Building a Factorial Treatment Table

Why Build a Treatment Table?

When a study has two factors, each with several levels, a table makes the design easier to inspect. Its rows list the levels of one factor, its columns list the levels of the other, and each interior cell marks one combination of levels. The table is a planning and organization tool: it shows which conditions belong in the design before any outcome values are entered.

A carefully labelled grid helps prevent a practical error: leaving out a combination or recording a combination under the wrong label. It also makes it easy to point to a specific condition. To read a cell, use its row label and column label together; neither label alone identifies the full combination.

The Basic Layout

Suppose Factor A has \(a\) levels and Factor B has \(b\) levels. Choose one factor for the rows and the other for the columns. For example, if Factor A has levels \(A_1\) and \(A_2\), and Factor B has levels \(B_1\), \(B_2\), and \(B_3\), the grid has two data rows and three data columns. Each cell names the row level and the column level together.

The corner label tells the reader what the row and column labels mean. The ordered pair inside a cell is a compact way to show both levels in that combination. In this example, the cell in row \(A_2\) and column \(B_3\) represents \((A_2,B_3)\). A table could instead use descriptive words rather than symbols, as long as the factor and level labels remain unambiguous.

The number of interior cells is the number of row levels multiplied by the number of column levels. Thus, a complete table with \(a\) levels of Factor A and \(b\) levels of Factor B has \(ab\) cells. This is the same count as the number of distinct pairs: each of the \(a\) row levels can be paired with each of the \(b\) column levels.

A Reliable Construction Method

Build the grid in a consistent order. First list the levels of both factors exactly as they are defined for the design. Next choose which factor labels the rows and which labels the columns. Add one row for each level of the row factor and one interior column for each level of the column factor. Finally, fill each cell with the pair formed from its row and column labels, then check the count and labels.

Reading and Checking Cells

A cell should be read by combining both of its margin labels. Reading only the row tells you the level of one factor but not the level of the other. Reading only the column has the complementary problem. In the study-time example, “40 minutes” alone does not specify the setting, and “background music” alone does not specify study time.

A useful completeness check is to count cells and then scan their labels. The count detects a missing or extra cell, while the scan checks that the actual pairs are right. Both checks matter: a table can have six cells and still repeat one pair while omitting another.

Changing the Orientation

Either factor can be placed in rows. Swapping the row and column factors changes the appearance of the table, but it does not change the set of combinations. For example, the pair “standard fertilizer, daily watering” remains the same pair whether fertilizer labels a row or a column. The most important requirements are to label both margins and to use the same orientation consistently throughout the table.

The choice of orientation is often made for readability. Put the factor with fewer levels in the rows if that makes the table easier to fit on a page, or use the order that matches a study's usual presentation. Neither choice changes the design. If a reader cannot tell which factor a label belongs to, however, the table is not clear enough.

What the Table Does—and Does Not—Show

A treatment table lists the design's combinations; it does not report the measured outcome. For example, the cell “40 minutes, background music” identifies a condition in the study-time example, not an observed score. A table of outcome data would need additional information, such as the recorded responses for experimental units assigned to that condition. Keeping the two purposes separate prevents condition labels from being mistaken for results.

The grid also does not, by itself, say how many experimental units receive each combination or how they are assigned. Those are design details beyond the table's basic role. A complete set of cells establishes which combinations are included; it does not establish equal allocation or any particular randomization procedure.

Common Mistakes

• Using only one margin to identify a condition. “The 40-minute group” leaves the study setting unspecified. Name the row and column levels together.
• Forgetting the corner heading. A table with unlabeled margins may contain the right words but still leave readers unsure which factor each margin represents. Label both factors.
• Counting levels instead of cells. With 3 levels of one factor and 2 of the other, there are \(3+2=5\) margin labels but \(3\times2=6\) treatment combinations. Multiply the numbers of levels to check the interior cells.
• Assuming the table gives the outcome or replication count. A cell names a combination; it does not say what responses were observed or how many units were assigned there.
• Thinking a reversed layout changes the combinations. Switching rows and columns only changes the display. The same level pairs must still appear once each.

The central habit is simple: label each factor clearly, let each cell inherit one level from each margin, and verify the full grid. That makes the treatment combinations easy to identify without implying anything about outcome values or allocation.