Transformation Methods (MANOVA: Multivariate command)

To specify a transformation method, use one of the following keywords available on the TRANSFORM subcommand. Note that these are identical to the keywords available for the CONTRAST subcommand (see CONTRAST Subcommand (MANOVA: Univariate command)). However, in univariate designs, they are applied to the different levels of a factor. Here they are applied to the continuous variables in the analysis. This reflects the fact that the different dependent variables in a multivariate MANOVA setup can often be thought of as corresponding to different levels of some factor.

  • The transformation keyword (and its specifications, if any) must follow all other specifications on the TRANSFORM subcommand.

DEVIATION. Deviations from the mean of the variables being transformed. The first transformed variable is the mean of all variables in the transformation. Other transformed variables represent deviations of individual variables from the mean. One of the original variables (by default, the last) is omitted as redundant. To omit a variable other than the last, specify the number of the variable to be omitted in parentheses after the DEVIATION keyword. For example, /TRANSFORM (A B C) = DEVIATION(1) omits A and creates variables representing the mean, the deviation of B from the mean, and the deviation of C from the mean. A DEVIATION transformation is not orthogonal.

DIFFERENCE. Difference or reverse Helmert transformation. The first transformed variable is the mean of the original variables. Each of the original variables except the first is then transformed by subtracting the mean of those (original) variables that precede it. A DIFFERENCE transformation is orthogonal.

HELMERT. Helmert transformation. The first transformed variable is the mean of the original variables. Each of the original variables except the last is then transformed by subtracting the mean of those (original) variables that follow it. A HELMERT transformation is orthogonal.

SIMPLE. Each original variable, except the last, is compared to the last of the original variables. To use a variable other than the last as the omitted reference variable, specify its number in parentheses following the keyword SIMPLE. For example, /TRANSFORM(A B C) = SIMPLE(2) specifies the second variable, B, as the reference variable. The three transformed variables represent the mean of A, B, and C, the difference between A and B, and the difference between C and B. A SIMPLE transformation is not orthogonal.

POLYNOMIAL. Orthogonal polynomial transformation. The first transformed variable represents the mean of the original variables. Other transformed variables represent the linear, quadratic, and higher-degree components. By default, values of the original variables are assumed to represent equally spaced points. You can specify unequal spacing by entering a metric consisting of one integer for each variable in parentheses after the keyword POLYNOMIAL. For example, /TRANSFORM(RESP1 RESP2 RESP3) = POLYNOMIAL(1,2,4) might indicate that three response variables correspond to levels of some stimulus that are in the proportion 1:2:4. The default metric is always (1,2,..., k), where k variables are involved. Only the relative differences between the terms of the metric matter: (1,2,4) is the same metric as (2,3,5) or (20,30,50) because in each instance the difference between the second and third numbers is twice the difference between the first and second.

REPEATED. Comparison of adjacent variables. The first transformed variable is the mean of the original variables. Each additional transformed variable is the difference between one of the original variables and the original variable that followed it. Such transformed variables are often called difference scores. A REPEATED transformation is not orthogonal.

SPECIAL. A user-defined transformation. After the keyword SPECIAL, enter a square matrix in parentheses with as many rows and columns as there are variables to transform. MANOVA multiplies this matrix by the vector of original variables to obtain the transformed variables (see the examples below).

Example

MANOVA X1 TO X3 BY A(1,4)
  /TRANSFORM(X1 X2 X3) = SPECIAL( 1  1  1, 
                                  1  0 -1, 
                                  2 -1 -1)
  /DESIGN.
  • The given matrix will be post-multiplied by the three continuous variables (considered as a column vector) to yield the transformed variables. The first transformed variable will therefore equal X1 + X2 + X3, the second will equal X1 − X3, and the third will equal 2X1 − X2 − X3.
  • The variable list is optional in this example since all three interval-level variables are transformed.
  • You do not need to enter the matrix one row at a time, as shown above. For example,

    /TRANSFORM = SPECIAL(1 1 1 1 0 -1 2 -1 -1)

    is equivalent to the TRANSFORM specification in the above example.

  • You can specify a repetition factor followed by an asterisk to indicate multiple consecutive elements of a SPECIAL transformation matrix. For example,

    /TRANSFORM = SPECIAL (4*1 0 -1 2 2*-1)

    is again equivalent to the TRANSFORM specification above.

Example

MANOVA X1 TO X3, Y1 TO Y3 BY A(1,4)
 /TRANSFORM (X1 X2 X3/Y1 Y2 Y3) = SPECIAL( 1  1  1, 
                                           1  0 -1, 
                                           2 -1 -1)
 /DESIGN.
  • Here the same transformation shown in the previous example is applied to X1, X2, X3 and to Y1, Y2, Y3.