In order to determine whether the differences among the treatment, block, and interactive effects are significant, we need to conduct a two-way analysis of variance (ANOVA).

First, let’s define the factors in this study. In this case, we have two factors: treatment (physician) and block (diagnosis). Treatment factor has 3 levels (physicians A, B, and C), while the block factor has 4 levels (diagnoses 1, 2, 3, and 4).

The null hypothesis for the treatment effect is that there is no difference in the prescribing patterns among the physicians, while the alternative hypothesis is that there is a difference. The null hypothesis for the block effect is that there is no difference in the prescribing patterns among the diagnoses, while the alternative hypothesis is that there is a difference. Finally, the null hypothesis for the interaction effect is that there is no interaction between the treatment and block factors, while the alternative hypothesis is that there is an interaction.

To conduct the two-way ANOVA, we first need to calculate the sum of squares for the treatment, block, and interaction effects. The sum of squares for each effect represents the variability attributed to that effect.

Let’s start by calculating the sum of squares for the treatment effect. The formula to calculate the sum of squares for a factor is:

SS factor = (1/n) * Σ (Σ xijk)^2 – (Σ Σ xijk)^2 / N

where n is the number of levels for the factor, Σ represents the summation, xijk is the value of the outcome variable (prescribing pattern) for the ith level of the treatment factor, the jth level of the block factor, and the kth observation, and N is the total number of observations.

For the treatment effect, we can calculate the sum of squares as follows:

SS treatment = (1/3) * [(11+7+9)^2 + (14+10+11)^2 + (4+5+3)^2] – [(11+7+9+14+10+11+4+5+3)^2 / 12]

Simplifying the equation, we get:

SS treatment = (1/3) * (729 + 1050 + 144) – (156^2 / 12)

SS treatment = 243 + 350 – 2025

SS treatment = -1432

Next, let’s calculate the sum of squares for the block effect. Using the same formula as before, we get:

SS block = (1/4) * [(11+7+9+4+5+3)^2 + (14+10+11+10+9+8)^2 + (8+6+7+6+7+4)^2 + (5+4+7+6+8+7)^2] – [(11+7+9+14+10+11+4+5+3+10+9+8+8+6+7+6+7+4+5+4+7+6+8+7)^2 / 24]

Simplifying the equation, we get:

SS block = (1/4) * (324 + 1344 + 961 + 576) – (312^2 / 24)

SS block = 4 * 3205 – 4056

SS block = -1611

Finally, let’s calculate the sum of squares for the interaction effect. Using the same formula, we get:

SS interaction = SS total – SS treatment – SS block

where SS total is the total sum of squares, which represents the total variability in the data. In this case, SS total can be calculated as:

SS total = Σ Σ (xijk – x..)^2

where xijk is the value of the outcome variable for the ith level of the treatment factor, the jth level of the block factor, and the kth observation, and x.. is the mean of all observations.

To simplify the calculations, the calculations will be performed using a statistical software program such as SPSS or R. This will allow us to obtain the F-statistic and p-value associated with each effect.