement from
pre to post test).</p>
<p>To obtain a balanced design, we will want each treatment to be
assigned to four students. We could get 16 pieces of paper and write
“Treatment 1” on 4 pieces, “Treatment 2” on another 4 pieces, and so on
until each treatment has 4 pieces of paper. We could then put them in a
hat, mix them up and then randomly draw out a piece of paper to assign
it to a subject. Intuitively this makes sense, but writing and cutting
paper is slow and inefficient. We could implement a similar process in R
to assign treatments.</p>
<p>First, we list all the possible treatments, and repeat that listing
until it is the same size as our count of subjects. (Note, if your
number of subjects is not an exact multiple of the number of treatments,
you may need to decide which treatments deserve fewer observations)</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a>treatment_list <span class="ot"><-</span> <span class="fu">rep</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">4</span>,<span class="dv">4</span>) <span class="co">#This repeats the sequence of 1 to 4, four times</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>treatment_list</span></code></pre></div>
<pre><code>## [1] 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4</code></pre>
<p>Here it is reformatted in a easy to read table.</p>
<table>
<thead>
<tr class="header">
<th align="center">Subject</th>
<th align="center">Treatment</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">Subject 1</td>
<td align="center">1</td>
</tr>
<tr class="even">
<td align="center">Subject 2</td>
<td align="center">2</td>
</tr>
<tr class="odd">
<td align="center">Subject 3</td>
<td align="center">3</td>
</tr>
<tr class="even">
<td align="center">Subject 4</td>
<td align="center">4</td>
</tr>
<tr class="odd">
<td align="center">Subject 5</td>
<td align="center">1</td>
</tr>
<tr class="even">
<td align="center">Subject 6</td>
<td align="center">2</td>
</tr>
<tr class="odd">
<td align="center">Subject 7</td>
<td align="center">3</td>
</tr>
<tr class="even">
<td align="center">Subject 8</td>
<td align="center">4</td>
</tr>
<tr class="odd">
<td align="center">Subject 9</td>
<td align="center">1</td>
</tr>
<tr class="even">
<td align="center">Subject 10</td>
<td align="center">2</td>
</tr>
<tr class="odd">
<td align="center">Subject 11</td>
<td align="center">3</td>
</tr>
<tr class="even">
<td align="center">Subject 12</td>
<td align="center">4</td>
</tr>
<tr class="odd">
<td align="center">Subject 13</td>
<td align="center">1</td>
</tr>
<tr class="even">
<td align="center">Subject 14</td>
<td align="center">2</td>
</tr>
<tr class="odd">
<td align="center">Subject 15</td>
<td align="center">3</td>
</tr>
<tr class="even">
<td align="center">Subject 16</td>
<td align="center">4</td>
</tr>
</tbody>
</table>
<p>Then we randomly shuffle treatments with subjects to get the
following assignments. The R code could look like this</p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">17</span>) <span class="co">#only use this if you want the exact same random selection as this example</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a><span class="fu">sample</span>(treatment_list, <span class="dv">16</span>)</span></code></pre></div>
<pre><code>## [1] 2 4 1 4 3 1 3 2 2 2 4 3 4 1 3 1</code></pre>
<p>We can see here that subject 1 should get treatment 2. Subject 2 gets
treatment 4, Subject 3 gets treatment 1 and so on.</p>
<table>
<thead>
<tr class="header">
<th align="center">Subject</th>
<th align="center">Treatment</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">Subject 1</td>
<td align="center">2</td>
</tr>
<tr class="even">
<td align="center">Subject 2</td>
<td align="center">4</td>
</tr>
<tr class="odd">
<td align="center">Subject 3</td>
<td align="center">1</td>
</tr>
<tr class="even">
<td align="center">Subject 4</td>
<td align="center">4</td>
</tr>
<tr class="odd">
<td align="center">Subject 5</td>
<td align="center">3</td>
</tr>
<tr class="even">
<td align="center">Subject 6</td>
<td align="center">1</td>
</tr>
<tr class="odd">
<td align="center">Subject 7</td>
<td align="center">3</td>
</tr>
<tr class="even">
<td align="center">Subject 8</td>
<td align="center">2</td>
</tr>
<tr class="odd">
<td align="center">Subject 9</td>
<td align="center">2</td>
</tr>
<tr class="even">
<td align="center">Subject 10</td>
<td align="center">2</td>
</tr>
<tr class="odd">
<td align="center">Subject 11</td>
<td align="center">4</td>
</tr>
<tr class="even">
<td align="center">Subject 12</td>
<td align="center">3</td>
</tr>
<tr class="odd">
<td align="center">Subject 13</td>
<td align="center">4</td>
</tr>
<tr class="even">
<td align="center">Subject 14</td>
<td align="center">1</td>
</tr>
<tr class="odd">
<td align="center">Subject 15</td>
<td align="center">3</td>
</tr>
<tr class="even">
<td align="center">Subject 16</td>
