3.2 Linear model for One-Way ANOVA (cell-means and reference-coding)

We introduced the statistical model \(y_{ij} = \mu_j+\varepsilon_{ij}\) in Chapter 2 for the situation with \(j = 1 \text{ or } 2\) to denote a situation where there were two groups and, for the model that is consistent with the alternative hypothesis, the means differed. Now there are seven groups and the previous model can be extended to this new situation by allowing \(j\) to be 1, 2, 3, …, 7. As before, the linear model assumes that the responses follow a normal distribution with the model defining the mean of the normal distributions and all observations have the same variance. *Linear models assume that the parameters for the mean in the model enter linearly. This last condition is hard to explain at this level of material – it is sufficient to know that there are models where the parameters enter the model nonlinearly and that they are beyond the scope of this function and this material and you won’t run into them in most statistical models. By employing this general “linear” modeling methodology, we will be able to use the same general modeling framework for the methods in Chapters ??, ??, ??, ??, and ??.

As in Chapter 2, the null hypothesis defines a situation (and model) where all the groups have the same mean. Specifically, the null hypothesis in the general situation with \(J\) groups (\(J\ge 2\)) is to have all the \(\underline{\text{true}}\) group means equal,

\[H_0:\mu_1 = \ldots = \mu_J.\]

This defines a model where all the groups have the same mean so it can be defined in terms of a single mean, \(\mu\), for the \(i^{th}\) observation from the \(j^{th}\) group as \(y_{ij} = \mu+\varepsilon_{ij}\). This is not the model that most researchers want to be the final description of their study as it implies no difference in the groups. There is more caution required to specify the alternative hypothesis with more than two groups. The alternative hypothesis needs to be the logical negation of this null hypothesis of all groups having equal means; to make the null hypothesis false, we only need one group to differ but more than one group could differ from the others. Essentially, there are many ways to “violate” the null hypothesis so we choose some delicate wording for the alternative hypothesis when there are more than 2 groups. Specifically, we state the alternative as

\[H_A: \text{ Not all } \mu_j \text{ are equal}\]

or, in words, at least one of the true means differs among the J groups. You might be attracted to trying to say that all means are different in the alternative but we do not put this strict a requirement in place to reject the null hypothesis. The alternative model allows all the true group means to differ but does require that they are actually all different with the model written as

\[y_{ij} = {\color{red}{\mu_j}}+\varepsilon_{ij}.\]

This linear model states that the response for the \(i^{th}\) observation in the \(j^{th}\) group, \(\mathbf{y_{ij}}\), is modeled with a group \(j\) (\(j=1, \ldots, J\)) population mean, \(\mu_j\), and a random error for each subject in each group, \(\varepsilon_{ij}\), that we assume follows a normal distribution and that all the random errors have the same variance, \(\sigma^2\). We can write the assumption about the random errors, often called the normality assumption, as \(\varepsilon_{ij} \sim N(0,\sigma^2)\). There is a second way to write out this model that allows extension to more complex models discussed below, so we need a name for this version of the model. The model written in terms of the \({\color{red}{\mu_j}}\text{'s}\) is called the cell means model and is the easier version of this model to understand.

One of the reasons we learned about pirate-plots is that it helps us visually consider all the aspects of this model. In Figure 2.28, we can see the bold horizontal lines that provide the estimated (sample) group means. The bigger the differences in the sample means (especially relative to the variability around the means), the more evidence we will find against the null hypothesis. You can also see the null model on the plot that assumes all the groups have the same mean as displayed in the dashed horizontal line at 117.1 cm (the R code below shows the overall mean of Distance is 117.1). While the hypotheses focus on the means, the model also contains assumptions about the distribution of the responses – specifically that the distributions are normal and that all the groups have the same variability, which do not appear to be clearly violated in this situation.

mean(dd$Distance)

## [1] 117.126

There is a second way to write out the One-Way ANOVA model that provides a framework for extensions to more complex models described in Chapter ?? and beyond. The other parameterization (way of writing out or defining) of the model is called the reference-coded model since it writes out the model in terms of a baseline group and deviations from that baseline or reference level. The reference-coded model for the \(i^{th}\) subject in the \(j^{th}\) group is \(y_{ij} ={\color{purple}{\boldsymbol{\alpha + \tau_j}}}+\varepsilon_{ij}\) where \(\color{purple}{\boldsymbol{\alpha}}\) (“alpha”) is the true mean for the baseline group (usually first alphabetically) and the \(\color{purple}{\boldsymbol{\tau_j}}\) (tau \(j\)) are the deviations from the baseline group for group \(j\). The deviation for the baseline group, \(\color{purple}{\boldsymbol{\tau_1}}\), is always set to 0 so there are really just deviations for groups 2 through \(J\). The equivalence between the reference-coded and cell-means models can be seen by considering the mean for the first, second, and \(J^{th}\) groups in both models:

\[\begin{array}{lccc} & \textbf{Cell means:} && \textbf{Reference-coded:}\\ \textbf{Group } 1: & \color{red}{\mu_1} && \color{purple}{\boldsymbol{\alpha}} \\ \textbf{Group } 2: & \color{red}{\mu_2} && \color{purple}{\boldsymbol{\alpha + \tau_2}} \\ \ldots & \ldots && \ldots \\ \textbf{Group } J: & \color{red}{\mu_J} && \color{purple}{\boldsymbol{\alpha +\tau_J}} \end{array}\]

