Constructing SAS Contrast/Estimate Statements S. R. Bowley, University of Guelph 2013

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Reynold Harrell
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1 onstructing SS ontrast/estimate Statements S. R. owley, University of Guelph 2013 The coefficients for contrast/estimate statements for single factors are easily created. oefficients for contrasts involving interaction means of two or three factors are more complex to construct. The following will guide you through the process of designing statements suitable for interaction means. Step 1. Identify the order of levels within each factor. This will either be alphanumeric or in order of first occurrence in the datafile if order=data is specified. Either review the order of levels given in the first page of the Proc Mixed/GLM output or request the lsmeans the order listed is the order of levels within each factor. Step 2. Identify the order of factors. This is specified by the order they are listed in the class statement. If you have three factors,,, and, and the class statement is: class ; then for interactions, the ** interaction means will be arranged as. If instead you have them listed as class, then the ** means will be arranged as. Step 3. Identify the specific means, or group of means, you want to contrast. ssume you have an experiment with Factor with two levels (a1 & a2), Factor with three levels (b1, b2, & b3) and Factor with four levels (c1, c2, c3, & c4). Factors a1 a2 b1 b2 b3 c1 c2 c3 c4 The following examples assume the class statement specifies the order of factors to be --: class ; I. EXMPLE 1. ontrast of two means. ontrast of the mean of b1 vs the mean of b2 (averaged over all levels of & ). estimate ' b1 vs b2' 1-1 0; The sum of the coefficients must be zero. If more means are involved, decimal coefficients, or use the /divisor=n option to use whole numbers and avoid messy decimals. Recommended practice: dd the option e to the estimate statement. This will list the model effects & their levels, and the coefficients applied to each level (called the L matrix). You can use this matrix to double-check you have the coefficients correctly specified. estimate ' b1 vs b2' / e ; Page 1 of 10

2 II. EXMPLE 2. ontrast of two means at a set value of another factor. ontrast of the mean of b1 vs the mean of b2 when the level of is a1. i) reate a table with the levels of as rows and the levels of as columns. eside/below these, enter the coefficients associated with the omparison of interest. In this case it is b1 vs b2 (1-1 0), and for when the level of is a1 (1 0). a1 1 a2 0 b1 b2 b ii) Multiply the coefficients in each row-column combination and enter the value in the respective cell in the centre of the table. b1 b2 b a a iii) ompute the total across each rows and within each column of the cells computed in step ii). lso compute the grand total. b1 b2 b a a The coefficients can be obtained from the above table. 1. The intercept value is the grand total. In this case it is 0, so the intercept drops out of the computation. 2. The row totals (shaded) give the coefficients for the factor. In this case they are all 0 so the factor drops out of the computation. 3. The column totals (shaded) give the coefficients for the factor. In this case they are The computed cells are the coefficients for the x factor; these are in bold. Note that you collate the coefficients by reading across each row. In this case, the x interaction coefficients are The estimate (or contrast) statement for comparing the mean of b1 vs mean of b2 when the level of is a1 would be specified as: estimate ' b1 vs b2 when =a1' * ; Page 2 of 10

3 III. EXMPLE 3. ontrast of two means at a set value of two other factors. ontrast of the mean of b1 vs the mean of b2 when the level of is a1 and the level of is c1. With three factors, there are eight possible terms that may be involved in specifying an estimate/contrast statement; namely, the intercept,,,, x, x, x, and xx interactions. i) With three factors, this will involve main, two-factor, and three-factor interaction coefficients. egin by creating a table with the levels of as rows and the levels of as columns. ontinue as for Example 2 to obtain the coefficients for the x interaction. However, don t bother summing the rows and columns in this step. b1 b2 b a a ii) Now create a table of the x interaction coefficients against (x against ). a1 b1 1 b2-1 b3 0 a2 b1 0 b2 0 b iii) Multiply the coefficients in each row-column combination and enter the value in the respective cell in the centre of the table a1 b b b a2 b b b iv) dd across the rows and along the columns of the cells computed in step iii). lso compute the grand total. Page 3 of 10

4 a1 b b b a2 b b b This table gives the coefficients for the xx interaction, the x interaction (last column), the factor (last row), and the intercept (grand total). For this example, the xx interaction coefficients are the x interaction coefficients are and the factor and the intercept drop out. However, for a three-factor, there are two additional tables to compute: x against and x against. x against. i) Obtain the x coefficients: a a ii) Obtain the x against coefficients: b1 b2 b a1 c c c c a2 c c c c This table gives the coefficients for the x interaction (last column) and the factor. In this example, the x interaction drops out, the factor remains with coefficients Page 4 of 10

5 x against. i) Obtain the x coefficients: b b b ii) Obtain the x against coefficients: a1 a2 1 0 b1 c c c c b2 c c c c b3 c c c c This table gives the coefficients for the x interaction (last column) and the factor. In this example, the factor drops out, the x interaction remains with coefficients ombining the information from the xx, and the x against, and x against tables, the intercept,,, & x terms drop out. The estimate statement requires coefficients for, x, x, and xx. The estimate (or contrast) statement for comparing the mean of b1 vs mean of b2 when the level of is a1 and the level of is c1 would be specified as: estimate ' b1 vs b2 when =a1 & =c1' * * ** ; [Note: The trailing zeros can be dropped in the above coefficients.] Similar estimates statements for other levels can also be generated by shifting the 1 & -1 to the appropriate location within the sequence. For example, if factor was time, the comparison at at time=4 (the endpoint) would be: estimate ' b1 vs b2 when =a1 & =c4' * * ** ; Page 5 of 10

