## Applied Statistics Lesson of the Day – Additive Models vs. Interaction Models in 2-Factor Experimental Designs

In a recent “Machine Learning Lesson of the Day“, I discussed the difference between a supervised learning model in machine learning and a regression model in statistics.  In that lesson, I mentioned that a statistical regression model usually consists of a systematic component and a random component.  Today’s lesson strictly concerns the systematic component.

An additive model is a statistical regression model in which the systematic component is the arithmetic sum of the individual effects of the predictors.  Consider the simple case of an experiment with 2 factors.  If $Y$ is the response and $X_1$ and $X_2$ are the 2 predictors, then an additive linear model for the relationship between the response and the predictors is

$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon$

In other words, the effect of $X_1$ on $Y$ does not depend on the value of $X_2$, and the effect of $X_2$ on $Y$ does not depend on the value of $X_1$.

In contrast, an interaction model is a statistical regression model in which the systematic component is not the arithmetic sum of the individual effects of the predictors.  In other words, the effect of $X_1$ on $Y$ depends on the value of $X_2$, or the effect of $X_2$ on $Y$ depends on the value of $X_1$.  Thus, such a regression model would have 3 effects on the response:

1. $X_1$
2. $X_2$
3. the interaction effect of $X_1$ and $X_2$

full factorial design with 2 factors uses the 2-factor ANOVA model, which is an example of an interaction model.  It assumes a linear relationship between the response and the above 3 effects.

$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2 + \varepsilon$

Note that additive models and interaction models are not confined to experimental design; I have merely used experimental design to provide examples for these 2 types of models.

## Applied Statistics Lesson of the Day – The Full Factorial Design

An experimenter may seek to determine the causal relationships between $G$ factors and the response, where $G > 1$.  On first instinct, you may be tempted to conduct $G$ separate experiments, each using the completely randomized design with 1 factor.  Often, however, it is possible to conduct 1 experiment with $G$ factors at the same time.  This is better than the first approach because

• it is faster
• it uses less resources to answer the same questions
• the interactions between the $G$ factors can be examined

Such an experiment requires the full factorial design; in this design, the treatments are all possible combinations of all levels of all factors.  After controlling for confounding variables and choosing the appropriate range and number of levels of the factor, the different treatments are applied to the different groups, and data on the resulting responses are collected.

The simplest full factorial experiment consists of 2 factors, each with 2 levels.  Such an experiment would result in $2 \times 2 = 4$ treatments, each being a combination of 1 level from the first factor and 1 level from the second factor.  Since this is a full factorial design, experimental units are independently assigned to all treatments.  The 2-factor ANOVA model is commonly used to analyze data from such designs.

In later lessons, I will discuss interactions and 2-factor ANOVA in more detail.

## Applied Statistics Lesson of the Day – The Matched-Pair (or Paired) t-Test

My last lesson introduced the matched pairs experimental design, which is a special type of the randomized blocked design.  Let’s now talk about how to analyze the data from such a design.

Since the experimental units are organized in pairs, the units between pairs (blocks) are not independently assigned.  (The units within each pair are independently assigned – returning to the glove example, one hand is randomly chosen to wear the nitrile glove, while the other is randomly chosen to wear the latex glove.)  Because of this lack of independence between pairs, the independent 2-sample t-test is not applicable.  Instead, use the matched pair t-test (also called the paired or the paired difference t-test).  This is really a 1-sample t-test that tests the difference between the responses of the experimental and the control groups.

## Applied Statistics Lesson of the Day – Blocking and the Randomized Complete Blocked Design (RCBD)

A completely randomized design works well for a homogeneous population – one that does not have major differences between any sub-populations.  However, what if a population is heterogeneous?