The hypotheses for the reference-coded model are similar to those in the cell-means coding except that they are defined in terms of the deviations, \({\color{purple}{\boldsymbol{\tau_j}}}\). The null hypothesis is that there is no deviation from the baseline for any group – that all the \({\color{purple}{\boldsymbol{\tau_j\text{'s}}}}=0\),

\[\boldsymbol{H_0: \tau_2=\ldots=\tau_J=0}.\]

The alternative hypothesis is that at least one of the deviations is not 0,

\[\boldsymbol{H_A:} \textbf{ Not all } \boldsymbol{\tau_j} \textbf{ equal } \bf{0}.\]

In this chapter, you are welcome to use either version (unless we instruct you otherwise) but we have to use the reference-coding in subsequent chapters. The next task is to learn how to use R’s linear model, lm, function to get estimates of the parameters⁵⁰ in each model, but first a quick review of these new ideas:

Cell Means Version

• \(H_0: {\color{red}{\mu_1= \ldots = \mu_J}}\)             \(H_A: {\color{red}{\text{ Not all } \mu_j \text{ equal}}}\)

• Null hypothesis in words: No difference in the true means among the groups.

• Null model: \(y_{ij} = \mu+\varepsilon_{ij}\)

• Alternative hypothesis in words: At least one of the true means differs among the groups.

• Alternative model: \(y_{ij} = \color{red}{\mu_j}+\varepsilon_{ij}.\)

Reference-coded Version

• \(H_0: \color{purple}{\boldsymbol{\tau_2 = \ldots = \tau_J = 0}}\)          \(H_A: \color{purple}{\text{ Not all } \tau_j \text{ equal 0}}\)

• Null hypothesis in words: No deviation of the true mean for any groups from the baseline group.

• Null model: \(y_{ij} =\boldsymbol{\alpha} +\varepsilon_{ij}\)

• Alternative hypothesis in words: At least one of the true deviations is different from 0 or that at least one group has a different true mean than the baseline group.

• Alternative model: \(y_{ij} =\color{purple}{\boldsymbol{\alpha + \tau_j}}+\varepsilon_{ij}\)

In order to estimate the models discussed above, the lm function is used⁵¹. The lm function continues to use the same format as previous functions and in Chapter 2 , lm(Y~X, data=datasetname). It ends up that lm generates the reference-coded version of the model by default (The developers of R thought it was that important!). But we want to start with the cell-means version of the model, so we have to override the standard technique and add a “-1” to the formula interface to tell R that we want to the cell-means coding. Generally, this looks like lm(Y~X-1, data=datasetname). Once we fit a model in R, the summary function run on the model provides a useful “summary” of the model coefficients and a suite of other potentially interesting information. For the moment, we will focus on the estimated model coefficients, so only those lines are provided. When fitting the cell-means version of the One-Way ANOVA model, you will find a row of output for each group relating estimating the \(\mu_j\text{'s}\). The output contains columns for an estimate (Estimate), standard error (Std. Error), \(t\)-value (t value), and p-value (Pr(>|t|)). We’ll explore which of these are of interest in these models below, but focus on the estimates of the parameters that the function provides in the first column (“Estimate”) of the coefficient table and compare these results to what was found using favstats.

lm1 <- lm(Distance~Condition-1, data=dd)
summary(lm1)$coefficients

##                  Estimate Std. Error  t value Pr(>|t|)
## Conditioncasual  117.6110   1.071873 109.7248        0
## Conditioncommute 114.6079   1.021931 112.1484        0
## Conditionhiviz   118.4383   1.101992 107.4765        0
## Conditionnovice  116.9405   1.053114 111.0426        0
## Conditionpolice  122.1215   1.064384 114.7344        0
## Conditionpolite  114.0518   1.015435 112.3182        0
## Conditionracer   116.7559   1.024925 113.9164        0

In general, we denote estimated parameters with a hat over the parameter of interest to show that it is an estimate. For the true mean of group \(j\), \(\mu_j\), we estimate it with \(\hat{\mu}_j\), which is just the sample mean for group \(j\), \(\bar{x}_j\). The model suggests an estimate for each observation that we denote as \(\hat{y}_{ij}\) that we will also call a fitted value based on the model being considered. The same estimate is used for all observations in the each group in this model. R tries to help you to sort out which row of output corresponds to which group by appending the group name with the variable name. Here, the variable name was Condition and the first group alphabetically was casual, so R provides a row labeled Conditioncasual with an estimate of 117.61. The sample means from the seven groups can be seen to directly match the favstats results presented previously.