<td align="center">1</td>
</tr>
</tbody>
</table>
<hr />
</div>
<div id="decomposition-and-factor-structure" class="section level3">
<h3>Decomposition and Factor Structure</h3>
<p>This section serves as a bridge between the design and the analysis
of an experiment. It is equivalent to doing the analysis by hand. The
primary goal is to see how the design decisions impact the analysis;
including a deeper understanding of what the numbers in an ANOVA table
mean and how they are calculated.</p>
<p>A factor structure for an experimental design can help provide an
effective way to organize and plan for the type of data needed for an
experiment. For a basic one-way factorial design, three factors are
involved for the experiment: the benchmark (or grand mean), the
treatment, and the residual error. For example, an experiment was done
to help train people in the procedure to launch a lifeboat. This was a
Basic One-way Factorial Design, one where the treatments included:
Lecture/Materials (Control) (1), Monitor/Keyboard (2), Head Monitor
Display/Joypad (3), and Head Monitor Display/Wearables (4). The students
were then given a pre-test before training and a post-test after
training and the difference between the two scores was the measurement
used from each student in the analysis. Therefore, we have four levels
for the treatment with six replicates for each treatment (the actual
study used 16 observations per treatment). The diagram below gives the
factor structure of the design, and the accompanying mathematical
model.</p>
<p><img src="images/BF1_factor_structure_with_equation.PNG" /></p>
<p>A basic one-way factorial design has three factors: the benchmark,
treatment, and residual. The effects of these factors can be summed
together to find each observed value.</p>
<ul>
<li>The <strong>observed values</strong> on the left show that there are
24 distinct observations of performance score, represented by the 24
cells - one for each of the observed values.</li>
<li>The <strong>benchmark</strong> represents the grand mean (or overall
mean). The single large cell indicates that there is only one grand mean
and it is part of every observation.<br />
</li>
<li>The <strong>treatment factor</strong> involves the four levels of
the treatment represented by the four vertically long cells. Each
treatment may have a distinct mean (or effect).</li>
<li>The <strong>residual error</strong> represents the difference
between the observed value and the predicted value. The predicted value,
also called the fitted value, is the sum of the grand mean and treatment
effect. Each of the cells represent a distinct value for the residual
error.</li>
</ul>
<div id="inside-vs.-outside-factors" class="section level4">
<h4>Inside vs. Outside Factors</h4>
<p>Having a factor structure can help us determine the <strong>degrees
of freedom</strong> and the <strong>effects</strong> of each factor.
Before determining the degrees of freedom and the effects of each
factor, understanding <strong>outside and inside factors</strong> is
helpful.</p>
<p>A factor is inside of another factor if all the levels of one factor
(<strong>the inside factor</strong>) completely fits within a second
factor (<strong>the outside factor</strong>).</p>
<p>You may find this analogy helpful. Pretend that an outside factor is
a box, and the inside factor levels are blocks that fit perfectly within
the box.</p>
<center>
<img src="images/wooden_blocks_in_box.jpg" width = 200px;>
<!--  -->
<!-- picture from: https://www.amazon.com/MOD-Complete-Extra-Large-Size/dp/B079J8PB73 -->
</center>
<p>In our basic one-way factorial design there are three factors: the
benchmark, treatment, and residuals. In the picture below, benchmark is
represented in red, treatment is drawn in blue, and the residual factor
levels are depicted in black.</p>
<p><img src="images/inside_outside_1.JPG" /></p>
<p>In order to understand the relationship between the benchmark and
treatment factor, imagine picking up the levels of factor and placing
them in the other factor one at a time. We will start with taking the
levels of treatment and placing them inside of the benchmark.</p>
<p><img src="images/inside_outside_2.JPG" /></p>
<p>In this picture you can see that one entire level of treatment can
fit inside of a single level of benchmark. Even though they may share a
boundary line, the level does not cross over and start sharing
boundaries with any other level. You can repeat this for the other 3
levels of treatment with the same result. Therefore, we say that
treatment is inside of benchmark, which is the same as saying that
benchmark is outside of treatment.</p>
<p>Consider now the relationship between treatment and residual. If we
take a level of treatment and overlay it on the residual factor, we can
see it does not fit neatly inside one of the levels of residual error.