Consider an example that commonly occurs in medical studies.  An experiment seeks to determine the effectiveness of a drug on curing a disease, and 100 patients are recruited for this double-blinded study – 50 are men, and 50 are women.  An abundance of biological knowledge tells us that men and women have significantly physiologies, and this is a heterogeneous population with respect to gender.  If a completely randomized design is used for this study, gender could be a confounding variable; this is especially true if the experimental group has a much higher proportion of one gender, and the control group has a much higher proportion of the other gender.  (For instance, purely due to the randomness, 45 males may be assigned to the experimental group, and 45 females may be assigned to the control group.)  If a statistically significant difference in the patients’ survival from the disease is observed between such a pair of experimental and control groups, this effect could be attributed to the drug or to gender, and that would ruin the goal of determining the cause-and-effect relationship between the drug and survival from the disease.

To overcome this heterogeneity and control for the effect of gender, a randomized blocked design could be used.  Blocking is the division of the experimental units into homogeneous sub-populations before assigning treatments to them.  A randomized blocked design for our above example would divide the males and females into 2 separate sub-populations, and then each of these 2 groups is split into the experimental and control group.  Thus, the experiment actually has 4 groups:

1. 25 men take the drug (experimental)
2. 25 men take a placebo (control)
3. 25 women take the drug (experimental)
4. 25 women take a placebo (control)

Essentially, the population is divided into blocks of homogeneous sub-populations, and a completely randomized design is applied to each block.  This minimizes the effect of gender on the response and increases the precision of the estimate of the effect of the drug.

## Applied Statistics Lesson of the Day – The Completely Randomized Design with 1 Factor

The simplest experimental design is the completely randomized design with 1 factor.  In this design, each experimental unit is randomly assigned to a factor level.  This design is most useful for a homogeneous population (one that does not have major differences between any sub-populations).  It is appealing because of its simplicity and flexibility – it can be used for a factor with any number of levels, and different treatments can have different sample sizes.  After controlling for confounding variables and choosing the appropriate range and number of levels of the factor, the different treatments are applied to the different groups, and data on the resulting responses are collected.  The means of the response variable in the different groups are compared; if there are significant differences, then there is evidence to suggest that the factor and the response have a causal relationship.  The single-factor analysis of variance (ANOVA) model is most commonly used to analyze the data in such an experiment, but it does assume that the data in each group have a normal distribution, and that all groups have equal variance.  The Kruskal-Wallis test is a non-parametric alternative to ANOVA in analyzing data from single-factor completely randomized experiments.

If the factor has 2 levels, you may think that an independent 2-sample t-test with equal variance can also be used to analyze the data.  This is true, but the square of the t-test statistic in this case is just the F-test statistic in a single-factor ANOVA with 2 groups.  Thus, the results of these 2 tests are the same.  ANOVA generalizes the independent 2-sample t-test with equal variance to more than 2 groups.

Some textbooks state that “random assignment” means random assignment of experimental units to treatments, whereas other textbooks state that it means random assignment of treatments to experimental units.  I don’t think that there is any difference between these 2 definitions, but I welcome your thoughts in the comments.

## Applied Statistics Lesson of the Day – Choosing the Range of Levels for Quantitative Factors in Experimental Design

In addition to choosing the number of levels for a quantitative factor in designing an experiment, the experimenter must also choose the range of the levels of the factor.

• If the levels are too close together, then there may not be a noticeable difference in the corresponding responses.
• If the levels are too far apart, then an important trend in the causal relationship could be missed.

Consider the following example of making sourdough bread from Gänzle et al. (1998).  The experimenters sought to determine the relationship between temperature and the growth rates of 2 strains of bacteria and 1 strain of yeast, and they used mathematical models and experimental data to study this relationship.  The plots below show the results for Lactobacillus sanfranciscensis LTH2581 (Panel A) and LTH1729 (Panel B), and Candida milleri LTH H198 (Panel C).  The figures contain the predicted curves (solid and dashed lines) and the actual data (circles).  Notice that, for all 3 organisms,

• the relationship is relatively “flat” in the beginning, so choosing temperatures that are too close together at low temperatures (e.g. 1 and 2 degrees Celsius) would not yield noticeably different growth rates
• the overall relationship between growth rate and temperature is rather complicated, and choosing temperatures that are too far apart might miss important trends.

