1.2.1 Simple model to assess the effects of treatment (Model A)

We introduce in this section a very simple example of a statistical model using this data. A model can be thought of as an attempt to describe quantitatively the effect of a number of factors on each observation. Any model we describe is likely to be a gross oversimplification of reality. In developing models, we are seeking ones which are as simple as possible but which contain enough truth to ask questions of interest. In this first simple model, we will deliberately be oversimplistic in order to introduce our notation. We just describe the effect of the two treatments. The model may be expressed as

where

1. j = A or B,
   
2. y_ij = observation for treatment j on the ith patient,
   
3. μ = overall mean,
   
4. t_j = effect of treatment j,
   
5. e_ij = error for treatment j on the ith patient.

The constant μ represents the overall mean of the observations. μ + t_A corresponds to the mean in the treatment group A, while μ + t_B corresponds to the mean in the treatment group B. The constants μ, t_A and t_B can thus be estimated from the data. In our example, we can estimate the value of μ to be 20.75, the overall mean value. From the mean value in the first treatment group, we can estimate μ + t_A as 22.83, and hence our estimate of t_A is 22.83 − 20.75 = 2.08. Similarly, from the mean of the second treatment group, we estimate t_B as −2.08. The term t_j can therefore be thought of as a measure of the relative effect that treatment j has had on our outcome variable.

The error term, e_ij, or residual is what remains for each patient in each period when μ + t_j is deducted from their observed measurement. This represents random variation about the mean value for each treatment. As such, the residuals can be regarded as the result of drawing random samples from a distribution. We will assume that the distribution is Gaussian or normal, with standard deviation σ, and that the samples drawn from the distribution are independent of each other. The mean of the distribution can be taken as zero, since any other value would simply cause a corresponding change in the value of μ. Thus, we will write this as

where σ² is the variance of the residuals. In practice, checks should be made to determine whether this assumption of normally distributed residuals is reasonable. Suitable checking methods will be considered in Section 2.4.6. As individual observations are modelled as the sum of μ + t_j, which are both constants, and the residual term, it follows that the variance of individual observations equals the residual variance:

The covariance of any two separate observations y_ij and can be written as

Since all the residuals are assumed independent (i.e. uncorrelated), it follows that

The residual variance, σ², can be estimated using a standard technique known as analysis of variance (ANOVA). The essence of the method is that the total variation in the data is decomposed into components that are associated with possible causes of this variation, for example, that one treatment may be associated with higher observations, with the other being associated with lower observations. For this first model, using this technique, we obtain the following ANOVA table:

Note: F = value for the F test (ratio of mean square for treatments to mean square for residual).

p = significance level corresponding to the F test.

The residual mean square of 19.42 is our estimate of the residual variance, σ², for this model. The key question often arising from this type of study is as follows: ‘do the treatment effects differ significantly from each other?’ This can be assessed by the F test, which assesses the null hypothesis of no mean difference between the treatments (the larger the treatment difference, the larger the treatment mean square and the higher the value of F). The p value of 0.13 is greater than the conventionally used cutoff point for statistical significance of 0.05. Therefore, we cannot conclude that the treatment effects are significantly different. The difference between the treatment effects and the SE of this difference provides a measure of the size of the treatment difference and the accuracy with which it is estimated:

The SE of the difference is given by the formula

Note that a t test can also be constructed from this difference and SE, giving t = 4.16/2.54 = 1.63. This is the square root of our F statistic of 2.68 and gives an identical t test p value of 0.13.

1.2.2 A model taking patient effects into account (Model B)

Model A as discussed previously did not utilise the fact that pairs of observations were taken on the same patients. It is possible, and indeed likely, that some patients will tend to have systematically higher measurements than others, and we may be able to improve the model by making allowance for this. This can be done by additionally including patient effects into the model:

where p_i are constants representing the ith patient effect.

The ANOVA table arising from this model is as follows:

The estimate of the residual variance, σ², is now 7.88. It is lower than in Model A because it represents the ‘within-patient’ variation, as we have taken account of patient effects. The F test p value of 0.05 indicates that the treatment effects are now significantly different. The difference between the treatment effects is the same as in Model A, 4.16, but its SE is now as follows:

(Note that the SE of the treatment difference could alternatively have been obtained directly from the differences in patient observations.)

Model B is perhaps the ‘obvious’ one to think of for this dataset. However, even in this simple case, by comparison with Model A we can see that the statistical modeller has some flexibility in his/her choice of model. In most situations, there is no single ‘correct’ model, and, in fact, models are rarely completely adequate. The job of the statistical modeller is to choose that model which most closely achieves the objectives of the study.