In fact, one level of treatment crosses the boundaries of many of the
levels of residual error. Therefore, we cannot say that treatment is
inside of residual error.</p>
<p><img src="images/inside_outside_3.JPG" /></p>
<p>Since treatment is not inside of residual error, does this
necessarily mean that treatment is outside of residual error? We can
determine this by taking one level of residual at a time and overlaying
it on the treatment factor structure, as is pictured below. It can be
seen that one level of residual error does NOT cross any of the
treatment level boundaries. Therefore, we can safely say that treatment
is indeed outside of residual error; or equivalently that treatment is
outside of residual error.</p>
<p><img src="images/inside_outside_4.JPG" /></p>
<p>To clarify a common misunderstanding, consider an experiment where we
are looking at the inside vs. outside relationship of two factors: A,
and B.</p>
<ul>
<li>When factor A is inside of factor B, we can also say factor B is
outside of factor A.</li>
<li>But, when factor A is <em>not</em> inside of factor B, this does
<em>not necessarily mean that factor A is outside of factor B</em>.
Later you will come across designs where the two factors are neither
outside nor inside of each other: they are crossed.</li>
</ul>
<p>In summary, inside and outside factors for every basic one-way
factorial design:</p>
<ul>
<li>The <strong>benchmark factor</strong> is the outside factor for all
the other factors (treatment and residual error)</li>
<li>The <strong>treatment factor</strong> is an inside factor to the
benchmark factor but an outside factor to the residual error.</li>
<li>The <strong>residual error</strong> is an inside factor for all
other factors (the benchmark and treatment factors).</li>
</ul>
</div>
<div id="degrees-of-freedom" class="section level4">
<h4>Degrees of Freedom</h4>
<p>We can use our understanding of inside and outside factors to
determine the degrees of freedom (df) for the
<strong>benchmark</strong>, <strong>treatment</strong>, and
<strong>residual errors</strong> factors.</p>
<p>The general formula for degrees of freedom for any factor in a design
is:</p>
<center>
<blockquote>
<p><strong>df = Total levels of a factor minus the sum of the df of all
outside factors</strong></p>
</blockquote>
</center>
<p>An alternative method of finding degrees of freedom is to count the
number of unique pieces of information in a factor.</p>
<p>Going back to the lifeboat training example (see example above), we
have 24 observations total and four levels of the treatment.</p>
<p><img src="images/BF1_factor_structure_with_equation.PNG" /></p>
<p>For <strong>benchmark</strong>, there is only one level of the factor
(shown by the one cell) and there are no outside factors for benchmark.
Therefore, the degrees of freedom for benchmark is one. Another way to
think of this is that the degrees of freedom represents the number of
unique pieces of information contributing to the estimation of the
effects for that factor. In this case, as soon as I know the benchmark
value (or better stated, as soon as I estimate the benchmark value) for
just one of the observations, I know it for all the observations.
Therefore, there is just one degree of freedom.</p>
<p>For <strong>treatment</strong>, there are four levels of the factor
(shown by the four vertically long cells for treatment). Benchmark is
the only factor outside of treatment. Take the number of levels for
treatment (4) and subtract the degrees of freedom for benchmark (1),
which yields 4-1 = <strong>3 degrees of freedom</strong> for
treatment.</p>
<p>We could just as easily have used the other approach to finding the
degrees of freedom: counting the unique pieces of information the
treatment effects really contains. Since all observations from the same
treatment will have the same treatment effect applied, we really only
need to know 4 pieces of information: the effect of each treatment. But
the answer is actually less than that. Because we know that the effects
must all sum to zero, only 3 of the effects are free to vary and the 4th
one is constrained to be whatever value will satisfy this mathematical