Once again, the experimenter’s prior knowledge and hypothesis can be very useful in making this decision.  In this case, the experimenters had the benefit of their mathematical models in guiding their hypothesis and choosing the range of temperatures for collecting the data on the growth rates.

#### Reference:

Gänzle, Michael G., Michaela Ehmann, and Walter P. Hammes. “Modeling of growth of Lactobacillus sanfranciscensis and Candida milleri in response to process parameters of sourdough fermentation.” Applied and environmental microbiology 64.7 (1998): 2616-2623.

## Applied Statistics Lesson of the Day: Sample Size and Replication in Experimental Design

The goal of an experiment is to determine

1. whether or not there is a cause-and-effect relationship between the factor and the response
2. the strength of the causal relationship, should such a relationship exist.

To answer these questions, the response variable is measured in both the control group and the experimental group.  If there is a difference between the 2 responses, then there is evidence to suggest that the causal relationship exists, and the difference can be measured and quantified.

However, in most* experiments, there is random variation in the response.  Random variation exists in the natural sciences, and there is even more of it in the social sciences.  Thus, an observed difference between the control and experimental groups could be mistakenly attributed to a cause-and-effect relationship when the source of the difference is really just random variation.  In short, the difference may simply be due to the noise rather than the signal.

To detect an actual difference beyond random variation (i.e. to obtain a higher signal-to-noise ratio), it is important to use replication to obtain a sufficiently large sample size in the experiment.  Replication is the repeated application of the treatments to multiple independently assigned experimental units.  (Recall that randomization is an important part of controlling for confounding variables in an experiment.  Randomization ensures that the experimental units are independently assigned to the different treatments.)  The number of independently assigned experimental units that receive the same treatment is the sample size.

*Deterministic computer experiments are unlike most experiments; they do not have random variation in the responses.

## Applied Statistics Lesson of the Day – Basic Terminology in Experimental Design #1

The word “experiment” can mean many different things in various contexts.  In science and statistics, it has a very particular and subtle definition, one that is not immediately familiar to many people who work outside of the field of experimental design. This is the first of a series of blog posts to clarify what an experiment is, how it is conducted, and why it is so central to science and statistics.

Experiment: A procedure to determine the causal relationship between 2 variables – an explanatory variable and a response variable.  The value of the explanatory variable is changed, and the value of the response variable is observed for each value of the explantory variable.

• An experiment can have 2 or more explanatory variables and 2 or more response variables.
• In my experience, I find that most experiments have 1 response variable, but many experiments have 2 or more explanatory variables.  The interactions between the multiple explanatory variables are often of interest.
• All other variables are held constant in this process to avoid confounding.

Explanatory Variable or Factor: The variable whose values are set by the experimenter.  This variable is the cause in the hypothesis.  (*Many people call this the independent variable.  I discourage this usage, because “independent” means something very different in statistics.)

Response Variable: The variable whose values are observed by the experimenter as the explanatory variable’s value is changed.  This variable is the effect in the hypothesis.  (*Many people call this the dependent variable.  Further to my previous point about “independent variables”, dependence means something very different in statistics, and I discourage using this usage.)

Factor Level: Each possible value of the factor (explanatory variable).  A factor must have at least 2 levels.

Treatment: Each possible combination of factor levels.

• If the experiment has only 1 explanatory variable, then each treatment is simply each factor level.
• If the experiment has 2 explanatory variables, X and Y, then each treatment is a combination of 1 factor level from X and 1 factor level from Y.  Such combining of factor levels generalizes to experiments with more than 2 explanatory variables.

Experimental Unit: The object on which a treatment is applied.  This can be anything – person, group of people, animal, plant, chemical, guitar, baseball, etc.