6 IV. EXMPLE 4. ontrast of means involving nested factors. ssume you have an experiment with Factor with three levels (a1, a2 & a3), Factor is nested within and has two levels (b1, b2) in level a1 of, three levels (b3, b4, b5) in level a2 of, and two levels (b6, b7) in level a3 of. The model statement would be: model y= (); Factors a1 b1, b2, a2 a3 b3, b4, b5 b6, b7 Example 4a: ontrast of the mean of b1 within a1 vs the mean of b2 within a1. i) egin by creating an x matrix and cross out the factor combinations that do not exist. a1 a2 a3 () ii) Enter the factor combinations you wish to contrast. () a1 1-1 a a3 0 0 iii) Total the rows and columns. () a a a The total of the rows are the coefficients for the factor, the total of the columns are the coefficients for the () factor. In this example, the coefficients for are zero so it does not need to be specified. The estimate (or contrast) statement for comparing the mean of b1 vs mean of b2 when the level of is a1 would be specified as: estimate ' b1(a1) vs b2(a1)' () ; Page 6 of 10

7 Example 4b: ontrast of the mean of b1 within a1 vs the mean of b6 and b7 within a3. i) egin by creating an x matrix and cross out the factor combinations that do not exist. a1 a2 a3 () ii) Enter the factor combinations you wish to contrast. () a1 2 0 a a iii) Total the rows and columns. () a a a The total of the rows are the coefficients for the factor, the total of the columns are the coefficients for the () factor. The estimate statement for comparing the mean of b1 within a1 vs the mean of b6 & b7 within a3 would be specified as: estimate ' b1(a1) vs b6 & b7(a3)' () / divisor=2 ; The contrast statement would be specified as : contrast ' b1(a1) vs b6 & b7(a3)' () ; Page 7 of 10

8 Similarly, if you wanted to calculate the best linear unbiased estimate (LUE) of the mean of factors b6 & b7 within a3, the completed matrix and totals would be: () a a a The estimate statement for computing the mean of b6 & b7 within a3 would be specified as: estimate ' mean of b6 & b7(a3)' intercept () / divisor=2 ; V. EXMPLE 5. Subdividing factors and interactions into regression forms. ssume you have an experiment with Factor with three levels (b1, b2 & b3) and Factor with four levels (c1, c2, c3, & c4). Factor is a quantitative variable such as a time or a chemical application rate. Hence, Factor and interactions with Factor can be subpartitioned into regression forms. Example 5a) omparison of regression responses for Factor. Since Factor is quantitative and has four levels, it can be divided into three orthogonal regression forms: linear, quadratic, and cubic. If the levels of are equally spaced, the coefficients in Table 6.1 in the text (owley 2008) can be used. If they were unequally spaced, then a Proc IML method (see Example 6.1 in owley 2008) would need to be used. For this example, we will assume the levels of c1, c2, c3, & c4 are equally spaced. The linear coefficients for would be and the quadratic coefficients for would be The contrast statements for the partition of Factor - linear and of -quadratic would be specified as : contrast ' Factor linear' ; contrast ' Factor quadratic' ; Example 5b) omparison of two levels of Factor for their difference in regression response. Factor has three levels, assume level b1 is a control and we wish to determine if level b3 has a different response (slope) over levels of compared to the control. gain, construct the x interaction matrix to compare b1 & b3 for their regression response over levels of. Here is the matrix to compare their linear response: b b b Page 8 of 10

9 nd sum across rows and columns: b b b The contrast statement for comparing the mean of b1 vs b3 for their linear response over levels of would be specified as: estimate ' b1 vs b3 x linear' * ; This contrast is a comparison of the slope of b1 to the slope of b3 for the linear regression over levels of. similar method can be used to compare their quadratic response. Example 5c) omparison of all levels of Factor for their difference in regression response. Factor has three levels and we wish to determine if the group has differences in response (slope) over levels of. In this case, we need to combine all possible pairs of the levels of Factor for their response. (b1 vs b2, b1 vs b3, and b2 vs b3). Obtain the coefficients for each of the possible pairs of the Factor levels. b1 vs b2 x Factor linear b b b b1 vs b3 x Factor linear b b b vs b2 x Factor linear b b b The contrast statement for comparing all Factor levels for their linear response over levels of would be specified as: estimate ' among x linear' * , * , * ; ommas are used to combine the sets into one pooled value. This contrast is a comparison of the slope of all levels of Factor for the linear regression over levels of. If this was significant, that would indicate that there were differences among the levels for their linear response (slope). One would need to apply the method in Example 5b to determine which pairs differed. similar method can be used to compare their quadratic form. Page 9 of 10

10 VI. LUP & Group option If you have specified a model with heterogeneity among groups using the group= option in a random statement and you are wanting to generate a LUP estimate involving the grouping variable, you will need to add the group option and the coefficients for it. Example, there are three treatments (trt) and two experiments (expt) and heterogeneity among the random experiments has been specified by group=expt. To compute the mean of the second treatment level, within the second experiment, the following statement could be used: estimate 'Mean trt 2 ' intercept 1 trt expt 0 1 trt*expt /e group 0 1; Page 10 of 10

Research Design - - Topic 13a Split Plot Design with Either a Continuous or Categorical Between Subjects Factor 2008 R.C. Gardner, Ph.D.

Research Design - - Topic 13a Split Plot Design with Either a Continuous or Categorical Between Subjects Factor 2008 R.C. Gardner, Ph.D. esearch Design - - Topic 3a Split Plot Design with Either a ontinuous or ategorical etween Subjects Factor 8.. Gardner Ph.D. General Description Example Purpose Two ategorical Factors Using Multiple egression