The reference-coded version of the same model is more complicated but ends up giving the same results once we understand what it is doing. It uses a different parameterization to accomplish this, so has different model output. Here is the model summary:

lm2 <- lm(Distance~Condition, data=dd)
summary(lm2)$coefficients

##                     Estimate Std. Error     t value    Pr(>|t|)
## (Intercept)      117.6110398   1.071873 109.7247845 0.000000000
## Conditioncommute  -3.0031051   1.480964  -2.0278039 0.042626835
## Conditionhiviz     0.8272234   1.537302   0.5381008 0.590528548
## Conditionnovice   -0.6705193   1.502651  -0.4462242 0.655452292
## Conditionpolice    4.5104792   1.510571   2.9859423 0.002839115
## Conditionpolite   -3.5591965   1.476489  -2.4105807 0.015958695
## Conditionracer    -0.8551713   1.483032  -0.5766371 0.564207492

The estimated model coefficients are \(\hat{\alpha} = 117.61\) cm, \(\hat{\tau}_2 =-3.00\) cm, \(\hat{\tau}_3=0.83\) cm, and so on up to \(\hat{\tau}_7=-0.86\) cm, where R selected group 1 for casual, 2 for commute, 3 for hiviz, all the way up to group 7 for racer. The way you can figure out the baseline group (group 1 is casual here) is to see which category label is not present in the reference-coded output. The baseline level is typically the first group label alphabetically, but you should always check this⁵². Based on these definitions, there are interpretations available for each coefficient. For \(\hat{\alpha} = 117.61\) cm, this is an estimate of the mean overtake distance for the casual outfit group. \(\hat{\tau}_2 =-3.00\) cm is the deviation of the commute group’s mean from the causal group’s mean (specifically, it is \(3.00\) cm lower and was a quantity we explored in detail in Chapter 2 when we just focused on comparing casual and commute groups). \(\hat{\tau}_3=0.83\) cm tells us that the hiviz group mean distance is 0.83 cm higher than the casual group mean and \(\hat{\tau}_3=-0.86\) says that the racer sample mean was 0.86 cm lower than for the casual group. These interpretations are interesting as they directly relate to comparisons of groups with the baseline and lead directly to reconstructing the estimated means for each group by combining the baseline and a pertinent deviation as shown in Table 2.3.

(ref:tab3-1) Constructing group mean estimates from the reference-coded linear model estimates.

We can also visualize the results of our linear models using what are called term-plots or effect-plots (from the effects package; (Fox et al. 2019)) as displayed in Figure 2.29. We don’t want to use the word “effect” for these model components unless we have random assignment in the study design so we generically call these term-plots as they display terms or components from the model in hopefully useful ways to aid in model interpretation even in the presence of complicated model parameterizations. The word “effect” has a causal connotation that we want to avoid as much as possible in non-causal (so non-randomly assigned) situations. Term-plots take an estimated model and show you its estimates along with 95% confidence intervals generated by the linear model. These confidence intervals may differ from the confidence intervals in the pirate-plots since the pirate-plots make them for each group separately and term-plots are combining information across groups via the estimated model and then doing inferences for individual group means. To make term-plots, you need to install and load the effects package and then use plot(allEffects(...)) functions together on the lm object called lm2 that was estimated above. You can find the correspondence between the displayed means and the estimates that were constructed in Table 2.3.

require(effects)
plot(allEffects(lm2))

Figure 2.29: Plot of the estimated group mean distances from the reference-coded model for the overtake data from the effects package.

In order to assess overall evidence against having the same means for the all groups (vs not having the same means), we compare either of the previous models (cell-means or reference-coded) to a null model based on the null hypothesis of \(H_0: \mu_1 = \ldots = \mu_J\), which implies a model of \(\color{red}{y_{ij} = \mu+\varepsilon_{ij}}\) in the cell-means version where \({\color{red}{\mu}}\) is a common mean for all the observations. We will call this the mean-only model since it only has a single mean in it. In the reference-coded version of the model, we have a null hypothesis of \(H_0: \tau_2 = \ldots = \tau_J = 0\), so the “mean-only” model is \(\color{purple}{y_{ij} =\boldsymbol{\alpha}+\varepsilon_{ij}}\) with \(\color{purple}{\boldsymbol{\alpha}}\) having the same definition as \(\color{red}{\mu}\) for the cell-means model – it forces a common value for the mean for all the groups. Moving from the reference-coded model to the mean-only model is also an example of a situation where we move from a “full” model to a “reduced” model by setting some coefficients in the “full” model to 0 and, by doing this, get a simpler or “reduced” model. Simple models can be good as they are easier to interpret, but having a model for \(J\) groups that suggests no difference in the groups is not a very exciting result in most, but not all, situations⁵³. In order for R to provide results for the mean-only model, we remove the grouping variable, Condition, from the model formula and just include a “1”. The (Intercept) row of the output provides the estimate for the mean-only model as a reduced model from either the cell-means or reference-coded models when we assume that the mean is the same for all groups:

lm3 <- lm(Distance~1, data=dd)
summary(lm3)$coefficients

##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  117.126  0.3977533 294.469        0

This model provides an estimate of the common mean for all observations of \(117.13 = \hat{\mu} = \hat{\alpha}\) cm. This value also is the dashed horizontal line in the pirate-plot in Figure 2.28. Some people call this mean-only model estimate the “grand” or “overall” mean and notationally is represented as \(\bar{\bar{y}}\).